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The carbon-carbon bond length in diamond is $154.45 \mathrm{pm}$. If diamond were considered to be a close-packed structure of hard spheres with radii equal to half the bond length, what would be its expected density? The diamond lattice is face-centred cubic and its actual density is $3.516 \mathrm{~g} \mathrm{~cm}^{-3}$.

The atomic packing factor of the diamond cubic structure (the proportion of space that would be filled by spheres that are centered on the vertices of the structure and are as large as possible without overlapping) is ≈ 0.34,. significantly smaller (indicating a less dense structure) than the packing factors for the face- centered and body-centered cubic lattices.. The toughness of natural diamond has been measured as 7.5–10 MPa·m1/2. It also has a high density, ranging from 3150 to 3530 kilograms per cubic metre (over three times the density of water) in natural diamonds and 3520 kg/m in pure diamond. St Edmundsbury Press Ltd, Bury St Edwards. ==External links== *Properties of diamond *Properties of diamond (S. Sque, PhD thesis, 2005, University of Exeter, UK) Category:Diamond Category:Allotropes of carbon Diamond Category:Superhard materials Diamond is the allotrope of carbon in which the carbon atoms are arranged in the specific type of cubic lattice called diamond cubic. Diamond is extremely strong owing to its crystal structure, known as diamond cubic, in which each carbon atom has four neighbors covalently bonded to it. thumb|upright=1.25|Main diamond producing countries Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. A lattice of 3×3×3 unit cells thumb|245px|Molar volume vs. pressure at room temperature. thumb|3D ball-and- stick model of a diamond lattice The precise tensile strength of diamond is unknown, though strength up to has been observed, and theoretically it could be as high as depending on the sample volume/size, the perfection of diamond lattice and on its orientation: Tensile strength is the highest for the [100] crystal direction (normal to the cubic face), smaller for the [110] and the smallest for the [111] axis (along the longest cube diagonal). thumb|250px|3D ball-and-stick model of a diamond lattice The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. The following issues are considered: * The hardness of diamond and its ability to cleave strongly depend on the crystal orientation. A similar proportion of diamonds comes from the lower mantle at depths between 660 and 800 km. Diamond is thermodynamically stable at high pressures and temperatures, with the phase transition from graphite occurring at greater temperatures as the pressure increases. The Allnatt Diamond is a diamond measuring 101.29 carats (20.258 g) with a cushion cut, rated in color as Fancy Vivid Yellow by the Gemological Institute of America. Thermal conductivity of natural diamond was measured to be about 2200 W/(m·K), which is five times more than silver, the most thermally conductive metal. Diamonds are carbon crystals that form under high temperatures and extreme pressures such as deep within the Earth. Diamonds crystallize in the diamond cubic crystal system (space group Fdm) and consist of tetrahedrally, covalently bonded carbon atoms. The diamond crystal lattice is exceptionally strong, and only atoms of nitrogen, boron, and hydrogen can be introduced into diamond during the growth at significant concentrations (up to atomic percents). Some extrasolar planets may be almost entirely composed of diamond. Thus, graphite is much softer than diamond. Unlike many other minerals, the specific gravity of diamond crystals (3.52) has rather small variation from diamond to diamond. ==Hardness and crystal structure== Known to the ancient Greeks as (, 'proper, unalterable, unbreakable') and sometimes called adamant, diamond is the hardest known naturally occurring material, and serves as the definition of 10 on the Mohs scale of mineral hardness. At normal temperature and pressure, and , the stable phase of carbon is graphite, but diamond is metastable and its rate of conversion to graphite is negligible. The Amsterdam Diamond is a black diamond weighing , and has 145 facets. This was determined by comparing the ratios of carbon isotopes present. == Optical and electronic properties == The optical absorption for all diamondoids lies deep in the ultraviolet spectral region with optical band gaps around 6 electronvolts and higher.

7.654

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A swimmer enters a gloomier world (in one sense) on diving to greater depths. Given that the mean molar absorption coefficient of seawater in the visible region is $6.2 \times 10^{-3} \mathrm{dm}^3 \mathrm{~mol}^{-1} \mathrm{~cm}^{-1}$, calculate the depth at which a diver will experience half the surface intensity of light.

To calculate this coefficient, light energy is measured at a series of depths from the surface to the depth of 1% illumination. As the depth increases, more light is absorbed by the water. By using artificial light, it is possible to view an object in full color at greater depths. ==Need== Water attenuates light by absorption, so use of a dive light will improve a diver's underwater vision at depth. Maximum Operating Depth (MOD) in metres sea water (msw) for pO2 1.2 to 1.6 MOD (msw) % oxygen 4 8 12 16 20 21 24 28 32 36 40 50 60 70 80 90 100 Maximum pO2 (bar) 1.6 390 190 123 90 70 66 57 47 40 34 30 22 17 13 10 8 6 1.5 365 178 115 84 65 61 53 44 37 32 28 20 15 11 9 7 5 1.4 340 165 107 78 60 57 48 40 34 29 25 18 13 10 8 6 4 1.3 315 153 98 71 55 52 44 36 31 26 23 16 12 9 6 4 3 1.2 290 140 90 65 50 47 40 33 28 23 20 14 10 7 5 3 2 These depths are rounded to the nearest metre. ==See also== * == Notes == == References == == Sources == * * * * Category:Dive planning Category:Underwater diving safety Category:Breathing gases By directing the same type of beam downwards, the depth to the bottom of the ocean could be calculated. A dive light is a light source carried by an underwater diver to illuminate the underwater environment. The United States Navy Experimental Diving Unit continues to evaluate dive lights for wet and dry illumination output, battery duration, watertight integrity, as well as maximum operating depth. Then, the exponential decline in light is calculated using Beer’s Law with the equation: {I_z \over I_0}= e^{-kz} where k is the light attenuation coefficient, Iz is the intensity of light at depth z, and I0 is the intensity of light at the ocean surface.Idso, Sherwood B. and Gilbert, R. Gene (1974) On the Universality of the Poole and Atkins Secchi Disk: Light Extinction Equation British Ecological Society. Of this total pressure which can be tolerated by the diver, 1 atmosphere is due to surface pressure of the Earth's air, and the rest is due to the depth in water. For example, if a gas contains 36% oxygen and the maximum pO2 is 1.4 bar, the MOD (msw) is 10 msw/bar x [(1.4 bar / 0.36) − 1] = 28.9 msw. == Tables == Maximum Operating Depth (MOD) in feet sea water (fsw) for pO2 1.2 to 1.6 MOD (fsw) % oxygen 4 8 12 16 20 21 24 28 32 36 40 50 60 70 80 90 100 Maximum pO2 (bar) 1.6 1287 627 407 297 231 218 187 156 132 114 99 73 55 42 33 26 20 1.5 1205 586 380 276 215 203 173 144 122 105 91 66 50 38 29 22 17 1.4 1122 545 352 256 198 187 160 132 111 95 83 59 44 33 25 18 13 1.3 1040 503 325 235 182 171 146 120 101 86 74 53 39 28 21 15 10 1.2 957 462 297 215 165 156 132 108 91 77 66 46 33 24 17 11 7 These depths are rounded to the nearest foot. 300px|thumb|right|A canister style dive lightNight diving is underwater diving done during the hours of darkness. In addition to light penetration, the term water clarity is also often used to describe underwater visibility. In underwater diving activities such as saturation diving, technical diving and nitrox diving, the maximum operating depth (MOD) of a breathing gas is the depth below which the partial pressure of oxygen (pO2) of the gas mix exceeds an acceptable limit. For example, if a gas contains 36% oxygen (FO2 = 0.36) and the limiting maximum pO2 is chosen at 1.4 atmospheres absolute, the MOD in feet of seawater (fsw) is 33 fsw/atm x [(1.4 ata / 0.36) − 1] = 95.3 fsw. The diver can experience a different underwater environment at night, because many marine animals are nocturnal. The pressure produced by depth in water, is converted to pressure in feet sea water (fsw) or metres sea water (msw) by multiplying with the appropriate conversion factor, 33 fsw per atm, or 10 msw per bar. thumb|upright=1.5|A diver enters crystal clear water in Lake Huron. The depth at which the disk is no longer visible is taken as a measure of the transparency of the water. A dive light is routinely used during night dives and cave dives, when there is little or no natural light, but also has a useful function during the day, as water absorbs the longer (red) wavelengths first then the yellow and green with increasing depth. Bright dive lights have values from about 2500 lumens. This corresponds to a sound intensity 5.4 dB, or 3.5 times, higher than the threshold in air (see Measurements above). ====Safety thresholds==== High levels of underwater sound create a potential hazard to human divers. A modern dive light usually has an output of at least about 100 lumens.

556

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E

Calculate the molar energy required to reverse the direction of an $\mathrm{H}_2 \mathrm{O}$ molecule located $100 \mathrm{pm}$ from a $\mathrm{Li}^{+}$ ion. Take the magnitude of the dipole moment of water as $1.85 \mathrm{D}$.

thumb|Lewis Structure of H2O indicating bond angle and bond length Water () is a simple triatomic bent molecule with C2v molecular symmetry and bond angle of 104.5° between the central oxygen atom and the hydrogen atoms. 150px|thumb|Shape of water molecule showing that the real bond angle 104.5° deviates from the ideal sp3 angle of 109.5°. The bond angle for water is 104.5°. These tables list values of molar ionization energies, measured in kJ⋅mol−1. In predicting the bond angle of water, Bent's rule suggests that hybrid orbitals with more s character should be directed towards the lone pairs, while that leaves orbitals with more p character directed towards the hydrogens, resulting in deviation from idealized O(sp3) hybrid orbitals with 25% s character and 75% p character. The first molar ionization energy applies to the neutral atoms. The bond angles in those molecules are 104.5° and 107° respectively, which are below the expected tetrahedral angle of 109.5°. In predicting the bond angle of water, Bent's rule suggests that hybrid orbitals with more s character should be directed towards the very electropositive lone pairs, while that leaves orbitals with more p character directed towards the hydrogens. thumb|200px|Model of the hydrogen molecule and its axial projection In addition to the model of the atom, Niels Bohr also proposed a model of the chemical bond. A particularly well known example is water, where the angle between the two O-H bonds is only 104.5°. The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. thumb|Example of bent electron arrangement (water molecule). However, the bond angle between the two O–H bonds is only 104.5°, rather than the 109.5° of a regular tetrahedron, because the two lone pairs (whose density or probability envelopes lie closer to the oxygen nucleus) exert a greater mutual repulsion than the two bond pairs. thumb|300px|Calculated structure of a (H2O)100 icosahedral water cluster. In the case of water, with its 104.5° HOH angle, the OH bonding orbitals are constructed from O(~sp4.0) orbitals (~20% s, ~80% p), while the lone pairs consist of O(~sp2.3) orbitals (~30% s, ~70% p). The four hydrogen atoms are positioned at the vertices of a tetrahedron, and the bond angle is cos−1(−) ≈ 109° 28′."Angle Between 2 Legs of a Tetrahedron" – Maze5.net This is referred to as an AX4 type of molecule. Oxygen difluoride 103.8° As one moves down the table, the substituents become more electronegative and the bond angle between them decreases. For simple molecules, pictorially generating their MO diagram can be achieved without extensive knowledge of point group theory and using reducible and irreducible representations.thumb|330x330px|Hybridized MO of H2O Note that the size of the atomic orbitals in the final molecular orbital are different from the size of the original atomic orbitals, this is due to different mixing proportions between the oxygen and hydrogen orbitals since their initial atomic orbital energies are different. By directing hybrid orbitals of more p character towards the fluorine, the energy of that bond is not increased very much. This is in open agreement with the true bond angle of 104.45°. In other words, if water was formed from two identical O-H bonds and two identical sp3 lone pairs on the oxygen atom as predicted by valence bond theory, then its photoelectron spectrum (PES) would have two (degenerate) peaks and energy, one for the two O-H bonds and the other for the two sp3 lone pairs. The bond angles between substituents are ~109.5°, ~120°, and 180°.

1.07

4.49

205.0

4

2.14

A

In an industrial process, nitrogen is heated to $500 \mathrm{~K}$ at a constant volume of $1.000 \mathrm{~m}^3$. The gas enters the container at $300 \mathrm{~K}$ and $100 \mathrm{~atm}$. The mass of the gas is $92.4 \mathrm{~kg}$. Use the van der Waals equation to determine the approximate pressure of the gas at its working temperature of $500 \mathrm{~K}$. For nitrogen, $a=1.39 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}, b=0.0391 \mathrm{dm}^3 \mathrm{~mol}^{-1}$.

Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A.

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The chemical shift of the $\mathrm{CH}_3$ protons in acetaldehyde (ethanal) is $\delta=2.20$ and that of the $\mathrm{CHO}$ proton is 9.80 . What is the difference in local magnetic field between the two regions of the molecule when the applied field is $1.5 \mathrm{~T}$

In this way the acetylenic protons are located in the cone-shaped shielding zone hence the upfield shift. :thumb|none|400px|Induced magnetic field of alkynes in external magnetic fields, field lines in grey. ==Magnetic properties of most common nuclei== 1H and 13C are not the only nuclei susceptible to NMR experiments. When a signal is found with a higher chemical shift: * the applied effective magnetic field is lower, if the resonance frequency is fixed (as in old traditional CW spectrometers) * the frequency is higher, when the applied magnetic field is static (normal case in FT spectrometers) * the nucleus is more deshielded * the signal or shift is downfield or at low field or paramagnetic Conversely a lower chemical shift is called a diamagnetic shift, and is upfield and more shielded. ==Diamagnetic shielding== In real molecules protons are surrounded by a cloud of charge due to adjacent bonds and atoms. The induced magnetic field lines are parallel to the external field at the location of the alkene protons which therefore shift downfield to a 4.5 ppm to 7.5 ppm range. Although the absolute resonance frequency depends on the applied magnetic field, the chemical shift is independent of external magnetic field strength. Acetaldehyde (IUPAC systematic name ethanal) is an organic chemical compound with the formula CH3CHO, sometimes abbreviated by chemists as MeCHO (Me = methyl). The nucleus is said to be experiencing a diamagnetic shielding. ==Factors causing chemical shifts== Important factors influencing chemical shift are electron density, electronegativity of neighboring groups and anisotropic induced magnetic field effects. Field effects are relatively weak, and diminish rapidly with distance, but have still been found to alter molecular properties such as acidity.thumb|160x160px|Field effect on a carbonyl arising from the dipole in a C-F bond. == Field sources == thumb|A bicycloheptane acid with an electron- withdrawing substituent, X, at the 4-position experiences a field effect on the acidic proton from the C-X bond dipole.|left|180x180pxleft|thumb|180x180px|A bicyclooctance acid with an electron-withdrawing substituent, X, at the 4-position experiences the same field effect on the acidic proton from the C-X bond dipole as the related bicylcoheptane. In nuclear magnetic resonance (NMR) spectroscopy, the chemical shift is the resonant frequency of an atomic nucleus relative to a standard in a magnetic field. As noted above, a consensus CSI method that filters upfield/downfield chemical shift changes in 13Cα, 13Cβ, and 13C' atoms in a similar manner to 1Hα shifts has also been developed. The protons in aromatic compounds are shifted downfield even further with a signal for benzene at 7.73 ppm as a consequence of a diamagnetic ring current. Field effects have also been shown in substituted arenes to dominate the electrostatic potential maps, which are maps of electron density used to explain intermolecular interactions. == Evidence for field effects == thumb|This octane derivative has only a single linker between the electron-withdrawing substituent and the acidic group.|180x180px Localized electronic effects are a combination of inductive and field effects. While chemical shift is referenced in order that the units are equivalent across different field strengths, the actual frequency separation in Hertz scales with field strength (). The variations of nuclear magnetic resonance frequencies of the same kind of nucleus, due to variations in the electron distribution, is called the chemical shift. This can be attributed to a field effect because in the same compound with the chlorines pointed away from the acidic group the pKa is lower, and if the effect were inductive the conformational position would not matter.thumb|The dichloroethano-bridged anthroic acid isomer with the C-Cl bond dipole oriented over the carboxylic acid has pKa of 6.07.|left|200x200px thumb|The isomer of dichloroethano- bridged anthroic acid in which the C-Cl dipole points away from the carboxylic acid has a pKa of 5.67.|left|200x200px == References == Category:Chemical properties Category:Chemistry Category:Electrostatics Category:Electromagnetism Category:Molecular physics Category:Molecules Category:Physical chemistry In carbon NMR the chemical shift of the carbon nuclei increase in the same order from around −10 ppm to 70 ppm. The total magnetic field experienced by a nucleus includes local magnetic fields induced by currents of electrons in the molecular orbitals (electrons have a magnetic moment themselves). In proton NMR of methyl halides (CH3X) the chemical shift of the methyl protons increase in the order from 2.16 ppm to 4.26 ppm reflecting this trend. Anisotropic induced magnetic field effects are the result of a local induced magnetic field experienced by a nucleus resulting from circulating electrons that can either be paramagnetic when it is parallel to the applied field or diamagnetic when it is opposed to it. However, the experimental data shows that effect on acidity in related octanes and cubanes is very similar, and therefore the dominant effect must be through space.thumb|This octane derivative has two linkers between the electron- withdrawing substituent and the acidic group.|179x179px thumb|This cubane derivative has four linkers but the acidic proton still feels the same effect from the C-X dipole because the interaction is a field effect.|180x180pxIn the cis-11,12-dichloro-9,10-dihydro-9,10-ethano-2-anthroic acid syn and anti isomers seen below and to the left, the chlorines provide a field effect. As is the case for NMR the chemical shift reflects the electron density at the atomic nucleus. ==See also== * EuFOD, a shift agent * MRI * Nuclear magnetic resonance * Nuclear magnetic resonance spectroscopy of carbohydrates * Nuclear magnetic resonance spectroscopy of nucleic acids * Nuclear magnetic resonance spectroscopy of proteins * Protein NMR * Random coil index * Relaxation (NMR) * Solid-state NMR * TRISPHAT, a chiral shift reagent for cations * Zeeman effect ==References== ==External links== *chem.wisc.edu *BioMagResBank *NMR Table *Proton chemical shifts *Carbon chemical shifts * Online tutorials (these generally involve combined use of IR, 1H NMR, 13C NMR and mass spectrometry) **Problem set 1 (see also this link for more background information on spin-spin coupling) **Problem set 2 **Problem set 4 **Problem set 5 **Combined solutions to problem set 5 (Problems 1-32) and (Problems 33-64) Category:Nuclear chemistry Category:Nuclear physics Category:Nuclear magnetic resonance spectroscopy pl:Spektroskopia NMR#Przesunięcie chemiczne The size of the chemical shift is given with respect to a reference frequency or reference sample (see also chemical shift referencing), usually a molecule with a barely distorted electron distribution. ==Operating frequency== The operating (or Larmor) frequency of a magnet is calculated from the Larmor equation : \omega_{0} = \gamma B_0\,, where is the actual strength of the magnet in units like Teslas or Gauss, and is the gyromagnetic ratio of the nucleus being tested which is in turn calculated from its magnetic moment and spin number with the nuclear magneton and the Planck constant : : \gamma = \frac{\mu\,\mu_\mathrm{N}}{hI}\,. SHIFTCOR identifies potential chemical shift referencing problems by comparing the difference between the average value of each set of observed backbone (1Hα, 13Cα, 13Cβ, 13CO, 15N and 1HN) shifts and their corresponding predicted chemical shifts.

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B

Suppose that the junction between two semiconductors can be represented by a barrier of height $2.0 \mathrm{eV}$ and length $100 \mathrm{pm}$. Calculate the transmission probability of an electron with energy $1.5 \mathrm{eV}$.

Note that the probabilities and amplitudes as written are for any energy (above/below) the barrier height. === E = V0 === The transmission probability at E=V_0 is T=\frac{1}{1+ma^2V_0/2\hbar^2}. ==Remarks and applications== The calculation presented above may at first seem unrealistic and hardly useful. However, according to quantum mechanics, the electron has a non-zero wave amplitude in the barrier, and hence it has some probability of passing through the barrier. Note that these expressions hold for any energy If then so there is a singularity in both of these expressions. ==Analysis of the obtained expressions== ===E < V0=== thumb|350x350px|Transmission probability through a finite potential barrier for \sqrt{2m V_0} a / \hbar = 1, 3, and 7. The surprising result is that for energies less than the barrier height, E < V_0 there is a non-zero probability T=|t|^2= \frac{1}{1+\frac{V_0^2\sinh^2(k_1 a)}{4E(V_0-E)}} for the particle to be transmitted through the barrier, with .}} Classically, the electron has zero probability of passing through the barrier. For electrons, the barrier height \Phi_{B_{n}}can be easily calculated as the difference between the metal work function and the electron affinity of the semiconductor: \Phi_{B_n}=\Phi_M-\chi While the barrier height for holes is equal to the difference between the energy gap of the semiconductor and the energy barrier for electrons: \Phi_{B_p}=E_\text{gap}-\Phi_{B_n} In reality, what can happen is that charged interface states can pin the Fermi level at a certain energy value no matter the work function values, influencing the barrier height for both carriers. thumb|right|Schematic representation of an electron tunneling through a barrier In electronics/spintronics, a tunnel junction is a barrier, such as a thin insulating layer or electric potential, between two electrically conducting materials. To a first approximation, the barrier between a metal and a semiconductor is predicted by the Schottky–Mott rule to be proportional to the difference of the metal- vacuum work function and the semiconductor-vacuum electron affinity. The current-voltage relationship is qualitatively the same as with a p-n junction, however the physical process is somewhat different. === Conduction values === The thermionic emission can be formulated as following: J_{th}= A^{**}T^2e^{-\frac{\Phi_{B_{n,p}}}{k_bT}}\biggl(e^{\frac{qV}{k_bT}}-1\biggr) While the tunneling current density can be expressed, for a triangular shaped barrier (considering WKB approximation) as: J_{T_{n,p}}= \frac{q^3E^2}{16\pi^2\hbar\Phi_{B_{n,p}}} e^{\frac{-4\Phi_{B_{n,p}}^{3/2}\sqrt{2m^*_{n,p}}}{3q\hbar E}} From both formulae it is clear that the current contributions are related to the barrier height for both electrons and holes. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. The Monte Carlo method for electron transport is a semiclassical Monte Carlo (MC) approach of modeling semiconductor transport. The transmission is exponentially suppressed with the barrier width, which can be understood from the functional form of the wave function: Outside of the barrier it oscillates with wave vector whereas within the barrier it is exponentially damped over a distance If the barrier is much wider than this decay length, the left and right part are virtually independent and tunneling as a consequence is suppressed. === E > V0 === In this case T=|t|^2= \frac{1}{1+\frac{V_0^2\sin^2(k_1 a)}{4E(E-V_0)}}, where Equally surprising is that for energies larger than the barrier height, E > V_0, the particle may be reflected from the barrier with a non-zero probability R=|r|^2=1-T. Electrons (or quasiparticles) pass through the barrier by the process of quantum tunnelling. A classical particle with energy E larger than the barrier height V_0 would always pass the barrier, and a classical particle with E < V_0 incident on the barrier would always get reflected. An earlier version containing some of the material appeared in the Bell Labs Technical Journal in 1949. ==Contents== The arrangement of chapters is as follows: Part I INTRODUCTION TO TRANSISTOR ELECTRONICS *Chapter 1: THE BULK PROPERTIES OF SEMICONDUCTORS *Chapter 2: THE TRANSISTOR AS A CIRCUIT ELEMENT *Chapter 3: QUANTITATIVE STUDIES OF INJECTION OF HOLES AND ELECTRONS *Chapter 4: ON THE PHYSICAL THEORY OF TRANSISTORS *Chapter 5: QUANTUM STATES, ENERGY BANDS, AND BRILLOUIN ZONES PART II DESCRIPTIVE THEORY OF SEMICONDUCTORS *Chapter 6: VELOCITIES AND CURRENTS FOR ELECTRONS I N CRYSTALS *Chapter 7: ELECTRONS AND HOLES IN ELECTRIC AND MAGNETIC FIELDS *Chapter 8: INTRODUCTORY THEORY OF CONDUCTIVITY AND HALL EFFECT *Chapter 9: DISTRIBUTIONS OF QUANTUM STATES IN ENERGY *Chapter 10: FERMI-DIRAC STATISTICS FOR SEMICONDUCTORS *Chapter 11: MATHEMATICAL THEORY OF CONDUCTIVITY AND HALL EFFECT *Chapter 12: APPLICATIONS TO TRANSISTOR ELECTRONICS PART III QUANTUM MECHANICAL FOUNDATIONS *Chapter 13: INTRODUCTION TO PART III *Chapter 14: ELEMENTARY QUANTUM MECHANICS WITH CIRCUIT THEORY ANALOGUES *Chapter 15: THEORY OF ELECTRON AND HOLE VELOCITIES, CURRENTS AND ACCELERATIONS *Chapter 16: STATISTICAL MECHANICS FOR SEMICONDUCTORS *Chapter 17: THE THEORY OF TRANSITION PROBABILITIES FOR HOLES AND ELECTRONS *APPENDIX A Units *APPENDIX B Periodic Table *APPENDIX C Values of the physical constants *APPENDIX D Energy conversion chart *NAME INDEX *SUBJECT INDEX Category:1950 non-fiction books Category:1950 in science Category:Physics textbooks Under large voltage bias, the electric current flowing through the barrier is essentially governed by the laws of thermionic emission, combined with the fact that the Schottky barrier is fixed relative to the metal's Fermi level.This interpretation is due to Hans Bethe, after the incorrect theory of Schottky, see * Under forward bias, there are many thermally excited electrons in the semiconductor that are able to pass over the barrier. The scattering events and the duration of particle flight is determined through the use of random numbers. == Background == === Boltzmann transport equation === The Boltzmann transport equation model has been the main tool used in the analysis of transport in semiconductors. Within these approximations, Fermi's Golden Rule gives, to the first order, the transition probability per unit time for a scattering mechanism from a state |k \rangle to a state |k' \rangle: : S(k,k') = \frac{2\pi}{\hbar} \left | \langle k|H'|k' \rangle \right |^2 \cdot \delta(E - E') where H' is the perturbation Hamiltonian representing the collision and E and E′ are respectively the initial and final energies of the system constituted of both the carrier and the electron and phonon gas. To study the quantum case, consider the following situation: a particle incident on the barrier from the left side It may be reflected or transmitted To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations A_r = 1 (incoming particle), A_l = r (reflection), C_l = 0 (no incoming particle from the right), and C_r = t (transmission). 350px|right|thumb In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. A Schottky barrier, named after Walter H. Schottky, is a potential energy barrier for electrons formed at a metal–semiconductor junction. The value of ΦB depends on the combination of metal and semiconductor.Schottky barrier tutorial.

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The diffusion coefficient of a particular kind of t-RNA molecule is $D=1.0 \times 10^{-11} \mathrm{~m}^2 \mathrm{~s}^{-1}$ in the medium of a cell interior. How long does it take molecules produced in the cell nucleus to reach the walls of the cell at a distance $1.0 \mu \mathrm{m}$, corresponding to the radius of the cell?

The rate of diffusion NA, is usually expressed as the number of moles diffusing across unit area in unit time. Of mass transport mechanisms, molecular diffusion is known as a slower one. === Biology === In cell biology, diffusion is a main form of transport for necessary materials such as amino acids within cells. The rate of diffusion of A, NA, depend on concentration gradient and the average velocity with which the molecules of A moves in the x direction. The source term in the diffusion equation becomes S(\vec{r},t, \vec{r'},t')=\delta(\vec{r}-\vec{r'})\delta(t-t'), where \vec{r} is the position at which fluence rate is measured and \vec{r'} is the position of the source. For the vector form, the RTE is multiplied by direction \hat{s} before evaluation.): : \frac{\partial \Phi(\vec{r},t)}{c\partial t} + \mu_a\Phi(\vec{r},t) + abla \cdot \vec{J}(\vec{r},t) = S(\vec{r},t) : \frac{\partial \vec{J}(\vec{r},t)}{c\partial t} + (\mu_a+\mu_s')\vec{J}(\vec{r},t) + \frac{1}{3} abla \Phi(\vec{r},t) = 0 The diffusion approximation is limited to systems where reduced scattering coefficients are much larger than their absorption coefficients and having a minimum layer thickness of the order of a few transport mean free path. ===The diffusion equation=== Using the second assumption of diffusion theory, we note that the fractional change in current density \vec{J}(\vec{r},t) over one transport mean free path is negligible. Diffusion explains the net flux of molecules from a region of higher concentration to one of lower concentration. In practice, Knudsen diffusion applies only to gases because the mean free path for molecules in the liquid state is very small, typically near the diameter of the molecule itself. == Mathematical description == The diffusivity for Knudsen diffusion is obtained from the self-diffusion coefficient derived from the kinetic theory of gases: :{D_{AA*}} = {{\lambda u} \over {3}} = {{\lambda}\over{3}} \sqrt{{8R T}\over {\pi M_{A}}} For Knudsen diffusion, path length λ is replaced with pore diameter d, as species A is now more likely to collide with the pore wall as opposed with another molecule. A common approximation summarized here is the diffusion approximation. The nuclear lamina is a dense (~30 to 100 nm thick) fibrillar network inside the nucleus of eukaryote cells. The result of diffusion is a gradual mixing of material such that the distribution of molecules is uniform. Nuclear collision length is the mean free path of a particle before undergoing a nuclear reaction, for a given particle in a given medium. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale. Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. Molecular diffusion is typically described mathematically using Fick's laws of diffusion. == Applications == Diffusion is of fundamental importance in many disciplines of physics, chemistry, and biology. For an attractive interaction between particles, the diffusion coefficient tends to decrease as concentration increases. In physiology, transport maximum (alternatively Tm or Tmax) refers to the point at which increase in concentration of a substance does not result in an increase in movement of a substance across a cell membrane. For a repulsive interaction between particles, the diffusion coefficient tends to increase as concentration increases. Substituting Fick's law into the scalar representation of the RTE gives the diffusion equation: : \frac{1}{c}\frac{\partial \Phi(\vec{r},t)}{\partial t} + \mu_a\Phi(\vec{r},t) - abla \cdot [D abla\Phi(\vec{r},t)] = S(\vec{r},t) D=\frac{1}{3(\mu_a+\mu_s')} is the diffusion coefficient and μ's=(1-g)μs is the reduced scattering coefficient. * Chemical diffusion occurs in a presence of concentration (or chemical potential) gradient and it results in net transport of mass. This is the process described by the diffusion equation. thumb|300px|Schematic drawing of a molecule in a cylindrical pore in the case of Knudsen diffusion; are indicated the pore diameter () and the free path of the particle (). Generally, the diffusion approximation is less accurate as the absorption coefficient μa increases and the scattering coefficient μs decreases.

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At what pressure does the mean free path of argon at $20^{\circ} \mathrm{C}$ become comparable to the diameter of a $100 \mathrm{~cm}^3$ vessel that contains it? Take $\sigma=0.36 \mathrm{~nm}^2$

New York:McGraw-Hill, , pg 108 thumb|300px|Stress in the cylinder body of a pressure vessel. The formulae of pressure vessel design standards are extension of Lamé's theorem by putting some limit on ratio of inner radius and thickness. Stress in a thin-walled pressure vessel in the shape of a cylinder is :\sigma_\theta = \frac{pr}{t}, :\sigma_{\rm long} = \frac{pr}{2t}, where: * \sigma_\theta is hoop stress, or stress in the circumferential direction * \sigma_{long} is stress in the longitudinal direction * p is internal gauge pressure * r is the inner radius of the cylinder * t is thickness of the cylinder wall. The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. upright=1.4|thumb|A welded steel pressure vessel constructed as a horizontal cylinder with domed ends. PV Publishing, Inc. Oklahoma City, OK ==Further reading== * Megyesy, Eugene F. (2008, 14th ed.) Pressure Vessel Handbook. Therefore, pressure vessels are designed to have a thickness proportional to the radius of tank and the pressure of the tank and inversely proportional to the maximum allowed normal stress of the particular material used in the walls of the container. The normal (tensile) stress in the walls of the container is proportional to the pressure and radius of the vessel and inversely proportional to the thickness of the walls. (Pa)For a sphere the thickness d = rP/2σ, where r is the radius of the tank. thumb|upright=1.5|In inverse depth parametrization, a point is identified by its inverse depth \rho = \frac{1}{\left\Vert \mathbf{p} - \mathbf{c}_0\right\Vert} along the ray, with direction v = (\cos \phi \sin \theta, -\sin \phi, \cos \phi \cos \theta), from which it was first observed. Hydraulic (filled with water) testing pressure is usually 50% higher than the working pressure. ==== Vessel thread ==== Until 1990, high pressure cylinders were produced with conical (tapered) threads. Construction methods and materials may be chosen to suit the pressure application, and will depend on the size of the vessel, the contents, working pressure, mass constraints, and the number of items required. ISBN 978-0-626-23561-1 == See also == * * * * * * * * * * * * * - a small, inexpensive, disposable metal gas cylinder for providing pneumatic power * * * * – a device for measuring leaf water potentials * * * or Knock-out drum * * * ==Notes== ==References== * A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed. * E.P. Popov, Engineering Mechanics of Solids, 1st ed. * Megyesy, Eugene F. "Pressure Vessel Handbook, 14th Edition." File:Ресивер хладагента FP-LR-100.png|Cylindrical pressure vessel. For example, the ASME Boiler and Pressure Vessel Code (BPVC) (UG-27) formulas are: Spherical shells: Thickness has to be less than 0.356 times inner radius :\sigma_\theta = \sigma_{\rm long} = \frac{p(r + 0.2t)}{2tE} Cylindrical shells: Thickness has to be less than 0.5 times inner radius :\sigma_\theta = \frac{p(r + 0.6t)}{tE} :\sigma_{\rm long} = \frac{p(r - 0.4t)}{2tE} where E is the joint efficiency, and all others variables as stated above. However, a spherical shape is difficult to manufacture, and therefore more expensive, so most pressure vessels are cylindrical with 2:1 semi-elliptical heads or end caps on each end. For cylindrical vessels with a diameter up to 600 mm (NPS of 24 in), it is possible to use seamless pipe for the shell, thus avoiding many inspection and testing issues, mainly the nondestructive examination of radiography for the long seam if required. A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the ambient pressure. A disadvantage of these vessels is that greater diameters are more expensive, so that for example the most economic shape of a , pressure vessel might be a diameter of and a length of including the 2:1 semi-elliptical domed end caps. ===Construction materials=== thumb|200px|Composite overwrapped pressure vessel with titanium liner. For a stored gas, PV is proportional to the mass of gas at a given temperature, thus :M = {3 \over 2} nRT {\rho \over \sigma}. (see gas law) The other factors are constant for a given vessel shape and material. A vessel can be considered "thin-walled" if the diameter is at least 10 times (sometimes cited as 20 times) greater than the wall thickness.Richard Budynas, J. Nisbett, Shigley's Mechanical Engineering Design, 8th ed., For these reasons, the definition of a pressure vessel varies from country to country.

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A

The equilibrium pressure of $\mathrm{O}_2$ over solid silver and silver oxide, $\mathrm{Ag}_2 \mathrm{O}$, at $298 \mathrm{~K}$ is $11.85 \mathrm{~Pa}$. Calculate the standard Gibbs energy of formation of $\mathrm{Ag}_2 \mathrm{O}(\mathrm{s})$ at $298 \mathrm{~K}$.

Silver oxide is the chemical compound with the formula Ag2O. Silver sulfate is the inorganic compound with the formula Ag2SO4. Silver sulfite is the chemical compound with the formula Ag2SO3. In 1993, AgF2 cost between 1000-1400 US dollars per kg. ==Composition and structure== AgF2 is a white crystalline powder, but it is usually black/brown due to impurities. The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. For some time, it was doubted that silver was actually in the +2 oxidation state, rather than some combination of states such as AgI[AgIIIF4], which would be similar to silver(I,III) oxide. Such reactions often work best when the silver oxide is prepared in situ from silver nitrate and alkali hydroxide. ==References== ==External links== * Annealing of Silver Oxide – Demonstration experiment: Instruction and video Category:Silver compounds Category:Transition metal oxides The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). This reaction does not afford appreciable amounts of silver hydroxide due to the favorable energetics for the following reaction:Holleman, A. F.; Wiberg, E. "Inorganic Chemistry" Academic Press: San Diego, 2001. . :2 AgOH -> Ag2O + H2O (pK = 2.875) With suitably controlled conditions, this reaction can be used to prepare Ag2O powder with properties suitable for several uses including as a fine grained conductive paste filler. ==Structure and properties== Ag2O features linear, two-coordinate Ag centers linked by tetrahedral oxides. Silver(II) fluoride is a chemical compound with the formula AgF2. Silver sulfate and anhydrous sodium sulfate adopt the same structure. ==Silver(II) sulfate== The synthesis of silver(II) sulfate (AgSO4) with a divalent silver ion instead of a monovalent silver ion was first reported in 2010 by adding sulfuric acid to silver(II) fluoride (HF escapes). :AgNO3 + NaCl -> AgCl(v) + NaNO3 :2 AgNO3 + CoCl2 -> 2 AgCl(v) + Co(NO3)2 It can also be produced by the reaction of silver metal and aqua regia, however, the insolubility of silver chloride decelerates the reaction. Silver chloride is a chemical compound with the chemical formula AgCl. AgF and AgBr crystallize similarly.Wells, A.F. (1984) Structural Inorganic Chemistry, Oxford: Clarendon Press. . The AgI[AgIIIF4] was found to be present at high temperatures, but it was unstable with respect to AgF2. The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. In the gas phase, AgF2 is believed to have D∞h symmetry. This unstable silver compound when heated and/or in light it decomposes to silver dithionate and silver sulfate. ==Preparation== Silver sulfite can be prepared by dissolving silver nitrate with the stoichiometric quantity of sodium sulfite solution, yielding a precipitation of silver sulfite by the following reaction: :2 AgNO3 \+ Na2SO3 Ag2SO3 \+ 2 NaNO3 After precipitation then filtering silver sulfite, washing it using well-boiled water, and drying it in vacuum. ==References== Category:Silver compounds Category:Sulfites Silver usually exists in its +1 oxidation state. It is formed as an intermediate in the catalysis of gaseous reactions with fluorine by silver. It is a rare example of a silver(II) compound.

152.67

0

0.36

-11.2

6.9

D

When alkali metals dissolve in liquid ammonia, their atoms each lose an electron and give rise to a deep-blue solution that contains unpaired electrons occupying cavities in the solvent. These 'metal-ammonia solutions' have a maximum absorption at $1500 \mathrm{~nm}$. Supposing that the absorption is due to the excitation of an electron in a spherical square well from its ground state to the next-higher state (see the preceding problem for information), what is the radius of the cavity?

The problems stems from the fact that a cavity is an open non- Hermitian system with leakage and absorption. Because of these boundary conditions that must be satisfied at resonance (tangential electric fields must be zero at cavity walls), it follows that cavity length must be an integer multiple of half- wavelength at resonance.David Pozar, Microwave Engineering, 2nd edition, Wiley, New York, NY, 1998. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. Cavity perturbation theory has been initially proposed by Bethe-Schwinger in optics , and Waldron in the radio frequency domain. Because the ions overlap, their separation in the crystal will be less than the sum of their soft-sphere radii. That is, the distance between two neighboring iodides in the crystal is assumed to be twice the radius of the iodide ion, which was deduced to be 214 pm. Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. The Q factor of a resonant cavity can be calculated using cavity perturbation theory and expressions for stored electric and magnetic energy. The polarizability is related the van der Waals volume by the relation V_{\rm w} = {1\over{4\pi\varepsilon_0}}\alpha , so the van der Waals volume of helium V = = 0.2073 Å by this method, corresponding to r = 0.37 Å. The microwaves bounce back and forth between the walls of the cavity. Ionic radius, rion, is the radius of a monatomic ion in an ionic crystal structure. External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis.Montgomery, C. G. & Dicke, Robert H. & Edward M. Purcell, Principles of microwave circuits / edited by C.G. Montgomery, R.H. Dicke, E.M. Purcell, Peter Peregrinus on behalf of the Institution of Electrical Engineers, London, U.K., 1987. === Resonant frequencies === The resonant frequencies of a cavity are a function of its geometry. ==== Rectangular cavity ==== thumb|Rectangular cavity Resonance frequencies of a rectangular microwave cavity for any \scriptstyle TE_{mnl} or \scriptstyle TM_{mnl} resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. The process of gas or liquid which penetrate into the body of adsorbent is commonly known as absorption. 550px|link=https://doi.org/10.1351/goldbook.A00036|thumb|right|alt=IUPAC definition for absorption|[https://doi.org/10.1351/goldbook.A00036 https://doi.org/10.1351/goldbook.A00036]. ==Equation== If absorption is a physical process not accompanied by any other physical or chemical process, it usually follows the Nernst distribution law: :"the ratio of concentrations of some solute species in two bulk phases when it is equilibrium and in contact is constant for a given solute and bulk phases": :: \frac{[x]_{1}}{[x]_{2}} = \text{constant} = K_{N(x,12)} The value of constant KN depends on temperature and is called partition coefficient. A microwave cavity or radio frequency cavity (RF cavity) is a special type of resonator, consisting of a closed (or largely closed) metal structure that confines electromagnetic fields in the microwave or RF region of the spectrum. Cavity perturbation measurement techniques for material characterization are used in many fields ranging from physics and material science to medicine and biology.Wenquan Che; Zhanxian Wang; Yumei Chang; Russer, P.; "Permittivity Measurement of Biological Materials with Improved Microwave Cavity Perturbation Technique," Microwave Conference, 2008. In Schrödinger's quantum-mechanical theory of the hydrogen atom, the Bohr radius is the value of the radial coordinate for which the radial probability density of the electron position is highest. ;TM modes:T. Wangler, RF linear accelerators, Wiley (2008) f_{mnp}=\frac{c}{2\pi\sqrt{\mu_r\epsilon_r}} \sqrt{\left(\frac{X_{mn}}{R}\right)^2 + \left(\frac{p \pi}{L}\right)^2} ;TE modes: f_{mnp}=\frac{c}{2\pi\sqrt{\mu_r\epsilon_r}} \sqrt{\left(\frac{X'_{mn}}{R}\right)^2 + \left(\frac{p \pi}{L}\right)^2} Here, \scriptstyle X_{mn} denotes the \scriptstyle n-th zero of the \scriptstyle m-th Bessel function, and \scriptstyle X'_{mn} denotes the \scriptstyle n-th zero of the derivative of the \scriptstyle m-th Bessel function. === Quality factor === The quality factor \scriptstyle Q of a cavity can be decomposed into three parts, representing different power loss mechanisms. *\scriptstyle Q_c, resulting from the power loss in the walls which have finite conductivity *\scriptstyle Q_d, resulting from the power loss in the lossy dielectric material filling the cavity. *\scriptstyle Q_{ext}, resulting from power loss through unclosed surfaces (holes) of the cavity geometry. Ideally, the material to be measured is introduced into the cavity at the position of maximum electric or magnetic field. This frequency is given by \cdot k_{mnl}\\\ &= \frac{c}{2\pi\sqrt{\mu_r\epsilon_r}}\sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2 + \left(\frac{l\pi}{d}\right)^2}\\\ &= \frac{c}{2\sqrt{\mu_r\epsilon_r}}\sqrt{\left( \frac{m}{a}\right) ^2+\left(\frac{n}{b}\right) ^2 + \left(\frac{l}{d}\right) ^2} \end{align}|}} where \scriptstyle k_{mnl} is the wavenumber, with \scriptstyle m, \scriptstyle n, \scriptstyle l being the mode numbers and \scriptstyle a, \scriptstyle b, \scriptstyle d being the corresponding dimensions; c is the speed of light in vacuum; and \scriptstyle \mu_r and \scriptstyle \epsilon_r are relative permeability and permittivity of the cavity filling respectively. ==== Cylindrical cavity ==== thumb|Cylindrical cavity The field solutions of a cylindrical cavity of length \scriptstyle L and radius \scriptstyle R follow from the solutions of a cylindrical waveguide with additional electric boundary conditions at the position of the enclosing plates. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . The concept can be extended to solvated ions in liquid solutions taking into consideration the solvation shell. == Trends == X− NaX AgX F 464 492 Cl 564 555 Br 598 577 Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides.

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Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \mathrm{pm}$ ?

Fortunately as long as the electromagnetic field traversed by the electron changes only slowly compared with this wavelength (see typical values in matter wave#Applications of matter waves), Kirchhoff's diffraction formula applies. Electron diffraction occurs due to elastic scattering, when there is no change in the energy of the electrons during their interactions with atoms. The electron pulse undergoes diffraction as a result of interacting with the sample. For electron diffraction the electrons behave as if they are non-relativistic particles of mass m^* in terms of how they interact with the atoms. For context, the typical energy of a chemical bond is a few eV; electron diffraction involves electrons up to 5,000,000 eV. *The development of new approaches to reduce dynamical effects such as precession electron diffraction and three-dimensional diffraction methods. A fast electron beam is generated in an electron gun, enters a diffraction chamber typically at a vacuum of 10−7 mbar. *Fast and accurate methods to calculate intensities for low-energy electron diffraction so it could be used to determine atomic positions, for instance references. Electron diffraction refers to changes in the direction of electron beams due to interactions with atoms. Principles of Electron Optics. The wavelength of the electrons \lambda in vacuum is : \lambda = \frac1k = \frac{h}{\sqrt{2m^* E}}=\frac{h c}{\sqrt{E(2 m_0 c^2 + E)}}, and can range from about 0.1 nanometers, roughly the size of an atom, down to a thousandth of that. framed|Geometry of electron beam in precession electron diffraction. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. Strictly, the term electron diffraction refers to how electrons are scattered by atoms, a process that is mathematically modelled by solving forms of Schrödinger equation. # The first level of more accuracy where it is approximated that the electrons are only scattered once, which is called kinematical diffraction and is also a far-field or Fraunhofer approach. The electron particle trajectory formula matches the formula for geometrical optics with a suitable electron-optical index of refraction. The theory of precession electron diffraction is still an active area of research, and efforts to improve on the ability to correct measured intensities without a priori knowledge are ongoing. == Historical development == The first precession electron diffraction system was developed by Vincent and Midgley in Bristol, UK and published in 1994. Scheme 1 shows the schematic procedure of an electron diffraction experiment. These have efficiencies and accuracies that can be a thousand or more times that of the photographic film used in the earliest experiments, with the information available in real time rather than requiring photographic processing after the experiment. ==Basics== === Geometrical considerations === What is seen in an electron diffraction pattern depends upon the sample and also the energy of the electrons. Electron diffraction also plays a major role in the contrast of images in electron microscopes. UED can provide a wealth of dynamics on charge carriers, atoms, and molecules. ==History== The design of early ultrafast electron diffraction instruments was based on x-ray streak cameras, the first reported UED experiment demonstrating an electron pulse length of 100 ps. ==Electron Pulse Production== The electron pulses are typically produced by the process of photoemission in which a fs optical pulse is directed toward a photocathode. However, with Cs corrected microscopes, the probe can be made much smaller. === Practical considerations === Precession electron diffraction is typically conducted using accelerating voltages between 100-400 kV.

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Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \mathrm{pm}$ ?

Fortunately as long as the electromagnetic field traversed by the electron changes only slowly compared with this wavelength (see typical values in matter wave#Applications of matter waves), Kirchhoff's diffraction formula applies. For electron diffraction the electrons behave as if they are non-relativistic particles of mass m^* in terms of how they interact with the atoms. The wavelength of the electrons \lambda in vacuum is : \lambda = \frac1k = \frac{h}{\sqrt{2m^* E}}=\frac{h c}{\sqrt{E(2 m_0 c^2 + E)}}, and can range from about 0.1 nanometers, roughly the size of an atom, down to a thousandth of that. Electron diffraction occurs due to elastic scattering, when there is no change in the energy of the electrons during their interactions with atoms. For context, the typical energy of a chemical bond is a few eV; electron diffraction involves electrons up to 5,000,000 eV. Principles of Electron Optics. *Fast and accurate methods to calculate intensities for low-energy electron diffraction so it could be used to determine atomic positions, for instance references. Electron diffraction refers to changes in the direction of electron beams due to interactions with atoms. *The development of new approaches to reduce dynamical effects such as precession electron diffraction and three-dimensional diffraction methods. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. framed|Geometry of electron beam in precession electron diffraction. The electron particle trajectory formula matches the formula for geometrical optics with a suitable electron-optical index of refraction. Strictly, the term electron diffraction refers to how electrons are scattered by atoms, a process that is mathematically modelled by solving forms of Schrödinger equation. # The first level of more accuracy where it is approximated that the electrons are only scattered once, which is called kinematical diffraction and is also a far-field or Fraunhofer approach. However, with Cs corrected microscopes, the probe can be made much smaller. === Practical considerations === Precession electron diffraction is typically conducted using accelerating voltages between 100-400 kV. The theory of precession electron diffraction is still an active area of research, and efforts to improve on the ability to correct measured intensities without a priori knowledge are ongoing. == Historical development == The first precession electron diffraction system was developed by Vincent and Midgley in Bristol, UK and published in 1994. These have efficiencies and accuracies that can be a thousand or more times that of the photographic film used in the earliest experiments, with the information available in real time rather than requiring photographic processing after the experiment. ==Basics== === Geometrical considerations === What is seen in an electron diffraction pattern depends upon the sample and also the energy of the electrons. These were rapidly followed by the first non-relativistic diffraction model for electrons by Hans Bethe based upon the Schrödinger equation, which is very close to how electron diffraction is now described. Given sufficient voltage, the electron can be accelerated sufficiently fast to exhibit measurable relativistic effects. Then, the rule is that the amount of energy absorbed by an electron may allow for the electron to be promoted from a vibrational and electronic ground state to a vibrational and electronic excited state. While the wavevector increases as the energy increases, the change in the effective mass compensates this so even at the very high energies used in electron diffraction there are still significant interactions. Relativistic electron beams are streams of electrons moving at relativistic speeds.

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Nelson, et al. (Science 238, 1670 (1987)) examined several weakly bound gas-phase complexes of ammonia in search of examples in which the $\mathrm{H}$ atoms in $\mathrm{NH}_3$ formed hydrogen bonds, but found none. For example, they found that the complex of $\mathrm{NH}_3$ and $\mathrm{CO}_2$ has the carbon atom nearest the nitrogen (299 pm away): the $\mathrm{CO}_2$ molecule is at right angles to the $\mathrm{C}-\mathrm{N}$ 'bond', and the $\mathrm{H}$ atoms of $\mathrm{NH}_3$ are pointing away from the $\mathrm{CO}_2$. The magnitude of the permanent dipole moment of this complex is reported as $1.77 \mathrm{D}$. If the $\mathrm{N}$ and $\mathrm{C}$ atoms are the centres of the negative and positive charge distributions, respectively, what is the magnitude of those partial charges (as multiples of $e$ )?

Theoretically, the bond strength of the hydrogen bonds can be assessed using NCI index, non-covalent interactions index, which allows a visualization of these non-covalent interactions, as its name indicates, using the electron density of the system. Another study found a much smaller number of hydrogen bonds: 2.357 at 25 °C. Quantum chemical calculations of the relevant interresidue potential constants (compliance constants) revealed large differences between individual H bonds of the same type. van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 van der Waals radii taken from Bondi's compilation (1964). Van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 Van der Waals radii taken from Bondi's compilation (1964). Defining and counting the hydrogen bonds is not straightforward however. Scott Rowland and Robin Taylor re-examined these 1964 figures in the light of more recent crystallographic data: on the whole, the agreement was very good, although they recommend a value of 1.09 Å for the van der Waals radius of hydrogen as opposed to Bondi's 1.20 Å. As well, it has been argued that the Van der Waals radius is not a fixed property of an atom in all circumstances, rather, that it will vary with the chemical environment of the atom. == Gallery == File:3D ammoniac.PNG|alt=Ammonia, van-der-Waals-based model|Ammonia, NH3, space- filling, Van der Waal's-based representation, nitrogen (N) in blue, hydrogen (H) in white. This is slightly different from the intramolecular bound states of, for example, covalent or ionic bonds; however, hydrogen bonding is generally still a bound state phenomenon, since the interaction energy has a net negative sum. This description of the hydrogen bond has been proposed to describe unusually short distances generally observed between or . ===Structural details=== The distance is typically ≈110 pm, whereas the distance is ≈160 to 200 pm. In weaker hydrogen bonds,Desiraju, G. R. and Steiner, T. Some of Bondi's figures are given in the table at the top of this article, and they remain the most widely used "consensus" values for the van der Waals radii of the elements. ==Hydrogen bonds in small molecules== thumb|right|Crystal structure of hexagonal ice. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The coefficient E_{0} is positive and of the order V\alpha^{3}, where V is the ionization energy and \alpha is the mean atomic polarizability; the exact value of E_{0} depends on the magnitudes of the dipole matrix elements and on the energies of the p orbitals. ==References== * Category:Chemical bonding Category:Quantum mechanical potentials The ideal bond angle depends on the nature of the hydrogen bond donor. Structural details, in particular distances between donor and acceptor which are smaller than the sum of the van der Waals radii can be taken as indication of the hydrogen bond strength. 100px 120px 100px From top to bottom, azides, nitrones, and nitro compounds are examples of 1,3-dipoles. The Hydrogen Bond Franklin Classics, 2018), Jeffrey, G. A.; In organic chemistry, a 1,3-dipolar compound or 1,3-dipole is a dipolar compound with delocalized electrons and a separation of charge over three atoms. Generally, the hydrogen bond is characterized by a proton acceptor that is a lone pair of electrons in nonmetallic atoms (most notably in the nitrogen, and chalcogen groups). The strength of intramolecular hydrogen bonds can be studied with equilibria between conformers with and without hydrogen bonds.

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The NOF molecule is an asymmetric rotor with rotational constants $3.1752 \mathrm{~cm}^{-1}, 0.3951 \mathrm{~cm}^{-1}$, and $0.3505 \mathrm{~cm}^{-1}$. Calculate the rotational partition function of the molecule at $25^{\circ} \mathrm{C}$.

Nitrosyl fluoride (NOF) is a covalently bonded nitrosyl compound. ==Reactions== NOF is a highly reactive fluorinating agent that converts many metals to their fluorides, releasing nitric oxide in the process: :n NOF + M → MFn \+ n NO NOF also fluorinates fluorides to form adducts that have a salt- like character, such as NOBF4. In terms of these constants, the rotational partition function can be written in the high temperature limit as G. Herzberg, ibid, Equation (V,29) \zeta^\text{rot} \approx \frac{\sqrt{\pi} }{\sigma} \sqrt{ \frac{ (k_\text{B} T)^3 }{ A B C }} with \sigma again known as the rotational symmetry number G. Herzberg, ibid; see Table 140 for values for common molecular point groups which in general equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms. In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. == Definition == The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions \zeta:Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 Z = \frac{ \zeta^N }{ N! } with: \zeta = \sum_j g_j e^{ -E_j / k_\text{B} T} , where g_j is the degeneracy of the j-th quantum level of an individual particle, k_\text{B} is the Boltzmann constant, and T is the absolute temperature of system. NF4F.jpg|NF4+F− R3m structure NF4-2NF6F.jpg|(NF4+)2NF6−F− I4/m structure NF4NF6.jpg|NF4+NF6− P4/n structure ==Covalent molecule== thumb|left|upright=1.55|Possible structures of NF5 and analogous fluorohydrides For a NF5 molecule to form, five fluorine atoms have to be arranged around a nitrogen atom. In the high temperature limit, it is traditional to correct for the missing nuclear spin states by dividing the rotational partition function by a factor \sigma = 2 with \sigma known as the rotational symmetry number which is 2 for linear molecules with a center of symmetry and 1 for linear molecules without. ==Nonlinear molecules== A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written A, B, and C , which can often be determined by rotational spectroscopy. Nitrogen pentafluoride (NF5) is a theoretical compound of nitrogen and fluorine that is hypothesized to exist based on the existence of the pentafluorides of the atoms below nitrogen in the periodic table, such as phosphorus pentafluoride. Calculations show that the NF5 molecule is thermodynamically favourably inclined to form NF4 and F radicals with energy 36 kJ/mol and a transition barrier around 67–84 kJ/mol. Nitrogen pentafluoride also violates the octet rule in which compounds with eight outer shell electrons are particularly stable. ==References== Category:Nitrogen fluorides Category:Hypothetical chemical compounds Category:Nitrogen(V) compounds For a diatomic molecule like CO or HCl, or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are E_J^\text{rot} = \frac{\mathbf{J}^2}{2I} = \frac{J(J+1)\hbar^2}{2I} = J(J+1)B. J is the quantum number for total rotational angular momentum and takes all integer values starting at zero, i.e., J = 0,1,2, \ldots, B = \frac{\hbar^2}{2I} is the rotational constant, and I is the moment of inertia. N-Fluoropyridinium triflate is an organofluorine compound with the formula [C5H5NF]O3SCF3. In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is k_\text{B} T . == Quantum symmetry effects == For a diatomic molecule with a center of symmetry, such as \rm H_2, N_2, CO_2, or \mathrm{ H_2 C_2} (i.e. D_{\infty h} point group), rotation of a molecule by \pi radian about an axis perpendicular to the molecule axis and going through the center of mass will interchange pairs of equivalent atoms. Theoretical models of the nitrogen pentafluoride molecule are either a trigonal bipyramidal covalently bound molecule with symmetry group D3h, or NFF−, which would be an ionic solid. ==Ionic solid== A variety of other tetrafluoroammonium salts are known (NFX−), as are fluoride salts of other ammonium cations ). Let each rotating molecule be associated with a unit vector \hat{n}; for example, \hat{n} might represent the orientation of an electric or magnetic dipole moment. Another convenient expression for the rotational partition function for symmetric and asymmetric tops is provided by Gordy and Cook: \zeta^\text{rot} \approx \frac{5.34 \times 10^6}{\sigma} \sqrt{ \frac{T^3}{A B C}} where the prefactor comes from \sqrt{\frac{(\pi k_\text{B}) ^3}{h^3}} = 5.34 \times 10^6 when A, B, and C are expressed in units of MHz. The expressions for \zeta^\text{rot} works for asymmetric, symmetric and spherical top rotors. ==References== ==See also== * Translational partition function * Vibrational partition function * Partition function (mathematics) Category:Equations of physics Category:Partition functions The mean thermal rotational energy per molecule can now be computed by taking the derivative of \zeta^\text{rot} with respect to temperature T. Molecule \theta_{\mathrm{R}} (K)P. Atkins and J. de Paula "Physical Chemistry", 9th edition (W.H. Freeman 2010), Table 13.2, Data section in appendix H2 87.6 N2 2.88 O2 2.08 F2 1.27 HF 30.2 HCl 15.2 CO2 0.561P. Dominik Kurzydłowski and Patryk Zaleski-Ejgierd predict that a mixture of fluorine and nitrogen trifluoride under pressure between 10 and 33 GPa forms NFF− with space group R3m. This has CAS number 71485-49-9.Tetrafluoroammonium bifluoride I. J. Solomon believed that nitrogen pentafluoride was produced by the thermal decomposition of NF4AsF6, but experimental results were not reproduced. thumb|A molecule with a red cross on its front undergoing 3 dimensional rotational diffusion. Therefore, there should be three rotational diffusion constants - the eigenvalues of the rotational diffusion tensor - resulting in five rotational time constants. It is a salt, consisting of the N-fluoropyridinium cation ([C5H5NF]+) and the triflate anion. The characteristic rotational temperature ( or ) is commonly used in statistical thermodynamics to simplify the expression of the rotational partition function and the rotational contribution to molecular thermodynamic properties.

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Suppose that $2.5 \mathrm{mmol} \mathrm{N}_2$ (g) occupies $42 \mathrm{~cm}^3$ at $300 \mathrm{~K}$ and expands isothermally to $600 \mathrm{~cm}^3$. Calculate $\Delta G$ for the process.

For an ideal gas, the change in entropyTipler, P., and Mosca, G. Physics for Scientists and Engineers (with modern physics), 6th edition, 2008. pages 602 and 647. is the same as for isothermal expansion where all heat is converted to work: \Delta S = \int_\text{i}^\text{f} dS = \int_{V_\text{i}}^{V_\text{f}} \frac{P\,dV}{T} = \int_{V_\text{i}}^{V_\text{f}} \frac{n R\,dV}{V} = n R \ln \frac{V_\text{f}}{V_\text{i}} = N k_\text{B} \ln \frac{V_\text{f}}{V_\text{i}}. For the above defined path we have that and thus , and hence the increase in entropy for the Joule expansion is \Delta S=\int_i^f\mathrm{d}S=\int_{V_0}^{2V_0} \frac{P\,\mathrm{d}V}{T}=\int_{V_0}^{2V_0} \frac{n R\,\mathrm{d}V}{V}=n R\ln 2. As a result, the change in internal energy, \Delta U, is zero. During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. Negative and positive thermal expansion hereby compensate each other to a certain amount if the temperature is changed. At room temperature (25 °C, or 298.15 K) 1 kJ·mol−1 is approximately equal to 0.4034 k_B T. == References == Category:SI derived units In geochemistry, hydrology, paleoclimatology and paleoceanography, δ15N (pronounced "delta fifteen n") or delta-N-15 is a measure of the ratio of the two stable isotopes of nitrogen, 15N:14N. ==Formulas== Two very similar expressions for are in wide use in hydrology. Thermodynamics, p. 414. The best method (i.e. the method involving the least work) is that of a reversible isothermal compression, which would take work given by W = -\int_{2V_0}^{V_0} P\,\mathrm{d}V = - \int_{2V_0}^{V_0} \frac{nRT}{V} \mathrm{d}V = nRT\ln 2 = T \Delta S_\text{gas}. From these initial measurements, Gibbs free energy changes (\Delta G) and entropy changes (\Delta S) can be determined using the relationship: ::: \Delta G = -RT\ln{K_a} = \Delta H -T\Delta S (where R is the gas constant and T is the absolute temperature). In thermodynamics, Stefan's formula says that the specific surface energy at a given interface is determined by the respective enthalpy difference \scriptstyle \Delta H^*. : \sigma = \gamma_0 \left( \frac{\Delta H^*}{N_\text{A}^{1/3}V_\text{m}^{2/3}}\right), where σ is the specific surface energy, NA is the Avogadro constant, \gamma_0 is a steric dimensionless coefficient, and Vm is the molar volume. ==References== Category:Thermodynamic equations Category:Chemical thermodynamics For example, the Gibbs free energy of a compound in the area of thermochemistry is often quantified in units of kilojoules per mole (symbol: kJ·mol−1 or kJ/mol), with 1 kilojoule = 1000 joules. Since 1 mole = 6.02214076 particles (atoms, molecules, ions etc.), 1 joule per mole is equal to 1 joule divided by 6.02214076 particles, ≈1.660539 joule per particle. The joule per mole (symbol: J·mol−1 or J/mol) is the unit of energy per amount of substance in the International System of Units (SI), such that energy is measured in joules, and the amount of substance is measured in moles. Negative thermal expansion (NTE) is an unusual physicochemical process in which some materials contract upon heating, rather than expand as most other materials do. thumb|240px|The Joule expansion, in which a volume is expanded to a volume in a thermally isolated chamber. thumb|180px|A free expansion of a gas can be achieved by moving the piston out faster than the fastest molecules in the gas. It is also an SI derived unit of molar thermodynamic energy defined as the energy equal to one joule in one mole of substance. Thus, at low temperatures the Joule expansion process provides information on intermolecular forces. ===Ideal gases=== If the gas is ideal, both the initial (T_{\mathrm{i}}, P_{\mathrm{i}}, V_{\mathrm{i}}) and final (T_{\mathrm{f}}, P_{\mathrm{f}}, V_{\mathrm{f}}) conditions follow the Ideal Gas Law, so that initially P_{\mathrm{i}} V_{\mathrm{i}} = n R T_{\mathrm{i}} and then, after the tap is opened, P_{\mathrm{f}} V_{\mathrm{f}} = n R T_{\mathrm{f}}. Thus in 2D and 3D negative thermal expansion in close-packed systems with pair interactions is realized even when the third derivative of the potential is zero or even negative. For a monatomic ideal gas , with the molar heat capacity at constant volume. Isothermal titration calorimetry for chiral chemistry. Therefore, the sign of thermal expansion coefficient is determined by the sign of the third derivative of the potential.

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Calculate the standard potential of the $\mathrm{Ce}^{4+} / \mathrm{Ce}$ couple from the values for the $\mathrm{Ce}^{3+} / \mathrm{Ce}$ and $\mathrm{Ce}^{4+} / \mathrm{Ce}^{3+}$ couples.

Cerium(III) sulfate, also called cerous sulfate, is an inorganic compound with the formula Ce2(SO4)3. 94 Ceti (HD 19994) is a trinary star system approximately 73 light-years away in the constellation Cetus. 94 Ceti A is a yellow-white dwarf star with about 1.3 times the mass of the Sun while 94 Ceti B and C are red dwarf stars. Cerianite-(Ce) is a relatively rare oxide mineral, belonging to uraninite group with the formula .Graham, A.R., 1955. The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . * After dividing by the number of electrons, the standard potential E° is related to the standard Gibbs free energy of formation ΔGf° by: E = \frac{\sum \Delta G_\text{left}-\sum \Delta G_\text{right}}{F} where F is the Faraday constant. thumb|upright=0.4|right|Ceres-1 Galactic Energy () is a Chinese private space launch enterprise flying the Ceres-1 and developing the Pallas-1 and 2 orbital rockets. {{Chembox | ImageFile = | ImageSize = | ImageAlt = | IUPACName = Triiodocerium | OtherNames = Cerous triiodide, Cerium triiodide | Section1 = | Section2 = | Section3 = | Section4 = }} Cerium(III) iodide (CeI3) is the compound formed by cerium(III) cations and iodide anions. == Preparation == Cerium metal reacts with iodine when heated to form cerium(III) iodide: : It is also formed when cerium reacts with mercury(II) iodide at high temperatures: : == Structure == Cerium(III) iodide adopts the plutonium(III) bromide crystal structure. For example: : \+ Cu(s) ( = +0.520 V) Cu + 2 Cu(s) ( = +0.337 V) Cu + ( = +0.159 V) :Calculating the potential using Gibbs free energy ( = 2 – ) gives the potential for as 0.154 V, not the experimental value of 0.159 V. : __TOC__ ==Table of standard electrode potentials== Legend: (s) - solid; (l) - liquid; (g) - gas; (aq) - aqueous (default for all charged species); (Hg) - amalgam; bold - water electrolysis equations. Cerianite-(Ce) associates with minerals of the apatite group, bastnäsite-group minerals, calcite, feldspar, "fluocerite", "hydromica", ilmenite, nepheline, magnetite, "törnebohmite" and tremolite. It is most stable if its inclination is either 65 or 115, ± 3. ==See also== * 79 Ceti * 81 Ceti * Lists of exoplanets ==References== ==External links== * SolStation: 94 Ceti 2 + orbits * 94 Ceti by Professor Jim Kaler. It contains 8-coordinate bicapped trigonal prismatic Ce3+ ions. == Uses == Cerium(III) iodide is used as a pharmaceutical intermediate and as a starting material for organocerium compounds. ==References== Category:Cerium(III) compounds Category:Iodides Category:Lanthanide halides The temperature of this dust is 40 K. ==Stellar system== This system is a hierarchical triple star system with 94 Ceti A being orbited by 94 Ceti BC, a pair of M dwarfs, in 2000 years. 94 Ceti B and C meanwhile orbit each other in a 1-year orbit. ==Planetary system== On 7 August 2000, a planet was announced by the Geneva Extrasolar Planet Search team as a result of radial velocity measurements taken with the Swiss 1.2-metre Leonhard Euler Telescope at La Silla Observatory in Chile. Cerianite CeO2: a new rare-earth oxide mineral. It is the most simple cerium mineral known. ==Notes on chemistry== Beside thorium cerianite-(Ce) may contain trace niobium, yttrium, lanthanum, ytterbium, zirconium and tantalum. ==Crystal structure== For details on crystal structure see cerium(IV) oxide. American Mineralogist 40, 560-564 It is one of a few currently known minerals containing essential tetravalent cerium, the other examples being stetindite and dyrnaesite-(La). ==Occurrence and association== Cerianite-(Ce) is associated with alkaline rocks, mostly nepheline syenites. :The Nernst equation will then give potentials at concentrations, pressures, and temperatures other than standard. The first launch of Ceres-1 took place at 7 November 2020, successfully placing the Tianqi 11 (also transcribed Tiange, also known as TQ 11, and Scorpio 1, COSPAR 2020-080A) satellite in orbit. On 6 December 2021, Galactic Energy launched its second Ceres-1 rocket, becoming the first Chinese private firm to reach orbit twice. All of the reactions should be divided by the stoichiometric coefficient for the electron to get the corresponding corrected reaction equation. The relation in electrode potential of metals in saltwater (as electrolyte) is given in the galvanic series. Using three Pallas-1 booster cores as its first stage, Pallas-2 will be capable of putting a 14-tonne payload into low Earth orbit. == Marketplace == Galactic Space is in competition with several other Chinese space rocket startups, being LandSpace, LinkSpace, ExPace, i-Space, OneSpace and Deep Blue Aerospace. == Launches == Rocket & Serial Date Payload Orbit Launch Site Outcome Notes Ceres-1 Y1 7 November 2020, 07:12 UTC Tianqi-11 (Scorpio-1) SSO Jiuquan First flight of Ceres-1. Cerium (III) sulfate tetrahydrate is a white solid that releases its water of crystallisation at 220 °C.

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An effusion cell has a circular hole of diameter $1.50 \mathrm{~mm}$. If the molar mass of the solid in the cell is $300 \mathrm{~g} \mathrm{~mol}^{-1}$ and its vapour pressure is $0.735 \mathrm{~Pa}$ at $500 \mathrm{~K}$, by how much will the mass of the solid decrease in a period of $1.00 \mathrm{~h}$ ?

If a small area A on the container is punched to become a small hole, the effusive flow rate will be \begin{align} Q_\text{effusion} &= J_\text{impingement} \times A \\\ &= \frac{P A}{\sqrt{2 \pi m k_{B} T}} \\\ &= \frac{P A N_A}{\sqrt{2 \pi M R T}} \end{align} where M is the molar mass, N_A is the Avogadro constant, and R = N_A k_B is the gas constant. The vapor slowly effuses through a pinhole, and the loss of mass is proportional to the vapor pressure and can be used to determine this pressure. Assuming the pressure difference between the two sides of the barrier is much smaller than P_{\rm avg}, the average absolute pressure in the system (i.e. \Delta P\ll P_{\rm avg}), it is possible to express effusion flow as a volumetric flow rate as follows: :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi k_BT}{32m}} or :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi RT}{32M}} where \Phi_V is the volumetric flow rate of the gas, P_{\rm avg} is the average pressure on either side of the orifice, and d is the hole diameter. ==Effect of molecular weight== At constant pressure and temperature, the root-mean-square speed and therefore the effusion rate are inversely proportional to the square root of the molecular weight. The rate \Phi_N at which a gas of molar mass M effuses (typically expressed as the number of molecules passing through the hole per second) is thenPeter Atkins and Julio de Paula, Physical Chemistry (8th ed., W.H.Freeman 2006) p.756 : \Phi_N = \frac{\Delta PAN_A}{\sqrt{2\pi MRT}}. The change in mass is the amount that flows after crossing the boundary for some time duration, not the initial amount of mass at the boundary minus the final amount at the boundary, since the change in mass flowing through the area would be zero for steady flow. ==Alternative equations== thumb|245x245px|Illustration of volume flow rate. Thus, the faster the gas particles are moving, the more likely they are to pass through the effusion orifice. ==Knudsen effusion cell== The Knudsen effusion cell is used to measure the vapor pressures of a solid with very low vapor pressure. The effusion rate for a gas depends directly on the average velocity of its particles. Considering flow through porous media, a special quantity, superficial mass flow rate, can be introduced. Hence, a relation exists between the Spalding mass transfer number and the Spalding heat transfer number and writes: :B_T=\left( 1+B_M\right)^{\frac{1}{Le}\frac{C_{p,F}}{C_{p,g}}}-1 where: * Le is the gas film Lewis number (-) * C_{p,g} is the gas film specific heat at constant pressure (J.Kg−1.K−1) The droplet vaporization rate can be expressed as a function of the Sherwood number. As a consequence, the vaporization rate increases with the droplet Reynolds number. In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the mean free path of the molecules.K.J. Laidler and J.H. Meiser, Physical Chemistry, Benjamin/Cummings 1982, p.18. Such a solid forms a vapor at low pressure by sublimation. Gases with a lower molecular weight effuse more rapidly than gases with a higher molecular weight, so that the number of lighter molecules passing through the hole per unit time is greater. ===Graham's law=== Scottish chemist Thomas Graham (1805–1869) found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles. ==Effusion into vacuum== Effusion from an equilibrated container into outside vacuum can be calculated based on kinetic theory. Droplet vaporization model for spray combustion calculations, Int. J. Heat Mass Transfer, Vol. 32, No. 9, pp. 1605-1618. In other words, the ratio of the rates of effusion of two gases at the same temperature and pressure is given by the inverse ratio of the square roots of the masses of the gas particles. : {\mbox{Rate of effusion of gas}_1 \over \mbox{Rate of effusion of gas}_2}=\sqrt{M_2 \over M_1} where M_1 and M_2 represent the molar masses of the gases. Here \Delta P is the gas pressure difference across the barrier, A is the area of the hole, N_A is the Avogadro constant, R is the gas constant and T is the absolute temperature. The average velocity of effused particles is \begin{align} \overline{v_x}&=\overline{v_y}=0\\\ \overline{v_z}&=\sqrt{\frac{\pi k_BT}{2m}}. \end{align} Combined with the effusive flow rate, the recoil/thrust force on the system itself is F=m\overline{v_z}{\times}Q_\text{effusion}=\frac{PA}{2}. As the radius R of the sphere shrinks, the diameter of the cylinder must also shrink in order that h can remain the same. The conservation equation of mass simplifies to: :\rho_g r^2 u = cte = \left( \rho_g r^2 u\right)_s = \frac{\dot{m}_F}{4\pi} Combining the conservation equations for mass and fuel vapor mass fraction the following differential equation for the fuel vapor mass fraction Y_F(r) is obtained: :4 \pi r^2 \rho_g \mathcal{D} \frac{\mathrm{d}Y_F(r)}{\mathrm{d}r} = \dot{m}_F \left( Y_F(r)-1\right) Integrating this equation between r and the ambient gas phase region r = \infty and applying the boundary condition at r=r_d gives the expression for the droplet vaporization rate: :\dot{m}_F = 4 \pi \rho_g \mathcal{D} r_d \ln \left(1+ B_M \right) and :B_M=\frac{Y_{F,\infty}-Y_{F,s}}{Y_{F,s}-1} where: * B_M is the Spalding mass transfer number Phase equilibrium is assumed at the droplet surface and the mole fraction of fuel vapor at the droplet surface is obtained via the use of the Clapeyron's equation. A comparison of vaporization models in spray calculations, AIAA Journal, Vol. 22, No 10, p. 1448. for its balance between computational costs and accuracy. The vaporizing droplet (droplet vaporization) problem is a challenging issue in fluid dynamics.

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The speed of a certain proton is $6.1 \times 10^6 \mathrm{~m} \mathrm{~s}^{-1}$. If the uncertainty in its momentum is to be reduced to 0.0100 per cent, what uncertainty in its location must be tolerated?

However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. One way to quantify the precision of the position and momentum is the standard deviation σ. H_x+H_p \approx 0.69 + 0.53 = 1.22 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.31 ==Uncertainty relation with three angular momentum components== For a particle of spin-j the following uncertainty relation holds \sigma_{J_x}^2+\sigma_{J_y}^2+\sigma_{J_z}^2\ge j, where J_l are angular momentum components. H_p = -\sum_{j=-\infty}^\infty \operatorname P[p_j] \ln \operatorname P[p_j] = -\operatorname P[p_0] \ln \operatorname P[p_0]-2 \cdot \sum_{j=1}^{\infty} \operatorname P[p_j] \ln \operatorname P[p_j] \approx 0.53 The entropic uncertainty is indeed larger than the limiting value. For all n=1, \, 2, \, 3,\, \ldots, the quantity \sqrt{\frac{n^2\pi^2}{3}-2} is greater than 1, so the uncertainty principle is never violated. The uncertainty principle also says that eliminating uncertainty about position maximizes uncertainty about momentum, and eliminating uncertainty about momentum maximizes uncertainty about position. Uncertainty of measurement results. For numerical concreteness, the smallest value occurs when n = 1, in which case \sigma_x \sigma_p = \frac{\hbar}{2} \sqrt{\frac{\pi^2}{3}-2} \approx 0.568 \hbar > \frac{\hbar}{2}. ===Constant momentum=== 360 px|thumb|right|Position space probability density of an initially Gaussian state moving at minimally uncertain, constant momentum in free space Assume a particle initially has a momentum space wave function described by a normal distribution around some constant momentum p0 according to \varphi(p) = \left(\frac{x_0}{\hbar \sqrt{\pi}} \right)^{1/2} \exp\left(\frac{-x_0^2 (p-p_0)^2}{2\hbar^2}\right), where we have introduced a reference scale x_0=\sqrt{\hbar/m\omega_0}, with \omega_0>0 describing the width of the distribution—cf. nondimensionalization. * Grabe, M ., Measurement Uncertainties in Science and Technology, Springer 2005. Evaluating the Uncertainty of Measurement. In his celebrated 1927 paper, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik" ("On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics"), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, but he did not give a precise definition for the uncertainties Δx and Δp. Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Under the above definition, the entropic uncertainty relation is H_x + H_p > \ln\left(\frac{e}{2}\right)-\ln\left(\frac{\delta x \delta p}{h} \right). Entropic uncertainty of the normal distribution We demonstrate this method on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. In doing so, we find the momentum of the particle to > arbitrary accuracy by conservation of momentum. In physics, the proton-to-electron mass ratio, μ or β, is the rest mass of the proton (a baryon found in atoms) divided by that of the electron (a lepton found in atoms), a dimensionless quantity, namely: :μ = The number in parentheses is the measurement uncertainty on the last two digits, corresponding to a relative standard uncertainty of ==Discussion== μ is an important fundamental physical constant because: * Baryonic matter consists of quarks and particles made from quarks, like protons and neutrons. * Introduction to evaluating uncertainty of measurement * JCGM 200:2008. In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. The measurement uncertainty strongly depends on the size of the measured value itself, e.g. amplitude-proportional. == See also == * measurement uncertainty * uncertainty * accuracy and precision * confidence Interval Category:Measurement The combination of these trade-offs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant. The formula for cutting off the calculated value of is :U=\min \left\\{ ~9, ~ \max \Bigl\\{ \; 0, \; \left\lfloor 9\cdot\frac{\log r}{\;\log 648{,}000\;} \right\rfloor + 1 \; \Bigr\\} ~ \right\\} For instance: As of 10 September 2016, Ceres technically has an uncertainty of around −2.6, but is instead displayed as the minimal 0. The U value should not be used as a predictor for the uncertainty in the future motion of near-Earth objects. == Orbital uncertainty == Classical Kuiper belt objects 40–50 AU from the Sun JPL SBDB Uncertainty parameter Horizons January 2018 Uncertainty in distance from the Sun (millions of kilometers) Object Reference Ephemeris Location: @sun Table setting: 39 0 ±0.01 (134340) Pluto E2022-J69 1 ±0.04 2 ±0.14 20000 Varuna 3 ±0.84 19521 Chaos 4 ±1.4 5 ±8.2 6 ±70. 7 ±190. 8 ±590. 9 ±1,600. 1995 GJ ‘D’ Data insufficient for orbit determination.

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It is possible to produce very high magnetic fields over small volumes by special techniques. What would be the resonance frequency of an electron spin in an organic radical in a field of $1.0 \mathrm{kT}$ ?

More fundamental to the radical-pair mechanism, however, is the fact that radical-pair electrons both have spin, short for spin angular momentum, which gives each separate radical a magnetic moment. Magnetic isotope effects arise when a chemical reaction involves spin- selective processes, such as the radical pair mechanism. "Normal" high field NMR relies on the detection of spin- precession with inductive detection with a simple coil. Most commonly demonstrated in reactions of organic compounds involving radical intermediates, a magnetic field can speed up a reaction by decreasing the frequency of reverse reactions. === History === The radical-pair mechanism emerged as an explanation to CIDNP and CIDEP and was proposed in 1969 by Closs; Kaptein and Oosterhoff. === Radicals and radical- pairs === thumb|Example radical: Structure of Hydrocarboxyl radical, lone electron indicated as single black dot |145x145px A radical is a molecule with an odd number of electrons, and is induced in a variety of ways, including ultra-violet radiation. Therefore, spin states can be altered by magnetic fields. === Singlet and triplet spin states === The radical-pair is characterized as triplet or singlet by the spin state of the two lone electrons, paired together. The radical-pair mechanism explains how external magnetic fields can prevent radical-pair recombination with Zeeman interactions, the interaction between spin and an external magnetic field, and shows how a higher occurrence of the triplet state accelerates radical reactions because triplets can proceed only to products, and singlets are in equilibrium with the reactants as well as with the products. Spin chemistry is a sub-field of chemistry and physics, positioned at the intersection of chemical kinetics, photochemistry, magnetic resonance and free radical chemistry, that deals with magnetic and spin effects in chemical reactions. Low field NMR spans a range of different nuclear magnetic resonance (NMR) modalities, going from NMR conducted in permanent magnets, supporting magnetic fields of a few tesla (T), all the way down to zero field NMR, where the Earth's field is carefully shielded such that magnetic fields of nanotesla (nT) are achieved where nuclear spin precession is close to zero. Spinhenge@home was a volunteer computing project on the BOINC platform, which performs extensive numerical simulations concerning the physical characteristics of magnetic molecules. Spin chemistry concerns phenomena such as chemically induced dynamic nuclear polarization (CIDNP), chemically induced electron polarization (CIDEP), magnetic isotope effects in chemical reactions, and it is hypothesized to be key in the underlying mechanism for avian magnetoreception and consciousness. == Radical-pair mechanism == The radical-pair mechanism explains how a magnetic field can affect reaction kinetics by affecting electron spin dynamics. {{Chembox | ImageFile = File:Kölsch Radical V.1.svg | ImageCaption = Two resonance forms showing the predominant locations of the unpaired electron at the 1 and 3 positions | ImageSize = | ImageAlt = | PIN = 9-[(9H-Fluoren-9-ylidene)(phenyl)methyl]-9H-fluoren-9-yl | OtherNames = | Section1 = | Section2 = | Section3 = }} The Koelsch radical (also known as Koelsch's radical and 1,3-bisdiphenylene-2-phenylallyl or α,γ-bisdiphenylene- β-phenylallyl, abbreviated BDPA) is a chemical compound that is an unusually stable carbon-centered radical, due to its resonance structures. ==Properties== BDPA is an unusually stable radical compound due to the extent to which its electrons are delocalized through resonance structures. Zeeman interactions can “flip” only one of the radical's electron's spin if the radical-pair is anisotropic, thereby converting singlet radical-pairs to triplets. thumb|Typical Reaction Scheme of the Radical-pair Mechanism, which shows the effect of alternative product formation from singlet versus triplet radical-pairs. The spin relationship is such that the two unpaired electrons, one in each radical molecule, may have opposite spin (singlet; anticorrelated), or the same spin (triplet; correlated). Electron spin resonance can be employed to quantify the probe's concentration. ==References== Category:Molecular physics A spin probe is a molecule with stable free radical character that carries a functional group. Furthermore, the spin of each electron previously involved in the bond is conserved, which means that the radical- pair now formed is a singlet (each electron has opposite spin, as in the origin bond). Low field NMR also includes Earth's field NMR where simply the Earth's magnetic field is exploited to cause nuclear spin-precession which is detected. The Zeeman and Hyperfine Interactions take effect in the yellow box, denoted as step 4 in the process|367x367px The Zeeman interaction is an interaction between spin and external magnetic field, and is given by the equation :\Delta E=h u_L=g\mu_BB, where \Delta E is the energy of the Zeeman interaction, u_L is the Larmor frequency, B is the external magnetic field, \mu_B is the Bohr magneton, h is Planck's constant, and g is the g-factor of a free electron, 2.002319, which is slightly different in different radicals. In a broad sense, Low-field NMR is the branch of NMR that is not conducted in superconducting high-field magnets. It has been observed that migratory birds lose their navigational abilities in such conditions where the Zeeman interaction is obstructed in radical-pairs. == External links == * Spin chemistry portal ==References== Category:Physical chemistry Category:Nuclear magnetic resonance It is common to see the Zeeman interaction formulated in other ways. === Hyperfine interactions === Hyperfine interactions, the internal magnetic fields of local magnetic isotopes, play a significant role as well in the spin dynamics of radical-pairs. === Zeeman interactions and magnetoreception === Because the Zeeman interaction is a function of magnetic field and Larmor frequency, it can be obstructed or amplified by altering the external magnetic or the Larmor frequency with experimental instruments that generate oscillating fields. The project began beta testing on September 1, 2006 and used the Metropolis Monte Carlo algorithm to calculate and simulate spin dynamics in nanoscale molecular magnets.

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A particle of mass $1.0 \mathrm{~g}$ is released near the surface of the Earth, where the acceleration of free fall is $g=8.91 \mathrm{~m} \mathrm{~s}^{-2}$. What will be its kinetic energy after $1.0 \mathrm{~s}$. Ignore air resistance?

The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. For example, for a speed of the correction to the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a Newtonian kinetic energy of 5 GJ/kg). Kinetic energy is the movement energy of an object. In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion., Chapter 1, p. 9 It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. The kinetic energy of an object is related to its momentum by the equation: :E_\text{k} = \frac{p^2}{2m} where: *p is momentum *m is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass m, whose center of mass is moving in a straight line with speed v, as seen above is equal to : E_\text{t} = \frac{1}{2} mv^2 where: *m is the mass of the body *v is the speed of the center of mass of the body. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. Factoring out the rest energy gives: :E_\text{k} = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is \frac{1}{2}mv^2. This is done by binomial approximation or by taking the first two terms of the Taylor expansion for the reciprocal square root: :E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right) - m c^2 = \frac{1}{2} m v^2 So, the total energy E_k can be partitioned into the rest mass energy plus the Newtonian kinetic energy at low speeds. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e., :Fs = \frac{1}{2} mv^2 Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. "Falling" is a song by industrial rock band Gravity Kills from the album Perversion, released by TVT Records in 1998. ==Release== "Falling" reached No. 35 on Billboard's Mainstream Rock chart on July 4, 1998. The energy is not destroyed; it has only been converted to another form by friction. Notice that this can be obtained by replacing p by \hat p in the classical expression for kinetic energy in terms of momentum, :E_\text{k} = \frac{p^2}{2m}. The mathematical by- product of this calculation is the mass–energy equivalence formula—the body at rest must have energy content :E_\text{rest} = E_0 = m c^2 At a low speed (v ≪ c), the relativistic kinetic energy is approximated well by the classical kinetic energy. Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. Thus for a stationary observer (v = 0) :u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{tt}}} and thus the kinetic energy takes the form :E_\text{k} = -m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,. Sir William Thomson and Professor Tait have lately substituted the word 'kinetic' for 'actual. ==Overview== Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. Then kinetic energy is the total relativistic energy minus the rest energy: E_K = E - m_{0}c^2 = \sqrt{(p \textrm c)^2 + \left(m_0 \textrm c^2\right)^2}- m_{0}c^2 At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy. thumb|450px|center| Log of relativistic kinetic energy versus log relativistic momentum, for many objects of vastly different scales. For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as :E_\text{k} = \frac{1}{2} \cdot 80 \,\text{kg} \cdot \left(18 \,\text{m/s}\right)^2 = 12,960 \,\text{J} = 12.96 \,\text{kJ} When a person throws a ball, the person does work on it to give it speed as it leaves the hand. Substituting, we get:Physics notes - Kinetic energy in the CM frame .

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A particle of mass $1.0 \mathrm{~g}$ is released near the surface of the Earth, where the acceleration of free fall is $g=8.91 \mathrm{~m} \mathrm{~s}^{-2}$. What will be its kinetic energy after $1.0 \mathrm{~s}$. Ignore air resistance?

The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. For example, for a speed of the correction to the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a Newtonian kinetic energy of 5 GJ/kg). Kinetic energy is the movement energy of an object. In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion., Chapter 1, p. 9 It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. The kinetic energy of an object is related to its momentum by the equation: :E_\text{k} = \frac{p^2}{2m} where: *p is momentum *m is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass m, whose center of mass is moving in a straight line with speed v, as seen above is equal to : E_\text{t} = \frac{1}{2} mv^2 where: *m is the mass of the body *v is the speed of the center of mass of the body. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is \frac{1}{2}mv^2. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e., :Fs = \frac{1}{2} mv^2 Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. Factoring out the rest energy gives: :E_\text{k} = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,. This is done by binomial approximation or by taking the first two terms of the Taylor expansion for the reciprocal square root: :E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right) - m c^2 = \frac{1}{2} m v^2 So, the total energy E_k can be partitioned into the rest mass energy plus the Newtonian kinetic energy at low speeds. The energy is not destroyed; it has only been converted to another form by friction. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. "Falling" is a song by industrial rock band Gravity Kills from the album Perversion, released by TVT Records in 1998. ==Release== "Falling" reached No. 35 on Billboard's Mainstream Rock chart on July 4, 1998. Notice that this can be obtained by replacing p by \hat p in the classical expression for kinetic energy in terms of momentum, :E_\text{k} = \frac{p^2}{2m}. The mathematical by- product of this calculation is the mass–energy equivalence formula—the body at rest must have energy content :E_\text{rest} = E_0 = m c^2 At a low speed (v ≪ c), the relativistic kinetic energy is approximated well by the classical kinetic energy. Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. Thus for a stationary observer (v = 0) :u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{tt}}} and thus the kinetic energy takes the form :E_\text{k} = -m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,. Sir William Thomson and Professor Tait have lately substituted the word 'kinetic' for 'actual. ==Overview== Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. Then kinetic energy is the total relativistic energy minus the rest energy: E_K = E - m_{0}c^2 = \sqrt{(p \textrm c)^2 + \left(m_0 \textrm c^2\right)^2}- m_{0}c^2 At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy. thumb|450px|center| Log of relativistic kinetic energy versus log relativistic momentum, for many objects of vastly different scales. For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as :E_\text{k} = \frac{1}{2} \cdot 80 \,\text{kg} \cdot \left(18 \,\text{m/s}\right)^2 = 12,960 \,\text{J} = 12.96 \,\text{kJ} When a person throws a ball, the person does work on it to give it speed as it leaves the hand. Acceleration due to gravity, acceleration of gravity or gravity acceleration may refer to: *Gravitational acceleration, the acceleration caused by the gravitational attraction of massive bodies in general *Gravity of Earth, the acceleration caused by the combination of gravitational attraction and centrifugal force of the Earth *Standard gravity, or g, the standard value of gravitational acceleration at sea level on Earth ==See also== *g-force, the acceleration of a body relative to free-fall

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The flux of visible photons reaching Earth from the North Star is about $4 \times 10^3 \mathrm{~mm}^{-2} \mathrm{~s}^{-1}$. Of these photons, 30 per cent are absorbed or scattered by the atmosphere and 25 per cent of the surviving photons are scattered by the surface of the cornea of the eye. A further 9 per cent are absorbed inside the cornea. The area of the pupil at night is about $40 \mathrm{~mm}^2$ and the response time of the eye is about $0.1 \mathrm{~s}$. Of the photons passing through the pupil, about 43 per cent are absorbed in the ocular medium. How many photons from the North Star are focused onto the retina in $0.1 \mathrm{~s}$ ?

Choosing parameter values thought typical of normal dark-site observations (e.g. eye pupil 0.7cm and F=2) he found N=7.69.Crumey, op. cit., Eq. Various authorsCited in Crumey, op. cit., Sec. 3.2. have stated the limiting magnitude of a telescope with entrance pupil D centimetres to be of the form : m = 5 logD \+ N with suggested values for the constant N ranging from 6.8 to 8.7. The astronomer H.D. Curtis reported his naked-eye limit as 6.53, but by looking at stars through a hole in a black screen (i.e. against a totally dark background) was able to see one of magnitude 8.3, and possibly one of 8.9.Section=1.6.5 of Naked-eye magnitude limits can be modelled theoretically using laboratory data on human contrast thresholds at various background brightness levels. More generally, for situations where it is possible to raise a telescope's magnification high enough to make the sky background effectively black, the limiting magnitude is approximated by :m = 5 logD \+ 8 – 2.5 log (p^2F/T) where D and F are as stated above, p is the observer's pupil diameter in centimetres, and T is the telescope transmittance (e.g. 0.75 for a typical reflector).Crumey, A. Modelling the Visibility of Deep-Sky Objects. upright=1.6|thumb|Visual effect of night sky's brightness. The pupil magnification of an optical system is the ratio of the diameter of the exit pupil to the diameter of the entrance pupil. In the dark it will be the same at first, but will approach the maximum distance for a wide pupil 3 to 8 mm. thumb|The apparent position of a star viewed from the Earth depends on the Earth's velocity. The very darkest skies have a zenith surface brightness of approximately 22 mag arcsec−2, so it can be seen from the equation that such a sky would be expected to show stars approximately 0.4 mag fainter than one with a surface brightness of 21 mag arcsec−2. Crumey obtained a formula for N as a function of the sky surface brightness, telescope magnification, observer's eye pupil diameter and other parameters including the personal factor F discussed above. For example, the cat's slit pupil can change the light intensity on the retina 135-fold compared to 10-fold in humans. However, the limiting visibility is 7th magnitude for faint stars visible from dark rural areas located 200 kilometers from major cities. Crumey showed that for a sky background with surface brightness \mu_{sky} > 21 mag arcsec−2, the visual limit m could be expressed as: :m=0.4260\mu_{sky} –2.3650–2.5logF where F is a "field factor" specific to the observer and viewing situation.Crumey, op. cit., Eq. The image of the pupil as seen from outside the eye is the entrance pupil, which does not exactly correspond to the location and size of the physical pupil because it is magnified by the cornea. Peripheral Light Focusing (PLF) can be described as the focusing of Solar Ultraviolet Radiation (SUVR) at the nasal limbus of the cornea. Bowen did not record parameters such as his eye pupil diameter, naked-eye magnitude limit, or the extent of light loss in his telescopes; but because he made observations at a range of magnifications using three telescopes (with apertures 0.33 inch, 6 inch and 60 inch), Crumey was able to construct a system of simultaneous equations from which the remaining parameters could be deduced. A star's brightness (more precisely its illuminance) must exceed the sky's surface brightness (i.e. luminance) by a sufficient amount. In addition to dilation and contraction caused by light and darkness, it has been shown that solving simple multiplication problems affects the size of the pupil. The limiting magnitude will depend on the observer, and will increase with the eye's dark adaptation. From brightly lit Midtown Manhattan, the limiting magnitude is possibly 2.0, meaning that from the heart of New York City only approximately 15 stars will be visible at any given time. For example, at the peak age of 15, the dark- adapted pupil can vary from 4 mm to 9 mm with different individuals. This corresponds to roughly 250 visible stars, or one-tenth the number that can be perceived under perfectly dark skies.

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When ultraviolet radiation of wavelength $58.4 \mathrm{~nm}$ from a helium lamp is directed on to a sample of krypton, electrons are ejected with a speed of $1.59 \times 10^6 \mathrm{~m} \mathrm{~s}^{-1}$. Calculate the ionization energy of krypton.

The ionization energy will be the energy of photons hνi (h is the Planck constant) that caused a steep rise in the current: Ei = hνi. Krypton light has many spectral lines, and krypton plasma is useful in bright, high-powered gas lasers (krypton ion and excimer lasers), each of which resonates and amplifies a single spectral line. The average atmospheric concentration of krypton-85 was approximately 0.6 Bq/m3 in 1976, and has increased to approximately 1.3 Bq/m3 as of 2005. Krypton's concentration in the atmosphere is about 1 ppm. For hydrogen in the ground state Z=1 and n=1 so that the energy of the atom before ionization is simply E = - 13.6\ \mathrm{eV} After ionization, the energy is zero for a motionless electron infinitely far from the proton, so that the ionization energy is : I = E(\mathrm{H}^+) - E(\mathrm{H}) = +13.6\ \mathrm{eV}. The first measurements suggest an abundance of krypton in space. ==Applications== left|thumb|Krypton gas discharge tube Krypton's multiple emission lines make ionized krypton gas discharges appear whitish, which in turn makes krypton-based bulbs useful in photography as a white light source. In physics, ionization energy is usually expressed in electronvolts (eV) or joules (J). The first ionization energy is quantitatively expressed as :X(g) + energy ⟶ X+(g) + e− where X is any atom or molecule, X+ is the resultant ion when the original atom was stripped of a single electron, and e− is the removed electron. Krypton-85 (85Kr) is a radioisotope of krypton. The nth ionization energy refers to the amount of energy required to remove the most loosely bound electron from the species having a positive charge of (n − 1). There are two main ways in which ionization energy is calculated. The second way of calculating ionization energies is mainly used at the lowest level of approximation, where the ionization energy is provided by Koopmans' theorem, which involves the highest occupied molecular orbital or "HOMO" and the lowest unoccupied molecular orbital or "LUMO", and states that the ionization energy of an atom or molecule is equal to the negative value of energy of the orbital from which the electron is ejected. The energy of these electrons that gives rise to a sharp onset of the current of ions and freed electrons through the tube will match the ionization energy of the atoms. == Atoms: values and trends == Generally, the (N+1)th ionization energy of a particular element is larger than the Nth ionization energy (it may also be noted that the ionization energy of an anion is generally less than that of cations and neutral atom for the same element). In physics and chemistry, ionization energy (IE) (American English spelling), ionisation energy (British English spelling) is the minimum energy required to remove the most loosely bound electron of an isolated gaseous atom, positive ion, or molecule. In 1960, the International Bureau of Weights and Measures defined the meter as 1,650,763.73 wavelengths of light emitted in the vacuum corresponding to the transition between the 2p10 and 5d5 levels in the isotope krypton-86. Krypton-85 has a half-life of 10.756 years and a maximum decay energy of 687 keV. Most or all of this krypton-85 is retained in the spent nuclear fuel rods; spent fuel on discharge from a reactor contains between 0.13–1.8 PBq/Mg of krypton-85. thumb|right|Final amplifier of the Nike laser where laser beam energy is increased from 150 J to ~5 kJ by passing through a krypton/fluorine/argon gas mixture excited by irradiation with two opposing 670,000 volt electron beams. This in turn makes its ionization energies increase by 18 kJ/mol−1. That 2p electron is much closer to the nucleus than the 3s electrons removed previously. thumb|350x350px|Ionization energies peak in noble gases at the end of each period in the periodic table of elements and, as a rule, dip when a new shell is starting to fill. The krypton-86 definition lasted until the October 1983 conference, which redefined the meter as the distance that light travels in vacuum during 1/299,792,458 s.Unit of length (meter), NIST ==Characteristics== Krypton is characterized by several sharp emission lines (spectral signatures) the strongest being green and yellow. Krypton, like the other noble gases, is used in lighting and photography.

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If $125 \mathrm{~cm}^3$ of hydrogen gas effuses through a small hole in 135 seconds, how long will it take the same volume of oxygen gas to effuse under the same temperature and pressure?

Here \Delta P is the gas pressure difference across the barrier, A is the area of the hole, N_A is the Avogadro constant, R is the gas constant and T is the absolute temperature. In other words, the ratio of the rates of effusion of two gases at the same temperature and pressure is given by the inverse ratio of the square roots of the masses of the gas particles. : {\mbox{Rate of effusion of gas}_1 \over \mbox{Rate of effusion of gas}_2}=\sqrt{M_2 \over M_1} where M_1 and M_2 represent the molar masses of the gases. The effusion rate for a gas depends directly on the average velocity of its particles. The rate \Phi_N at which a gas of molar mass M effuses (typically expressed as the number of molecules passing through the hole per second) is thenPeter Atkins and Julio de Paula, Physical Chemistry (8th ed., W.H.Freeman 2006) p.756 : \Phi_N = \frac{\Delta PAN_A}{\sqrt{2\pi MRT}}. If a small area A on the container is punched to become a small hole, the effusive flow rate will be \begin{align} Q_\text{effusion} &= J_\text{impingement} \times A \\\ &= \frac{P A}{\sqrt{2 \pi m k_{B} T}} \\\ &= \frac{P A N_A}{\sqrt{2 \pi M R T}} \end{align} where M is the molar mass, N_A is the Avogadro constant, and R = N_A k_B is the gas constant. Assuming the pressure difference between the two sides of the barrier is much smaller than P_{\rm avg}, the average absolute pressure in the system (i.e. \Delta P\ll P_{\rm avg}), it is possible to express effusion flow as a volumetric flow rate as follows: :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi k_BT}{32m}} or :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi RT}{32M}} where \Phi_V is the volumetric flow rate of the gas, P_{\rm avg} is the average pressure on either side of the orifice, and d is the hole diameter. ==Effect of molecular weight== At constant pressure and temperature, the root-mean-square speed and therefore the effusion rate are inversely proportional to the square root of the molecular weight. Gases with a lower molecular weight effuse more rapidly than gases with a higher molecular weight, so that the number of lighter molecules passing through the hole per unit time is greater. ===Graham's law=== Scottish chemist Thomas Graham (1805–1869) found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles. The Lockman Hole is an area of the sky in which minimal amounts of neutral hydrogen gas are observed. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. In particular, the short average distances between molecules increases intermolecular forces between gas molecules enough to substantially change the pressure exerted by them, an effect not included in the ideal gas model. ==See also== * * * * * * * * * ==References== Category:Gas laws Category:Physical chemistry Category:Engineering thermodynamics de:Partialdruck#Dalton-Gesetz et:Daltoni seadus Such a hole is often described as a pinhole and the escape of the gas is due to the pressure difference between the container and the exterior. In this region, the typical column density of neutral hydrogen is NH = 0.6 x 1020 cm−2. In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the mean free path of the molecules.K.J. Laidler and J.H. Meiser, Physical Chemistry, Benjamin/Cummings 1982, p.18. Forming gas is a mixture of hydrogen (mole fraction varies) and nitrogen. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. The number of atomic or molecular collisions with a wall of a container per unit area per unit time (impingement rate) is given by: J_\text{impingement} = \frac{P}{\sqrt{2 \pi m k_{B} T}}. assuming mean free path is much greater than pinhole diameter and the gas can be treated as an ideal gas. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. Pressure vessels for gas storage may also be classified by volume. Conversely, when the diameter is larger than the mean free path of the gas, flow obeys the Sampson flow law. Forming gas is used as an atmosphere for processes that need the properties of hydrogen gas. Under these conditions, essentially all molecules which arrive at the hole continue and pass through the hole, since collisions between molecules in the region of the hole are negligible. Dalton's law is related to the ideal gas laws. ==Formula== Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation: p_\text{total} = \sum_{i=1}^n p_i = p_1+p_2+p_3+\cdots+p_n where p1, p2, ..., pn represent the partial pressures of each component. p_{i} = p_\text{total} x_i where xi is the mole fraction of the ith component in the total mixture of n components . ==Volume-based concentration== The relationship below provides a way to determine the volume- based concentration of any individual gaseous component p_i = p_\text{total} c_i where ci is the concentration of component i.

258.14

0.0029

15.0

537

52

D

The vibrational wavenumber of $\mathrm{Br}_2$ is $323.2 \mathrm{~cm}^{-1}$. Evaluate the vibrational partition function explicitly (without approximation) and plot its value as a function of temperature. At what temperature is the value within 5 per cent of the value calculated from the approximate formula?

The vibrational temperature is used commonly when finding the vibrational partition function. In terms of the vibrational wavenumbers we can write the partition function as Q_\text{vib}(T) = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c \tilde{ u}_j}{k_\text{B} T}} } It is convenient to define a characteristic vibrational temperature \Theta_{i,\text{vib}} = \frac{h u_i}{k_\text{B}} where u is experimentally determined for each vibrational mode by taking a spectrum or by calculation. Using this approximation we can derive a closed form expression for the vibrational partition function. The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom. ==Definition== For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by Q_\text{vib}(T) = \prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} where T is the absolute temperature of the system, k_B is the Boltzmann constant, and E_{j,n} is the energy of j-th mode when it has vibrational quantum number n = 0, 1, 2, \ldots . It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. The vibrational temperature is commonly used in thermodynamics, to simplify certain equations. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The vibrational partition functionDonald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of freedom of a system. In the Central England Temperature series, dating back to 1659, at the time it was the 2nd hottest July on record, the hottest since 1783. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics For an isolated molecule of n atoms, the number of vibrational modes (i.e. values of j) is 3n − 5 for linear molecules and 3n − 6 for non-linear ones.G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945 In crystals, the vibrational normal modes are commonly known as phonons. ==Approximations== ===Quantum harmonic oscillator=== The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators. Q_\text{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} = \prod_j e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}} \sum_n \left( e^{-\frac{\hbar \omega_j}{k_\text{B} T}} \right)^n = \prod_j \frac{e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}}}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } where E_\text{ZP} = \frac{1}{2} \sum_j \hbar \omega_j is total vibrational zero point energy of the system. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The heat equation can be efficiently solved numerically using the implicit Crank–Nicolson method of . By taking the zero point energy as the reference point to which other energies are measured, the expression for the partition function becomes Q_\text{vib}(T) = \prod_{i=1}^f \frac{1}{1-e^{-\Theta_{\text{vib},i}/T}} == References == ==See also== * Partition function (mathematics) Category:Partition functions The specific heat capacity has a sharp peak as the temperature approaches the lambda point. While the light is turned off, the value of q for the tungsten filament would be zero. == Solving the heat equation using Fourier series == right|thumb|300px|Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions. thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. Often the wavenumber, \tilde{ u} with units of cm−1 is given instead of the angular frequency of a vibrational mode and also often misnamed frequency. The 1808 United Kingdom heat wave was a period of exceptionally high temperatures during July 1808. Given a solution of the heat equation, the value of for a small positive value of may be approximated as times the average value of the function over a sphere of very small radius centered at . ===Character of the solutions=== right|frame|Solution of a 1D heat partial differential equation. A quantum harmonic oscillator has an energy spectrum characterized by: E_{j,n} = \hbar\omega_j\left(n_j + \frac{1}{2}\right) where j runs over vibrational modes and n_j is the vibrational quantum number in the j-th mode, \hbar is Planck's constant, h, divided by 2 \pi and \omega_j is the angular frequency of the j'th mode.

+37

432

'-5.0'

4500

-0.75

D

A thermodynamic study of $\mathrm{DyCl}_3$ (E.H.P. Cordfunke, et al., J. Chem. Thermodynamics 28, 1387 (1996)) determined its standard enthalpy of formation from the following information (1) $\mathrm{DyCl}_3(\mathrm{~s}) \rightarrow \mathrm{DyCl}_3(\mathrm{aq}$, in $4.0 \mathrm{M} \mathrm{HCl}) \quad \Delta_{\mathrm{r}} H^{\ominus}=-180.06 \mathrm{~kJ} \mathrm{~mol}^{-1}$ (2) $\mathrm{Dy}(\mathrm{s})+3 \mathrm{HCl}(\mathrm{aq}, 4.0 \mathrm{~m}) \rightarrow \mathrm{DyCl}_3(\mathrm{aq}$, in $4.0 \mathrm{M} \mathrm{HCl}(\mathrm{aq}))+\frac{3}{2} \mathrm{H}_2(\mathrm{~g})$ $\Delta_{\mathrm{r}} H^{\ominus}=-699.43 \mathrm{~kJ} \mathrm{~mol}^{-1}$ (3) $\frac{1}{2} \mathrm{H}_2(\mathrm{~g})+\frac{1}{2} \mathrm{Cl}_2(\mathrm{~g}) \rightarrow \mathrm{HCl}(\mathrm{aq}, 4.0 \mathrm{M}) \quad \Delta_{\mathrm{r}} H^{\ominus}=-158.31 \mathrm{~kJ} \mathrm{~mol}^{-1}$ Determine $\Delta_{\mathrm{f}} H^{\ominus}\left(\mathrm{DyCl}_3, \mathrm{~s}\right)$ from these data.

Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as : f = -kT \lim_{N \rightarrow \infty} \frac{1}{N}\log Z_N The magnetization M per site in the thermodynamic limit, depending on the external magnetic field H and temperature T is given by : M(T,H) \ \stackrel{\mathrm{def}}{=}\ \lim_{N \rightarrow \infty} \frac{1}{N} \left( \sum_i \sigma_i \right) = - \left( \frac{\partial f}{\partial H} \right)_T where \sigma_i is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively : \chi_T(T,H) = \left( \frac{\partial M}{\partial H} \right)_T and : c_H = -T \left( \frac{\partial^2 f}{\partial T^2} \right)_H. ==Definitions== The critical exponents \alpha, \alpha', \beta, \gamma, \gamma' and \delta are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows : M(t,0) \simeq (-t)^{\beta}\mbox{ for }t \uparrow 0 : M(0,H) \simeq |H|^{1/ \delta} \operatorname{sign}(H)\mbox{ for }H \rightarrow 0 : \chi_T(t,0) \simeq \begin{cases} (t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases} : c_H(t,0) \simeq \begin{cases} (t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases} where : t \ \stackrel{\mathrm{def}}{=}\ \frac{T-T_c}{T_c} measures the temperature relative to the critical point. ==Derivation== For the magnetic analogue of the Maxwell relations for the response functions, the relation : \chi_T (c_H -c_M) = T \left( \frac{\partial M}{\partial T} \right)_H^2 follows, and with thermodynamic stability requiring that c_H, c_M\mbox{ and }\chi_T \geq 0 , one has : c_H \geq \frac{T}{\chi_T} \left( \frac{\partial M}{\partial T} \right)_H^2 which, under the conditions H=0, t>0 and the definition of the critical exponents gives : (-t)^{-\alpha'} \geq \mathrm{constant}\cdot(-t)^{\gamma'}(-t)^{2(\beta-1)} which gives the Rushbrooke inequality : \alpha' + 2\beta + \gamma' \geq 2. This page provides supplementary chemical data on Ytterbium(III) chloride == Structure and properties data == Structure and properties Standard reduction potential, E° 1.620 eV Crystallographic constants a = 6.73 b = 11.65 c = 6.38 β = 110.4° Effective nuclear charge 3.290 Bond strength 1194±7 kJ/mol Bond length 2.434 (Yb-Cl) Bond angle 111.5° (Cl-Yb-Cl) Magnetic susceptibility 4.4 μB == Thermodynamic properties == Phase behavior Std enthalpy change of fusionΔfusH ~~o~~ 58.1±11.6 kJ/mol Std entropy change of fusionΔfusS ~~o~~ 50.3±10.1 J/(mol•K) Std enthalpy change of atomizationΔatH ~~o~~ 1166.5±4.3 kJ/mol Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −959.5±3.0 kJ/mol Standard molar entropy S ~~o~~ solid 163.5 J/(mol•K) Heat capacity cp 101.4 J/(mol•K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid −212.8 kJ/mol Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas −651.1±5.0 kJ/mol == Spectral data == UV-Vis λmax 268 nm Extinction coefficient 303 M−1cm−1 IR Major absorption bands ν1 = 368.0 cm−1 ν2 = 178.4 cm−1 ν3 = 330.7 cm−1 ν4 = 117.8 cm−1 MS Ionization potentials from electron impact YbCl3+ = 10.9±0.1 eV YbCl2+ = 11.6±0.1 eV YbCl+ = 14.3±0.1 eV ==References== Category:Chemical data pages Category:Chemical data pages cleanup J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u. ===Constant of motion=== The bicentric problem has a constant of motion, namely, : r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) - 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right], from which the problem can be solved using the method of the last multiplier. ==Derivation== ===New variables=== To eliminate the v functions, the variables are changed to an equivalent set : \varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})}, giving the relation : v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} = \dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F, which defines a new variable F. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. * J.A. Dean (ed) in Lange's Handbook of Chemistry, McGraw-Hill, New York, USA, 14th edition, 1992. This page provides supplementary chemical data on gold(III) chloride == Thermodynamic properties == Phase behavior Triple point ? It follows that : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2\frac{d}{dt} \left( E \chi_{r} - \omega_{r} \right), which may be integrated once to yield : \frac{1}{2} Y^{2} \dot{\varphi}_{r}^{2} = E \chi_{r} - \omega_{r} + \gamma_{r}, where the \gamma_{r} are constants of integration subject to the energy conservation : \sum_{r=1}^{s} \gamma_{r} = 0. This page provides supplementary chemical data on Lutetium(III) oxide == Thermodynamic properties == Phase behavior Triple point ?

0.318

-2.99

122.0

449

-994.3

E

Calculate $\Delta_{\mathrm{r}} G^{\ominus}(375 \mathrm{~K})$ for the reaction $2 \mathrm{CO}(\mathrm{g})+\mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_2(\mathrm{~g})$ from the values of $\Delta_{\mathrm{r}} G^{\ominus}(298 \mathrm{~K})$ : and $\Delta_{\mathrm{r}} H^{\ominus}(298 \mathrm{~K})$, and the GibbsHelmholtz equation.

For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. ==The Physical Properties of Greenhouse Gases== Gas Alternate Name Formula 1998 Level Increase since 1750 Radiative forcing (Wm−2) Specific heat at STP (J kg−1) Carbon dioxide (CO2) 365ppm 87 ppm 1.46 819 Methane (CH4) 1,745ppb 1,045ppb 0.48 2191 Nitrous oxide (N2O) 314ppb 44ppb 0.15 880 Tetrafluoromethane Carbon tetrafluoride (CF4) 80ppt 40ppt 0.003 1330 Hexafluoroethane (C2F6) 3 ppt 3ppt 0.001 Sulfur hexafluoride (SF6) 4.2ppt 4.2ppt 0.002 HFC-23* Trifluoromethane (CHF3) 14ppt 14ppt 0.002 HFC-134a* 1,1,1,2-tetrafluoroethane (C2H2F4) 7.5ppt 7.5ppt 0.001 HFC-152a* 1,1-Difluoroethane (C2H4F2) 0.5ppt 0.5ppt 0.000 Water vapour (H2O(G)) 2000 Category:Greenhouse gases The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. The molecular formula C17H14N2O2 (molar mass: 278.305 g/mol, exact mass: 278.1055 u) may refer to: * Bimakalim * Sudan Red G Category:Molecular formulas The molecular formula C16H10N2Na2O7S2 (molar mass: 452.369 g/mol) may refer to: * Orange G * Orange GGN * Sunset Yellow FCF Category:Molecular formulas The molecular formula C18H26O (molar mass: 258.40 g/mol, exact mass: 258.1984 u) may refer to: * Galaxolide (HHCB) * Xibornol The molecular formula C12H22O10 (molar mass: 326.29 g/mol, exact mass: 326.121297 u) may refer to: * Neohesperidose or 2-O-alpha-L-Rhamnopyranosyl-D- glucopyranose * Robinose * Rutinose or 6-O-alpha-L-Rhamnopyranosyl-D- glucupyranose A closely related technique is the use of an electroanalytical voltaic cell, which can be used to measure the Gibbs energy for certain reactions as a function of temperature, yielding K_\mathrm{eq}(T) and thereby \Delta_{\text {rxn}} H^\ominus . Integration of this equation permits the evaluation of the heat of reaction at one temperature from measurements at another temperature.Laidler K.J. and Meiser J.H., "Physical Chemistry" (Benjamin/Cummings 1982), p.62Atkins P. and de Paula J., "Atkins' Physical Chemistry" (8th edn, W.H. Freeman 2006), p.56 :\Delta H^\circ \\! \left( T \right) = \Delta H^\circ \\! \left( T^\circ \right) + \int_{T^\circ}^{T} \Delta C_P^\circ \, \mathrm{d} T Pressure variation effects and corrections due to mixing are generally minimal unless a reaction involves non-ideal gases and/or solutes, or is carried out at extremely high pressures. The enthalpy of formation of one mole of ethane gas refers to the reaction 2 C (graphite) + 3 H2 (g) → C2H6 (g). * Standard enthalpy of combustion is the enthalpy change when one mole of an organic compound reacts with molecular oxygen (O2) to form carbon dioxide and liquid water. Petrucci, Harwood and Herring, pages 241–243 Finally the reaction enthalpy may be estimated using bond energies for the bonds which are broken and formed in the reaction of interest. For example, the standard enthalpy of combustion of ethane gas refers to the reaction C2H6 (g) + (7/2) O2 (g) → 2 CO2 (g) + 3 H2O (l). This method is based on Hess's law, which states that the enthalpy change is the same for a chemical reaction which occurs as a single reaction or in several steps. If the enthalpies for each step can be measured, then their sum gives the enthalpy of the overall single reaction. This method is only approximate, however, because a reported bond energy is only an average value for different molecules with bonds between the same elements.Petrucci, Harwood and Herring, pages 422–423 ==References== Category:Enthalpy Category:Thermochemistry Category:Thermodynamics pl:Standardowe molowe ciepło tworzenia Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. One large class of reactions for which such measurements are common is the combustion of organic compounds by reaction with molecular oxygen (O2) to form carbon dioxide and water (H2O). We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. It is also possible to evaluate the enthalpy of one reaction from the enthalpies of a number of other reactions whose sum is the reaction of interest, and these not need be formation reactions.

+93.4

-501

0.000226

12

+2.9

B

The vapour pressure of benzene is $53.3 \mathrm{kPa}$ at $60.6^{\circ} \mathrm{C}$, but it fell to $51.5 \mathrm{kPa}$ when $19.0 \mathrm{~g}$ of an non-volatile organic compound was dissolved in $500 \mathrm{~g}$ of benzene. Calculate the molar mass of the compound.

The molecular formula C25H25NO4 (molar mass: 403.47 g/mol, exact mass: 403.1784 u) may refer to: * Benzhydrocodone * 7-Spiroindanyloxymorphone (SIOM) The molecular formula C25H25NO (molar mass: 355.47 g/mol, exact mass: 355.1936 u) may refer to: * JWH-007 * JWH-019 * JWH-047 * JWH-122 Category:Molecular formulas The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C25H27N (molar mass: 341.49 g/mol, exact mass: 341.2143 u) may refer to: * JWH-184 * JWH-196 The molecular formula C2H6ClO2PS (molar mass: 160.56 g/mol, exact mass: 159.9515 u) may refer to: * Dimethyl chlorothiophosphate * Dimethyl phosphorochloridothioate The molecular formula C24H23NO (molar mass: 341.44 g/mol, exact mass: 341.1780 u) may refer to: * JWH-018, also known as 1-pentyl-3-(1-naphthoyl)indole or AM-678 * JWH-148 Category:Molecular formulas The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633

5.5

0.5

85.0

3.38

13

C

J.G. Dojahn, et al. (J. Phys. Chem. 100, 9649 (1996)) characterized the potential energy curves of the ground and electronic states of homonuclear diatomic halogen anions. The ground state of $\mathrm{F}_2^{-}$is ${ }^2 \sum_{\mathrm{u}}^{+}$with a fundamental vibrational wavenumber of $450.0 \mathrm{~cm}^{-1}$ and equilibrium internuclear distance of $190.0 \mathrm{pm}$. The first two excited states are at 1.609 and $1.702 \mathrm{eV}$ above the ground state. Compute the standard molar entropy of $\mathrm{F}_2^{-}$at $298 \mathrm{~K}$.

Constants of Diatomic Molecules, by K. P. Huber and Gerhard Herzberg (Van nostrand Reinhold company, New York, 1979, ), is a classic comprehensive multidisciplinary reference text contains a critical compilation of available data for all diatomic molecules and ions known at the time of publication - over 900 diatomic species in all - including electronic energies, vibrational and rotational constants, and observed transitions. Extensive footnotes discuss the reliability of these data and additional detailed informationon potential energy curves, spin- coupling constants, /\\-type doubling, perturbations between electronic states, hyperfine structure, rotational g factors, dipole moments, radiative lifetimes, oscillator strengths, dissociation energies and ionization potentials when available, and other aspects. {\Gamma( u/2)}\,x^{- u/2-1} e^{-1/(2 x)}\\!| cdf =\Gamma\\!\left(\frac{ u}{2},\frac{1}{2x}\right) \bigg/\, \Gamma\\!\left(\frac{ u}{2}\right)\\!| mean =\frac{1}{ u-2}\\! for u >2\\!| median = \approx \dfrac{1}{ u\bigg(1-\dfrac{2}{9 u}\bigg)^3}| mode =\frac{1}{ u+2}\\!| variance =\frac{2}{( u-2)^2 ( u-4)}\\! for u >4\\!| skewness =\frac{4}{ u-6}\sqrt{2( u-4)}\\! for u >6\\!| kurtosis =\frac{12(5 u-22)}{( u-6)( u-8)}\\! for u >8\\!| entropy =\frac{ u}{2} \\!+\\!\ln\\!\left(\frac{ u}{2}\Gamma\\!\left(\frac{ u}{2}\right)\right) \\!-\\!\left(1\\!+\\!\frac{ u}{2}\right)\psi\\!\left(\frac{ u}{2}\right)| mgf =\frac{2}{\Gamma(\frac{ u}{2})} \left(\frac{-t}{2i}\right)^{\\!\\!\frac{ u}{4}} K_{\frac{ u}{2}}\\!\left(\sqrt{-2t}\right); does not exist as real valued function| char =\frac{2}{\Gamma(\frac{ u}{2})} \left(\frac{-it}{2}\right)^{\\!\\!\frac{ u}{4}} K_{\frac{ u}{2}}\\!\left(\sqrt{-2it}\right)| }} In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distributionBernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory ,Wiley (pages 119, 431) ) is a continuous probability distribution of a positive-valued random variable. The Rydberg–Klein–Rees method is a procedure used in the analysis of rotational-vibrational spectra of diatomic molecules to obtain a potential energy curve from the experimentally-known line positions. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. ==Earth bulk continental crust and upper continental crust== *C1 — Crust: CRC Handbook *C2 — Crust: Kaye and Laby *C3 — Crust: Greenwood *C4 — Crust: Ahrens (Taylor) *C5 — Crust: Ahrens (Wänke) *C6 — Crust: Ahrens (Weaver) *U1 — Upper crust: Ahrens (Taylor) *U2 — Upper crust: Ahrens (Shaw) Element C1 C2 C3 C4 C5 C6 U1 U2 01 H hydrogen 1.40×10−3 1.52 _0_ ×10−3 02 He helium 8×10−9 03 Li lithium 2.0×10−5 2.0×10−5 1.8×10−5 1.3×10−5 1.37×10−5 2. _0_ ×10−5 2.2×10−5 04 Be beryllium 2.8×10−6 2.0×10−6 2×10−6 1.5 _00_ ×10−6 3. _000_ ×10−6 05 B boron 1.0×10−5 7.0×10−6 9×10−6 1. _0000_ ×10−5 1.5 _000_ ×10−5 06 C carbon 2.00×10−4 1.8 _0_ ×10−4 3.76×10−3 07 N nitrogen 1.9×10−5 2.0×10−5 1.9×10−5 08 O oxygen 4.61×10−1 3.7×10−1 4.55 _000_ ×10−1 09 F fluorine 5.85×10−4 4.6×10−4 5.44×10−4 5.25×10−4 10 Ne neon 5×10−9 11 Na sodium 2.36×10−2 2.3×10−2 2.27 _00_ ×10−2 2.3 _000_ ×10−2 2.44 _00_ ×10−2 3.1 _000_ ×10−2 2.89×10−2 2.57×10−2 12 Mg magnesium 2.33×10−2 2.8×10−2 2.764 _0_ ×10−2 3.20×10−2 2.37×10−2 1.69×10−2 1.33×10−2 1.35×10−2 13 Al aluminium 8.23×10−2 8.0×10−2 8.3 _000_ ×10−2 8.41 _00_ ×10−2 8.305 _0_ ×10−2 8.52 _00_ ×10−2 8.04 _00_ ×10−2 7.74 _00_ ×10−2 14 Si silicon 2.82×10−1 2.7×10−1 2.72 _000_ ×10−1 2.677×10−1 2.81×10−1 2.95×10−1 3.08×10−1 3.04×10−1 15 P phosphorus 1.05×10−3 1.0×10−3 1.12 _0_ ×10−3 7.63×10−4 8.3 _0_ ×10−4 16 S sulfur 3.50×10−4 3.0×10−4 3.4 _0_ ×10−4 8.81×10−4 17 Cl chlorine 1.45×10−4 1.9×10−4 1.26×10−4 1.9 _00_ ×10−3 18 Ar argon 3.5×10−6 19 K potassium 2.09×10−2 1.7×10−2 1.84 _00_ ×10−2 9.1 _00_ ×10−3 1.76 _00_ ×10−2 1.7 _000_ ×10−2 2.8 _000_ ×10−2 2.57 _00_ ×10−2 20 Ca calcium 4.15×10−2 5.1×10−2 4.66 _00_ ×10−2 5.29 _00_ ×10−2 4.92 _00_ ×10−2 3.4 _000_ ×10−2 3. _0000_ ×10−2 2.95 _00_ ×10−2 21 Sc scandium 2.2×10−5 2.2×10−5 2.5×10−5 3. _0_ ×10−5 2.14×10−5 1.1×10−5 7×10−6 22 Ti titanium 5.65×10−3 8.6×10−3 6.32 _0_ ×10−3 5.4 _00_ ×10−3 5.25 _0_ ×10−3 3.6 _00_ ×10−3 3. _000_ ×10−3 3.12 _0_ ×10−3 23 V vanadium 1.20×10−4 1.7×10−4 1.36×10−4 2.3 _0_ ×10−4 1.34×10−4 6. _0_ ×10−5 5.3×10−5 24 Cr chromium 1.02×10−4 9.6×10−5 1.22×10−4 1.85×10−4 1.46×10−4 5.6×10−5 3.5×10−5 3.5×10−5 25 Mn manganese 9.50×10−4 1.0×10−3 1.06 _0_ ×10−3 1.4 _00_ ×10−3 8.47×10−4 1. _000_ ×10−3 6. _00_ ×10−4 5.27×10−4 26 Fe iron 5.63×10−2 5.8×10−2 6.2 _000_ ×10−2 7.07×10−2 4.92×10−2 3.8×10−2 3.50×10−2 3.09×10−2 27 Co cobalt 2.5×10−5 2.8×10−5 2.9×10−5 2.9×10−5 2.54×10−5 1. _0_ ×10−5 1.2×10−5 28 Ni nickel 8.4×10−5 7.2×10−5 9.9×10−5 1.05×10−4 6.95×10−5 3.5×10−5 2×10−5 1.9×10−5 29 Cu copper 6.0×10−5 5.8×10−5 6.8×10−5 7.5×10−5 4.7×10−5 2.5×10−5 1.4×10−5 30 Zn zinc 7.0×10−5 8.2×10−5 7.6×10−5 8. _0_ ×10−5 7.6×10−5 7.1×10−5 5.2×10−5 31 Ga gallium 1.9×10−5 1.7×10−5 1.9×10−5 1.8×10−5 1.86×10−5 1.7×10−5 1.4×10−5 32 Ge germanium 1.5×10−6 1.3×10−6 1.5×10−6 1.6×10−6 1.32×10−6 1.6×10−6 33 As arsenic 1.8×10−6 2.0×10−6 1.8×10−6 1.0×10−6 2.03×10−6 1.5×10−6 34 Se selenium 5×10−8 5×10−8 5×10−8 5×10−8 1.53×10−7 5×10−8 35 Br bromine 2.4×10−6 4.0×10−6 2.5×10−6 6.95×10−6 36 Kr krypton 1×10−10 37 Rb rubidium 9.0×10−5 7.0×10−5 7.8×10−5 3.2×10−5 7.90×10−5 6.1×10−5 1.12×10−4 1.1 _0_ ×10−4 38 Sr strontium 3.70×10−4 4.5×10−4 3.84×10−4 2.6 _0_ ×10−4 2.93×10−4 5.03×10−4 3.5 _0_ ×10−4 3.16×10−4 39 Y yttrium 3.3×10−5 3.5×10−7 3.1×10−5 2. _0_ ×10−5 1.4×10−5 2.2×10−5 2.1×10−5 40 Zr zirconium 1.65×10−4 1.4×10−4 1.62×10−4 1. _00_ ×10−4 2.1 _0_ ×10−4 1.9 _0_ ×10−4 2.4 _0_ ×10−4 41 Nb niobium 2.0×10−5 2.0×10−5 2. _0_ ×10−5 1.1 _000_ ×10−5 1.3 _000_ ×10−5 2.5 _000_ ×10−5 2.6 _000_ ×10−5 42 Mo molybdenum 1.2×10−6 1.2×10−6 1.2×10−6 1. _000_ ×10−6 1.5 _00_ ×10−6 43 Tc technetium 44 Ru ruthenium 1×10−9 1×10−10 45 Rh rhodium 1×10−9 1×10−10 46 Pd palladium 1.5×10−8 3×10−9 1.5×10−8 1.0×10−9 5×10−10 47 Ag silver 7.5×10−8 8×10−8 8×10−8 8. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 48 Cd cadmium 1.5×10−7 1.8×10−7 1.6×10−7 9.8×10−8 1. _00_ ×10−7 9.8×10−8 49 In indium 2.5×10−7 2×10−7 2.4×10−7 5. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 50 Sn tin 2.3×10−6 1.5×10−6 2.1×10−6 2.5 _00_ ×10−6 5.5 _00_ ×10−6 51 Sb antimony 2×10−7 2×10−7 2×10−7 2. _00_ ×10−7 2.03×10−7 2. _00_ ×10−7 52 Te tellurium 1×10−9 1×10−9 2.03×10−9 53 I iodine 4.5×10−7 5×10−7 4.6×10−7 1.54 _0_ ×10−6 54 Xe xenon 3×10−11 55 Cs caesium 3×10−6 1.6×10−6 2.6×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 3.7 _00_ ×10−6 56 Ba barium 4.25×10−4 3.8×10−4 3.9 _0_ ×10−4 2.5 _0000_ ×10−4 5.42 _000_ ×10−4 7.07 _000_ ×10−4 5.5 _0000_ ×10−4 1.07 _0000_ ×10−3 57 La lanthanum 3.9×10−5 5.0×10−5 3.5×10−5 1.6 _000_ ×10−5 2.9 _000_ ×10−5 2.8 _000_ ×10−5 3. _0000_ ×10−5 3.2 _00_ ×10−6 58 Ce cerium 6.65×10−5 8.3×10−5 6.6×10−5 3.3 _000_ ×10−5 5.42 _00_ ×10−5 5.7 _000_ ×10−5 6.4 _000_ ×10−5 6.5 _000_ ×10−5 59 Pr praseodymium 9.2×10−6 1.3×10−5 9.1×10−6 3.9 _00_ ×10−6 7.1 _00_ ×10−6 60 Nd neodymium 4.15×10−5 4.4×10−5 4. _0_ ×10−5 1.6 _000_ ×10−5 2.54 _00_ ×10−5 2.3 _000_ ×10−5 2.6 _000_ ×10−5 2.6 _000_ ×10−5 61 Pm promethium 62 Sm samarium 7.05×10−6 7.7×10−6 7.0×10−6 3.5 _00_ ×10−6 5.59 _0_ ×10−6 4.1 _00_ ×10−6 4.5 _00_ ×10−6 4.5 _00_ ×10−6 63 Eu europium 2.0×10−6 2.2×10−6 2.1×10−6 1.1 _00_ ×10−6 1.407×10−6 1.09 _0_ ×10−6 8.8 _0_ ×10−7 9.4 _0_ ×10−7 64 Gd gadolinium 6.2×10−6 6.3×10−6 6.1×10−6 3.3 _00_ ×10−6 8.14 _0_ ×10−6 3.8 _00_ ×10−6 2.8 _00_ ×10−6 65 Tb terbium 1.2×10−6 1.0×10−6 1.2×10−6 6. _00_ ×10−7 1.02 _0_ ×10−6 5.3 _0_ ×10−7 6.4 _0_ ×10−7 4.8 _0_ ×10−7 66 Dy dysprosium 5.2×10−6 8.5×10−6 3.7 _00_ ×10−6 6.102×10−6 3.5 _00_ ×10−6 67 Ho holmium 1.3×10−6 1.6×10−6 1.3×10−6 7.8 _0_ ×10−7 1.86 _0_ ×10−6 8. _00_ ×10−7 6.2 _0_ ×10−7 68 Er erbium 3.5×10−6 3.6×10−6 3.5×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 2.3 _00_ ×10−6 69 Tm thulium 5.2×10−7 5.2×10−7 5×10−7 3.2 _0_ ×10−7 2.4 _0_ ×10−7 3.3 _0_ ×10−7 70 Yb ytterbium 3.2×10−6 3.4×10−6 3.1×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 1.53 _0_ ×10−6 2.2 _00_ ×10−6 1.5 _00_ ×10−6 71 Lu lutetium 8×10−7 8×10−7 3. _00_ ×10−7 5.76×10−7 2.3 _0_ ×10−7 3.2 _0_ ×10−7 2.3 _0_ ×10−7 72 Hf hafnium 3.0×10−6 4×10−6 2.8×10−6 3. _000_ ×10−6 3.46 _0_ ×10−6 4.7 _00_ ×10−6 5.8 _00_ ×10−6 5.8 _00_ ×10−6 73 Ta tantalum 2.0×10−6 2.4×10−6 1.7×10−6 1. _000_ ×10−6 2.203×10−6 2.2 _00_ ×10−6 74 W tungsten 1.25×10−6 1.0×10−6 1.2×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 2. _000_ ×10−6 75 Re rhenium 7×10−10 4×10−10 7×10−10 5×10−10 1.02×10−9 5×10−10 76 Os osmium 1.5×10−9 2×10−10 5×10−9 1.02×10−9 77 Ir iridium 1×10−9 2×10−10 1×10−9 1×10−10 1.02×10−9 2×10−11 78 Pt platinum 5×10−9 1×10−8 79 Au gold 4×10−9 2×10−9 4×10−9 3.0×10−9 4.07×10−9 1.8×10−9 80 Hg mercury 8.5×10−8 2×10−8 8×10−8 81 Tl thallium 8.5×10−7 4.7×10−7 7×10−7 3.6 _0_ ×10−7 7.5 _0_ ×10−7 5.2 _0_ ×10−7 82 Pb lead 1.4×10−5 1.0×10−5 1.3×10−5 8. _000_ ×10−6 1.5 _000_ ×10−5 2. _0000_ ×10−5 1.7 _000_ ×10−5 83 Bi bismuth 8.5×10−9 4×10−9 8×10−9 6. _0_ ×10−8 1.27×10−7 84 Po polonium 2×10−16 85 At astatine 86 Rn radon 4×10−19 87 Fr francium 88 Ra radium 9×10−13 89 Ac actinium 5.5×10−16 90 Th thorium 9.6×10−6 5.8×10−6 8.1×10−6 3.5 _00_ ×10−6 5.7 _00_ ×10−6 1.07 _00_ ×10−5 1. _0000_ ×10−5 91 Pa protactinium 1.4×10−12 92 U uranium 2.7×10−6 1.6×10−6 2.3×10−6 9.1 _0_ ×10−7 1.2 _00_ ×10−6 1.3 _00_ ×10−6 2.8 _00_ ×10−6 2.5 _00_ ×10−6 93 Np neptunium 94 Pu plutonium ==Urban soils== The established abundances of chemical elements in urban soils can be considered a geochemical (ecological and geochemical) characteristic, the accumulated impact of technogenic and natural processes at the beginning of the 21st century. Element S1 Y1 Y2 01 H hydrogen 2.8×104 2.8×104* 2.79×104 02 He helium 2.7×103 2.7×103* 2.72×103 03 Li lithium 4.0×10−7 5.7×10−5 5.71×10−5 (9.2%) 04 Be beryllium 4.0×10−7 7.0×10−7 7.30×10−7 (9.5%) 05 B boron 1.1×10−5 2.1×10−5 2.12×10−5 (10%) 06 C carbon 1.0×101 1.0×101* 1.01×101 07 N nitrogen 3.1×100 3.1×100* 3.13×100 08 O oxygen 2.4×101 2.4×101* 2.38×101 (10%) 09 F fluorine about 1.0×10−3 8.5×10−4 8.43×10−4 (15%) 10 Ne neon 3.0×100 3.0×100* 3.44×100 (14%) 11 Na sodium 6.0×10−2 5.7×10−2 5.74×10−2 (7.1%) 12 Mg magnesium 1.0×100 1.1×100 1.074×100 (3.8%) 13 Al aluminium 8.3×10−2 8.5×10−2 8.49×10−2 (3.6%) 14 Si silicon 1.0×100 1.0×100 1.0×100 (4.4%) 15 P phosphorus 8.0×10−3 1.0×10−2 1.04×10−2 (10%) 16 S sulfur 4.5×10−1 5.2×10−1 5.15×10−1 (13%) 17 Cl chlorine about 9.0×10−3 5.2×10−3 5.24×10−3 (15%) 18 Ar argon 1.0×10−1* 1.0×10−1* 1.01×10−1 (6%) 19 K potassium 3.7×10−3 3.8×10−3 3.77×10−3 (7.7%) 20 Ca calcium 6.4×10−2 6.1×10−2 6.11×10−2 (7.1%) 21 Sc scandium 3.5×10−5 3.4×10−5 3.42×10−5 (8.6%) 22 Ti titanium 2.7×10−3 2.4×10−3 2.40×10−3 (5.0%) 23 V vanadium 2.8×10−4 2.9×10−4 2.93×10−4 (5.1%) 24 Cr chromium 1.3×10−2 1.3×10−2 1.35×10−2 (7.6%) 25 Mn manganese 6.9×10−3 9.5×10−3 9.55×10−3 (9.6%) 26 Fe iron 9.0×10−1 9.0×10−1 9.00×10−1 (2.7%) 27 Co cobalt 2.3×10−3 2.3×10−3 2.25×10−3 (6.6%) 28 Ni nickel 5.0×10−2 5.0×10−2 4.93×10−2 (5.1%) 29 Cu copper 4.5×10−4 5.2×10−4 5.22×10−4 (11%) 30 Zn zinc 1.1×10−3 1.3×10−3 1.26×10−3 (4.4%) 31 Ga gallium 2.1×10−5 3.8×10−5 3.78×10−5 (6.9%) 32 Ge germanium 7.2×10−5 1.2×10−4 1.19×10−4 (9.6%) 33 As arsenic 6.6×10−6 6.56×10−6 (12%) 34 Se selenium 6.3×10−5 6.21×10−5 (6.4%) 35 Br bromine 1.2×10−5 1.18×10−5 (19%) 36 Kr krypton 4.8×10−5 4.50×10−5 (18%) 37 Rb rubidium 1.1×10−5 7.0×10−6 7.09×10−6 (6.6%) 38 Sr strontium 2.2×10−5 2.4×10−5 2.35×10−5 (8.1%) 39 Y yttrium 4.9×10−6 4.6×10−6 4.64×10−6 (6.0%) 40 Zr zirconium 1.12×10−5 1.14×10−5 1.14×10−5 (6.4%) 41 Nb niobium 7.0×10−7 7.0×10−7 6.98×10−7 (1.4%) 42 Mo molybdenum 2.3×10−6 2.6×10−6 2.55×10−6 (5.5%) 43 Tc technetium 44 Ru ruthenium 1.9×10−6 1.9×10−6 1.86×10−6 (5.4%) 45 Rh rhodium 4.0×10−7 3.4×10−7 3.44×10−7 (8%) 46 Pd palladium 1.4×10−6 1.4×10−6 1.39×10−6 (6.6%) 47 Ag silver about 2.0×10−7 4.9×10−7 4.86×10−7 (2.9%) 48 Cd cadmium 2.0×10−6 1.6×10−6 1.61×10−6 (6.5%) 49 In indium about 1.3×10−6 1.9×10−7 1.84×10−7 (6.4%) 50 Sn tin about 3.0×10−6 3.9×10−6 3.82×10−6 (9.4%) 51 Sb antimony about 3.0×10−7 3.1×10−7 3.09×10−7 (18%) 52 Te tellurium 4.9×10−6 4.81×10−6 (10%) 53 I iodine 9.0×10−7 9.00×10−7 (21%) 54 Xe xenon 4.8×10−6 4.70×10−6 (20%) 55 Cs caesium 3.7×10−7 3.72×10−7 (5.6%) 56 Ba barium 3.8×10−6 4.5×10−6 4.49×10−6 (6.3%) 57 La lanthanum 5.0×10−7 4.4×10−7 4.46×10−7 (2.0%) 58 Ce cerium 1.0×10−6 1.1×10−6 1.136×10−6 (1.7%) 59 Pr praseodymium 1.4×10−7 1.7×10−7 1.669×10−7 (2.4%) 60 Nd neodymium 9.0×10−7 8.3×10−7 8.279×10−7 (1.3%) 61 Pm promethium 62 Sm samarium 3.0×10−7 2.6×10−7 2.582×10−7 (1.3%) 63 Eu europium 9.0×10−8 9.7×10−8 9.73×10−8 (1.6%) 64 Gd gadolinium 3.7×10−7 3.3×10−7 3.30×10−7 (1.4%) 65 Tb terbium about 2.0×10−8 6.0×10−8 6.03×10−8 (2.2%) 66 Dy dysprosium 3.5×10−7 4.0×10−7 3.942×10−7 (1.4%) 67 Ho holmium about 5.0×10−8 8.9×10−8 8.89×10−8 (2.4%) 68 Er erbium 2.4×10−7 2.5×10−7 2.508×10−7 (1.3%) 69 Tm thulium about 3.0×10−8 3.8×10−8 3.78×10−8 (2.3%) 70 Yb ytterbium 3.4×10−7 2.5×10−7 2.479×10−7 (1.6%) 71 Lu lutetium about 1.5×10−7 3.7×10−8 3.67×10−8 (1.3%) 72 Hf hafnium 2.1×10−7 1.5×10−7 1.54×10−7 (1.9%) 73 Ta tantalum 3.8×10−8 2.07×10−8 (1.8%) 74 W tungsten about 3.6×10−7 1.3×10−7 1.33×10−7 (5.1%) 75 Re rhenium 5.0×10−8 5.17×10−8 (9.4%) 76 Os osmium 8.0×10−7 6.7×10−7 6.75×10−7 (6.3%) 77 Ir iridium 6.0×10−7 6.6×10−7 6.61×10−7 (6.1%) 78 Pt platinum about 1.8×10−6 1.34×10−6 1.34×10−6 (7.4%) 79 Au gold about 3.0×10−7 1.9×10−7 1.87×10−7 (15%) 80 Hg mercury 3.4×10−7 3.40×10−7 (12%) 81 Tl thallium about 2.0×10−7 1.9×10−7 1.84×10−7 (9.4%) 82 Pb lead 2.0×10−6 3.1×10−6 3.15×10−6 (7.8%) 83 Bi bismuth 1.4×10−7 1.44×10−7 (8.2%) 84 Po polonium 85 At astatine 86 Rn radon 87 Fr francium 88 Ra radium 89 Ac actinium 90 Th thorium 5.0×10−8 4.5×10−8 3.35×10−8 (5.7%) 91 Pa protactinium 92 U uranium 1.8×10−8 9.00×10−9 (8.4%) 93 Np neptunium 94 Pu plutonium ==See also== *Chemical elements data references ==Notes== Due to the estimate nature of these values, no single recommendations are given. Because of this, the crystal is locked into a state with 2^N different corresponding microstates, giving a residual entropy of S=Nk\ln(2), rather than zero. :From this source with some modifications and additions of later data: :*W.S. Fyfe, Geochemistry, Oxford University Press, (1974). Molecular Spectra and Molecular Structure IV. Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero. In particular for integer valued degrees of freedom u we have: For u >1 even, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 5 \cdot 3} { 2 \sqrt{ u}( u -2)( u -4)\cdots 4 \cdot 2\,}\cdot For u >1 odd, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 4 \cdot 2} {\pi \sqrt{ u}( u -2)( u -4)\cdots 5 \cdot 3\,}\cdot\\! For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. Geometrically frustrated systems in particular often exhibit residual entropy. (The average density of sea water in the surface is 1.025 kg/L) Element W1 W2 01 H hydrogen 1.08×10−1 1.1×10−1 02 He helium 7×10−12 7.2×10−12 03 Li lithium 1.8×10−7 1.7×10−7 04 Be beryllium 5.6×10−12 6×10−13 05 B boron 4.44×10−6 4.4×10−6 06 C carbon 2.8×10−5 2.8×10−5 07 N nitrogen 5×10−7 1.6×10−5 08 O oxygen 8.57×10−1 8.8×10−1 09 F fluorine 1.3×10−6 1.3×10−6 10 Ne neon 1.2×10−10 1.2×10−10 11 Na sodium 1.08×10−2 1.1×10−2 12 Mg magnesium 1.29×10−3 1.3×10−3 13 Al aluminium 2×10−9 1×10−9 14 Si silicon 2.2×10−6 2.9×10−6 15 P phosphorus 6×10−8 8.8×10−8 16 S sulfur 9.05×10−4 9.0×10−4 17 Cl chlorine 1.94×10−2 1.9×10−2 18 Ar argon 4.5×10−7 4.5×10−7 19 K potassium 3.99×10−4 3.9×10−4 20 Ca calcium 4.12×10−4 4.1×10−4 21 Sc scandium 6×10−13 < 4×10−12 22 Ti titanium 1×10−9 1×10−9 23 V vanadium 2.5×10−9 1.9×10−9 24 Cr chromium 3×10−10 2×10−10 25 Mn manganese 2×10−10 1.9×10−9 26 Fe iron 2×10−9 3.4×10−9 27 Co cobalt 2×10−11 3.9×10−10 28 Ni nickel 5.6×10−10 6.6×10−9 29 Cu copper 2.5×10−10 2.3×10−8 30 Zn zinc 4.9×10−9 1.1×10−8 31 Ga gallium 3×10−11 3×10−11 32 Ge germanium 5×10−11 6×10−11 33 As arsenic 3.7×10−9 2.6×10−9 34 Se selenium 2×10−10 9.0×10−11 35 Br bromine 6.73×10−5 6.7×10−5 36 Kr krypton 2.1×10−10 2.1×10−10 37 Rb rubidium 1.2×10−7 1.2×10−7 38 Sr strontium 7.9×10−6 8.1×10−6 39 Y yttrium 1.3×10−11 1.3×10−12 40 Zr zirconium 3×10−11 2.6×10−11 41 Nb niobium 1×10−11 1.5×10−11 42 Mo molybdenum 1×10−8 1.0×10−8 43 Tc technetium 44 Ru ruthenium 7×10−13 45 Rh rhodium 46 Pd palladium 47 Ag silver 4×10−11 2.8×10−10 48 Cd cadmium 1.1×10−10 1.1×10−10 49 In indium 2×10−8 50 Sn tin 4×10−12 8.1×10−10 51 Sb antimony 2.4×10−10 3.3×10−10 52 Te tellurium 53 I iodine 6×10−8 6.4×10−8 54 Xe xenon 5×10−11 4.7×10−11 55 Cs caesium 3×10−10 3.0×10−10 56 Ba barium 1.3×10−8 2.1×10−8 57 La lanthanum 3.4×10−12 3.4×10−12 58 Ce cerium 1.2×10−12 1.2×10−12 59 Pr praseodymium 6.4×10−13 6.4×10−13 60 Nd neodymium 2.8×10−12 2.8×10−12 61 Pm promethium 62 Sm samarium 4.5×10−13 4.5×10−13 63 Eu europium 1.3×10−13 1.3×10−13 64 Gd gadolinium 7×10−13 7.0×10−13 65 Tb terbium 1.4×10−13 1.4×10−12 66 Dy dysprosium 9.1×10−13 9.1×10−13 67 Ho holmium 2.2×10−13 2.2×10−13 68 Er erbium 8.7×10−13 8.7×10−12 69 Tm thulium 1.7×10−13 1.7×10−13 70 Yb ytterbium 8.2×10−13 8.2×10−13 71 Lu lutetium 1.5×10−13 1.5×10−13 72 Hf hafnium 7×10−12 < 8×10−12 73 Ta tantalum 2×10−12 < 2.5×10−12 74 W tungsten 1×10−10 < 1×10−12 75 Re rhenium 4×10−12 76 Os osmium 77 Ir iridium 78 Pt platinum 79 Au gold 4×10−12 1.1×10−11 80 Hg mercury 3×10−11 1.5×10−10 81 Tl thallium 1.9×10−11 82 Pb lead 3×10−11 3×10−11 83 Bi bismuth 2×10−11 2×10−11 84 Po polonium 1.5×10−20 85 At astatine 86 Rn radon 6×10−22 87 Fr francium 88 Ra radium 8.9×10−17 89 Ac actinium 90 Th thorium 1×10−12 1.5×10−12 91 Pa protactinium 5×10−17 92 U uranium 3.2×10−9 3.3×10−9 93 Np neptunium 94 Pu plutonium ==Sun and solar system== *S1 — Sun: Kaye & Laby *Y1 — Solar system: Kaye & Laby *Y2 — Solar system: Ahrens, with uncertainty s (%) Atom mole fraction relative to silicon = 1. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). A great deal of research has thus been undertaken into finding other systems that exhibit residual entropy. One of the interesting properties of geometrically frustrated magnetic materials such as spin ice is that the level of residual entropy can be controlled by the application of an external magnetic field. An alternative formula, valid for t^2 < u, is :\int_{-\infty}^t f(u)\,du = \tfrac{1}{2} + t\frac{\Gamma \left( \tfrac{1}{2}( u+1) \right)} {\sqrt{\pi u}\,\Gamma \left(\tfrac{ u}{2}\right)} \, {}_2F_1 \left( \tfrac{1}{2}, \tfrac{1}{2}( u+1); \tfrac{3}{2}; -\tfrac{t^2}{ u} \right), where 2F1 is a particular case of the hypergeometric function. This material is thus analogous to water ice, with the exception that the spins on the corners of the tetrahedra can point into or out of the tetrahedra, thereby producing the same 2-in, 2-out rule as in water ice, and therefore the same residual entropy. However, it turns out that for a large number of water molecules in this configuration, the hydrogen atoms have a large number of possible configurations that meet the 2-in 2-out rule (each oxygen atom must have two 'near' (or 'in') hydrogen atoms, and two far (or 'out') hydrogen atoms). The first definition yields a probability density function given by : f_1(x; u) = \frac{2^{- u/2}}{\Gamma( u/2)}\,x^{- u/2-1} e^{-1/(2 x)}, while the second definition yields the density function : f_2(x; u) = \frac{( u/2)^{ u/2}}{\Gamma( u/2)} x^{- u/2-1} e^{- u/(2 x)} .

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The duration of a $90^{\circ}$ or $180^{\circ}$ pulse depends on the strength of the $\mathscr{B}_1$ field. If a $180^{\circ}$ pulse requires $12.5 \mu \mathrm{s}$, what is the strength of the $\mathscr{B}_1$ field?

This minimum value depends on the definition used for the duration and on the shape of the pulse. The intensity functions—temporal I(t) and spectral S(\omega) —determine the time duration and spectrum bandwidth of the pulse. A pulsed field gradient is a short, timed pulse with spatial-dependent field intensity. The interval between the 50% points of the final amplitude is usually used to determine or define pulse duration, and this is understood to be the case unless otherwise specified. 500px|thumb|right|The duration-bandwidth product depends on the shape of the power spectrum of the pulse. PSR B1919+21 is a pulsar with a period of 1.3373 seconds and a pulse width of 0.04 seconds. For different pulse shapes, the minimum duration-bandwidth product is different. right|thumb|300px|Pulse duration using 50% peak amplitude. thumb|300px|DECT phone pulduration measurement (100 Hz / 10 mS) on channel 8 In signal processing and telecommunication, pulse duration is the interval between the time, during the first transition, that the amplitude of the pulse reaches a specified fraction (level) of its final amplitude, and the time the pulse amplitude drops, on the last transition, to the same level. Other fractions of the final amplitude, e.g., 90% or 1/e, may also be used, as may the root mean square (rms) value of the pulse amplitude. In radar, the pulse duration is the time the radar's transmitter is energized during each cycle. ==References== * * Category:Signal processing Category:Telecommunication theory The length of a pulse thereby is determined by its complex spectral components, which include not just their relative intensities, but also the relative positions (spectral phase) of these spectral components. For example, \mathrm{sech^2} pulses have a minimum duration-bandwidth product of 0.315 while gaussian pulses have a minimum value of 0.441. Usually the attribute 'ultrashort' applies to pulses with a duration of a few tens of femtoseconds, but in a larger sense any pulse which lasts less than a few picoseconds can be considered ultrashort. Intensity autocorrelation gives the pulse width when a particular pulse shape is assumed. The signal had a -second period (not in 1967, but in 1991) and 0.04-second pulsewidth. For a given spectrum, the minimum time-bandwidth product, and therefore the shortest pulse, is obtained by a transform-limited pulse, i.e., for a constant spectral phase \phi(\omega) . In the specialized literature, "ultrashort" refers to the femtosecond (fs) and picosecond (ps) range, although such pulses no longer hold the record for the shortest pulses artificially generated. The terms in \gamma_x and \gamma_y describe the walk-off of the pulse; the coefficient \gamma_x ~ (\gamma_y ) is the ratio of the component of the group velocity x ~ (y) and the unit vector in the direction of propagation of the pulse (z-axis). In optics, an ultrashort pulse, also known as an ultrafast event, is an electromagnetic pulse whose time duration is of the order of a picosecond (10−12 second) or less. A pulse shaper can be used to make more complicated alterations on both the phase and the amplitude of ultrashort pulses. There is no standard definition of ultrashort pulse. A bandwidth-limited pulse (also known as Fourier- transform-limited pulse, or more commonly, transform-limited pulse) is a pulse of a wave that has the minimum possible duration for a given spectral bandwidth.

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In 1976 it was mistakenly believed that the first of the 'superheavy' elements had been discovered in a sample of mica. Its atomic number was believed to be 126. What is the most probable distance of the innermost electrons from the nucleus of an atom of this element? (In such elements, relativistic effects are very important, but ignore them here.)

* Bohr radius: the radius of the lowest-energy electron orbit predicted by Bohr model of the atom (1913). The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. In this case, it is the poor shielding capacity of the 3d-electrons which affects the atomic radii and chemistries of the elements immediately following the first row of the transition metals, from gallium (Z = 31) to bromine (Z = 35). ==Calculated atomic radius== The following table shows atomic radii computed from theoretical models, as published by Enrico Clementi and others in 1967. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. These dimensions are much smaller than the diameter of the atom itself (nucleus + electron cloud), by a factor of about 26,634 (uranium atomic radius is about ())"Uranium" IDC Technologies. to about 60,250 (hydrogen atomic radius is about ).26,634 derives from x / ; 60,250 derives from x / The branch of physics concerned with the study and understanding of the atomic nucleus, including its composition and the forces that bind it together, is called nuclear physics. ==Introduction== ===History=== The nucleus was discovered in 1911, as a result of Ernest Rutherford's efforts to test Thomson's "plum pudding model" of the atom. The elements immediately following the lanthanides have atomic radii which are smaller than would be expected and which are almost identical to the atomic radii of the elements immediately above them. The diameter of the nucleus is in the range of () for hydrogen (the diameter of a single proton) to about for uranium. van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 van der Waals radii taken from Bondi's compilation (1964). Accuracy of ±5 pm.The way the atomic radius varies with increasing atomic number can be explained by the arrangement of electrons in shells of fixed capacity. Similarly, the distance from shell-closure explains the unusual instability of isotopes which have far from stable numbers of these particles, such as the radioactive elements 43 (technetium) and 61 (promethium), each of which is preceded and followed by 17 or more stable elements. HE 1327-2326, discovered in 2005 by Anna Frebel and collaborators, was the star with the lowest known iron abundance until SMSS J031300.36−670839.3 was discovered. These trends of the atomic radii (and of various other chemical and physical properties of the elements) can be explained by the electron shell theory of the atom; they provided important evidence for the development and confirmation of quantum theory. ==Atomic radius== Note: All measurements given are in picometers (pm). The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. Under most definitions the radii of isolated neutral atoms range between 30 and 300 pm (trillionths of a meter), or between 0.3 and 3 ångströms. Bohemium was the name assigned to the element with atomic number 93, now known as neptunium, when its discovery was first incorrectly alleged. The atomic radius of a chemical element is a measure of the size of its atom, usually the mean or typical distance from the center of the nucleus to the outermost isolated electron. Although the model itself is now obsolete, the Bohr radius for the hydrogen atom is still regarded as an important physical constant. ==Empirically measured atomic radius== The following table shows empirically measured covalent radii for the elements, as published by J. C. Slater in 1964. Therefore, the radius of an atom is more than 10,000 times the radius of its nucleus (1–10 fm), and less than 1/1000 of the wavelength of visible light (400–700 nm). The largest known completely stable nucleus (i.e. stable to alpha, beta, and gamma decay) is lead-208 which contains a total of 208 nucleons (126 neutrons and 82 protons). Such predictions are especially useful for elements whose radii cannot be measured experimentally (e.g. those that have not been discovered, or that have too short of a half- life). ==References== Category:Atomic radius Category:Properties of chemical elements The value of the radius may depend on the atom's state and context. Data derived from other sources with different assumptions cannot be compared. * † to an accuracy of about 5 pm * (b) 12 coordinate * (c) gallium has an anomalous crystal structure * (d) 10 coordinate * (e) uranium, neptunium and plutonium have irregular structures *Triple bond mean-square deviation 3pm. ==References== Data is as quoted at http://www.webelements.com/ from these sources: ===Covalent radii (single bond)=== * * * * * ===Metallic radius=== Category:Properties of chemical elements Category:Chemical element data pages Category:Atomic radius

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The ground level of $\mathrm{Cl}$ is ${ }^2 \mathrm{P}_{3 / 2}$ and a ${ }^2 \mathrm{P}_{1 / 2}$ level lies $881 \mathrm{~cm}^{-1}$ above it. Calculate the electronic contribution to the molar Gibbs energy of $\mathrm{Cl}$ atoms at $500 \mathrm{~K}$.

Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . These tables list values of molar ionization energies, measured in kJ⋅mol−1. :ΔG˚ = -RTln(Keq) :ΔG˚ = (2.303)RT(pKeq) :ΔG˚ = -nFE˚1/2 Useful conversion factors: :-23.06 (kcal/mol)(e−)−1(V)−1 :1.37(pKeq) kcal/mol :1.37[-log(Keq)] kcal/mol == References == Category:Thermodynamic cycles The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena. The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. Since the Gibbs free energy is the Legendre transformation of the internal energy, the derivatives can be replaced by their definitions, transforming the above equation into: :\mathrm{d}\mathbf{G} =V \mathrm{d}p-S \mathrm{d}T +\sum_{i=1}^I \mu_i \mathrm{d}N_i The chemical potential is simply another name for the partial molar Gibbs free energy (or the partial Gibbs free energy, depending on whether N is in units of moles or particles). Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:A to Z of Thermodynamics Pierre Perrot :\sum_{i=1}^I N_i \mathrm{d}\mu_i = - S \mathrm{d}T + V \mathrm{d}p where N_i is the number of moles of component i, \mathrm{d}\mu_i the infinitesimal increase in chemical potential for this component, S the entropy, T the absolute temperature, V volume and p the pressure. Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. The first molar ionization energy applies to the neutral atoms. If multiple phases of matter are present, the chemical potentials across a phase boundary are equal.Fundamentals of Engineering Thermodynamics, 3rd Edition Michael J. Moran and Howard N. Shapiro, p. 710 Combining expressions for the Gibbs–Duhem equation in each phase and assuming systematic equilibrium (i.e. that the temperature and pressure is constant throughout the system), we recover the Gibbs' phase rule. A Bordwell thermodynamic cycle use experimentally determined and reasonable estimates of Gibbs free energy (ΔG˚) values to determine unknown and experimentally inaccessible values. == Overview == Analogous to Hess's Law which deal with the summation of enthalpy (ΔH) values, Bordwell thermodynamic cycles deal with the summation of Gibbs free energy (ΔG) values. When pressure and temperature are variable, only I-1 of I components have independent values for chemical potential and Gibbs' phase rule follows. One particularly useful expression arises when considering binary solutions.The Properties of Gases and Liquids, 5th Edition Poling, Prausnitz and O'Connell, p. 8.13, At constant P (isobaric) and T (isothermal) it becomes: :0= N_1 \mathrm{d}\mu_1 + N_2 \mathrm{d}\mu_2 or, normalizing by total number of moles in the system N_1 + N_2, substituting in the definition of activity coefficient \gamma and using the identity x_1 + x_2 = 1 : :0= x_1 \mathrm{d}\ln(\gamma_1) + x_2 \mathrm{d}\ln(\gamma_2) Chemical Thermodynamics of Materials, 2004 Svein Stølen, p. 79, This equation is instrumental in the calculation of thermodynamically consistent and thus more accurate expressions for the vapor pressure of a fluid mixture from limited experimental data. == Ternary and multicomponent solutions and mixtures== Lawrence Stamper Darken has shown that the Gibbs-Duhem equation can be applied to the determination of chemical potentials of components from a multicomponent system from experimental data regarding the chemical potential \bar {G_2} of only one component (here component 2) at all compositions. The law can also be written as a function of the total number of atoms N in the sample: :C/N = 3k_{\rm B}, where kB is Boltzmann constant. ==Application limits== thumb|upright=2.2|The molar heat capacity plotted of most elements at 25°C plotted as a function of atomic number. This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. In modern terms, Dulong and Petit found that the heat capacity of a mole of many solid elements is about 3R, where R is the universal gas constant. The value of 3R is about 25 joules per kelvin, and Dulong and Petit essentially found that this was the heat capacity of certain solid elements per mole of atoms they contained.

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Calculate the melting point of ice under a pressure of 50 bar. Assume that the density of ice under these conditions is approximately $0.92 \mathrm{~g} \mathrm{~cm}^{-3}$ and that of liquid water is $1.00 \mathrm{~g} \mathrm{~cm}^{-3}$.

The pressure melting point is nearly a constant 0 °C at pressures above the triple point at 611.7 Pa, where water can exist in only the solid or liquid phases, through atmospheric pressure (100 kPa) until about 10 MPa. The pressure melting point of ice is the temperature at which ice melts at a given pressure. thumb|Phase diagram of water Ice VI is a form of ice that exists at high pressure at the order of about 1 GPa (= 10 000 bar) and temperatures ranging from 130 up to 355 Kelvin (−143 °C up to 82 °C); see also the phase diagram of water. As the pressure increases with depth in a glacier from the weight of the ice above, the pressure melting point of ice decreases within bounds, as shown in the diagram. The level where ice can start melting is where the pressure melting point equals the actual temperature. The triple point of ice VI with ice VII and liquid water is at about 82 °C and 2.22 GPa and its triple point with ice V and liquid water is at 0.16 °C and 0.6324 GPa = 6324 bar.Water Phase Diagram www1.lsbu.ac.uk, version of 9 September 2019, retrieved 3 October 2019 Ice VI undergoes phase transitions into ices XV and XIX upon cooling depending on pressure as hydrochloric acid is doped. == See also == * Ice phases (overview) == References == == External links == * Physik des Eises (PDF in German, iktp.tu-dresden.de) * Ice phases (www.idc- online.com) Category:Water ice With increasing pressure above 10 MPa, the pressure melting point decreases to a minimum of −21.9 °C at 209.9 MPa. Thereafter, the pressure melting point rises rapidly with pressure, passing back through 0 °C at 632.4 MPa. ==Pressure melting point in glaciers== Glaciers are subject to geothermal heat flux from below and atmospheric warming or cooling from above. Water vapor (H2O) 200px Liquid state Water Solid state Ice Properties Molecular formula H2O Molar mass 18.01528(33) g/mol Melting point Vienna Standard Mean Ocean Water (VSMOW), used for calibration, melts at 273.1500089(10) K (0.000089(10) °C) and boils at 373.1339 K (99.9839 °C) Boiling point specific gas constant 461.5 J/(kg·K) Heat of vaporization 2.27 MJ/kg Heat capacity 1.864 kJ/(kg·K) Water vapor, water vapour or aqueous vapor is the gaseous phase of water. A variety of empirical formulas exist for this quantity; the most used reference formula is the Goff-Gratch equation for the SVP over liquid water below zero degrees Celsius: :\begin{align} \log_{10} \left( p \right) = & -7.90298 \left( \frac{373.16}{T}-1 \right) + 5.02808 \log_{10} \frac{373.16}{T} \\\ & \- 1.3816 \times 10^{-7} \left( 10^{11.344 \left( 1-\frac{T}{373.16} \right)} -1 \right) \\\ & \+ 8.1328 \times 10^{-3} \left( 10^{-3.49149 \left( \frac{373.16}{T}-1 \right)} -1 \right) \\\ & \+ \log_{10} \left( 1013.246 \right) \end{align} where , temperature of the moist air, is given in units of kelvin, and is given in units of millibars (hectopascals). Humidity ranges from 0 grams per cubic metre in dry air to 30 grams per cubic metre (0.03 ounce per cubic foot) when the vapor is saturated at 30 °C. === Sublimation === Sublimation is the process by which water molecules directly leave the surface of ice without first becoming liquid water. The temperature range that is determined can then be averaged to gain the melting point of the sample being examined. Melting occurs on both the top and the bottom of the ice. Ice X, within physical chemistry, is a cubic crystalline form of ice formed in the same manner as ice VII, but at pressures as high as about 70 GPa. The formula is valid from about −50 to 102 °C; however there are a very limited number of measurements of the vapor pressure of water over supercooled liquid water. It would also be the level of the base of an ice shelf, or the ice-water interface of a subglacial lake. ==References== Category:Glaciology To estimate ice area, scientists calculate the percentage of sea ice in each pixel, multiply by the pixel area, and total the amounts. Sea ice extent is the area of sea with a specified amount of ice, usually 15%. thumb|A Fisher–Johns apparatus A melting-point apparatus is a scientific instrument used to determine the melting point of a substance. Sea ice rejects salt over time and becomes less salty resulting in a higher melting point. If there is a net increase of heat, then the ice will thin. In static equilibrium conditions, this would be the highest level where water can exist in a glacier.

0.318

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What is the temperature of a two-level system of energy separation equivalent to $400 \mathrm{~cm}^{-1}$ when the population of the upper state is one-third that of the lower state?

At equilibrium, only a thermally isolating boundary can support a temperature difference. ==See also== * Closed system * Dynamical system * Mechanically isolated system * Open system * Thermodynamic system * Isolated system ==References== Category:Thermodynamic systems Equilibrium isotope fractionation is the partial separation of isotopes between two or more substances in chemical equilibrium. thumb|right|250 px|Vertical cross-section of a thermal low Thermal lows, or heat lows, are non-frontal low-pressure areas that occur over the continents in the subtropics during the warm season, as the result of intense heating when compared to their surrounding environments.Glossary of Meteorology (2009). thumb|The approximate temperature in the solar atmosphere plotted against height The solar transition region is a region of the Sun's atmosphere between the upper chromosphere and corona. The vibrational temperature is used commonly when finding the vibrational partition function. This is a collection of temperature conversion formulas and comparisons among eight different temperature scales, several of which have long been obsolete. The opposite of a thermally isolated system is a thermally open system, which allows the transfer of heat energy and entropy. Thermally open systems may vary, however, in the rate at which they equilibrate, depending on the nature of the boundary of the open system. Thermal lows which develop near sea level can build in height during the warm season, or summer, to the elevation of the 700 hPa pressure surface,David R. Rowson and Stephen J. Colucci (1992). At equilibrium, the temperatures on both sides of a thermally open boundary are equal. Thermal Low. Some numbers in this table have been rounded. === Graphical representation === :File:Comparison of temperature scales blank.svg|845x580px| circle 40 330 4 0 K / 0 °R (−273.15 °C) circle 504 330 4 0 °F (−17.78 °C) circle 537 270 4 150 °D circle 537 317 4 32 °F circle 537 325 4 7.5 °Rø circle 537 332 4 0 °C / 0 °Ré / 0 °N circle 718 245 4 212 °F circle 718 290 4 100 °C circle 718 298 4 80 °Ré circle 718 306 4 60 °Rø circle 718 317 4 33 °N circle 718 330 4 0 °D rect 0 0 845 580 :File:Comparison of temperature scales blank.svg desc none Rankine (°R) Kelvin (K) Fahrenheit (°F) Celsius (°C) Réaumur (°Ré) Rømer (°Rø) Newton (°N) Delisle (°D) Absolute zero Lowest recorded surface temperature on Earth Fahrenheit's ice/water/salt mixture Melting point of ice (at standard pressure) Average surface temperature on Earth (15 °C) Average human body temperature (37 °C) Highest recorded surface temperature on Earth Boiling point of water (at standard pressure) ==Conversion table between the different temperature units== ==See also== * Degree of frost * Conversion of units * Gas mark == Notes and references== Category:Scales of temperature Category:Conversion of units of measurement Temperatures on scales that either do not share a numeric zero or are nonlinearly related cannot correctly be mathematically equated (related using the symbol =), and thus temperatures on different scales are more correctly described as corresponding (related using the symbol ≘). == Celsius scale == == Kelvin scale == == Fahrenheit scale == == Rankine scale == == Delisle scale == == Sir Isaac Newton's degree of temperature == == Réaumur scale == == Rømer scale == ==Comparison values chart== Celsius Fahrenheit Kelvin Rankine Delisle Newton Réaumur Rømer 500.00 932.00 773.15 1391.67 −600.00 165.00 400.00 270.00 490.00 914.00 763.15 1373.67 −585.00 161.70 392.00 264.75 480.00 896.00 753.15 1355.67 −570.00 158.40 384.00 259.50 470.00 878.00 743.15 1337.67 −555.00 155.10 376.00 254.25 460.00 860.00 733.15 1319.67 −540.00 151.80 368.00 249.00 450.00 842.00 723.15 1301.67 −525.00 148.50 360.00 243.75 440.00 824.00 713.15 1283.67 −510.00 145.20 352.00 238.50 430.00 806.00 703.15 1265.67 −495.00 141.90 344.00 233.25 420.00 788.00 693.15 1247.67 −480.00 138.60 336.00 228.00 410.00 770.00 683.15 1229.67 −465.00 135.30 328.00 222.75 400.00 752.00 673.15 1211.67 −450.00 132.00 320.00 217.50 390.00 734.00 663.15 1193.67 −435.00 128.70 312.00 212.25 380.00 716.00 653.15 1175.67 −420.00 125.40 304.00 207.00 370.00 698.00 643.15 1157.67 −405.00 122.10 296.00 201.75 360.00 680.00 633.15 1139.67 −390.00 118.80 288.00 196.50 350.00 662.00 623.15 1121.67 −375.00 115.50 280.00 191.25 340.00 644.00 613.15 1103.67 −360.00 112.20 272.00 186.00 330.00 626.00 603.15 1085.67 −345.00 108.90 264.00 180.75 320.00 608.00 593.15 1067.67 −330.00 105.60 256.00 175.50 310.00 590.00 583.15 1049.67 −315.00 102.30 248.00 170.25 300.00 572.00 573.15 1031.67 −300.00 99.00 240.00 165.00 290.00 554.00 563.15 1013.67 −285.00 95.70 232.00 159.75 280.00 536.00 553.15 995.67 −270.00 92.40 224.00 154.50 270.00 518.00 543.15 977.67 −255.00 89.10 216.00 149.25 260.00 500.00 533.15 959.67 −240.00 85.80 208.00 144.00 250.00 482.00 523.15 941.67 −225.00 82.50 200.00 138.75 240.00 464.00 513.15 923.67 −210.00 79.20 192.00 133.50 230.00 446.00 503.15 905.67 −195.00 75.90 184.00 128.25 220.00 428.00 493.15 887.67 −180.00 72.60 176.00 123.00 210.00 410.00 483.15 869.67 −165.00 69.30 168.00 117.75 200.00 392.00 473.15 851.67 −150.00 66.00 160.00 112.50 190.00 374.00 463.15 833.67 −135.00 62.70 152.00 107.25 180.00 356.00 453.15 815.67 −120.00 59.40 144.00 102.00 170.00 338.00 443.15 797.67 −105.00 56.10 136.00 96.75 160.00 320.00 433.15 779.67 −90.00 52.80 128.00 91.50 150.00 302.00 423.15 761.67 −75.00 49.50 120.00 86.25 140.00 284.00 413.15 743.67 −60.00 46.20 112.00 81.00 130.00 266.00 403.15 725.67 −45.00 42.90 104.00 75.75 120.00 248.00 393.15 707.67 −30.00 39.60 96.00 70.50 110.00 230.00 383.15 689.67 −15.00 36.30 88.00 65.25 100.00 212.00 373.15 671.67 0.00 33.00 80.00 60.00 90.00 194.00 363.15 653.67 15.00 29.70 72.00 54.75 80.00 176.00 353.15 635.67 30.00 26.40 64.00 49.50 70.00 158.00 343.15 617.67 45.00 23.10 56.00 44.25 60.00 140.00 333.15 599.67 60.00 19.80 48.00 39.00 50.00 122.00 323.15 581.67 75.00 16.50 40.00 33.75 40.00 104.00 313.15 563.67 90.00 13.20 32.00 28.50 30.00 86.00 303.15 545.67 105.00 9.90 24.00 23.25 20.00 68.00 293.15 527.67 120.00 6.60 16.00 18.00 10.00 50.00 283.15 509.67 135.00 3.30 8.00 12.75 0.00 32.00 273.15 491.67 150.00 0.00 0.00 7.50 −10.00 14.00 263.15 473.67 165.00 −3.30 −8.00 2.25 -14.26 6.29 258.86 465.96 171.43 -4.71 -11.43 0.00 -17.78 0.00 255.37 459.67 176.67 -5.87 -14.22 -1.83 −20.00 −4.00 253.15 455.67 180.00 −6.60 −16.00 −3.00 −30.00 −22.00 243.15 437.67 195.00 −9.90 −24.00 −8.25 −40.00 −40.00 233.15 419.67 210.00 −13.20 −32.00 −13.50 −50.00 −58.00 223.15 401.67 225.00 −16.50 −40.00 −18.75 −60.00 −76.00 213.15 383.67 240.00 −19.80 −48.00 −24.00 −70.00 −94.00 203.15 365.67 255.00 −23.10 −56.00 −29.25 −80.00 −112.00 193.15 347.67 270.00 −26.40 −64.00 −34.50 −90.00 −130.00 183.15 329.67 285.00 −29.70 −72.00 −39.75 −100.00 −148.00 173.15 311.67 300.00 −33.00 −80.00 −45.00 −110.00 −166.00 163.15 293.67 315.00 −36.30 −88.00 −50.25 −120.00 −184.00 153.15 275.67 330.00 −39.60 −96.00 −55.50 −130.00 −202.00 143.15 257.67 345.00 −42.90 −104.00 −60.75 −140.00 −220.00 133.15 239.67 360.00 −46.20 −112.00 −66.00 −150.00 −238.00 123.15 221.67 375.00 −49.50 −120.00 −71.25 −160.00 −256.00 113.15 203.67 390.00 −52.80 −128.00 −76.50 −170.00 −274.00 103.15 185.67 405.00 −56.10 −136.00 −81.75 −180.00 −292.00 93.15 167.67 420.00 −59.40 −144.00 −87.00 −190.00 −310.00 83.15 149.67 435.00 −62.70 −152.00 −92.25 −200.00 −328.00 73.15 131.67 450.00 −66.00 −160.00 −97.50 −210.00 −346.00 63.15 113.67 465.00 −69.30 −168.00 −102.75 −220.00 −364.00 53.15 95.67 480.00 −72.60 −176.00 −108.00 −230.00 −382.00 43.15 77.67 495.00 −75.90 −184.00 −113.25 −240.00 −400.00 33.15 59.67 510.00 −79.20 −192.00 −118.50 −250.00 −418.00 23.15 41.67 525.00 −82.50 −200.00 −123.75 −260.00 −436.00 13.15 23.67 540.00 −85.80 −208.00 −129.00 −270.00 −454.00 3.15 5.67 555.00 −89.10 −216.00 −134.25 −273.15 −459.67 0.00 0.00 559.725 −90.1395 −218.52 −135.90375 Celsius Fahrenheit Kelvin Rankine Delisle Newton Réaumur Rømer ==Comparison of temperature scales== Comparison of temperature scales Comment Kelvin Celsius Fahrenheit Rankine Delisle Newton Réaumur Rømer Absolute zero 0.00 −273.15 −459.67 0.00 559.73 −90.14 −218.52 −135.90 Lowest recorded surface temperature on EarthThe Coldest Inhabited Places on Earth; researchers of the Vostok Station recorded the coldest known temperature on Earth on July 21st 1983: −89.2 °C (−128.6 °F). 184 −89.2 −128.6 331 284 −29 −71 −39 Fahrenheit's ice/salt mixture 255.37 −17.78 0.00 459.67 176.67 −5.87 −14.22 −1.83 Ice melts (at standard pressure) 273.15 0.00 32.00 491.67 150.00 0.00 0.00 7.50 Triple point of water 273.16 0.01 32.018 491.688 149.985 0.0033 0.008 7.50525 Average surface temperature on Earth 288 15 59 519 128 5 12 15 Average human body temperature* 310 37 98 558 95 12 29 27 Highest recorded surface temperature on Earth 331 58 136.4 596 63 19 46 38 Water boils (at standard pressure) 373.1339 99.9839 211.97102 671.64102 0.00 33.00 80.00 60.00 Titanium melts 1941 1668 3034 3494 −2352 550 1334 883 The surface of the Sun 5800 5500 9900 10400 −8100 1800 4400 2900 * Normal human body temperature is 36.8 °C ±0.7 °C, or 98.2 °F ±1.3 °F. The commonly given value 98.6 °F is simply the exact conversion of the nineteenth-century German standard of 37 °C. Horita J. and Wesolowski D.J. (1994) Liquid- vapor fractionation of oxygen and hydrogen isotopes of water from the freezing to the critical temperature. The internal energy of a thermally isolated system may therefore change due to the exchange of work energy. This condition holds at the top of the chromosphere, where the equilibrium temperature is a few tens of thousands of kelvins. Equilibrium fractionation is strongest at low temperatures, and (along with kinetic isotope effects) forms the basis of the most widely used isotopic paleothermometers (or climate proxies): D/H and 18O/16O records from ice cores, and 18O/16O records from calcium carbonate. Thermal lows occur near the Sonoran Desert, on the Mexican plateau, in California's Great Central Valley, in the Sahara, in the Kalahari, over north- west Argentina, in South America, over the Kimberley region of north-west Australia, over the Iberian peninsula, and over the Tibetan plateau. An example of equilibrium isotope fractionation is the concentration of heavy isotopes of oxygen in liquid water, relative to water vapor, :{H2{^{16}O}{(l)}} + {H2{^{18}O}{(g)}} <=> {H2{^{18}O}{(l)}} + {H2{^{16}O}{(g)}} At 20 °C, the equilibrium fractionation factor for this reaction is :\alpha = \frac\ce{(^{18}O/^{16}O)_{Liquid}}\ce{(^{18}O/^{16}O)_{Vapor}} = 1.0098 Equilibrium fractionation is a type of mass-dependent isotope fractionation, while mass-independent fractionation is usually assumed to be a non- equilibrium process. * Animated explanation of the temperature of the Transition Region (and Chromosphere) (University of South Wales). In thermodynamics, a thermally isolated system can exchange no mass or heat energy with its environment.

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At $300 \mathrm{~K}$ and $20 \mathrm{~atm}$, the compression factor of a gas is 0.86 . Calculate the volume occupied by $8.2 \mathrm{mmol}$ of the gas under these conditions.

For an ideal gas the compressibility factor is Z=1 per definition. The compression ratio is the ratio between the volume of the cylinder and combustion chamber in an internal combustion engine at their maximum and minimum values. The reduced specific volume is defined by, : u_R = \frac{ u_\text{actual}}{RT_\text{cr}/P_\text{cr}} where u_\text{actual} is the specific volume. page 140 Once two of the three reduced properties are found, the compressibility chart can be used. thumb|right|225px|Static compression ratio is determined using the cylinder volume when the piston is at the top and bottom of its travel. The compressibility factor is defined in thermodynamics and engineering frequently as: :Z = \frac{p}{\rho R_\text{specific} T}, where p is the pressure, \rho is the density of the gas and R_\text{specific} = \frac{R}{M} is the specific gas constant, page 327 M being the molar mass, and the T is the absolute temperature (kelvin or Rankine scale). Alternatively, the compressibility factor for specific gases can be read from generalized compressibility charts that plot Z as a function of pressure at constant temperature. The compressibility factor should not be confused with the compressibility (also known as coefficient of compressibility or isothermal compressibility) of a material, which is the measure of the relative volume change of a fluid or solid in response to a pressure change. ==Definition and physical significance== thumb|A graphical representation of the behavior of gases and how that behavior relates to compressibility factor. For example, methyl chloride, a highly polar molecule and therefore with significant intermolecular forces, the experimental value for the compressibility factor is Z=0.9152 at a pressure of 10 atm and temperature of 100 °C. page 3-268 For air (small non-polar molecules) at approximately the same conditions, the compressibility factor is only Z=1.0025 (see table below for 10 bars, 400 K). ===Compressibility of air=== Normal air comprises in crude numbers 80 percent nitrogen and 20 percent oxygen . For a gas that is a mixture of two or more pure gases (air or natural gas, for example), the gas composition must be known before compressibility can be calculated. The closer the gas is to its critical point or its boiling point, the more Z deviates from the ideal case. === Fugacity === The compressibility factor is linked to the fugacity by the relation: : f = P \exp\left(\int \frac{Z-1}{P} dP\right) ==Generalized compressibility factor graphs for pure gases== thumb|400px|Generalized compressibility factor diagram. For example, if the static compression ratio is 10:1, and the dynamic compression ratio is 7.5:1, a useful value for cylinder pressure would be 7.51.3 × atmospheric pressure, or 13.7 bar (relative to atmospheric pressure). In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. thumb|400px|Definition of formation volume factor Bo and gas/oil ratio Rs for oil When oil is produced to surface temperature and pressure it is usual for some natural gas to come out of solution. Experimental values for the compressibility factor confirm this. As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature, T_r, and reduced pressure, P_r, should have the same compressibility factor. Racing engines burning methanol and ethanol fuel often have a compression ratio of 14:1 to 16:1. == Mathematical formula == In a piston engine, the static compression ratio (CR) is the ratio between the volume of the cylinder and combustion chamber when the piston is at the bottom of its stroke, and the volume of the combustion chamber when the piston is at the top of its stroke. * Real Gases includes a discussion of compressibility factors. If either the reduced pressure or temperature is unknown, the reduced specific volume must be found. The alveolar gas equation is the method for calculating partial pressure of alveolar oxygen (PAO2). In order to obtain a generalized graph that can be used for many different gases, the reduced pressure and temperature, P_r and T_r, are used to normalize the compressibility factor data. This is the volume of the space in the cylinder left at the end of the compression stroke. The quantum gases hydrogen, helium, and neon do not conform to the corresponding-states behavior and the reduced pressure and temperature for those three gases should be redefined in the following manner to improve the accuracy of predicting their compressibility factors when using the generalized graphs: : T_r = \frac{T}{T_c + 8} and P_r = \frac{P}{P_c + 8} where the temperatures are in kelvins and the pressures are in atmospheres. ==Reading a generalized compressibility chart== In order to read a compressibility chart, the reduced pressure and temperature must be known.

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A very crude model of the buckminsterfullerene molecule $\left(\mathrm{C}_{60}\right)$ is to treat it as a collection of electrons in a cube with sides of length equal to the mean diameter of the molecule $(0.7 \mathrm{~nm})$. Suppose that only the $\pi$ electrons of the carbon atoms contribute, and predict the wavelength of the first excitation of $\mathrm{C}_{60}$. (The actual value is $730 \mathrm{~nm}$.)

The nucleus to nucleus diameter of a buckminsterfullerene molecule is about 0.71 nm. A related fullerene molecule, named buckminsterfullerene (C60 fullerene), consists of 60 carbon atoms. The buckminsterfullerene molecule has two bond lengths. The van der Waals diameter of a buckminsterfullerene molecule is about 1.1 nanometers (nm). C70 fullerene is the fullerene molecule consisting of 70 carbon atoms. thumb|Model of the C60 fullerene (buckminsterfullerene).|alt= thumb|Model of the C20 fullerene.|alt= thumb|right|Model of a carbon nanotube. thumb|C60 fullerite (bulk solid C60).|alt= A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The empirical formula of buckminsterfullerene is and its structure is a truncated icosahedron, which resembles an association football ball of the type made of twenty hexagons and twelve pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge. C60 fullerene has 60 π electrons but a closed shell configuration requires 72 electrons. Its average bond length is 1.4 Å. ====Other fullerenes==== Another fairly common fullerene has empirical formula , but fullerenes with 72, 76, 84 and even up to 100 carbon atoms are commonly obtained. Buckminsterfullerene-2D-skeletal numbered.svg|(-Ih)[5,6]fullerene Carbon numbering. In 2019, ionized C60 molecules were detected with the Hubble Space Telescope in the space between those stars. ==Types== There are two major families of fullerenes, with fairly distinct properties and applications: the closed buckyballs and the open-ended cylindrical carbon nanotubes. In the mathematical field of graph theory, the 26-fullerene graph is a polyhedral graph with V = 26 vertices and E = 39 edges. The number of fullerenes with a given even number of vertices grows quickly in the number of vertices; 26 is the largest number of vertices for which the fullerene structure is unique. The vertices of the 26-fullerene graph can be labeled with sequences of 12 bits, in such a way that distance in the graph equals half of the Hamming distance between these bitvectors. The 26-fullerene graph is one of only five fullerenes with such an embedding.. right|200px|Fullerene C60 Fullerene chemistry is a field of organic chemistry devoted to the chemical properties of fullerenes. In a humorously speculative 1966 column for New Scientist, David Jones suggested the possibility of making giant hollow carbon molecules by distorting a plane hexagonal net with the addition of impurity atoms. ==See also== *Buckypaper *Carbocatalysis *Dodecahedrane *Fullerene ligand *Goldberg–Coxeter construction *Lonsdaleite *Triumphene *Truncated rhombic triacontahedron ==References== ==External links== * Nanocarbon: From Graphene to Buckyballs Interactive 3D models of cyclohexane, benzene, graphene, graphite, chiral & non-chiral nanotubes, and C60 Buckyballs - WeCanFigureThisOut.org. *Properties of fullerene *Richard Smalley's autobiography at Nobel.se *Sir Harry Kroto's webpage *Simple model of Fullerene *Introduction to fullerites *Bucky Balls, a short video explaining the structure of by the Vega Science Trust *Giant Fullerenes, a short video looking at Giant Fullerenes *Graphene, 15 September 2010, BBC Radio program Discovery Category:Emerging technologies The 26-fullerene graph has many perfect matchings. Fullerenes with fewer than 60 carbons do not obey isolated pentagon rule (IPR). C70fullerene-2D-skeletal numbered.svg|(-D5h(6))[5,6]fullerene Carbon numbering. Note that only one form of , buckminsterfullerene, has no pair of adjacent pentagons (the smallest such fullerene). The family is named after buckminsterfullerene (C60), the most famous member, which in turn is named after Buckminster Fuller.

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Consider the half-cell reaction $\operatorname{AgCl}(s)+\mathrm{e}^{-} \rightarrow$ $\operatorname{Ag}(s)+\mathrm{Cl}^{-}(a q)$. If $\mu^{\circ}(\mathrm{AgCl}, s)=-109.71 \mathrm{~kJ} \mathrm{~mol}^{-1}$, and if $E^{\circ}=+0.222 \mathrm{~V}$ for this half-cell, calculate the standard Gibbs energy of formation of $\mathrm{Cl}^{-}(a q)$.

The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena. Since the Gibbs free energy is the Legendre transformation of the internal energy, the derivatives can be replaced by their definitions, transforming the above equation into: :\mathrm{d}\mathbf{G} =V \mathrm{d}p-S \mathrm{d}T +\sum_{i=1}^I \mu_i \mathrm{d}N_i The chemical potential is simply another name for the partial molar Gibbs free energy (or the partial Gibbs free energy, depending on whether N is in units of moles or particles). Silver chloride is a chemical compound with the chemical formula AgCl. \left( \sqrt{\lambda}, \sqrt{x} \right) with Marcum Q-function Q_M(a,b) | mean =k+\lambda\,| median =| mode =| variance =2(k+2\lambda)\,| skewness =\frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}| kurtosis =\frac{12(k+4\lambda)}{(k+2\lambda)^2}| entropy =| mgf =\frac{\exp\left(\frac{\lambda t}{1-2t }\right)}{(1-2 t)^{k/2}} \text{ for }2t<1| char =\frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}} }} In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. Above 7.5 GPa, silver chloride transitions into a monoclinic KOH phase. The first few central moments are: :\mu_2=2(k+2\lambda)\, :\mu_3=8(k+3\lambda)\, :\mu_4=12(k+2\lambda)^2+48(k+4\lambda)\, The nth cumulant is :\kappa_n=2^{n-1}(n-1)!(k+n\lambda).\, Hence :\mu'_n = 2^{n-1}(n-1)!(k+n\lambda)+\sum_{j=1}^{n-1} \frac{(n-1)!2^{j-1}}{(n-j)!}(k+j\lambda )\mu'_{n-j}. === Cumulative distribution function === Again using the relation between the central and noncentral chi-squared distributions, the cumulative distribution function (cdf) can be written as :P(x; k, \lambda ) = e^{-\lambda/2}\; \sum_{j=0}^\infty \frac{(\lambda/2)^j}{j!} The research presented at the Gibbs Conference is focused on understanding biological process through quantitative thermodynamic analysis. "Standard potential of the silver-silver chloride electrode". This reaction is used in photography and film and is the following: :Cl− \+ hν → Cl + e− (excitation of the chloride ion, which gives up its extra electron into the conduction band) :Ag+ \+ e− → Ag (liberation of a silver ion, which gains an electron to become a silver atom) The process is not reversible because the silver atom liberated is typically found at a crystal defect or an impurity site so that the electron's energy is lowered enough that it is "trapped". ==Uses== ===Silver chloride electrode=== Silver chloride is a constituent of the silver chloride electrode which is a common reference electrode in electrochemistry. Most complexes derived from AgCl are two-, three-, and, in rare cases, four-coordinate, adopting linear, trigonal planar, and tetrahedral coordination geometries, respectively. :3AgCl(s) + Na3AsO3(aq) -> Ag3AsO3(s) + 3NaCl(aq) :3AgCl(s) +Na3AsO4(aq) -> Ag3AsO4(s) + 3NaCl(aq) The above 2 reactions are particularly important in the qualitative analysis of AgCl in labs as AgCl is white, which changes to Ag3AsO3 (silver arsenite) which is yellow, or Ag3AsO4(Silver arsenate) which is reddish brown. ==Chemistry== thumb|right|Silver chloride decomposes over time with exposure to UV light In one of the most famous reactions in chemistry, the addition of colorless aqueous silver nitrate to an equally colorless solution of sodium chloride produces an opaque white precipitate of AgCl:More info on Chlorine test :Ag+ (aq) + Cl^- (aq) -> AgCl (s) This conversion is a common test for the presence of chloride in solution. The equation can also be expressed in terms of the thermal wavelength \Lambda: : \frac{S}{k_{\rm B}N} = \ln\left(\frac{V}{N\Lambda^3}\right)+\frac{5}{2} , For a derivation of the Sackur–Tetrode equation, see the Gibbs paradox. Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. The indices in the series are (1 + 2i) + (k − 1) = k + 2i as required. == Properties == === Moment generating function === The moment-generating function is given by :M(t;k,\lambda)=\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}. === Moments === The first few raw moments are: :\mu'_1=k+\lambda :\mu'_2=(k+\lambda)^2 + 2(k + 2\lambda) :\mu'_3=(k+\lambda)^3 + 6(k+\lambda)(k+2\lambda)+8(k+3\lambda) :\mu'_4=(k+\lambda)^4+12(k+\lambda)^2(k+2\lambda)+4(11k^2+44k\lambda+36\lambda^2)+48(k+4\lambda). Silver chloride is also produced as a byproduct of the Miller process where silver metal is reacted with chlorine gas at elevated temperatures. ==History== Silver chloride was known since ancient times. The entropy predicted by the Sackur–Tetrode equation approaches negative infinity as the temperature approaches zero. ==Sackur–Tetrode constant== The Sackur–Tetrode constant, written S0/R, is equal to S/kBN evaluated at a temperature of T = 1 kelvin, at standard pressure (100 kPa or 101.325 kPa, to be specified), for one mole of an ideal gas composed of particles of mass equal to the atomic mass constant (). It remains to plug in the MGF for the non-central chi square distributions into the product and compute the new MGF – this is left as an exercise. For AgBr and AgI, the Ksp values are 5.2 x 10−13 and 8.3 x 10−17, respectively.

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$\mathrm{N}_2 \mathrm{O}_3$ dissociates according to the equilibrium $\mathrm{N}_2 \mathrm{O}_3(\mathrm{~g}) \rightleftharpoons \mathrm{NO}_2(\mathrm{~g})+\mathrm{NO}(\mathrm{g})$. At $298 \mathrm{~K}$ and one bar pressure, the degree of dissociation defined as the ratio of moles of $\mathrm{NO}_2(g)$ or $\mathrm{NO}(g)$ to the moles of the reactant assuming no dissociation occurs is $3.5 \times 10^{-3}$. Calculate $\Delta G_R^{\circ}$ for this reaction.

The molecular formula C19H22N2O3 (molar mass: 326.39 g/mol, exact mass: 326.1630 u) may refer to: * Bumadizone * 25CN-NBOMe right|thumb|Nitrogen dioxide Nitryl is the nitrogen dioxide (NO2) moiety when it occurs in a larger compound as a univalent fragment. An alternative method is reaction of Nb2O5 with Nb powder at 1100 °C.Pradyot Patnaik (2002), Handbook of Inorganic Chemicals,McGraw-Hill Professional, == Properties == The room temperature form of NbO2 has a tetragonal, rutile-like structure with short Nb-Nb distances, indicating Nb-Nb bonding.Wells A.F. (1984) Structural Inorganic Chemistry 5th edition Oxford Science Publications The high temperature form also has a rutile-like structure with short Nb-Nb distances. {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup An algorithm for computing a 4/3-approximation of the dissociation number in bipartite graphs was published in 2022. The problem of computing diss(G) (dissociation number problem) was firstly studied by Yannakakis. Niobium dioxide, is the chemical compound with the formula NbO2. The molecular formula C21H29NO3 (molar mass: 343.46 g/mol, exact mass: 343.2147 u) may refer to: * CAR-226,086 * CAR-301,060 * 25iP-NBOMe * 25P-NBOMe Category:Molecular formulas The number of vertices in a maximum cardinality dissociation set in G is called the dissociation number of G, denoted by diss(G). thumb|Examples for the definition of the dissociation number In the mathematical discipline of graph theory, a subset of vertices in a graph G is called dissociation if it induces a subgraph with maximum degree 1. NbO2 is insoluble in water and is a powerful reducing agent, reducing carbon dioxide to carbon and sulfur dioxide to sulfur. It is a bluish-black non-stoichiometric solid with a composition range of NbO1.94-NbO2.09. It can be prepared by reducing Nb2O5 with H2 at 800–1350 °C. Like nitrogen dioxide, the nitryl moiety contains a nitrogen atom with two bonds to the two oxygen atoms, and a third bond shared equally between the nitrogen and the two oxygen atoms. The dissociation number is a special case of the more general Maximum k-dependent Set Problem for k=1. In an industrial process for the production of niobium metal, NbO2 is produced as an intermediate, by the hydrogen reduction of Nb2O5.Patent EP1524252, Sintered bodies based on niobium suboxide, Schnitter C, Wötting G The NbO2 is subsequently reacted with magnesium vapor to produce niobium metal.Method for producing tantalum/niobium metal powders by the reduction of their oxides by gaseous magnesium, US patent 6171363 (2001), Shekhter L.N., Tripp T.B., Lanin L.L. (H. C. Starck, Inc.) ==References== Category:Niobium(IV) compounds Category:Non-stoichiometric compounds Category:Transition metal oxides Examples include nitryl fluoride (NO2F) and nitryl chloride (NO2Cl). Two high-pressure phases have been reported: one with a rutile-like structure (again, with short Nb-Nb distances); and a higher pressure with baddeleyite-related structure. The nitrogen-centred radical is then free to form a bond with another univalent fragment (X) to produce an N−X bond, where X can be F, Cl, OH, etc. In organic nomenclature, the nitryl moiety is known as the nitro group. For instance, nitryl benzene is normally called nitrobenzene (PhNO2). ==See also== * Dinitrogen tetroxide * Nitro compound * Nitrosyl (R−N=O) * Isocyanide (R−N≡C) * Nitryl fluoride * Nitrate ==References== Category:Inorganic nitrogen compounds Category:Oxides Category:Free radicals Category:Nitrogen–oxygen compounds The problem asks for the size of a largest subset S of the vertices of a graph G, so that the induced subgraph G[S] has maximum degree k. == Notes == == References == * * * Category:Graph invariants

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Approximately how many oxygen molecules arrive each second at the mitochondrion of an active person with a mass of $84 \mathrm{~kg}$ ? The following data are available: Oxygen consumption is about $40 . \mathrm{mL}$ of $\mathrm{O}_2$ per minute per kilogram of body weight, measured at $T=300 . \mathrm{K}$ and $P=1.00 \mathrm{~atm}$. In an adult there are about $1.6 \times 10^{10}$ cells per kg body mass. Each cell contains about 800 . mitochondria.

By convention, 1 MET is considered equivalent to the consumption of 3.5 ml O2·kg−1·min−1 (or 3.5 ml of oxygen per kilogram of body mass per minute) and is roughly equivalent to the expenditure of 1 kcal per kilogram of body weight per hour. Within aerobic respiration, the P/O ratio continues to be debated; however, current figures place it at 2.5 ATP per 1/2(O2) reduced to water, though some claim the ratio is 3. Mitochondria are commonly between 0.75 and 3 μm in cross section, but vary considerably in size and structure. Other definitions which roughly produce the same numbers have been devised, such as: : \text{1 MET}\ = 1 \, \frac{\text{kcal}}{\text{kg}\times\text{h}}\ = 4.184 \, \frac{\text{kJ}}{\text{kg}\times\text{h}} = 1.162 \, \frac{\text{W}}{\text{kg}} where * kcal = kilocalorie * kg = kilogram * h = hour * kJ = kilojoule * W = watt ===Based on watts produced and body surface area=== Still another definition is based on the body surface area, BSA, and energy itself, where the BSA is expressed in m2: : \text{1 MET}\ = 58.2 \, \frac{\text{J}}{\text{s}\times\text{BSA}}\ = 58.2 \, \frac{\text{W}}{\text{m}^2} = 18.4 \, \frac{\text{Btu}}{\text{h}\times\text{ft}^2} which is equal to the rate of energy produced per unit surface area of an average person seated at rest. The theoretical maximum value of is 21%, because the efficiency of glucose oxidation is only 42%, and half of the ATP so produced is wasted. == Criticism of explanations == Kozłowski and Konarzewski have argued that attempts to explain Kleiber's law via any sort of limiting factor is flawed, because metabolic rates vary by factors of 4-5 between rest and activity. Health and fitness studies often bracket cohort activity levels in MET⋅hours/week. ==Quantitative definitions== ===Based on oxygen utilization and body mass=== The original definition of metabolic equivalent of task is the oxygen used by a person in milliliters per minute per kilogram body mass divided by 3.5. The resulting P/O ratio would be the ratio of H/O and H/P; which is 10/3.67 or 2.73 for NADH-linked respiration, and 6/3.67 or 1.64 for UQH2-linked respiration, with actual values being somewhere between. == Notes == == References == *Garrett RH & Grisham CM (2010). Air is typically around 21% oxygen, and at sea level, the PO2 of air is typically around 159 mmHg. A MET also is defined as oxygen uptake in ml/kg/min with one MET equal to the oxygen cost of sitting quietly, equivalent to 3.5 ml/kg/min. A single mitochondrion is often found in unicellular organisms, while human liver cells have about 1000–2000 mitochondria per cell, making up 1/5 of the cell volume. Mitochondria stripped of their outer membrane are called mitoplasts. ===Outer membrane=== The outer mitochondrial membrane, which encloses the entire organelle, is 60 to 75 angstroms (Å) thick. The metabolic equivalent of task (MET) is the objective measure of the ratio of the rate at which a person expends energy, relative to the mass of that person, while performing some specific physical activity compared to a reference, currently set by convention at an absolute 3.5 mL of oxygen per kg per minute, which is the energy expended when sitting quietly by a reference individual, chosen to be roughly representative of the general population, and thereby suited to epidemiological surveys. This value was first experimentally derived from the resting oxygen consumption of a particular subject (a healthy 40-year-old, 70 kg man) and must therefore be treated as a convention. Such studies estimate that at the MAM, which may comprise up to 20% of the mitochondrial outer membrane, the ER and mitochondria are separated by a mere 10–25 nm and held together by protein tethering complexes. The number of mitochondria in a cell can vary widely by organism, tissue, and cell type. The MAM thus offers a perspective on mitochondria that diverges from the traditional view of this organelle as a static, isolated unit appropriated for its metabolic capacity by the cell.Csordás et al., Trends Cell Biol. 2018 Jul;28(7):523-540. . If the oxygen level is too low, mitochondria cannot metabolize nutrients for energy via aerobic metabolism. Mitochondria 10-0 The partial pressure of oxygen in mitochondria is generally assumed to be lower than the surroundings because the mitochondria consume oxygen. Taking this into account, it takes 8/3 +1 or 3.67 protons for vertebrate mitochondria to synthesize one ATP in the cytoplasm from ADP and Pi in the cytoplasm. Since the RMR of a person depends mainly on lean body mass (and not total weight) and other physiological factors such as health status and age, actual RMR (and thus 1-MET energy equivalents) may vary significantly from the kcal/(kg·h) rule of thumb. In respiratory physiology, the oxygen cascade describes the flow of oxygen from air to mitochondria, where it is consumed in aerobic respiration to release energy. RMR measurements by calorimetry in medical surveys have shown that the conventional 1-MET value overestimates the actual resting O2 consumption and energy expenditures by about 20% to 30% on the average; body composition (ratio of body fat to lean body mass) accounted for most of the variance. ==Standardized definition for research== The Compendium of Physical Activities was developed for use in epidemiologic studies to standardize the assignment of MET intensities in physical activity questionnaires.

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In a FRET experiment designed to monitor conformational changes in T4 lysozyme, the fluorescence intensity fluctuates between 5000 and 10,000 counts per second. Assuming that 7500 counts represents a FRET efficiency of 0.5 , what is the change in FRET pair separation distance during the reaction? For the tetramethylrhodamine/texas red FRET pair employed $r_0=50 . Å$.

In practice, FRET systems are characterized by the Förster's radius (R0): the distance between the fluorophores at which FRET efficiency is 50%. For many FRET fluorophore pairs, R0 lies between 20 and 90 Å, depending on the acceptor used and the spatial arrangements of the fluorophores within the assay. Single-molecule FRET measurements are typically performed on fluorescence microscopes, either using surface-immobilized or freely-diffusing molecules. The high time resolution of confocal single-molecule FRET measurements allows observers to potentially detect dynamics on time scales as low as 10 μs. FRET studies calculate corresponding FRET efficiencies as a result of time-resolved observation of protein folding events. Once the single molecule intensities vs. time are available the FRET efficiency can be computed for each FRET pair as a function of time and thereby it is possible to follow kinetic events on the single molecule scale and to build FRET histograms showing the distribution of states in each molecule. These FRET efficiencies can then be used to infer distances between molecules as a function of time. :FRET= \frac {\tfrac {I_A} {\eta_A Q_A} }{\tfrac {I_A} {\eta_A Q_A} + \tfrac {I_D} {\eta_D Q_D}}. where FRET is the FRET efficiency of the two-dye system at a period of time, I_A and I_D are measured photon counts of the acceptor and donor channel respectively at the same period of time, \eta_A and \eta_D are the photon collection efficiencies of the two channels, and Q_A and Q_D are quantum yield of the two dyes. The FRET efficiency is the number of photons emitted from the acceptor dye over the sum of the emissions of the donor and the acceptor dye. The FRET signal is weaker than with fluorescence, but has the advantage that there is only signal during a reaction (aside from autofluorescence). ===Scanning FCS === In Scanning fluorescence correlation spectroscopy (sFCS) the measurement volume is moved across the sample in a defined way. Time-resolved fluorescence energy transfer (TR-FRET) is the practical combination of time-resolved fluorometry (TRF) with Förster resonance energy transfer (FRET) that offers a powerful tool for drug discovery researchers. Using two acceptor fluorophores rather than one, FRET can observe multiple sites for correlated movements and spatial changes in any complex molecule. Single FRET pairs are illuminated using intense light sources, typically lasers, in order to generate sufficient fluorescence signals to enable single-molecule detection. In order to obtain statistical confidence of the FRET values, tens to hundreds of photons are required, which put the best possible time resolution to the order of 1 microsecond. This issue, however, is not particularly relevant when the distance estimation of the two fluorophores does not need to be determined with exact and absolute precision. The average molecular brightness (\langle \epsilon\rangle) is related to the variance (\sigma^2) and the average intensity (\langle I\rangle ) as follows: : \ \langle \varepsilon\rangle =\frac{\sigma^2 - \langle I\rangle}{\langle I\rangle} = \sum_i f_i \varepsilon_i Here f_i and \epsilon_i are the fractional intensity and molecular brightness, respectively, of species i. ===FRET-FCS=== Another FCS based approach to studying molecular interactions uses fluorescence resonance energy transfer (FRET) instead of fluorescence, and is called FRET-FCS. FRET involves two fluorophores, a donor and an acceptor. Unlike ensemble FRET, single-molecule FRET allows real-time monitoring of target binding events. The FRET aspect of the technology is driven by several factors, including spectral overlap and the proximity of the fluorophores involved, wherein energy transfer occurs only when the distance between the donor and the acceptor is small enough. Normally, the fluorescent emission of both donor and acceptor fluorophores is detected by two independent detectors and the FRET signal is computed from the ratio of intensities in the two channels. Fluorescent dye \ D [10−10 m2 s−1] T [°C] Excitation wavelength [nm] Reference Rhodamine 6G 2.8, 3.0, 4.14 ± 0.05, 4.20 ± 0.06 25 514 Rhodamine 110 2.7 488 Tetramethyl rhodamine 2.6 543 Cy3 2.8 543 Cy5 2.5, 3.7 ± 0.15 25 633 carboxyfluorescein 3.2 488 Alexa 488 1.96, 4.35 22.5±0.5 488 Atto 655-maleimide 4.07 ± 0.1 25 663 Atto 655-carboxylicacid 4.26 ± 0.08 25 663 2′, 7′-difluorofluorescein (Oregon Green 488) 4.11 ± 0.06 25 498 ==Variations of FCS== FCS almost always refers to the single point, single channel, temporal autocorrelation measurement, although the term "fluorescence correlation spectroscopy" out of its historical scientific context implies no such restriction. However, because the donor species used in a TR-FRET assay has a fluorescent lifetime that is many orders of magnitude longer than background fluorescence or scattered light, emission signal resulting from energy transfer can be measured after any interfering signal has completely decayed.

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An air conditioner is a refrigerator with the inside of the house acting as the cold reservoir and the outside atmosphere acting as the hot reservoir. Assume that an air conditioner consumes $1.70 \times 10^3 \mathrm{~W}$ of electrical power, and that it can be idealized as a reversible Carnot refrigerator. If the coefficient of performance of this device is 3.30 , how much heat can be extracted from the house in a day?

Performance of absorption refrigerator chillers is typically much lower, as they are not heat pumps relying on compression, but instead rely on chemical reactions driven by heat. == Equation == The equation is: :{\rm COP} = \frac{|Q|}{ W} where * Q \ is the useful heat supplied or removed by the considered system (machine). Heat pumps are measured by the efficiency with which they give off heat to the hot reservoir, COPheating; refrigerators and air conditioners by the efficiency with which they take up heat from the cold space, COPcooling: :\mathrm{COP}_{\mathrm{heating}} \equiv \frac{|Q_{\rm H}|}{W_{\rm in}} = \frac{Q_{\rm C} + W_{\rm in}}{W_{\rm in}} = \mathrm{COP}_{\mathrm{cooling}}+1\, :\mathrm{COP}_{\mathrm{cooling}} \equiv \frac{Q_{\rm C}}{W_{\rm in}}\, The reason the term "coefficient of performance" is used instead of "efficiency" is that, since these devices are moving heat, not creating it, the amount of heat they move can be greater than the input work, so the COP can be greater than 1 (100%). Less work is required to move heat than for conversion into heat, and because of this, heat pumps, air conditioners and refrigeration systems can have a coefficient of performance greater than one. The limiting value of the Carnot 'efficiency' for these processes, with the equality theoretically achievable only with an ideal 'reversible' cycle, is: :\mathrm{COP}_{\mathrm{heating}} \le \frac{T_{\rm H}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{heating,Carnot} :\mathrm{COP}_{\mathrm{cooling}} \le \frac{T_{\rm C}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{cooling,Carnot} The same device used between the same temperatures is more efficient when considered as a heat pump than when considered as a refrigerator since :\mathrm{COP}_{\mathrm{heating}} = \mathrm{COP}_{\mathrm{cooling}} + 1 This is because when heating, the work used to run the device is converted to heat and adds to the desired effect, whereas if the desired effect is cooling the heat resulting from the input work is just an unwanted by-product. For example, if an automobile engine burns gasoline at a temperature of T_{\rm H} = 816^\circ \text{C} = 1500^\circ \text{F} = 1089 \text{K} and the ambient temperature is T_{\rm C} = 21^\circ \text{C} = 70^\circ \text{F} = 294 \text{K}, then its maximum possible efficiency is: :\eta_{\rm th} \le \left (1 - \frac{294 K}{1089 K} \right ) 100\% = 73.0\% It can be seen that since T_{\rm C} is fixed by the environment, the only way for a designer to increase the Carnot efficiency of an engine is to increase T_{\rm H}, the temperature at which the heat is added to the engine. The coefficient of performance or COP (sometimes CP or CoP) of a heat pump, refrigerator or air conditioning system is a ratio of useful heating or cooling provided to work (energy) required. Not stating whether an efficiency is HHV or LHV renders such numbers very misleading. ==Heat pumps and refrigerators== Heat pumps, refrigerators and air conditioners use work to move heat from a colder to a warmer place, so their function is the opposite of a heat engine. At maximum theoretical efficiency, therefore : {\rm COP}_{\rm heating}=\frac{T_{\rm H}}{T_{\rm H}-T_{\rm C}} which is equal to the reciprocal of the thermal efficiency of an ideal heat engine, because a heat pump is a heat engine operating in reverse.Borgnakke, C., & Sonntag, R. (2013). For energy- conversion heating devices their peak steady-state thermal efficiency is often stated, e.g., 'this furnace is 90% efficient', but a more detailed measure of seasonal energy effectiveness is the annual fuel use efficiency (AFUE).HVAC Systems and Equipment volume of the ASHRAE Handbook, ASHRAE, Inc., Atlanta, GA, US, 2004 ===Heat exchangers=== A counter flow heat exchanger is the most efficient type of heat exchanger in transferring heat energy from one circuit to the other. Similarly, the COP of a refrigerator or air conditioner operating at maximum theoretical efficiency, : {\rm COP}_{\rm cooling}=\frac{Q_{\rm C}}{- \ Q_{\rm H}-Q_{\rm C}} =\frac{T_{\rm C}}{T_{\rm H}-T_{\rm C}} {\rm COP}_{\rm heating} applies to heat pumps and {\rm COP}_{\rm cooling} applies to air conditioners and refrigerators. The work energy (Win) that is applied to them is converted into heat, and the sum of this energy and the heat energy that is taken up from the cold reservoir (QC) is equal to the magnitude of the total heat energy given off to the hot reservoir (|QH|) :|Q_{\rm H}| = Q_{\rm C} + W_{\rm in} Their efficiency is measured by a coefficient of performance (COP). However, for a more complete picture of heat exchanger efficiency, exergetic considerations must be taken into account. Thermal efficiency is defined as :\eta_{\rm th} \equiv \frac{|W_{\rm out}|}{Q_{\rm in}} = \frac{ {Q_{\rm in}} - |Q_{\rm out}|} {Q_{\rm in}} = 1 - \frac{|Q_{\rm out}|}{Q_{\rm in}} The efficiency of even the best heat engines is low; usually below 50% and often far below. An absorption refrigerator is a refrigerator that uses a heat source (e.g., solar energy, a fossil-fueled flame, waste heat from factories, or district heating systems) to provide the energy needed to drive the cooling process. However, for heating, the COP is the ratio of the magnitude of the heat given off to the hot reservoir (which is the heat taken up from the cold reservoir plus the input work) to the input work: : {\rm COP}_{\rm cooling}=\frac{|Q_{\rm C}|}{ W}=\frac{Q_{\rm C}}{ W} : {\rm COP}_{\rm heating}=\frac{| Q_{\rm H}|}{ W}=\frac{Q_{\rm C} + W}{ W} = {\rm COP}_{\rm cooling} + 1 where * Q_{\rm C} > 0 \ is the heat removed from the cold reservoir and added to the system; * Q_{\rm H} < 0 \ is the heat given off to the hot reservoir; it is lost by the system and therefore negative. (see heat). So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. Sometimes, the term efficiency is used for the ratio of the achieved COP to the Carnot COP, which can not exceed 100%. ==Energy efficiency== The 'thermal efficiency' is sometimes called the energy efficiency. Examples are: *friction of moving parts *inefficient combustion *heat loss from the combustion chamber *departure of the working fluid from the thermodynamic properties of an ideal gas *aerodynamic drag of air moving through the engine *energy used by auxiliary equipment like oil and water pumps. *inefficient compressors and turbines *imperfect valve timing These factors may be accounted when analyzing thermodynamic cycles, however discussion of how to do so is outside the scope of this article. ==Energy conversion== For a device that converts energy from another form into thermal energy (such as an electric heater, boiler, or furnace), the thermal efficiency is :\eta_{\rm th} \equiv \frac{|Q_{\rm out}|}{Q_{\rm in}} where the Q quantities are heat-equivalent values. They cannot do this task perfectly, so some of the input heat energy is not converted into work, but is dissipated as waste heat Qout < 0 into the surroundings: :Q_{in} = |W_{\rm out}| + |Q_{\rm out}| The thermal efficiency of a heat engine is the percentage of heat energy that is transformed into work. An electric resistance heater has a thermal efficiency close to 100%. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants, in which a steam turbine is powered by exhaust heat from a gas turbine, it can approach 60%.

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An air conditioner is a refrigerator with the inside of the house acting as the cold reservoir and the outside atmosphere acting as the hot reservoir. Assume that an air conditioner consumes $1.70 \times 10^3 \mathrm{~W}$ of electrical power, and that it can be idealized as a reversible Carnot refrigerator. If the coefficient of performance of this device is 3.30 , how much heat can be extracted from the house in a day?

Performance of absorption refrigerator chillers is typically much lower, as they are not heat pumps relying on compression, but instead rely on chemical reactions driven by heat. == Equation == The equation is: :{\rm COP} = \frac{|Q|}{ W} where * Q \ is the useful heat supplied or removed by the considered system (machine). Heat pumps are measured by the efficiency with which they give off heat to the hot reservoir, COPheating; refrigerators and air conditioners by the efficiency with which they take up heat from the cold space, COPcooling: :\mathrm{COP}_{\mathrm{heating}} \equiv \frac{|Q_{\rm H}|}{W_{\rm in}} = \frac{Q_{\rm C} + W_{\rm in}}{W_{\rm in}} = \mathrm{COP}_{\mathrm{cooling}}+1\, :\mathrm{COP}_{\mathrm{cooling}} \equiv \frac{Q_{\rm C}}{W_{\rm in}}\, The reason the term "coefficient of performance" is used instead of "efficiency" is that, since these devices are moving heat, not creating it, the amount of heat they move can be greater than the input work, so the COP can be greater than 1 (100%). Less work is required to move heat than for conversion into heat, and because of this, heat pumps, air conditioners and refrigeration systems can have a coefficient of performance greater than one. The limiting value of the Carnot 'efficiency' for these processes, with the equality theoretically achievable only with an ideal 'reversible' cycle, is: :\mathrm{COP}_{\mathrm{heating}} \le \frac{T_{\rm H}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{heating,Carnot} :\mathrm{COP}_{\mathrm{cooling}} \le \frac{T_{\rm C}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{cooling,Carnot} The same device used between the same temperatures is more efficient when considered as a heat pump than when considered as a refrigerator since :\mathrm{COP}_{\mathrm{heating}} = \mathrm{COP}_{\mathrm{cooling}} + 1 This is because when heating, the work used to run the device is converted to heat and adds to the desired effect, whereas if the desired effect is cooling the heat resulting from the input work is just an unwanted by-product. For example, if an automobile engine burns gasoline at a temperature of T_{\rm H} = 816^\circ \text{C} = 1500^\circ \text{F} = 1089 \text{K} and the ambient temperature is T_{\rm C} = 21^\circ \text{C} = 70^\circ \text{F} = 294 \text{K}, then its maximum possible efficiency is: :\eta_{\rm th} \le \left (1 - \frac{294 K}{1089 K} \right ) 100\% = 73.0\% It can be seen that since T_{\rm C} is fixed by the environment, the only way for a designer to increase the Carnot efficiency of an engine is to increase T_{\rm H}, the temperature at which the heat is added to the engine. The coefficient of performance or COP (sometimes CP or CoP) of a heat pump, refrigerator or air conditioning system is a ratio of useful heating or cooling provided to work (energy) required. For energy- conversion heating devices their peak steady-state thermal efficiency is often stated, e.g., 'this furnace is 90% efficient', but a more detailed measure of seasonal energy effectiveness is the annual fuel use efficiency (AFUE).HVAC Systems and Equipment volume of the ASHRAE Handbook, ASHRAE, Inc., Atlanta, GA, US, 2004 ===Heat exchangers=== A counter flow heat exchanger is the most efficient type of heat exchanger in transferring heat energy from one circuit to the other. Not stating whether an efficiency is HHV or LHV renders such numbers very misleading. ==Heat pumps and refrigerators== Heat pumps, refrigerators and air conditioners use work to move heat from a colder to a warmer place, so their function is the opposite of a heat engine. At maximum theoretical efficiency, therefore : {\rm COP}_{\rm heating}=\frac{T_{\rm H}}{T_{\rm H}-T_{\rm C}} which is equal to the reciprocal of the thermal efficiency of an ideal heat engine, because a heat pump is a heat engine operating in reverse.Borgnakke, C., & Sonntag, R. (2013). Similarly, the COP of a refrigerator or air conditioner operating at maximum theoretical efficiency, : {\rm COP}_{\rm cooling}=\frac{Q_{\rm C}}{- \ Q_{\rm H}-Q_{\rm C}} =\frac{T_{\rm C}}{T_{\rm H}-T_{\rm C}} {\rm COP}_{\rm heating} applies to heat pumps and {\rm COP}_{\rm cooling} applies to air conditioners and refrigerators. The work energy (Win) that is applied to them is converted into heat, and the sum of this energy and the heat energy that is taken up from the cold reservoir (QC) is equal to the magnitude of the total heat energy given off to the hot reservoir (|QH|) :|Q_{\rm H}| = Q_{\rm C} + W_{\rm in} Their efficiency is measured by a coefficient of performance (COP). However, for a more complete picture of heat exchanger efficiency, exergetic considerations must be taken into account. Thermal efficiency is defined as :\eta_{\rm th} \equiv \frac{|W_{\rm out}|}{Q_{\rm in}} = \frac{ {Q_{\rm in}} - |Q_{\rm out}|} {Q_{\rm in}} = 1 - \frac{|Q_{\rm out}|}{Q_{\rm in}} The efficiency of even the best heat engines is low; usually below 50% and often far below. An absorption refrigerator is a refrigerator that uses a heat source (e.g., solar energy, a fossil-fueled flame, waste heat from factories, or district heating systems) to provide the energy needed to drive the cooling process. Sometimes, the term efficiency is used for the ratio of the achieved COP to the Carnot COP, which can not exceed 100%. ==Energy efficiency== The 'thermal efficiency' is sometimes called the energy efficiency. Examples are: *friction of moving parts *inefficient combustion *heat loss from the combustion chamber *departure of the working fluid from the thermodynamic properties of an ideal gas *aerodynamic drag of air moving through the engine *energy used by auxiliary equipment like oil and water pumps. *inefficient compressors and turbines *imperfect valve timing These factors may be accounted when analyzing thermodynamic cycles, however discussion of how to do so is outside the scope of this article. ==Energy conversion== For a device that converts energy from another form into thermal energy (such as an electric heater, boiler, or furnace), the thermal efficiency is :\eta_{\rm th} \equiv \frac{|Q_{\rm out}|}{Q_{\rm in}} where the Q quantities are heat-equivalent values. So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. They cannot do this task perfectly, so some of the input heat energy is not converted into work, but is dissipated as waste heat Qout < 0 into the surroundings: :Q_{in} = |W_{\rm out}| + |Q_{\rm out}| The thermal efficiency of a heat engine is the percentage of heat energy that is transformed into work. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants, in which a steam turbine is powered by exhaust heat from a gas turbine, it can approach 60%. An electric resistance heater has a thermal efficiency close to 100%. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. Seasonal efficiency gives an indication on how efficiently a heat pump operates over an entire cooling or heating season. ==See also== * Seasonal energy efficiency ratio (SEER) * Seasonal thermal energy storage (STES) * Heating seasonal performance factor (HSPF) * Power usage effectiveness (PUE) * Thermal efficiency * Vapor-compression refrigeration * Air conditioner * HVAC ==Notes== == External links == *Discussion on changes to COP of a heat pump depending on input and output temperatures *See COP definition in Cap XII of the book Industrial Energy Management - Principles and Applications Category:Heat pumps Category:Heating, ventilation, and air conditioning Category:Dimensionless numbers of thermodynamics Category:Engineering ratios

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You have collected a tissue specimen that you would like to preserve by freeze drying. To ensure the integrity of the specimen, the temperature should not exceed $-5.00{ }^{\circ} \mathrm{C}$. The vapor pressure of ice at $273.16 \mathrm{~K}$ is $624 \mathrm{~Pa}$. What is the maximum pressure at which the freeze drying can be carried out?

Usually, the freezing temperatures are between and . ===Primary drying=== During the primary drying phase, the pressure is lowered (to the range of a few millibars), and enough heat is supplied to the material for the ice to sublimate. It is important to note that, in this range of pressure, the heat is brought mainly by conduction or radiation; the convection effect is negligible, due to the low air density. ===Secondary drying=== thumb|A benchtop manifold freeze-drier The secondary drying phase aims to remove unfrozen water molecules, since the ice was removed in the primary drying phase. After the freeze-drying process is complete, the vacuum is usually broken with an inert gas, such as nitrogen, before the material is sealed. For increased efficiency, the condenser temperature should be 20 °C (36°F) less than the product during primary drying and have a defrosting mechanism to ensure that the maximum amount of water vapor in the air is condensed. ==== Shelf fluid ==== The amount of heat energy needed at times of the primary and secondary drying phase is regulated by an external heat exchanger. The freezing phase is the most critical in the whole freeze-drying process, as the freezing method can impact the speed of reconstitution, duration of freeze-drying cycle, product stability, and appropriate crystallization. In this phase, the temperature is raised higher than in the primary drying phase, and can even be above , to break any physico-chemical interactions that have formed between the water molecules and the frozen material. Therefore, freeze-drying is often reserved for materials that are heat- sensitive, such as proteins, enzymes, microorganisms, and blood plasma. At the end of the operation, the final residual water content in the product is extremely low, around 1–4%. == Applications of freeze drying == Freeze-drying causes less damage to the substance than other dehydration methods using higher temperatures. In bacteriology freeze-drying is used to conserve special strains. The frost point for a given parcel of air is always higher than the dew point, as breaking the stronger bonding between water molecules on the surface of ice compared to the surface of (supercooled) liquid water requires a higher temperature. == See also == * Bubble point * Carburetor heat * Hydrocarbon dew point * Psychrometrics * Thermodynamic diagrams ==References== ==External links== * Often Needed Answers about Temp, Humidity & Dew Point from the sci.geo.meteorology Category:Atmospheric thermodynamics Category:Temperature Category:Psychrometrics Category:Threshold temperatures Category:Gases Category:Meteorological data and networks Category:Humidity and hygrometry sv:Luftfuktighet#Daggpunkt In April 1966, the first human body was frozen—though it had been embalmed for two months—by being placed in liquid nitrogen and stored at just above freezing. Cryogenicists use the Kelvin or Rankine temperature scale, both of which measure from absolute zero, rather than more usual scales such as Celsius which measures from the freezing point of water at sea levelCelsius, Anders (1742) "Observationer om twänne beständiga grader på en thermometer" (Observations about two stable degrees on a thermometer), Kungliga Svenska Vetenskapsakademiens Handlingar (Proceedings of the Royal Swedish Academy of Sciences), 3 : 171–180 and Fig. 1.Don Rittner; Ronald A. Bailey (2005): Encyclopedia of Chemistry. Modern freeze drying began as early as 1890 by Richard Altmann who devised a method to freeze dry tissues (either plant or animal), but went virtually unnoticed until the 1930s. Because the final freeze dried product is porous, complete re-hydration can occur in the food. A significant turning point for freeze drying occurred during World War II when blood plasma and penicillin were needed to treat the wounded in the field. Freeze drying, also known as lyophilization or cryodesiccation, is a low temperature dehydration process that involves freezing the product and lowering pressure, removing the ice by sublimation. Cryonics uses temperatures below −130 °C, called cryopreservation, in an attempt to preserve enough brain information to permit the future revival of the cryopreserved person. thumb|Phase diagram of water Ice VI is a form of ice that exists at high pressure at the order of about 1 GPa (= 10 000 bar) and temperatures ranging from 130 up to 355 Kelvin (−143 °C up to 82 °C); see also the phase diagram of water. Freeze-drying is commonly used to preserve crustaceans, fish, amphibians, reptiles, insects, and smaller mammals. Freeze-drying is known to result in the highest quality of foods amongst all drying techniques because structural integrity is maintained along with preservation of flavors. The U.S. National Institute of Standards and Technology considers the field of cryogenics as that involving temperatures below -153 Celsius (120K; -243.4 Fahrenheit) Discovery of superconducting materials with critical temperatures significantly above the boiling point of nitrogen has provided new interest in reliable, low cost methods of producing high temperature cryogenic refrigeration. Hence, to avoid this issue, mass spectrometers are used to identify vapors released by silicone oil to immediately take corrective action and prevent contamination of the product. === Products === Mammalian cells generally do not survive freeze drying even though they still can be preserved. ==Equipment and types of freeze dryers== thumb|Unloading trays of freeze-dried material from a small cabinet-type freeze-dryer thumb|A residential freeze-dryer, along with the vacuum pump, and a cooling fan for the pump There are many types of freeze- dryers available, however, they usually contain a few essential components.

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The molar constant volume heat capacity for $\mathrm{I}_2(\mathrm{~g})$ is $28.6 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. What is the vibrational contribution to the heat capacity? You can assume that the contribution from the electronic degrees of freedom is negligible.

On the other hand, the vibration modes only start to become active around 350 K (77 °C) Accordingly, the molar heat capacity cP,m is nearly constant at 29.1 J⋅K−1⋅mol−1 from 100 K to about 300 °C. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1. The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. The molar heat capacity is an "intensive" property of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. So, for example, the atom-molar heat capacity of water is 1/3 of its molar heat capacity, namely 25.3 J⋅K−1⋅mol−1. Generally, the most notable constant parameter is the volumetric heat capacity (at least for solids) which is around the value of 3 megajoule per cubic meter per kelvin:Ashby, Shercliff, Cebon, Materials, Cambridge University Press, Chapter 12: Atoms in vibration: material and heat \rho c_p \simeq 3\,\text{MJ}/(\text{m}^3{\cdot}\text{K})\quad \text{(solid)} Note that the especially high molar values, as for paraffin, gasoline, water and ammonia, result from calculating specific heats in terms of moles of molecules. The molar heat capacity of a substance has the same dimension as the heat capacity of an object; namely, L2⋅M⋅T−2⋅Θ−1, or M(L/T)2/Θ. Said another way, the atom-molar heat capacity of a solid substance is expected to be 3R = 24.94 J⋅K−1⋅mol−1, where "amol" denotes an amount of the solid that contains the Avogadro number of atoms. As in the case f gases, some of the vibration modes will be "frozen out" at low temperatures, especially in solids with light and tightly bound atoms, causing the atom-molar heat capacity to be less than this theoretical limit. Here, it predicts higher heat capacities than are actually found, with the difference due to higher-energy vibrational modes not being populated at room temperatures in these substances. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . These extra degrees of freedom contribute to the molar heat capacity of the substance. The modern theory of the heat capacity of solids states that it is due to lattice vibrations in the solid. == History == Experimentally Pierre Louis Dulong and Alexis Thérèse Petit had found in 1819 that the heat capacity per weight (the mass-specific heat capacity) for 13 measured elements was close to a constant value, after it had been multiplied by a number representing the presumed relative atomic weight of the element. The molar heat capacity of the gas will then be determined only by the "active" degrees of freedom — that, for most molecules, can receive enough energy to overcome that quantum threshold.Quantum Physics and the Physics of large systems, Part 1A Physics, University of Cambridge, C.G. Smith, 2008. right|thumb|upright=1.25|Constant- volume specific heat capacity of a diatomic gas (idealised). According to Mayer's relation, the molar heat capacity at constant pressure would be :cP,m = cV,m \+ R = fR + R = (f + 2)R Thus, each additional degree of freedom will contribute R to the molar heat capacity of the gas (both cV,m and cP,m). Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. If specific heat is expressed per mole of atoms for these substances, none of the constant-volume values exceed, to any large extent, the theoretical Dulong–Petit limit of 25 J⋅mol−1⋅K−1 = 3 R per mole of atoms (see the last column of this table). Since the molar heat capacity of a substance is the specific heat c times the molar mass of the substance M/N its numerical value is generally smaller than that of the specific heat. This agreement is because in the classical statistical theory of Ludwig Boltzmann, the heat capacity of solids approaches a maximum of 3R per mole of atoms because full vibrational-mode degrees of freedom amount to 3 degrees of freedom per atom, each corresponding to a quadratic kinetic energy term and a quadratic potential energy term. The vibrational temperature is used commonly when finding the vibrational partition function. The following table shows the experimental molar heat capacities at constant pressure cP,m of the above polyatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* f Notes N≡N=O 38.6 7.3 51.8 10.5 62.0 12.9 13 "Nitrous oxide" NIST Chemistry WebBook, SRD 69, online.

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The diffusion coefficient for $\mathrm{CO}_2$ at $273 \mathrm{~K}$ and $1 \mathrm{~atm}$ is $1.00 \times 10^{-5} \mathrm{~m}^2 \mathrm{~s}^{-1}$. Estimate the collisional cross section of $\mathrm{CO}_2$ given this diffusion coefficient.

Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the "effective area" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. For a reaction between A and B, the collision frequency calculated with the hard-sphere model with the unit number of collisions per m3 per second is: : Z = n_\text{A} n_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}} = 10^6N_A^2\text{[A][B]} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}} where: *σAB is the reaction cross section (unit m2), the area when two molecules collide with each other, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where rA the radius of A and rB the radius of B in unit m. * kB is the Boltzmann constant unit J⋅K−1. Jixin Chen proposed a finite-time solution to the diffusion flux in 2022 which significantly changes the estimated collision frequency of two particles in a solution. ==Rate equations== The rate for a bimolecular gas-phase reaction, A + B → product, predicted by collision theory is : r(T) = kn_\text{A}n_\text{B}= Z \rho \exp \left( \frac{-E_\text{a}}{RT} \right) where: *k is the rate constant in units of (number of molecules)−1⋅s−1⋅m3. * nA is the number density of A in the gas in units of m−3. * nB is the number density of B in the gas in units of m−3. For the diffusive collision, at the infinite time limit when the molecular flux can be calculated from the Fick's laws of diffusion, in 1916 Smoluchowski derived a collision frequency between molecule A and B in a diluted solution: : Z_{AB} = 4 \pi R D_r C_A C_B where: * Z_{AB} is the collision frequency, unit #collisions/s in 1 m3 of solution. Thus the probability of collision should be calculated using the Brownian motion model, which can be approximated to a diffusive flux using various boundary conditions that yield different equations in the Smoluchowski model and the JChen Model. * [A] and [B] are the molar concentration of A and B respectively, unit mol/L. * k is the diffusive collision rate constant, unit L4/3 mol-4/3 s−1. ==See also== * Two-dimensional gas * Rate equation ==References== == External links == *Introduction to Collision Theory Category:Chemical kinetics * A is the area of the collision cross-section in unit m2. * \beta is the product of the unitless fractions of reactive surface area on A and B. * D_r is the relative diffusion constant between A and B, unit m2/s. * D_r is the relative diffusion constant between A and B, unit m2/s, and D_r = D_A + D_B. * C_A and C_B are the number concentrations of molecules A and B in the solution respectively, unit #molecule/m3. or : Z_{AB} = 1000 N_A * 4 \pi R D_r [A] [B] = k [A] [B] where: * Z_{AB} is in unit mole collisions/s in 1 L of solution. For a diluted solution in the gas or the liquid phase, the collision equation developed for neat gas is not suitable when diffusion takes control of the collision frequency, i.e., the direct collision between the two molecules no longer dominates. A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. * [A] and [B] are the molar concentration of A and B respectively, unit mol/L. * k is the diffusive collision rate constant, unit L mol−1 s−1. This hypothesis yields a fractal reaction kinetic rate equation of diffusive collision in a diluted solution: : Z_{AB} = (1000 N_A)^{4/3} * 8 \pi^{-1} A \beta D_r ([A] + [B])^{1/3}[A] [B] = k ([A] + [B])^{1/3}[A] [B] where: * Z_{AB} is in unit mole collisions/s in 1 L of solution. * R is the radius of the collision cross-section, unit m. A pair of colliding spheres of different sizes but of the same material have the same coefficient as below, but multiplied by \left(\frac{R_1}{R_2}\right)^{{3}/{8}} Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material. * e = coefficient of restitution * Sy = dynamic yield strength (dynamic "elastic limit") * E′ = effective elastic modulus * ρ = density * v = velocity at impact * μ = Poisson's ratio e = 3.1 \left(\frac{S_\text{y}}{1}\right)^{5/8} \left(\frac{1}{E'}\right)^{1/2} \left(\frac{1}{v}\right)^{1/4} \left(\frac{1}{\rho}\right)^{1/8} E' = \frac{E}{1-\mu^2} This equation overestimates the actual COR. : Experimental rate constants compared to the ones predicted by collision theory for reactions in solutionE.A. Moelwyn- Hughes, The kinetics of reactions in solution, 2nd ed, page 71. Reaction Solvent A, 1011 s−1⋅M−1 Z, 1011 s−1⋅M−1 Steric factor C2H5Br + OH− ethanol 4.30 3.86 1.11 C2H5O− \+ CH3I ethanol 2.42 1.93 1.25 ClCH2CO2− \+ OH− water 4.55 2.86 1.59 C3H6Br2 \+ I− methanol 1.07 1.39 0.77 HOCH2CH2Cl + OH− water 25.5 2.78 9.17 4-CH3C6H4O− \+ CH3I ethanol 8.49 1.99 4.27 CH3(CH2)2Cl + I− acetone 0.085 1.57 0.054 C5H5N + CH3I C2H2Cl4 — — 2.0 10 ==Alternative collision models for diluted solutions== Collision in diluted gas or liquid solution is regulated by diffusion instead of direct collisions, which can be calculated from Fick's laws of diffusion. Theoretical models to calculate the collision frequency in solutions have been proposed by Marian Smoluchowski in a seminal 1916 publication at the infinite time limit, and Jixin Chen in 2022 at a finite-time approximation. In this case, the change in Ek is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus: C_R = \sqrt{\frac{E_\text{p, (at bounce height)}}{E_\text{p, (at drop height)}}} = \sqrt{\frac{mgh}{mgH}} = \sqrt{\frac{h}{H}} == Speeds after impact == The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between. v_\text{a} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b}-u_\text{a})}{m_\text{a}+m_\text{b}} and v_\text{b} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{a} C_R(u_\text{a}-u_\text{b})}{m_\text{a}+m_\text{b}} where *v_\text{a} is the final velocity of the first object after impact *v_\text{b} is the final velocity of the second object after impact *u_\text{a} is the initial velocity of the first object before impact *u_\text{b} is the initial velocity of the second object before impact *m_\text{a} is the mass of the first object *m_\text{b} is the mass of the second object === Derivation === The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). The coefficient of restitution (COR, also denoted by e), is the ratio of the final to initial relative speed between two objects after they collide. For this reason, determining the COR of a material when there is not identical material for collision is best done by using a much harder material. We can then rearrange: :D(c^*) = - \frac{1}{2 t} \frac{\int^{c^*}_{c_R} x \mathrm{d}c}{(\mathrm{d}c/\mathrm{d}x)_{c=c^*}} Knowing the concentration profile c(x) at annealing time t, and assuming it is invertible as x(c), we can then calculate the diffusion coefficient for all concentrations between cR and cL. === The Matano interface === The last formula has one significant shortcoming: no information is given about the reference according to which x should be measured. A scheme of comparing the rate equations in pure gas and solution is shown in the right figure. thumb|A scheme comparing direct collision and diffusive collision, with corresponding rate equations.

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Benzoic acid, $1.35 \mathrm{~g}$, is reacted with oxygen in a constant volume calorimeter to form $\mathrm{H}_2 \mathrm{O}(l)$ and $\mathrm{CO}_2(g)$ at $298 \mathrm{~K}$. The mass of the water in the inner bath is $1.55 \times$ $10^3 \mathrm{~g}$. The temperature of the calorimeter and its contents rises $2.76 \mathrm{~K}$ as a result of this reaction. Calculate the calorimeter constant.

The theoretical bases of indirect calorimetry: a review." Another source gives :Energy (kcal/min) = (respiration in L/min times change in percentage oxygen) / 20 This corresponds to: :Metabolic rate (cal per minute) = 5 (VO2 in mL/min) ==References== ==Further reading== * Category:Calorimetry In SI units, the calorimeter constant is then calculated by dividing the change in enthalpy (ΔH) in joules by the change in temperature (ΔT) in kelvins or degrees Celsius: :C_\mathrm{cal} = \frac{\Delta{H}}{\Delta{T}} The calorimeter constant is usually presented in units of joules per degree Celsius (J/°C) or joules per kelvin (J/K). Another formula used is: citing :M=VO_2\left(\frac{RQ-0.7}{0.3}e_c+\frac{1-RQ}{0.3}e_f\right) where RQ is the respiratory quotient (ratio of volume CO2 produced to volume of O2 consumed), e_c is , the heat released per litre of oxygen by the oxidation of carbohydrate, and e_f is , the value for fat. The formula can also be written for units of calories per day where VO2 is oxygen consumption expressed in millilitres per minute and VCO2 is the rate of carbon dioxide production in millilitres per minute. Regardless of the specific chemical process, with a known calorimeter constant and a known change in temperature the heat added to the system may be calculated by multiplying the calorimeter constant by that change in temperature. ==See also== *Thermodynamics ==References== Category:Calorimetry Category:Thermochemistry Metabolism. 1988 Mar;37(3):287-301. ==Scientific background== Indirect calorimetry measures O2 consumption and CO2 production. The Weir formula is a formula used in indirect calorimetry, relating metabolic rate to oxygen consumption and carbon dioxide production. Indirect calorimetry estimates the type and rate of substrate utilization and energy metabolism in vivo starting from gas exchange measurements (oxygen consumption and carbon dioxide production during rest and steady-state exercise). thumb|Original RC1 Calorimeter A reaction calorimeter is a calorimeter that measures the amount of energy released (exothermic) or absorbed (endothermic) by a chemical reaction. Journal of Thermal Analysis and Calorimetry, 147(17), 9301–9351. Ambient and diluted fractions of O2 and CO2 are measured for a known ventilation rate, and O2 consumption and CO2 production are determined and converted into Resting Energy Expenditure.Academy of Nutrition and Dietetics "Measuring RMR with Indirect Calorimetry (IC)." Indirect calorimetry calculates heat that living organisms produce by measuring either their production of carbon dioxide and nitrogen waste (frequently ammonia in aquatic organisms, or urea in terrestrial ones), or from their consumption of oxygen. We know from the heat balance equation that: :Q = mf.Cpf.Tin \- Tout We also know that from the heat flow equation that :Q = U.A.LMTD We can therefore rearrange this such that :U = mf.Cpf.Tin \- Tout /A.LMTD This will allow us therefore to monitor U as a function of time. ==Continuous Reaction Calorimeter== thumb|Original Contiplant Calorimeter The Continuous Reaction Calorimeter is especially suitable to obtain thermodynamic information for a scale-up of continuous processes in tubular reactors. thumb|Indirect calorimetry metabolic cart measuring oxygen uptake (O2) and carbon dioxide production (CO2) of a spontaneously breathing subject (dilution method with canopy hood). According to original source, it says: :Metabolic rate (kcal per day) = 1440 (3.9 VO2 \+ 1.1 VCO2) where VO2 is oxygen consumption in litres per minute and VCO2 is the rate of carbon dioxide production in litres per minute. A calorimeter constant (denoted Ccal) is a constant that quantifies the heat capacity of a calorimeter. The profile of the curve is determined by the c-value, which is calculated using the equation: ::: c = n K_a M where n is the stoichiometry of the binding, K_a is the association constant and M is the concentration of the molecule in the cell.Quick Start: Isothermal Titration Calorimetry (ITC) (2016). The molecular formula C20H40O2 (molar mass: 312.53 g/mol, exact mass: 312.3028 u) may refer to: * Arachidic acid, also called eicosanoic acid * Phytanic acid Category:Molecular formulas It may be calculated by applying a known amount of heat to the calorimeter and measuring the calorimeter's corresponding change in temperature. The calorimeter has a gas collector that adapts to the subject and through a unidirectional valve minute by minute collects and quantifies the volume and concentration of O2 inspired and CO2 expired by the subject. This gives the same result as the Weir formula at RQ = 1 (burning only carbohydrates), and almost the same value at RQ = 0.7 (burning only fat). ⊅Σ′ ==History== Antoine Lavoisier noted in 1780 that heat production, in some cases, can be predicted from oxygen consumption, using multiple regression.

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The activation energy for a reaction is $50 . \mathrm{J} \mathrm{mol}^{-1}$. Determine the effect on the rate constant for this reaction with a change in temperature from $273 \mathrm{~K}$ to $298 \mathrm{~K}$.

The rate constant as a function of thermodynamic temperature is then given by: k(T) = Ae^{- E_\mathrm{a}/RT} The reaction rate is given by: r = Ae^{ - E_\mathrm{a}/RT}[\mathrm{A}]^m[\mathrm{B}]^n, where Ea is the activation energy, and R is the gas constant, and m and n are experimentally determined partial orders in [A] and [B], respectively. For the above reaction, one can expect the change of the reaction rate constant (based either on mole fraction or on molar concentration) with pressure at constant temperature to be: : \left(\frac{\partial \ln k_x}{\partial P} \right)_T = -\frac{\Delta V^{\ddagger}} {RT} In practice, the matter can be complicated because the partial molar volumes and the activation volume can themselves be a function of pressure. For a given reaction, the ratio of its rate constant at a higher temperature to its rate constant at a lower temperature is known as its temperature coefficient, (Q). For a one-step process taking place at room temperature, the corresponding Gibbs free energy of activation (ΔG‡) is approximately 23 kcal/mol. ==Dependence on temperature== The Arrhenius equation is an elementary treatment that gives the quantitative basis of the relationship between the activation energy and the reaction rate at which a reaction proceeds. This equation is of the form k = \frac{\kappa k_\mathrm{B}T}{h} e^{\frac{\Delta S^\ddagger }{R}} e^{-\frac{\Delta H^\ddagger}{RT}}where: * k = reaction rate constant * T = absolute temperature * \Delta H^\ddagger = enthalpy of activation * R = gas constant * \kappa = transmission coefficient * k_\mathrm{B} = Boltzmann constant = R/NA, NA = Avogadro constant * h = Planck's constant * \Delta S^\ddagger = entropy of activation This equation can be turned into the form \ln \frac{k}{T} = \frac{-\Delta H^\ddagger}{R} \cdot \frac{1}{T} + \ln \frac{\kappa k_\mathrm{B}}{h} + \frac{\Delta S^\ddagger}{R} The plot of \ln(k/T) versus 1/T gives a straight line with slope -\Delta H^\ddagger/ R from which the enthalpy of activation can be derived and with intercept \ln(\kappa k_\mathrm{B} / h) + \Delta S^\ddagger/ R from which the entropy of activation is derived. ==References== Category:Chemical kinetics The second molecule of H2 does not appear in the rate equation because it reacts in the third step, which is a rapid step after the rate-determining step, so that it does not affect the overall reaction rate. ==Temperature dependence== Each reaction rate coefficient k has a temperature dependency, which is usually given by the Arrhenius equation: : k = A e^{ - \frac{E_\mathrm{a}}{RT} }. It can be done with the help of computer simulation software. ==Rate constant calculations== Rate constant can be calculated for elementary reactions by molecular dynamics simulations. The temperature dependence of ΔG‡ is used to compute these parameters, the enthalpy of activation ΔH‡ and the entropy of activation ΔS‡, based on the defining formula ΔG‡ = ΔH‡ − TΔS‡. The reaction rate thus defined has the units of mol/L/s. In these equations k(T) is the reaction rate coefficient or rate constant, although it is not really a constant, because it includes all the parameters that affect reaction rate, except for time and concentration. In chemical kinetics, the entropy of activation of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation of the transition state theory. As temperature increases, the kinetic energy of the reactants increases. A rise of ten degrees Celsius results in approximately twice the reaction rate. For a reaction between reactants A and B to form a product C, where :A and B are reactants :C is a product :a, b, and c are stoichiometric coefficients, the reaction rate is often found to have the form: r = k[\mathrm{A}]^m [\mathrm{B}]^{n} Here is the reaction rate constant that depends on temperature, and [A] and [B] are the molar concentrations of substances A and B in moles per unit volume of solution, assuming the reaction is taking place throughout the volume of the solution. A=\frac{k}{e^{-E_a/RT}} The units of the pre-exponential factor A are identical to those of the rate constant and will vary depending on the order of the reaction. The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit time. As a rule of thumb, reaction rates for many reactions double for every ten degrees Celsius increase in temperature. For a unimolecular step the reaction rate is described by r = k_1[\mathrm{A}], where k_1 is a unimolecular rate constant. By using the mass balance for the system in which the reaction occurs, an expression for the rate of change in concentration can be derived. The rate ratio at a temperature increase of 10 degrees (marked by points) is equal to the Q10 coefficient. For a bimolecular step the reaction rate is described by r=k_2[\mathrm{A}][\mathrm{B}], where k_2 is a bimolecular rate constant. Substitution of this equation in the previous equation leads to a rate equation expressed in terms of the original reactants : v = k_2 K_1 [\ce{H2}] [\ce{NO}]^2 \, This agrees with the form of the observed rate equation if it is assumed that .

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How long will it take to pass $200 . \mathrm{mL}$ of $\mathrm{H}_2$ at $273 \mathrm{~K}$ through a $10 . \mathrm{cm}$-long capillary tube of $0.25 \mathrm{~mm}$ if the gas input and output pressures are 1.05 and $1.00 \mathrm{~atm}$, respectively?

This pressure difference can be calculated from Laplace's pressure equation, :\Delta P=\frac{2 \gamma}{R}. 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. This time the equation is solved for the height of the meniscus level which again can be used to give the capillary length. One can reorganize to show the capillary length as a function of surface tension and gravity. :\lambda_{\rm c}^2=\frac{hr}{2\cos\theta}, with h the height of the liquid, r the radius of the capillary tube, and \theta the contact angle. If the temperature is 20o then \lambda_c= 2.71mm The capillary length or capillary constant, is a length scaling factor that relates gravity and surface tension. On the other hand, the capillary length would be {\lambda \scriptscriptstyle c} = 6.68mm for water-air on the moon. For molecular fluids, the interfacial tensions and density differences are typically of the order of 10-100 mN m−1 and 0.1-1 g mL−1 respectively resulting in a capillary length of \sim3 mm for water and air at room temperature on earth. thumb|292x292px|The capillary length will vary for different liquids and different conditions. As above, the Laplace and hydrostatic pressure are equated resulting in :R= \frac{\gamma}{\Delta \rho g e_0}=\frac{\lambda_{\rm c}^2}{e_0}. Using the same premises of capillary rise, one can find the capillary length as a function of the volume increase, and wetting perimeter of the capillary walls. ====Association with a sessile droplet==== Another way to find the capillary length is using different pressure points inside a sessile droplet, with each point having a radius of curvature, and equate them to the Laplace pressure equation. 200px|right|thumb|Diagram of a balloon catheter. The equation for \lambda_{\rm c} can also be found with an extra \sqrt{2} term, most often used when normalising the capillary height. == Origin == ===Theoretical=== One way to theoretically derive the capillary length, is to imagine a liquid droplet at the point where surface tension balances gravity. There exists a pressure difference either side of this curvature, and when this balances out the pressure due to gravity, one can rearrange to find the capillary length. For a soap bubble, the surface tension must be divided by the mean thickness, resulting in a capillary length of about 3 meters in air! This was a mathematical explanation of the work published by James Jurin in 1719, where he quantified a relationship between the maximum height taken by a liquid in a capillary tube and its diameter – Jurin's law. The capillary length can then be worked out the same way except that the thickness of the film, e_0 must be taken into account as the bubble has a hollow center, unlike the droplet which is a solid. However if these conditions are known, the capillary length can be considered a constant for any given liquid, and be used in numerous fluid mechanical problems to scale the derived equations such that they are valid for any fluid. When a capillary tube is inserted into a liquid, the liquid will rise or fall in the tube, due to an imbalance in pressure. thumb|upright=0.4|right|Diagram of a Durham Tube Durham tubes are used in microbiology to detect production of gas by microorganisms. Therefore the bond number can be written as :\mathrm{Bo}=\left(\frac{L}{\lambda_{\rm c}}\right)^2, with \lambda_{\rm c} the capillary length. This property is usually used by physicists to estimate the height a liquid will rise in a particular capillary tube, radius known, without the need for an experiment. The pressure due to gravity (hydrostatic pressure) P_{\rm h} of a column of liquid is given by :P_{\rm h}=\rho g h=2\rho g\lambda_{\rm c} , where \rho is the droplet density, g the gravitational acceleration, and h=2\lambda_{\rm c} is the height of the droplet.

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Calculate the Debye-Hückel screening length $1 / \kappa$ at $298 \mathrm{~K}$ in a $0.0075 \mathrm{~m}$ solution of $\mathrm{K}_3 \mathrm{PO}_4$.

Today, \kappa^{-1} is called the Debye screening length. The extended Debye–Hückel equation provides accurate results for μ ≤ 0.1. In the context of solids, Thomas–Fermi screening length may be required instead of Debye length. == See also == * Bjerrum length * Debye–Falkenhagen effect * Plasma oscillation * Shielding effect * Screening effect == References == == Further reading == * * Category:Electricity Category:Electronics concepts Category:Colloidal chemistry Category:Plasma parameters Category:Electrochemistry Category:Length Category:Peter Debye In plasmas and electrolytes, the Debye length \lambda_{\rm D} (Debye radius or Debye–Hückel screening length), is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. The corresponding Debye screening wave vector k_{\rm D}=1/\lambda_{\rm D} for particles of density n, charge q at a temperature T is given by k_{\rm D}^2=4\pi n q^2/(k_{\rm B}T) in Gaussian units. P^\text{ex} = -\frac{k_\text{B} T \kappa_\text{cgs}^3}{24\pi} = -\frac{k_\text{B} T \left(\frac{4\pi \sum_j c_j q_j}{\varepsilon_0 \varepsilon_r k_\text{B} T }\right)^{3/2}}{24\pi}. This intuitive picture leads to an effective calculation of Debye shielding (see section II.A.2 of Meyer-Vernet N (1993) Aspects of Debye shielding. The first is what could be called the square of the reduced inverse screening length, (\kappa a)^2. The Debye–Hückel length may be expressed in terms of the Bjerrum length \lambda_{\rm B} as \lambda_{\rm D} = \left(4 \pi \, \lambda_{\rm B} \, \sum_{j = 1}^N n_j^0 \, z_j^2\right)^{-1/2}, where z_j = q_j/e is the integer charge number that relates the charge on the j-th ionic species to the elementary charge e. == In a plasma == For a weakly collisional plasma, Debye shielding can be introduced in a very intuitive way by taking into account the granular character of such a plasma. The equation is \ln(\gamma_i) = -\frac{z_i^2 q^2 \kappa}{8 \pi \varepsilon_r \varepsilon_0 k_\text{B} T} = -\frac{z_i^2 q^3 N^{1/2}_\text{A}}{4 \pi (\varepsilon_r \varepsilon_0 k_\text{B} T)^{3/2}} \sqrt{10^3\frac{I}{2}} = -A z_i^2 \sqrt{I}, where * z_i is the charge number of ion species i, * q is the elementary charge, * \kappa is the inverse of the Debye screening length \lambda_{\rm D} (defined below), * \varepsilon_r is the relative permittivity of the solvent, * \varepsilon_0 is the permittivity of free space, * k_\text{B} is the Boltzmann constant, * T is the temperature of the solution, * N_\mathrm{A} is the Avogadro constant, * I is the ionic strength of the solution (defined below), * A is a constant that depends on temperature. The Debye length of semiconductors is given: L_{\rm D} = \sqrt{\frac{\varepsilon k_{\rm B} T}{q^2 N_{\rm dop}}} where * ε is the dielectric constant, * kB is the Boltzmann constant, * T is the absolute temperature in kelvins, * q is the elementary charge, and * Ndop is the net density of dopants (either donors or acceptors). Debye length is an important parameter in plasma physics, electrolytes, and colloids (DLVO theory). It is possible that Debye–Hückel equation is not able to foresee this behavior because of the linearization of the Poisson–Boltzmann equation, or maybe not: studies about this have been started only during the last years of the 20th century because before it wasn't possible to investigate the 10−4 M region, so it is possible that during the next years new theories will be born. ==Extensions of the theory== A number of approaches have been proposed to extend the validity of the law to concentration ranges as commonly encountered in chemistry One such extended Debye–Hückel equation is given by: \- \log_{10}(\gamma) = \frac{A|z_+z_-|\sqrt{I}}{1 + Ba\sqrt{I}} where \gamma as its common logarithm is the activity coefficient, z is the integer charge of the ion (1 for H+, 2 for Mg2+ etc.), As the only characteristic length scale in the Debye–Hückel equation, \lambda_D sets the scale for variations in the potential and in the concentrations of charged species. Hence they developed the Debye–Hückel equation and Debye–Hückel limiting law. thumb|left|σ Persei in optical light Sigma Persei (Sigma Per, σ Persei, σ Per) is an orange K-type giant with an apparent magnitude of +4.36. The excess osmotic pressure obtained from Debye–Hückel theory is in cgs units:http://homepages.rpi.edu/~keblip/THERMO/chapters/Chapter33.pdf, page 9. Alternatively, \kappa^{-1} = \frac{1}{\sqrt{8\pi \lambda_{\rm B} N_{\rm A} \times 10^{-24} I}} where \lambda_{\rm B} is the Bjerrum length of the medium in nm, and the factor 10^{-24} derives from transforming unit volume from cubic dm to cubic nm. The molecular formula C20H28FN3O3 (molar mass: 377.453 g/mol, exact mass: 377.2115 u) may refer to: * 5F-ADB * 5F-EMB-PINACA Category:Molecular formulas Factor out the scalar potential and assign the leftovers, which are constant, to \kappa^2. The term in parentheses divided by \varepsilon, has the units of an inverse length squared and by dimensional analysis leads to the definition of the characteristic length scale \lambda_{\rm D} = \left(\frac{\varepsilon \, k_{\rm B} T}{\sum_{j = 1}^N n_j^0 \, q_j^2}\right)^{1/2} that commonly is referred to as the Debye–Hückel length. It has been solved by the Swedish mathematician Thomas Hakon Gronwall and his collaborators physicical chemists V. K. La Mer and Karl Sandved in a 1928 article from Physikalische Zeitschrift dealing with extensions to Debye–Huckel theory, which resorted to Taylor series expansion.

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A system consisting of $82.5 \mathrm{~g}$ of liquid water at $300 . \mathrm{K}$ is heated using an immersion heater at a constant pressure of 1.00 bar. If a current of $1.75 \mathrm{~A}$ passes through the $25.0 \mathrm{ohm}$ resistor for 100 .s, what is the final temperature of the water?

thumb |upright=1.35 |Upper ocean heat content (OHC) has been increasing because oceans have absorbed a large portion of the excess heat trapped in the atmosphere by human-caused global warming. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The partnership between Argo and Jason measurements has yielded ongoing improvements to estimates of OHC and other global ocean properties. ==Causes for heat uptake== thumb|320px|Over 90% of the thermal energy that has accumulated on Earth from global heating since 1970 is stored in the ocean. thumbnail|300px|Global Heat Content in the top 2000 meters of the ocean since 1958 The more abundant equatorial solar irradiance which is absorbed by Earth's tropical surface waters drives the overall poleward propagation of ocean heat. From this, Ohm determined his law of proportionality and published his results. thumb|Internal resistance model In modern notation we would write, I = \frac {\mathcal E}{r+R}, where \mathcal E is the open-circuit emf of the thermocouple, r is the internal resistance of the thermocouple and R is the resistance of the test wire. thumb|170px|A typical glass-tube immersion style aquarium heater An aquarium heater is a device used in the fishkeeping hobby to warm the temperature of water in aquariums. To calculate the ocean heat content, measurements of ocean temperature at many different locations and depths are required. To calculate the ocean heat content, measurements of ocean temperature at many different locations and depths are required. Ocean heat content (OHC) is the energy absorbed and stored by oceans. thumb|400px|Diagram showing pressure difference induced by a temperature difference. The increase in OHC accounts for 30–40% of global sea-level rise from 1900 to 2020 because of thermal expansion. Additionally, a study from 2022 on anthropogenic warming in the ocean indicates that 62% of the warming from the years between 1850 and 2018 in the North Atlantic along 25°N is kept in the water below 700 m, where a major percentage of the ocean's surplus heat is stored. In 2022, the world’s oceans, as given by OHC, were again the hottest in the historical record and exceeded the previous 2021 record maximum. 50x50px Text was copied from this source, which is available under a Creative Commons Attribution 4.0 International License Ocean heat content and sea level rise are important indicators of climate change. Between 1971 and 2018, the rise in OHC accounted for over 90% of Earth’s excess thermal energy from global heating. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. Then the heat per unit volume u satisfies an equation : \frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q. A study in 2015 concluded that ocean heat content increases by the Pacific Ocean were compensated by an abrupt distribution of OHC into the Indian Ocean. Thus instead of being for thermodynamic temperature (Kelvin - K), units of u should be J/L. The upper ocean (0–700 m) has warmed since 1971, while it is very likely that warming has occurred at intermediate depths (700–2000 m) and likely that deep ocean (below 2000 m) temperatures have increased. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). Measurements of temperature versus ocean depth generally show an upper mixed layer (0–200 m), a thermocline (200–1500 m), and a deep ocean layer (>1500 m). thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. While the light is turned off, the value of q for the tungsten filament would be zero. == Solving the heat equation using Fourier series == right|thumb|300px|Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions.

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For an ensemble consisting of a mole of particles having two energy levels separated by $1000 . \mathrm{cm}^{-1}$, at what temperature will the internal energy equal $3.00 \mathrm{~kJ}$ ?

Approximate values of kT at 298 K Units kT = J kT = pN⋅nm kT = cal kT = meV kT=-174 dBm/Hz kT/hc ≈ cm−1 kT/e = 25.7 mV RT = kT ⋅ NA = kJ⋅mol−1 RT = 0.593 kcal⋅mol−1 h/kT = 0.16 ps kT (also written as kBT) is the product of the Boltzmann constant, k (or kB), and the temperature, T. To calculate the transition temperature at any density, integrate, over all momentum states, the expression for maximum number of excited particles, : :\, N = V \int {d^3k \over (2\pi)^3} {p(k)\over 1-p(k)} = V \int {d^3k \over (2\pi)^3} {1 \over e^{k^2\over 2mT}-1} :\, p(k)= e^{-k^2\over 2mT}. Since 1995, many other atomic species have been condensed, and BECs have also been realized using molecules, quasi- particles, and photons. == Critical temperature == This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by: :T_{\rm c}=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi\hbar^2}{m k_{\rm B}} \approx 3.3125 \ \frac{\hbar^2 n^{2/3}}{m k_{\rm B}} where: : \,T_{\rm c} is the critical temperature, \,n the particle density, \,m the mass per boson, \hbar the reduced Planck constant, \,k_{\rm B} the Boltzmann constant and \,\zeta the Riemann zeta function; \,\zeta(3/2)\approx 2.6124. MJ/kg may refer to: * megajoules per kilogram * Specific kinetic energy * Heat of fusion * Heat of combustion kJ/kg may refer to: * kilojoules per kilogram * The SI derived units of specific energy * Specific Internal energy * Specific kinetic energy * Heat of fusion * Heat of combustion At temperature T, a particle will have a lesser probability to be in state |1\rangle by e^{-E/kT}. thumb|upright=1.5|Schematic Bose–Einstein condensation versus temperature of the energy diagram In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67 °F). In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. So a collection of enough Bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference. In the indistinguishable case, each value of K is a single state, which has its own separate Boltzmann probability. Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. Thermokinetics deals with the study of thermal decomposition kinetics. ==See also== *Thermogravimetry *Differential thermal analysis *Differential scanning calorimetry ==References== * * Category:Chemical kinetics If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. The high transition temperature (relative to atomic gases) is due to the magnons' small mass (near that of an electron) and greater achievable density. More fundamentally, kT is the amount of heat required to increase the thermodynamic entropy of a system by k. On the critical region it is possible to define a critical temperature and thermal wavelength: :\lambda_c^3=g_{3/2}(1)v=\zeta(3/2)v :T_c=\frac{2\pi \hbar^2 }{m k_B \lambda_c^2} recovering the value indicated on the previous section. The SI units for RT are joules per mole (J/mol). This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics. == Derivation == === Ideal Bose gas === For an ideal Bose gas we have the equation of state: :\frac{1}{v}=\frac{1}{\lambda^3}g_{3/2}(f)+\frac{1}{V}\frac{f}{1-f} where v=V/N is the per particle volume, \lambda the thermal wavelength, f the fugacity and :g_\alpha (f)=\sum \limits_{n=1}^\infty \frac{f^n}{n^\alpha} It is noticeable that g_{3/2} is a monotonically growing function of f in f \in [0, 1], which are the only values for which the series converge. For a system in equilibrium in canonical ensemble, the probability of the system being in state with energy E is proportional to . As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point, more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. It satisfies : \frac{n_{\rm u}}{n_{\rm l}} = \frac{g_{\rm u}}{g_{\rm l}} \exp{ \left(-\frac{\Delta E}{k T_{\rm ex}} \right) }, where * is the number of particles in an upper (e.g. excited) state; * is the statistical weight of those upper-state particles; * is the number of particles in a lower (e.g. ground) state; * is the statistical weight of those lower-state particles; * is the exponential function; * is the Boltzmann constant; * is the difference in energy between the upper and lower states. When the integral (also known as Bose–Einstein integral) is evaluated with factors of k_B and ℏ restored by dimensional analysis, it gives the critical temperature formula of the preceding section.

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A muscle fiber contracts by $3.5 \mathrm{~cm}$ and in doing so lifts a weight. Calculate the work performed by the fiber. Assume the muscle fiber obeys Hooke's law $F=-k x$ with a force constant $k$ of $750 . \mathrm{N} \mathrm{m}^{-1}$.

The actual force increased from 0 to F during the deformation; the work done can be computed by integration in distance. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. The work of the net force is calculated as the product of its magnitude and the particle displacement. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. Therefore, work need only be computed for the gravitational forces acting on the bodies. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . Notice that the work done by gravity depends only on the vertical movement of the object. This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. From Newton's second law, it can be shown that work on a free (no fields), rigid (no internal degrees of freedom) body, is equal to the change in kinetic energy corresponding to the linear velocity and angular velocity of that body, W = \Delta E_\text{k}. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. The work–energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. In [Erdemir, 2007] a list of possible cost functions with a brief rationale and the suggested model validation technique is available. ==== Clarification on the use of the maximum isometric force ==== Muscle contraction can be eccentric (velocity of contraction v<0), concentric (v >0) or isometric (v=0). If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. In physics, work is the energy transferred to or from an object via the application of force along a displacement. Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle. If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement).

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A gas sample is known to be a mixture of ethane and butane. A bulb having a $230.0 \mathrm{~cm}^3$ capacity is filled with the gas to a pressure of $97.5 \times 10^3 \mathrm{~Pa}$ at $23.1^{\circ} \mathrm{C}$. If the mass of the gas in the bulb is $0.3554 \mathrm{~g}$, what is the mole percent of butane in the mixture?

Butane is one of a group of liquefied petroleum gases (LP gases). Forming gas is a mixture of hydrogen (mole fraction varies) and nitrogen. For example, the density of liquid propane is 571.8±1 kg/m3 (for pressures up to 2MPa and temperature 27±0.2 °C), while the density of liquid butane is 625.5±0.7 kg/m3 (for pressures up to 2MPa and temperature -13±0.2 °C). alt=Density of liquid and vaporized butane|none|thumb|500x500px|Propane and butane density data == Isomers == Common name normal butane unbranched butane n-butane isobutane i-butane IUPAC name butane methylpropane Molecular diagram 150px 120px Skeletal diagram 120px 100px Rotation about the central C−C bond produces two different conformations (trans and gauche) for n-butane. == Reactions == When oxygen is plentiful, butane burns to form carbon dioxide and water vapor; when oxygen is limited, carbon (soot) or carbon monoxide may also be formed. The relative rates of the chlorination is partially explained by the differing bond dissociation energies, 425 and 411 kJ/mol for the two types of C-H bonds. == Uses == Normal butane can be used for gasoline blending, as a fuel gas, fragrance extraction solvent, either alone or in a mixture with propane, and as a feedstock for the manufacture of ethylene and butadiene, a key ingredient of synthetic rubber. When there is sufficient oxygen: : 2 C4H10 \+ 13 O2 → 8 CO2 \+ 10 H2O When oxygen is limited: : 2 C4H10 \+ 9 O2 → 8 CO + 10 H2O By weight, butane contains about or by liquid volume . The molecular formula C4H10 (molar mass: 58.12 g/mol, exact mass: 58.0783 u) may refer to: * Butane, or n-butane * Isobutane, also known as methylpropane or 2-methylpropane Butane is a highly flammable, colorless, easily liquefied gas that quickly vaporizes at room temperature and pressure. Butane is denser than air. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Butane did not have much practical use until the 1910s, when W. Snelling identified butane and propane as components in gasoline and found that, if they were cooled, they could be stored in a volume-reduced liquified state in pressurized containers. == Density == The density of butane is highly dependent on temperature and pressure in the reservoir. Butane is also used as lighter fuel for common lighters or butane torches and is sold bottled as a fuel for cooking, barbecues and camping stoves. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . For gasoline blending, n-butane is the main component used to manipulate the Reid vapor pressure (RVP). Butane is the most commonly abused volatile substance in the UK, and was the cause of 52% of solvent related deaths in 2000. Instead of a mole the constant can be expressed by considering the normal cubic meter. Butane () or n-butane is an alkane with the formula C4H10. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. File:Photo D2.jpg | Butane fuel canisters for use in camping stoves File:The Green Lighter 1 cropped.jpg | Butane lighter, showing liquid butane reservoir File:Aerosol.png | An aerosol spray can, which may be using butane as a propellant File:ButaneGasCylinder WhiteBack.jpg | Butane gas cylinder used for cooking == Effects and health issues == Inhalation of butane can cause euphoria, drowsiness, unconsciousness, asphyxia, cardiac arrhythmia, fluctuations in blood pressure and temporary memory loss, when abused directly from a highly pressurized container, and can result in death from asphyxiation and ventricular fibrillation. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. It provides a molar mass for air of 28.9625 g/mol, and provides a composition for standard dry air as a footnote. == References == Category:Gases When blended with propane and other hydrocarbons, the mixture may be referred to commercially as liquefied petroleum gas (LPG).

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One liter of fully oxygenated blood can carry 0.18 liters of $\mathrm{O}_2$ measured at $T=298 \mathrm{~K}$ and $P=1.00 \mathrm{~atm}$. Calculate the number of moles of $\mathrm{O}_2$ carried per liter of blood. Hemoglobin, the oxygen transport protein in blood has four oxygen binding sites. How many hemoglobin molecules are required to transport the $\mathrm{O}_2$ in $1.0 \mathrm{~L}$ of fully oxygenated blood?

Each hemoglobin molecule has the capacity to carry four oxygen molecules. The oxygen-carrying capacity of hemoglobin is determined by the type of hemoglobin present in the blood. Although binding of oxygen to hemoglobin continues to some extent for pressures about 50 mmHg, as oxygen partial pressures decrease in this steep area of the curve, the oxygen is unloaded to peripheral tissue readily as the hemoglobin's affinity diminishes. Venous blood with an oxygen concentration of 15 mL/100 mL would therefore lead to typical values of the a-vO2 diff at rest of around 5 mL/100 mL. To see the relative affinities of each successive oxygen as you remove/add oxygen from/to the hemoglobin from the curve compare the relative increase/decrease in p(O2) needed for the corresponding increase/decrease in s(O2). ==Factors that affect the standard dissociation curve== The strength with which oxygen binds to hemoglobin is affected by several factors. So, one will have a lesser hemoglobin saturation percentage for the same [O2] or a higher partial pressure of oxygen. The amount of oxygen bound to the hemoglobin at any time is related, in large part, to the partial pressure of oxygen to which the hemoglobin is exposed. The oxygen bound to the hemoglobin is released into the blood's plasma and absorbed into the tissues, and the carbon dioxide in the tissues is bound to the hemoglobin. The partial pressure of oxygen in the blood at which the hemoglobin is 50% saturated, typically about 26.6 mmHg (3.5 kPa) for a healthy person, is known as the P50. The binding affinity of hemoglobin to O2 is greatest under a relatively high pH. === Carbon dioxide === Carbon dioxide affects the curve in two ways. Arterial blood will generally contain an oxygen concentration of around 20 mL/100 mL. A hemoglobin molecule can bind up to four oxygen molecules in a reversible method. The T state has a lower affinity for oxygen than the R state, so with increased acidity, the hemoglobin binds less O2 for a given PO2 (and more H+). Specifically, the oxyhemoglobin dissociation curve relates oxygen saturation (SO2) and partial pressure of oxygen in the blood (PO2), and is determined by what is called "hemoglobin affinity for oxygen"; that is, how readily hemoglobin acquires and releases oxygen molecules into the fluid that surrounds it. thumb|Structure of oxyhemoglobin ==Background== Hemoglobin (Hb) is the primary vehicle for transporting oxygen in the blood. The a-vO2 diff is usually measured in millilitres of oxygen per 100 millilitres of blood (mL/100 mL).Malpeli, Physical Education, Chapter 4: Acute Responses to Exercise, p. 106. ==Measurement== The arteriovenous oxygen difference is usually taken by comparing the difference in the oxygen concentration of oxygenated blood in the femoral, brachial, or radial artery and the oxygen concentration in the deoxygenated blood from the mixed supply found in the pulmonary artery (as an indicator of the typical mixed venous supply). As the blood circulates to other body tissue in which the partial pressure of oxygen is less, the hemoglobin releases the oxygen into the tissue because the hemoglobin cannot maintain its full bound capacity of oxygen in the presence of lower oxygen partial pressures. ==Sigmoid shape== thumb|Hemoglobin saturation curve The curve is usually best described by a sigmoid plot, using a formula of the kind: :S(t) = \frac{1}{1 + e^{-t}}. In the capillaries, where carbon dioxide is produced, oxygen bound to the hemoglobin is released into the blood's plasma and absorbed into the tissues. Hemoglobin's affinity for oxygen increases as successive molecules of oxygen bind. HbF then delivers that bound oxygen to tissues that have even lower partial pressures where it can be released. ==See also== * Automated analyzer * Bohr effect ==Notes== ==References== ==External links== * * The Interactive Oxyhemoglobin Dissociation Curve * Simulation of the parameters CO2, pH and temperature on the oxygen–hemoglobin dissociation curve (left or right shift) Category:Respiratory physiology Category:Chemical pathology Category:Hematology Category:Oxygen The 'plateau' portion of the oxyhemoglobin dissociation curve is the range that exists at the pulmonary capillaries (minimal reduction of oxygen transported until the p(O2) falls 50 mmHg). Solid oxygen forms at normal atmospheric pressure at a temperature below 54.36 K (−218.79 °C, −361.82 °F). The phosphate/oxygen ratio, or P/O ratio, refers to the amount of ATP produced from the movement of two electrons through a defined electron transport chain, terminated by reduction of an oxygen atom.Garrett & Grisham 2010, p.620.

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Consider a collection of molecules where each molecule has two nondegenerate energy levels that are separated by $6000 . \mathrm{cm}^{-1}$. Measurement of the level populations demonstrates that there are exactly 8 times more molecules in the ground state than in the upper state. What is the temperature of the collection?

Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. So a collection of enough Bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference. To calculate the transition temperature at any density, integrate, over all momentum states, the expression for maximum number of excited particles, : :\, N = V \int {d^3k \over (2\pi)^3} {p(k)\over 1-p(k)} = V \int {d^3k \over (2\pi)^3} {1 \over e^{k^2\over 2mT}-1} :\, p(k)= e^{-k^2\over 2mT}. The molecular-scale temperature is the defining property of the U.S. Standard Atmosphere, 1962. thumb|upright=1.5|Schematic Bose–Einstein condensation versus temperature of the energy diagram In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67 °F). The high transition temperature (relative to atomic gases) is due to the magnons' small mass (near that of an electron) and greater achievable density. Average yearly temperature is 22.4°C, ranging from an average minimum of 12.2°C to a maximum of 29.9°C. In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments. At temperature T, a particle will have a lesser probability to be in state |1\rangle by e^{-E/kT}. Only 8% of atoms are in the ground state of the trap near absolute zero, rather than the 100% of a true condensate. Some of the warmest temperatures can be found in the thermosphere, due to its reception of strong ionizing radiation at the level of the Van Allen radiation belt. ==Temperature range== The variation in temperature that occurs from the highs of the day to the cool of nights is called diurnal temperature variation. The chemistry of systems at room temperature is determined by the electronic properties, which is essentially fermionic, since room temperature thermal excitations have typical energies much higher than the hyperfine values. == In fiction == * In the 2016 film Spectral, the US military battles mysterious enemy creatures fashioned out of Bose–Einstein condensates. Since 1995, many other atomic species have been condensed, and BECs have also been realized using molecules, quasi- particles, and photons. == Critical temperature == This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by: :T_{\rm c}=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi\hbar^2}{m k_{\rm B}} \approx 3.3125 \ \frac{\hbar^2 n^{2/3}}{m k_{\rm B}} where: : \,T_{\rm c} is the critical temperature, \,n the particle density, \,m the mass per boson, \hbar the reduced Planck constant, \,k_{\rm B} the Boltzmann constant and \,\zeta the Riemann zeta function; \,\zeta(3/2)\approx 2.6124. Atmospheric temperature is a measure of temperature at different levels of the Earth's atmosphere. Average maximum yearly temperature is 28.7°C and average minimum is 21.9°C. The average temperature range is 5.7°C only. Junction temperature, short for transistor junction temperature, is the highest operating temperature of the actual semiconductor in an electronic device. Cooling fermions to extremely low temperatures has created degenerate gases, subject to the Pauli exclusion principle. The average temperature range is 11.4 degrees. As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point, more particles will be forced into a single state than the maximum allowed for that state by statistical weighting.

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Calculate $\Delta S^{\circ}$ for the reaction $3 \mathrm{H}_2(g)+\mathrm{N}_2(g) \rightarrow$ $2 \mathrm{NH}_3(g)$ at $725 \mathrm{~K}$. Omit terms in the temperature-dependent heat capacities higher than $T^2 / \mathrm{K}^2$.

Since \Delta_r G^\ominus = - RT \ln K_{eq}, the temperature dependence of both terms can be described by Van t'Hoff equations as a function of T. Integration of this equation permits the evaluation of the heat of reaction at one temperature from measurements at another temperature.Laidler K.J. and Meiser J.H., "Physical Chemistry" (Benjamin/Cummings 1982), p.62Atkins P. and de Paula J., "Atkins' Physical Chemistry" (8th edn, W.H. Freeman 2006), p.56 :\Delta H^\circ \\! \left( T \right) = \Delta H^\circ \\! \left( T^\circ \right) + \int_{T^\circ}^{T} \Delta C_P^\circ \, \mathrm{d} T Pressure variation effects and corrections due to mixing are generally minimal unless a reaction involves non-ideal gases and/or solutes, or is carried out at extremely high pressures. The definite integral between temperatures and is then :\ln \frac{K_2}{K_1} = \frac{\Delta_r H^\ominus}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right). The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. Since in reality \Delta_r H^\ominus and the standard reaction entropy \Delta_r S^\ominus do vary with temperature for most processes, the integrated equation is only approximate. For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. For a change from reactants to products at constant temperature and pressure the equation becomes \Delta G = \Delta H - T\Delta S. Differentiation of this expression with respect to the variable while assuming that both \Delta_r H^\ominus and \Delta_r S^\ominus are independent of yields the Van 't Hoff equation. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. However, the relation loses its validity when the aggregation number is also temperature-dependent, and the following relation should be used instead: :RT^2\left(\frac{\partial}{\partial T}\ln\mathrm{CMC}\right)_P = -\Delta_r H^\ominus_\mathrm{m}(N) + T\left(\frac{\partial}{\partial N}\left(G_{N+1} - G_N\right)\right)_{T,P}\left(\frac{\partial N}{\partial T}\right)_P, with and being the free energies of the surfactant in a micelle with aggregation number and respectively. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. In this equation is the equilibrium constant at absolute temperature , and is the equilibrium constant at absolute temperature . ===Development from thermodynamics=== Combining the well-known formula for the Gibbs free energy of reaction : \Delta_r G^\ominus = \Delta_r H^\ominus - T\Delta_r S^\ominus, where is the entropy of the system, with the Gibbs free energy isotherm equation: :\Delta_r G^\ominus = -RT \ln K_\mathrm{eq}, we obtain :\ln K_\mathrm{eq} = -\frac{\Delta_r H^\ominus}{RT} + \frac{\Delta_r S^\ominus}{R}. The Van 't Hoff plot can be used to find the enthalpy and entropy change for each mechanism and the favored mechanism under different temperatures. :\begin{align} \Delta_r H_1 &= - R \times \text{slope}_1, & \Delta_r S_1 &= R \times \text{intercept}_1; \\\\[5pt] \Delta_r H_2 &= - R \times \text{slope}_2, & \Delta_r S_2 &= R \times \text{intercept}_2. \end{align} In the example figure, the reaction undergoes mechanism 1 at high temperature and mechanism 2 at low temperature. ===Temperature dependence=== thumb|right|x275px|Temperature-dependent Van 't Hoff plot The Van 't Hoff plot is linear based on the tacit assumption that the enthalpy and entropy are constant with temperature changes. The slope of the line may be multiplied by the gas constant to obtain the standard enthalpy change of the reaction, and the intercept may be multiplied by to obtain the standard entropy change. ===Van 't Hoff isotherm=== The Van 't Hoff isotherm can be used to determine the temperature dependence of the Gibbs free energy of reaction for non- standard state reactions at a constant temperature: :\left(\frac {dG}{d\xi}\right)_{T,p} = \Delta_\mathrm{r}G = \Delta_\mathrm{r}G^\ominus + RT \ln Q_\mathrm{r}, where \Delta_\mathrm{r}G is the Gibbs free energy of reaction under non-standard states at temperature T, \Delta_r G^\ominus is the Gibbs free energy for the reaction at (T,P^0), \xi is the extent of reaction, and is the thermodynamic reaction quotient. We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. In practice, the equation is often integrated between two temperatures under the assumption that the standard reaction enthalpy \Delta_r H^\ominus is constant (and furthermore, this is also often assumed to be equal to its value at standard temperature). Thus, for an exothermic reaction, the Van 't Hoff plot should always have a positive slope. === Error propagation === At first glance, using the fact that it would appear that two measurements of would suffice to be able to obtain an accurate value of : :\Delta_r H^\ominus = R \frac{\ln K_1 - \ln K_2}{\frac{1}{T_2} - \frac{1}{T_1}}, where and are the equilibrium constant values obtained at temperatures and respectively. Knowing the slope and intercept from the Van 't Hoff plot, the enthalpy and entropy of a reaction can be easily obtained using :\begin{align} \Delta_r H &= - R \times \text{slope}, \\\ \Delta_r S &= R \times \text{intercept}. \end{align} The Van 't Hoff plot can be used to quickly determine the enthalpy of a chemical reaction both qualitatively and quantitatively. When a reaction is at equilibrium, and \Delta_\mathrm{r}G = 0. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. A first-order approximation is to assume that the two different reaction products have different heat capacities. The data should have a linear relationship, the equation for which can be found by fitting the data using the linear form of the Van 't Hoff equation :\ln K_\mathrm{eq} = -\frac{\Delta_r H^\ominus}{RT} + \frac{\Delta_r S^\ominus}{R}.

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The thermal conductivities of acetylene $\left(\mathrm{C}_2 \mathrm{H}_2\right)$ and $\mathrm{N}_2$ at $273 \mathrm{~K}$ and $1 \mathrm{~atm}$ are 0.01866 and $0.0240 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~m}^{-1} \mathrm{~s}^{-1}$, respectively. Based on these data, what is the ratio of the collisional cross section of acetylene relative to $\mathrm{N}_2$ ?

Under these assumptions, an elementary calculation yields for the thermal conductivity : k = \beta \rho \lambda c_v \sqrt{\frac{2k_\text{B} T}{\pi m}}, where \beta is a numerical constant of order 1, k_\text{B} is the Boltzmann constant, and \lambda is the mean free path, which measures the average distance a molecule travels between collisions. In physics, the Wiedemann–Franz law states that the ratio of the electronic contribution of the thermal conductivity (κ) to the electrical conductivity (σ) of a metal is proportional to the temperature (T). : \frac \kappa \sigma = LT Theoretically, the proportionality constant L, known as the Lorenz number, is equal to : L = \frac \kappa {\sigma T} = \frac{\pi^2} 3 \left(\frac{k_{\rm B}} e \right)^2 = 2.44\times 10^{-8}\;\mathrm{V}^2\mathrm{K}^{-2}, where kB is Boltzmann's constant and e is the elementary charge. In summary, for a plate of thermal conductivity k, area A and thickness L, *thermal conductance = kA/L, measured in W⋅K−1. **thermal resistance = L/(kA), measured in K⋅W−1. *heat transfer coefficient = k/L, measured in W⋅K−1⋅m−2. **thermal insulance = L/k, measured in K⋅m2⋅W−1. "The Thermal Conductivity of Air at Reduced Pressures and Length Scales," Electronics Cooling, November 2002, http://www.electronics- cooling.com/2002/11/the-thermal-conductivity-of-air-at-reduced-pressures-and- length-scales/ Retrieved 05:20, 10 April 2016 (UTC). 273-293-298 300 600 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 hiAerosols2.95-loAerosols7.83×10−15 (78.03%N2,21%O2,+0.93%Ar,+0.04%CO2) (1 atm) The plate distance is one centimeter, the special conductivity values were calculated from the Lasance approximation formula in The Thermal conductivity of Air at Reduced Pressures and Length Scales and the primary values were taken from Weast at the normal pressure tables in the CRC handbook on page E2. From considerations of energy conservation, the heat flow between the two bodies in contact, bodies A and B, is found as: One may observe that the heat flow is directly related to the thermal conductivities of the bodies in contact, k_A and k_B, the contact area A, and the thermal contact resistance, 1/h_c, which, as previously noted, is the inverse of the thermal conductance coefficient, h_c. ==Importance== Most experimentally determined values of the thermal contact resistance fall between 0.000005 and 0.0005 m2 K/W (the corresponding range of thermal contact conductance is 200,000 to 2000 W/m2 K). For a plate of thermal conductivity k, area A and thickness L, the conductance is kA/L, measured in W⋅K−1.Bejan, p. 34 The relationship between thermal conductivity and conductance is analogous to the relationship between electrical conductivity and electrical conductance. The formation of an insulating boundary layer can also result in an apparent reduction in the thermal conductivity. ==Experimental values== thumb|upright=2.5|Experimental values of thermal conductivity The thermal conductivities of common substances span at least four orders of magnitude. This conductance, known as thermal boundary conductance, is due to the differences in electronic and vibrational properties between the contacting materials. Then : \kappa = L T \sigma = {\pi^2 k_{\rm B}^2 \over 3e^2}\times {n e^2 \over h} T = {\pi^2 k_{\rm B}^2 \over 3h} n T = g_0 n, where g_0 is the thermal conductance quantum defined above. ==References== ==See also== * Thermal properties of nanostructures Category:Mesoscopic physics Category:Nanotechnology Category:Quantum mechanics Category:Condensed matter physics Category:Physical quantities Category:Heat conduction Many pure metals have a peak thermal conductivity between 2 K and 10 K. In imperial units, thermal conductivity is measured in BTU/(h⋅ft⋅°F).1 Btu/(h⋅ft⋅°F) = 1.730735 W/(m⋅K) The dimension of thermal conductivity is M1L1T−3Θ−1, expressed in terms of the dimensions mass (M), length (L), time (T), and temperature (Θ). The reciprocal of the heat transfer coefficient is thermal insulance. This table shows thermal conductivity in SI units of watts per metre-kelvin (W·m−1·K−1). The thermal contact conductance coefficient, h_c, is a property indicating the thermal conductivity, or ability to conduct heat, between two bodies in contact. Therefore, :\frac \kappa \sigma = \frac{c_V m^2 \, \langle {v} \rangle^2}{3e^2} = \frac{8}{\pi} \frac{k_{\rm B}^2T}{e^2}, which is the Wiedemann–Franz law with an erroneous proportionality constant \frac{8}{\pi}\approx 2.55; ===Free electron model=== After taking into account the quantum effects, as in the free electron model, the heat capacity, mean free path and average speed of electrons are modified and the proportionality constant is then corrected to \frac{\pi^2} 3\approx3.29, which agrees with experimental values. ==Temperature dependence== The value L0 = 2.44×10−8 V2K−2 results from the fact that at low temperatures (T\rightarrow 0 K) the heat and charge currents are carried by the same quasi-particles: electrons or holes. *It happens that the online record has the thermal conductivity at 30 Kelvins and \parallel to the c axis posted at 1.36 W⋅cm−1 K−1 and 78.0 Btu hr−1 ft−1 F−1 which is incorrect. NBS 6.00 \parallel to c axis, 3.90 \perp to c axis 5.00 \parallel to c axis, 3.41 \perp to c axis 4.47 \parallel to c axis, 3.12 \perp to c axis 4.19 \parallel to c axis, 3.04 \perp to c axis List 311 366 422 500 600 700 800 The noted authorities have reported some values in three digits as cited here in metric translation but they have not demonstrated three digit measurement.R.W.Powell, C.Y.Ho and P.E.Liley, Thermal Conductivity of Selected Materials, NSRDS-NBS 8, Issued 25 November 1966, pages 69, 99>Link Text Errata: The numbered references in the NSRDS-NBS-8 pdf are found near the end of the TPRC Data Book Volume 2 and not somewhere in Volume 3 like it says. In a gas, thermal conduction is mediated by discrete molecular collisions. Finally, thermal diffusivity \alpha combines thermal conductivity with density and specific heat: :\alpha = \frac{ k }{ \rho c_{p} }. In many materials, q is observed to be directly proportional to the temperature difference and inversely proportional to the separation distance L: : q = -k \cdot \frac{T_2 - T_1}{L}. In the metallic phase, the electronic contribution to thermal conductivity was much smaller than what would be expected from the Wiedemann–Franz law. Following the Wiedemann–Franz law, thermal conductivity of metals is approximately proportional to the absolute temperature (in kelvins) times electrical conductivity.

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Consider the gas phase thermal decomposition of 1.0 atm of $\left(\mathrm{CH}_3\right)_3 \mathrm{COOC}\left(\mathrm{CH}_3\right)_3(\mathrm{~g})$ to acetone $\left(\mathrm{CH}_3\right)_2 \mathrm{CO}(\mathrm{g})$ and ethane $\left(\mathrm{C}_2 \mathrm{H}_6\right)(\mathrm{g})$, which occurs with a rate constant of $0.0019 \mathrm{~s}^{-1}$. After initiation of the reaction, at what time would you expect the pressure to be $1.8 \mathrm{~atm}$ ?

However, they observed that the thermodynamics became favourable for crystalline solid acetone at the melting point (−96 °C). One litre of acetone can dissolve around 250 litres of acetylene at a pressure of .Mine Safety and Health Administration (MSHA) – Safety Hazard Information – Special Hazards of Acetylene . At temperatures greater than acetone's flash point of , air mixtures of between 2.5% and 12.8% acetone, by volume, may explode or cause a flash fire. In acetone vapor at ambient temperature, only 2.4% of the molecules are in the enol form. :300px ===Aldol condensation=== In the presence of suitable catalysts, two acetone molecules also combine to form the compound diacetone alcohol , which on dehydration gives mesityl oxide . The synthesis involves the condensation of acetone with phenol: :(CH3)2CO + 2 C6H5OH -> (CH3)2C(C6H4OH)2 + H2O Many millions of kilograms of acetone are consumed in the production of the solvents methyl isobutyl alcohol and methyl isobutyl ketone. Since thermal decomposition is a kinetic process, the observed temperature of its beginning in most instances will be a function of the experimental conditions and sensitivity of the experimental setup. The flame temperature of pure acetone is 1980 °C.Haynes, p. 15.49 ===Toxicity=== Acetone has been studied extensively and is believed to exhibit only slight toxicity in normal use. thumb|280px|Temperature-dependency of the heats of vaporization for water, methanol, benzene, and acetone In thermodynamics, the enthalpy of vaporization (symbol ), also known as the (latent) heat of vaporization or heat of evaporation, is the amount of energy (enthalpy) that must be added to a liquid substance to transform a quantity of that substance into a gas. CH3)2CO -> (CH3)2C(OH)CH2C(O)CH3 Condensation with acetylene gives 2-methylbut-3-yn-2-ol, precursor to synthetic terpenes and terpenoids. ===Laboratory=== ====Chemical research==== In the laboratory, acetone is used as a polar, aprotic solvent in a variety of organic reactions, such as SN2 reactions. Octyl acetate, or octyl ethanoate, is an organic compound with the formula CH3(CH2)7O2CCH3. thumb|upright=1.75|Processes in the thermal degradation of organic matter at atmospheric pressure. In 1960, Soviet chemists observed that the thermodynamics of this process is unfavourable for liquid acetone, so that it (unlike thioacetone and formol) is not expected to polymerise spontaneously, even with catalysts. The technique, called acetone vapor bath smoothing, involves placing the printed part in a sealed chamber containing a small amount of acetone, and heating to around 80 degrees Celsius for 10 minutes. The use of acetone solvent is critical for the Jones oxidation. Acetone can be cooled with dry ice to −78 °C without freezing; acetone/dry ice baths are commonly used to conduct reactions at low temperatures. Msha.gov. Retrieved on 2012-11-26.History – Acetylene dissolved in acetone . As the liquid and gas are in equilibrium at the boiling point (Tb), ΔvG = 0, which leads to: :\Delta_\text{v} S = S_\text{gas} - S_\text{liquid} = \frac{\Delta_\text{v} H}{T_\text{b}} As neither entropy nor enthalpy vary greatly with temperature, it is normal to use the tabulated standard values without any correction for the difference in temperature from 298 K. The reaction is usually endothermic as heat is required to break chemical bonds in the compound undergoing decomposition. The decomposition temperature of a substance is the temperature at which the substance chemically decomposes. Thermokinetics deals with the study of thermal decomposition kinetics. ==See also== *Thermogravimetry *Differential thermal analysis *Differential scanning calorimetry ==References== * * Category:Chemical kinetics *The compound with the highest known decomposition temperature is carbon monoxide at ≈3870 °C (≈7000 °F). ===Decomposition of nitrates, nitrites and ammonium compounds=== * Ammonium dichromate on heating yields nitrogen, water and chromium(III) oxide. During World War I, Chaim Weizmann developed the process for industrial production of acetone (Weizmann Process).Chaim Weizmann chemistryexplained.com ==Production== In 2010, the worldwide production capacity for acetone was estimated at 6.7 million tonnes per year.

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Autoclaves that are used to sterilize surgical tools require a temperature of $120 .{ }^{\circ} \mathrm{C}$ to kill some bacteria. If water is used for this purpose, at what pressure must the autoclave operate?

The operator is required to manually perform steam pulsing at certain pressures as indicated by the gauge. == In medicine == thumb|Dental equipment in an autoclave to be sterilized for 2 hours at 150 to 180 degrees Celsius A medical autoclave is a device that uses steam to sterilize equipment and other objects. thumb|Cutaway illustration of a cylindrical-chamber autoclave An autoclave is a machine used to carry out industrial and scientific processes requiring elevated temperature and pressure in relation to ambient pressure and/or temperature. Many autoclaves are used to sterilize equipment and supplies by subjecting them to pressurized saturated steam at for around 30-60 minutes at a pressure of 15 psi (103 kPa or 1.02 atm) depending on the size of the load and the contents. There are physical, chemical, and biological indicators that can be used to ensure that an autoclave reaches the correct temperature for the correct amount of time. Autoclave tape works by changing color after exposure to temperatures commonly used in sterilization processes, typically 121°C in a steam autoclave. (A properly calibrated medical-grade autoclave uses thousands of gallons of water each day, independent of task, with correspondingly high electric power consumption.) ==In research== Autoclaves used in education, research, biomedical research, pharmaceutical research and industrial settings (often called "research-grade" autoclaves) are used to sterilize lab instruments, glassware, culture media, and liquid media. UCR's research-grade autoclaves performed the same tasks with equal effectiveness, but used 83% less energy and 97% less water. ==Quality assurance== In order to sterilize items effectively, it is important to use optimal parameters when running an autoclave cycle. Since exact temperature control is difficult, the temperature is monitored, and the sterilization time adjusted accordingly. ==Additional images== Image:Autoclave stove top.jpg|Stovetop autoclaves, also known as pressure cooker—the simplest of autoclaves File:Autoclave machine.jpg|The machine on the right is an autoclave used for processing substantial quantities of laboratory equipment prior to reuse, and infectious material prior to disposal. Autoclaves are found in many medical settings, laboratories, and other places that need to ensure the sterility of an object. Super heating conditions and steam generation are achieved by variable pressure control, which cycles between ambient and negative pressure within the sterilization vessel. Some computer-controlled autoclaves use an F0 (F-nought) value to control the sterilization cycle. The high heat and pressure that autoclaves generate help to ensure that the best possible physical properties are repeatable. Research-grade autoclaves—which are not approved for use in sterilizing instruments that will be directly used on humans—are primarily designed for efficiency, flexibility, and ease-of-use. Machines in this category largely operate under the same principles as conventional autoclaves in that they are able to neutralize potentially infectious agents by using pressurized steam and superheated water. Autoclaves are used before surgical procedures to perform sterilization and in the chemical industry to cure coatings and vulcanize rubber and for hydrothermal synthesis. With this process, waste enters and the product leaves the autoclave without the loss of temperature or pressure in the vessel. Some autoclaves, also referred to as waste converters, can operate in the atmospheric pressure range to achieve full sterilization of pathogenic waste. For steam sterilization to occur, the entire item must completely reach and maintain 121°C for 15–20 minutes with proper steam exposure to ensure sterilization. If the autoclave does not reach the right temperature, the spores will germinate when incubated and their metabolism will change the color of a pH-sensitive chemical. In most of the industrialized world medical-grade autoclaves are regulated medical devices. Other types of autoclaves are used to grow crystals under high temperatures and pressures. In dentistry, autoclaves provide sterilization of dental instruments.

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Imagine gaseous $\mathrm{Ar}$ at $298 \mathrm{~K}$ confined to move in a two-dimensional plane of area $1.00 \mathrm{~cm}^2$. What is the value of the translational partition function?

In statistical mechanics, the translational partition function, q_T is that part of the partition function resulting from the movement (translation) of the center of mass. The partition function is a function of the temperature T and the microstate energies E1, E2, E3, etc. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. The partition function has many physical meanings, as discussed in Meaning and significance. == Canonical partition function == === Definition === Initially, let us assume that a thermodynamically large system is in thermal contact with the environment, with a temperature T, and both the volume of the system and the number of constituent particles are fixed. Finding the partition function is also equivalent to performing a Laplace transform of the density of states function from the energy domain to the β domain, and the inverse Laplace transform of the partition function reclaims the state density function of energies. ==Grand canonical partition function== We can define a grand canonical partition function for a grand canonical ensemble, which describes the statistics of a constant-volume system that can exchange both heat and particles with a reservoir. When considering a set of N non-interacting but identical atoms or molecules, when QT ≫ N , or equivalently when ρ Λ ≪ 1 where ρ is the density of particles, the total translational partition function can be written : Q_T(T,N) = \frac{ q_T(T)^N }{N!} A plane partition may be represented visually by the placement of a stack of \pi_{i,j} unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture. In this case we must describe the partition function using an integral rather than a sum. The asymptotics for plane partitions were first calculated by E. M. Wright.E. M. Wright, Asymptotic partition formulae I. Plane partitions, The Quarterly Journal of Mathematics 1 (1931) 177–189. In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Plane partitions are a generalization of partitions of an integer. By taking the zero point energy as the reference point to which other energies are measured, the expression for the partition function becomes Q_\text{vib}(T) = \prod_{i=1}^f \frac{1}{1-e^{-\Theta_{\text{vib},i}/T}} == References == ==See also== * Partition function (mathematics) Category:Partition functions The sum of a plane partition is : n=\sum_{i,j} \pi_{i,j} . In classical statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms. In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. == Definition == The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions \zeta:Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 Z = \frac{ \zeta^N }{ N! } with: \zeta = \sum_j g_j e^{ -E_j / k_\text{B} T} , where g_j is the degeneracy of the j-th quantum level of an individual particle, k_\text{B} is the Boltzmann constant, and T is the absolute temperature of system. This partition function is closely related to the grand potential, \Phi_{\rm G}, by the relation : -k_B T \ln \mathcal{Z} = \Phi_{\rm G} = \langle E \rangle - TS - \mu \langle N\rangle. The usefulness of the partition function stems from the fact that the macroscopic thermodynamic quantities of a system can be related to its microscopic details through the derivatives of its partition function. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. From this point of view, a plane partition can be defined as a finite subset \mathcal{P} of positive integer lattice points (i, j, k) in \mathbb{N}^3, such that if (r, s, t) lies in \mathcal{P} and if (i, j, k) satisfies 1\leq i\leq r, 1\leq j\leq s, and 1\leq k\leq t, then (i, j, k) also lies in \mathcal{P}. The partition function can be related to thermodynamic properties because it has a very important statistical meaning. Using this approximation we can derive a closed form expression for the vibrational partition function. The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanics, and whether the spectrum of states is discrete or continuous. ====Classical discrete system==== For a canonical ensemble that is classical and discrete, the canonical partition function is defined as Z = \sum_i e^{-\beta E_i}, where * i is the index for the microstates of the system; * e is Euler's number; * \beta is the thermodynamic beta, defined as \tfrac{1}{k_\text{B} T} where k_\text{B} is Boltzmann's constant; * E_i is the total energy of the system in the respective microstate.

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Determine the equilibrium constant for the dissociation of sodium at $298 \mathrm{~K}: \mathrm{Na}_2(g) \rightleftharpoons 2 \mathrm{Na}(g)$. For $\mathrm{Na}_2$, $B=0.155 \mathrm{~cm}^{-1}, \widetilde{\nu}=159 \mathrm{~cm}^{-1}$, the dissociation energy is $70.4 \mathrm{~kJ} / \mathrm{mol}$, and the ground-state electronic degeneracy for $\mathrm{Na}$ is 2 .

Bioanalytical Chemistry Textbook De Gruyter 2021 https://doi.org/10.1515/9783110589160-206 For a general reaction: : A_\mathit{x} B_\mathit{y} <=> \mathit{x} A{} + \mathit{y} B in which a complex \ce{A}_x \ce{B}_y breaks down into x A subunits and y B subunits, the dissociation constant is defined as : K_D = \frac{[\ce A]^x [\ce B]^y}{[\ce A_x \ce B_y]} where [A], [B], and [Ax By] are the equilibrium concentrations of A, B, and the complex Ax By, respectively. right|thumb|The Madelung constant being calculated for the NaCl ion labeled 0 in the expanding spheres method. The dissociation constant for a particular ligand-protein interaction can change significantly with solution conditions (e.g., temperature, pH and salt concentration). Nonadiabatic transition state theory (NA-TST) is a powerful tool to predict rates of chemical reactions from a computational standpoint. (The symbol K_a, used for the acid dissociation constant, can lead to confusion with the association constant and it may be necessary to see the reaction or the equilibrium expression to know which is meant.) In chemistry, biochemistry, and pharmacology, a dissociation constant (K_D) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex falls apart into its component molecules, or when a salt splits up into its component ions. A molecule can have several acid dissociation constants. In the special case of salts, the dissociation constant can also be called an ionization constant. For example, for the ionic crystal NaCl, there arise two Madelung constants - one for Na and another for Cl. The molecular formula C17H14N2O2 (molar mass: 278.305 g/mol, exact mass: 278.1055 u) may refer to: * Bimakalim * Sudan Red G Category:Molecular formulas The dissociation constant is the inverse of the association constant. : Ab + Ag <=> AbAg : K_A = \frac{\left[ \ce{AbAg} \right]}{\left[ \ce{Ab} \right] \left[ \ce{Ag} \right]} = \frac{1}{K_D} This chemical equilibrium is also the ratio of the on-rate (kforward or ka) and off-rate (kback or kd) constants. Note also how the fluorite structure being intermediate between the caesium chloride and sphalerite structures is reflected in the Madelung constants. == Formula == A fast converging formula for the Madelung constant of NaCl is :12 \, \pi \sum_{m, n \geq 1, \, \mathrm{odd}} \operatorname{sech}^2\left(\frac{\pi}{2}(m^2+n^2)^{1/2}\right) == Generalization == It is assumed for the calculation of Madelung constants that an ion's charge density may be approximated by a point charge. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. == Formal expression == The Madelung constant allows for the calculation of the electric potential of all ions of the lattice felt by the ion at position :V_i = \frac{e}{4 \pi \varepsilon_0 } \sum_{j eq i} \frac{z_j}{r_{ij}}\,\\! where r_{ij} = |r_i-r_j| is the distance between the th and the th ion. The binding constant, or affinity constant/association constant, is a special case of the equilibrium constant K, and is the inverse of the dissociation constant. The dissociation constant has molar units (M) and corresponds to the ligand concentration [L] at which half of the proteins are occupied at equilibrium, i.e., the concentration of ligand at which the concentration of protein with ligand bound [LP] equals the concentration of protein with no ligand bound [P]. [B^{-3}] \over [H B^{-2}]} & \mathrm{p}K_3 &= -\log K_3 \end{align} ==Dissociation constant of water== The dissociation constant of water is denoted Kw: :K_\mathrm{w} = [\ce{H}^+] [\ce{OH}^-] The concentration of water, [H2O], is omitted by convention, which means that the value of Kw differs from the value of Keq that would be computed using that concentration. The properties of this ion are strongly related to the surface potential present on a corresponding solid. NA-TST can be reduced to the traditional TST in the limit of unit probability. ==References== Category:Chemical physics Examples of Madelung constants Ion in crystalline compound M (based on ) \overline{M} (based on ) Cl− and Cs+ in CsCl ±1.762675 ±2.035362 Cl− and Na+ in rocksalt NaCl ±1.747565 ±3.495129 S2− and Zn2+ in sphalerite ZnS ±3.276110 ±7.56585 F− in fluorite CaF2 1.762675 4.070723 Ca2+ in fluorite CaF2 -3.276110 −7.56585 The continuous reduction of with decreasing coordination number for the three cubic AB compounds (when accounting for the doubled charges in ZnS) explains the observed propensity of alkali halides to crystallize in the structure with highest compatible with their ionic radii. Sub-picomolar dissociation constants as a result of non-covalent binding interactions between two molecules are rare. For the binding of receptor and ligand molecules in solution, the molar Gibbs free energy ΔG, or the binding affinity is related to the dissociation constant Kd via :\Delta G = R T\ln{{K_{\rm d} \over c^{\ominus}}}, in which R is the ideal gas constant, T temperature and the standard reference concentration c ~~o~~ = 1 mol/L. == See also == * Binding coefficient Category:Equilibrium chemistry

−1.642876

24

2.25

6.9

537

C

At $298.15 \mathrm{~K}, \Delta G_f^{\circ}(\mathrm{HCOOH}, g)=-351.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $\Delta G_f^{\circ}(\mathrm{HCOOH}, l)=-361.4 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate the vapor pressure of formic acid at this temperature.

Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds", Hydrocarbon Processing, 68(10), Pages 65–68, 1989 ==See also== *Vapour pressure of water *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation *Raoult's law *Thermodynamic activity ==References== ==External links== * Gallica, scanned original paper * NIST Chemistry Web Book * Calculation of vapor pressures with the Antoine equation Category:Equations Category:Thermodynamic equations The second solution is switching to another vapor pressure equation with more than three parameters. Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. "A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The formula is valid from about −50 to 102 °C; however there are a very limited number of measurements of the vapor pressure of water over supercooled liquid water. (The small differences in the results are only caused by the used limited precision of the coefficients). ==Extension of the Antoine equations== To overcome the limits of the Antoine equation some simple extension by additional terms are used: : \begin{align} P &= \exp{\left( A + \frac{B}{C+T} + D \cdot T + E \cdot T^2 + F \cdot \ln \left( T \right) \right)} \\\ P &= \exp\left( A + \frac{B}{C+T} + D \cdot \ln \left( T \right) + E \cdot T^F\right). \end{align} The additional parameters increase the flexibility of the equation and allow the description of the entire vapor pressure curve. The (unattributed) constants are given as {| class="wikitable" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Regarding the measurement of upper-air humidity, this publication also reads (in Section 12.5.1): The saturation with respect to water cannot be measured much below –50 °C, so manufacturers should use one of the following expressions for calculating saturation vapour pressure relative to water at the lowest temperatures – Wexler (1976, 1977), reported by Flatau et al. (1992)., Hyland and Wexler (1983) or Sonntag (1994) – and not the Goff- Gratch equation recommended in earlier WMO publications. ==Experimental correlation== The original Goff–Gratch (1945) experimental correlation reads as follows: : \log\ e^*\ = -7.90298(T_\mathrm{st}/T-1)\ +\ 5.02808\ \log(T_\mathrm{st}/T) -\ 1.3816\times10^{-7}(10^{11.344(1-T/T_\mathrm{st})}-1) +\ 8.1328\times10^{-3}(10^{-3.49149(T_\mathrm{st}/T-1)}-1)\ +\ \log\ e^*_\mathrm{st} where: :log refers to the logarithm in base 10 :e* is the saturation water vapor pressure (hPa) :T is the absolute air temperature in kelvins :Tst is the steam-point (i.e. boiling point at 1 atm.) temperature (373.15 K) :e*st is e* at the steam-point pressure (1 atm = 1013.25 hPa) Similarly, the correlation for the saturation water vapor pressure over ice is: : \log\ e^*_i\ = -9.09718(T_0/T-1)\ -\ 3.56654\ \log(T_0/T) +\ 0.876793(1-T/T_0) +\ \log\ e^*_{i0} where: :log stands for the logarithm in base 10 :e*i is the saturation water vapor pressure over ice (hPa) :T is the air temperature (K) :T0 is the ice-point (triple point) temperature (273.16 K) :e*i0 is e* at the ice-point pressure (6.1173 hPa) ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Tetens equation *Lee–Kesler method ==References== * Goff, J. A., and Gratch, S. (1946) Low- pressure properties of water from −160 to 212 °F, in Transactions of the American Society of Heating and Ventilating Engineers, pp 95–122, presented at the 52nd annual meeting of the American Society of Heating and Ventilating Engineers, New York, 1946. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Water vapor (H2O) 200px Liquid state Water Solid state Ice Properties Molecular formula H2O Molar mass 18.01528(33) g/mol Melting point Vienna Standard Mean Ocean Water (VSMOW), used for calibration, melts at 273.1500089(10) K (0.000089(10) °C) and boils at 373.1339 K (99.9839 °C) Boiling point specific gas constant 461.5 J/(kg·K) Heat of vaporization 2.27 MJ/kg Heat capacity 1.864 kJ/(kg·K) Water vapor, water vapour or aqueous vapor is the gaseous phase of water. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. This doesn't affect the equation form. ==Sources for Antoine equation parameters== * NIST Chemistry WebBook * Dortmund Data Bank * Directory of reference books and data banks containing Antoine constants * Several reference books and publications, e. g. ** Lange's Handbook of Chemistry, McGraw-Hill Professional ** Wichterle I., Linek J., "Antoine Vapor Pressure Constants of Pure Compounds" ** Yaws C. L., Yang H.-C., "To Estimate Vapor Pressure Easily. The maximum partial pressure (saturation pressure) of water vapor in air varies with temperature of the air and water vapor mixture. The Antoine equation can also be transformed in a temperature-explicit form with simple algebraic manipulations: :T = \frac{B}{A-\log_{10}\, p} - C ==Validity range== Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the triple point to the critical point, because it is not flexible enough. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. The Goff–Gratch equation is one (arguably the first reliable in history) amongst many experimental correlation proposed to estimate the saturation water vapor pressure at a given temperature. The Antoine equation is a class of semi-empirical correlations describing the relation between vapor pressure and temperature for pure substances. A variety of empirical formulas exist for this quantity; the most used reference formula is the Goff-Gratch equation for the SVP over liquid water below zero degrees Celsius: :\begin{align} \log_{10} \left( p \right) = & -7.90298 \left( \frac{373.16}{T}-1 \right) + 5.02808 \log_{10} \frac{373.16}{T} \\\ & \- 1.3816 \times 10^{-7} \left( 10^{11.344 \left( 1-\frac{T}{373.16} \right)} -1 \right) \\\ & \+ 8.1328 \times 10^{-3} \left( 10^{-3.49149 \left( \frac{373.16}{T}-1 \right)} -1 \right) \\\ & \+ \log_{10} \left( 1013.246 \right) \end{align} where , temperature of the moist air, is given in units of kelvin, and is given in units of millibars (hectopascals).

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The collisional cross section of $\mathrm{N}_2$ is $0.43 \mathrm{~nm}^2$. What is the diffusion coefficient of $\mathrm{N}_2$ at a pressure of $1 \mathrm{~atm}$ and a temperature of $298 \mathrm{~K}$ ?

The diffusion profile therefore can be depicted by the following equation. (dln\bar{c}/dy^{6/5})^{5/3}=0.66(D_1/t)^{1/2}(1/D_b\delta) To further determine D_b , two common methods were used. The first equation represents diffusion in the volume, while the second shows diffusion along the grain boundary, respectively. \frac{\partial c}{\partial t}=D\left({\partial^2 c\over\partial x^2}+{\partial^2 c\over\partial y^2}\right) where |x|>\delta/2 \frac{\partial c_b}{\partial t}=D_b\left({\partial^2 c_b\over\partial y^2}\right)+\frac{2D}{\delta}\left(\frac{\partial c}{\partial x}\right)_{x=\delta/2} where c(x, y, t) is the volume concentration of the diffusing atoms and c_b(y, t) is their concentration in the grain boundary. On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System). thumb|Thermal diffusion coefficients vs. temperature, for air at normal pressure The Maxwell–Stefan diffusion (or Stefan–Maxwell diffusion) is a model for describing diffusion in multicomponent systems. * Method 2: To compare the length of penetration of a given concentration at the boundary \ \Delta y with the length of lattice penetration from the surface far from the boundary. == References == == See also == * Kirkendall effect * Phase transformations in solids * Mass diffusivity Category:Diffusion The grain boundary diffusion coefficient is the diffusion coefficient of a diffusant along a grain boundary in a polycrystalline solid.P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965. "On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System)". The basic assumption of the theory is that a deviation from equilibrium between the molecular friction and thermodynamic interactions leads to the diffusion flux.S. Rehfeldt, J. Stichlmair: Measurement and calculation of multicomponent diffusion coefficients in liquids, Fluid Phase Equilibria, 2007, 256, 99–104 The molecular friction between two components is proportional to their difference in speed and their mole fractions. Suppose that the thickness of the slab is \delta, the length is y, and the depth is a unit length, the diffusion process can be described as the following formula. The effective diffusion coefficient of a in atomic diffusion of solid polycrystalline materials like metal alloys is often represented as a weighted average of the grain boundary diffusion coefficient and the lattice diffusion coefficient.P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965. However, at temperatures below 700 °C, the values of D_b with polycrystal silver consistently lie above the values of D_b with a single crystal. == Measurement == The general way to measure grain boundary diffusion coefficients was suggested by Fisher. It dominates the effective diffusion rate at lower temperatures in metals and metal alloys. Stefan: Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 2te Abteilung a, 1871, 63, 63-124. for liquids. We can then rearrange: :D(c^*) = - \frac{1}{2 t} \frac{\int^{c^*}_{c_R} x \mathrm{d}c}{(\mathrm{d}c/\mathrm{d}x)_{c=c^*}} Knowing the concentration profile c(x) at annealing time t, and assuming it is invertible as x(c), we can then calculate the diffusion coefficient for all concentrations between cR and cL. === The Matano interface === The last formula has one significant shortcoming: no information is given about the reference according to which x should be measured. The Boltzmann–Matano method is used to convert the partial differential equation resulting from Fick's law of diffusion into a more easily solved ordinary differential equation, which can then be applied to calculate the diffusion coefficient as a function of concentration. Increasing temperature often allows for increased grain size, and the lattice diffusion component increases with increasing temperature, so often at 0.8 Tmelt (of an alloy), the grain boundary component can be neglected. ==Modeling== The effective diffusion coefficient can be modeled using Hart's equation when lattice diffusion is dominant (type A kinetics): : D_\text{eff} = f D_\text{gb} + (1-f) D_\ell where :D_\text{eff} = {}effective diffusion coefficient :D_\text{gb} = {}grain boundary diffusion coefficient :D_\ell = {}lattice diffusion coefficient :f = \frac{q \delta}{d} : q = {}value based on grain shape, 1 for parallel grains, 3 for square grains : d = {}average grain size :\delta = {}grain boundary width, often assumed to be 0.5 nm Grain boundary diffusion is significant in face-centered cubic metals below about 0.8 Tmelt (Absolute). The Mathematics of Diffusion. This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time (x \propto \sqrt t), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the distance (t \propto x^2); the square term gives the name parabolic law.See an animation of the parabolic law. == Matano’s method == Chuijiro Matano applied Boltzmann's transformation to obtain a method to calculate diffusion coefficients as a function of concentration in metal alloys. The ratio of the grain boundary diffusion activation energy over the lattice diffusion activation energy is usually 0.4–0.6, so as temperature is lowered, the grain boundary diffusion component increases. Assuming a diffusion coefficient D that is in general a function of concentration c, Fick's second law is :\frac{\partial c}{\partial t} = \frac{\partial}{\partial x} \underbrace{\left[ D(c)\frac{\partial c}{\partial x} \right]}_\text{flux}, where t is time, and x is distance. The methods used for solving two dimensional Diffusion problems are similar to those used for one dimensional problems. Another example of double diffusion is the formation of false bottoms at the interface of sea ice and under-ice meltwater layers.

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A vessel contains $1.15 \mathrm{~g}$ liq $\mathrm{H}_2 \mathrm{O}$ in equilibrium with water vapor at $30 .{ }^{\circ} \mathrm{C}$. At this temperature, the vapor pressure of $\mathrm{H}_2 \mathrm{O}$ is 31.82 torr. What volume increase is necessary for all the water to evaporate?

The rate of evaporation in an open system is related to the vapor pressure found in a closed system. The boiling temperature of water for atmospheric pressures can be approximated by the Antoine equation: :\log_{10}\left(\frac{P}{1\text{ Torr}}\right) = 8.07131 - \frac{1730.63\ {}^\circ\text{C}}{233.426\ {}^\circ\text{C} + T_b} or transformed into this temperature-explicit form: :T_b = \frac{1730.63\ {}^\circ\text{C}}{8.07131 - \log_{10}\left(\frac{P}{1\text{ Torr}}\right)} - 233.426\ {}^\circ\text{C} where the temperature T_b is the boiling point in degrees Celsius and the pressure P is in torr. ==Dühring's rule== Dühring's rule states that a linear relationship exists between the temperatures at which two solutions exert the same vapor pressure. ==Examples== The following table is a list of a variety of substances ordered by increasing vapor pressure (in absolute units). Applied Meteorology 6: 203-204. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2 ::P = 0.61078 \exp\left(\frac{21.875 T}{T + 265.5}\right). ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation ==References== Category:Meteorological concepts Category:Thermodynamic equations On the computation of saturation vapour pressure. The evaporation will continue until an equilibrium is reached when the evaporation of the liquid is equal to its condensation. (Alternate title: "Water Vapor Myths: A Brief Tutorial".) ==See also== * Absolute humidity * Antoine equation * Lee–Kesler method * Osmotic coefficient * Raoult's law: vapor pressure lowering in solution * Reid vapor pressure * Relative humidity * Relative volatility * Saturation vapor density * Triple point * True vapor pressure * Vapor–liquid equilibrium * Vapor pressures of the elements (data page) * Vapour pressure of water ==References== ==External links== *Fluid Characteristics Chart, Engineer's Edge *Vapor Pressure, Hyperphysics *Vapor Pressure, The MSDS HyperGlossary *Online vapor pressure calculation tool (Requires Registration) *Prediction of Vapor Pressures of Pure Liquid Organic Compounds Category:Engineering thermodynamics Category:Gases Category:Meteorological concepts Category:Pressure Category:Thermodynamic properties Air is given a vapour density of one. The equilibrium vapor pressure is an indication of a liquid's thermodynamic tendency to evaporate. Many of the molecules return to the liquid, with returning molecules becoming more frequent as the density and pressure of the vapor increases. At the normal boiling point of a liquid, the vapor pressure is equal to the standard atmospheric pressure defined as 1 atmosphere, 760Torr, 101.325kPa, or 14.69595psi. High concentration of the evaporating substance in the surrounding gas significantly slows down evaporation, such as when humidity affects rate of evaporation of water. Evaporation also tends to proceed more quickly with higher flow rates between the gaseous and liquid phase and in liquids with higher vapor pressure. Actually, as stated by Dalton's law (known since 1802), the partial pressure of water vapor or any substance does not depend on air at all, and the relevant temperature is that of the liquid. In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. ;Concentration of the substance evaporating in the air: If the air already has a high concentration of the substance evaporating, then the given substance will evaporate more slowly. The pressure exhibited by vapor present above a liquid surface is known as vapor pressure. According to Monteith and Unsworth, "Values of saturation vapour pressure from Tetens' formula are within 1 Pa of exact values up to 35 °C." Even at lower temperatures, individual molecules of a liquid can evaporate if they have more than the minimum amount of kinetic energy required for vaporization. == Factors influencing the rate of evaporation == Note: Air is used here as a common example of the surrounding gas; however, other gases may hold that role. With any incremental increase in that temperature, the vapor pressure becomes sufficient to overcome atmospheric pressure and cause the liquid to form vapor bubbles. When only a small proportion of the molecules meet these criteria, the rate of evaporation is low. In an enclosed environment, a liquid will evaporate until the surrounding air is saturated. For a system consisting of vapor and liquid of a pure substance, this equilibrium state is directly related to the vapor pressure of the substance, as given by the Clausius–Clapeyron relation: : \ln \left( \frac{ P_2 }{ P_1 } \right) = - \frac{ \Delta H_{\rm vap } }{ R } \left( \frac{ 1 }{ T_2 } - \frac{ 1 }{ T_1 } \right) where P1, P2 are the vapor pressures at temperatures T1, T2 respectively, ΔHvap is the enthalpy of vaporization, and R is the universal gas constant.

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A cell is roughly spherical with a radius of $20.0 \times 10^{-6} \mathrm{~m}$. Calculate the work required to expand the cell surface against the surface tension of the surroundings if the radius increases by a factor of three. Assume the cell is surrounded by pure water and that $T=298.15 \mathrm{~K}$.

In that case, in order to increase the surface area of a mass of liquid by an amount, , a quantity of work, , is needed (where is the surface energy density of the liquid). The work associated with the first step (unstrained) is W_1 = 2 \gamma_0 A_0, where \gamma_0 and A_0 are the excess free energy and area of each of new surfaces. For the second step, work (w_2), equals the work needed to elastically deform the total bulk volume and the four (two original and two newly formed) surfaces. A suggestion is surface stress define as association with the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface instead of up definition. The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk, or it is the work required to build an area of a particular surface. Another way to view the surface energy is to relate it to the work required to cut a bulk sample, creating two surfaces. Surface stress was first defined by Josiah Willard Gibbs (1839-1903) as the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface. Incorporating this value into the surface energy equation allows for the surface energy to be estimated. The driving force for a change in the surface concentration associated with a contraction of the surface is proportional to the difference between surface stress and surface free energy. In order to move a cube from the bulk of a material to the surface, energy is required. Based on the contact angle results and knowing the surface tension of the liquids, the surface energy can be calculated. A common approach to achieving this is known as the workcell. However, such a method cannot be used to measure the surface energy of a solid because stretching of a solid membrane induces elastic energy in the bulk in addition to increasing the surface energy. A workcell is an arrangement of resources in a manufacturing environment to improve the quality, speed and cost of the process. The following equation can be used as a reasonable estimate for surface energy: :\gamma \approx \frac{-\Delta_\text{sub} H\left(z_\sigma - z_\beta\right)}{a_0 N_\text{A} z_\beta} == Interfacial energy == The presence of an interface influences generally all thermodynamic parameters of a system. The surface energy of the facets can thus be found to within a scaling constant by measuring the relative sizes of the facets. ===Calculation=== ====Deformed solid==== In the deformation of solids, surface energy can be treated as the "energy required to create one unit of surface area", and is a function of the difference between the total energies of the system before and after the deformation: :\gamma = \frac{1}{A} \left(E_1 - E_0\right). This energy cost is incorporated into the surface energy of the material, which is quantified by: thumb|center|480x240px|Cube model. The surface wants to expand creating a compressive stress. Surface area can be determined by squaring the cube root of the volume of the molecule: :a_0 = V_\text{molecule}^\frac{2}{3} = \left(\frac{\bar{M}}{\rho N_\text{A}}\right)^\frac{2}{3} Here, corresponds to the molar mass of the molecule, corresponds to the density, and is the Avogadro constant. The cube model can be used to model pure, uniform materials or an individual molecular component to estimate their surface energy. :\gamma = \frac{\left(z_\sigma - z_\beta\right) \frac{1}{2}W_\text{AA}}{a_0} where and are coordination numbers corresponding to the surface and the bulk regions of the material, and are equal to 5 and 6, respectively; is the surface area of an individual molecule, and is the pairwise intermolecular energy. Experimental setup for measuring relative surface energy and its function can be seen in the video. ===Estimation from the heat of sublimation=== To estimate the surface energy of a pure, uniform material, an individual region of the material can be modeled as a cube. There are several different models for calculating the surface energy based on the contact angle readings.

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A vessel is filled completely with liquid water and sealed at $13.56^{\circ} \mathrm{C}$ and a pressure of 1.00 bar. What is the pressure if the temperature of the system is raised to $82.0^{\circ} \mathrm{C}$ ? Under these conditions, $\beta_{\text {water }}=2.04 \times 10^{-4} \mathrm{~K}^{-1}$, $\beta_{\text {vessel }}=1.42 \times 10^{-4} \mathrm{~K}^{-1}$, and $\kappa_{\text {water }}=4.59 \times 10^{-5} \mathrm{bar}^{-1}$.

Details of the calculation: \left( \frac{\partial P}{\partial T} \right)_{V} = -\left( \frac{\partial V}{\partial T} \right)_{p}\left( \frac{\partial P}{\partial V} \right)_{T} = - (V\alpha) \left(\frac{-1}{\kappa_T}\right) = \alpha\kappa_T \left( \frac{\partial P}{\partial T} \right)_{V} = \frac{\frac{1}{V}\left( \frac{\partial V}{\partial T} \right)_{p}}{\frac{-1}{V} \left( \frac{\partial V}{\partial P} \right)_{T}} = \frac{\alpha}{\beta} ==The utility of the thermal pressure== thumb|upright=1.6|Figure 1: Thermal pressure as a function of temperature normalized to A of the few compounds commonly used in the study of Geophysics. Specific heat capacity at constant pressure also increases with temperature, from 4.187 kJ/kg at 25 °C to 8.138 kJ/kg at 350 °C. thumb|400px|Diagram showing pressure difference induced by a temperature difference. Thus, the thermal pressure of a solid due to moderate temperature change above the Debye temperature can be approximated by assuming a constant value of \alpha and \kappa_T. Angel, Ross J., Miozzi Francesca, and Alvaro Matteo (2019). American Academy of Arts & Sciences. . ==Thermal pressure at high temperature== As mentioned above, \alpha\kappa_T is one of the most common formulations for the thermal pressure coefficient. Some formulations for the thermal pressure coefficient include: \left( \frac{\partial P}{\partial T} \right)_{v} = \alpha\kappa_T = \frac{\gamma}{V}C_V = \frac{\alpha}{\beta_T} Where \alpha is the volume thermal expansion, \kappa_T the isothermal bulk modulus, \gamma the Grüneisen parameter, \beta_T the compressibility and C_Vthe constant-volume heat capacity. Applied Meteorology 6: 203-204. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2 ::P = 0.61078 \exp\left(\frac{21.875 T}{T + 265.5}\right). ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation ==References== Category:Meteorological concepts Category:Thermodynamic equations This pressure is given by the saturated vapour pressure, and can be looked up in steam tables, or calculated. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. As a guide, the saturated vapour pressure at 121 °C is 200 kPa, 150 °C is 470 kPa, and 200 °C is 1,550 kPa. Int J Thermophys 43, 169 (2022). https://doi.org/10.1007/s10765-022-03089-8 authors demonstrated that,at ambient pressure, the pressure predicted of Au and MgO from a constant value of \alpha\kappa_T deviates from the experimental data, and the higher temperature, the more deviation. In thermodynamics, thermal pressure (also known as the thermal pressure coefficient) is a measure of the relative pressure change of a fluid or a solid as a response to a temperature change at constant volume. According to Monteith and Unsworth, "Values of saturation vapour pressure from Tetens' formula are within 1 Pa of exact values up to 35 °C." Above about 300 °C, water starts to behave as a near-critical liquid, and physical properties such as density start to change more significantly with pressure. The combined effect of a change in pressure and temperature is described by the strain tensor \varepsilon_{ij}:\varepsilon_{ij}= \alpha_{ij} dT- \beta_{ij} dP Where \alpha_{ij} is the volume thermal expansion tensor and \beta_{ij} is the compressibility tensor. Commonly the thermal pressure coefficient may be expressed as functions of temperature and volume. "A New Conception of Thermal Pressure and a Theory of Solutions". Thus, the study of the thermal pressure coefficient provides a useful basis for understanding the nature of liquid and solid. The thermal pressure coefficient is used to calculate results that are applied widely in industry, and they would further accelerate the development of thermodynamic theory. The Tetens equation is an equation to calculate the saturation vapour pressure of water over liquid and ice. For example, to heat water from 25 °C to steam at 250 °C at 1 atm requires 2869 kJ/kg. There are two main types of calculation of the thermal pressure coefficient: one is the Virial theorem and its derivatives; the other is the Van der Waals type and its derivatives.

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A crude model for the molecular distribution of atmospheric gases above Earth's surface (denoted by height $h$ ) can be obtained by considering the potential energy due to gravity: $$ P(h)=e^{-m g h / k T} $$ In this expression $m$ is the per-particle mass of the gas, $g$ is the acceleration due to gravity, $k$ is a constant equal to $1.38 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}$, and $T$ is temperature. Determine $\langle h\rangle$ for methane $\left(\mathrm{CH}_4\right)$ using this distribution function.

Atmospheric methane is the methane present in Earth's atmosphere. When methane reaches the surface and the atmosphere, it is known as atmospheric methane. Methane's heat of combustion is 55.5 MJ/kg.Energy Content of some Combustibles (in MJ/kg) . Methane has a boiling point of −161.5 °C at a pressure of one atmosphere. Using the values T=273 K and M=29 g/mol as characteristic of the Earth's atmosphere, H = RT/Mg = (8.315*273)/(29*9.8) = 7.99, or about 8 km, which coincidentally is approximate height of Mt. Everest. Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). [A] Net production of O3 CH4 \+ ·OH → CH3· + H2O CH3· + O2 \+ M → CH3O2· + M CH3O2· + NO → NO2 \+ CH3O· CH3O· + O2 → HO2· + HCHO HO2· + NO → NO2 \+ ·OH (2x) NO2 \+ hv → O(3P) + NO (2x) O(3P) + O2 \+ M → O3 \+ M [NET: CH4 \+ 4O2 → HCHO + 2O3 \+ H2O] [B] No net change of O3 CH4 \+ ·OH → CH3· + H2O CH3· + O2 \+ M → CH3O2· + M CH3O2· + HO2· + M → CH3O2H + O2 \+ M CH3O2H + hv → CH3O· + ·OH CH3O· + O2 → HO2· + HCHO [NET: CH4 \+ O2 → HCHO + H2O] ==See also== * Climate change * Global warming * Permafrost * Methane * Methane emissions ==Notes== ==References== ==External links== * * * * * * * * * * * Category:Methane Category:Atmosphere Category:Greenhouse gases A Chapman function describes the integration of atmospheric absorption along a slant path on a spherical earth, relative to the vertical case. The reaction of methane with hydroxyl in the troposphere or stratosphere creates the methyl radical ·CH3 and water vapor. Chapman–Enskog theory provides a framework in which equations of hydrodynamics for a gas can be derived from the Boltzmann equation. Methane is an important greenhouse gas with a global warming potential of 40 compared to (potential of 1) over a 100-year period, and above 80 over a 20-year period. The Chapman function is named after Sydney Chapman, who introduced the function in 1931. == Definition == In an isothermal model of the atmosphere, the density \varrho(h) varies exponentially with altitude h according to the Barometric formula: :\varrho(h) = \varrho_0 \exp\left(- \frac h H \right), where \varrho_0 denotes the density at sea level (h=0) and H the so-called scale height. The IPCC reports that the global warming potential (GWP) for methane is about 84 in terms of its impact over a 20-year timeframe See Table 8.7.—that means it traps 84 times more heat per mass unit than carbon dioxide (CO2) and 105 times the effect when accounting for aerosol interactions. The globally averaged concentration of methane in Earth's atmosphere increased by about 150% from 722 ± 25 ppb in 1750 to 1803.1 ± 0.6 ppb in 2011. Etminan et al. published their new calculations for methane's radiative forcing (RF) in a 2016 Geophysical Research Letters journal article which incorporated the shortwave bands of CH4 in measuring forcing, not used in previous, simpler IPCC methods. Image:Isothermal-barotropic atmosphere model.png ===The U.S. Standard Atmosphere=== The U.S. Standard Atmosphere model starts with many of the same assumptions as the isothermal-barotropic model, including ideal gas behavior, and constant molecular weight, but it differs by defining a more realistic temperature function, consisting of eight data points connected by straight lines; i.e. regions of constant temperature gradient. Methane is a strong GHG with a global warming potential 84 times greater than CO2 in a 20-year time frame. This global destruction of atmospheric methane mainly occurs in the troposphere. The annual average for methane (CH4) was 1866 ppb in 2019 and scientists reported with "very high confidence" that concentrations of CH4 were higher than at any time in at least 800,000 years. As air rises in the tropics, methane is carried upwards through the tropospherethe lowest portion of Earth's atmosphere which is to from the Earth's surface, into the lower stratospherethe ozone layerand then the upper portion of the stratosphere. (The total air mass below a certain altitude is calculated by integrating over the density function.) This geopotential altitude h is then used instead of geometric altitude z in the hydrostatic equations. ==Common models== * COSPAR International Reference Atmosphere * International Standard Atmosphere * Jacchia Reference Atmosphere, an older model still commonly used in spacecraft dynamics * Jet standard atmosphere * NRLMSISE-00 is a recent model from NRL often used in the atmospheric sciences * US Standard Atmosphere ==See also== * Standard temperature and pressure * Upper- atmospheric models ==References== ==External links== *Public Domain Aeronautical Software – Derivation of hydrostatic equations used in the 1976 US Standard Atmosphere *FORTRAN code to calculate the US Standard Atmosphere *NASA GSFC Atmospheric Models overview *Various models at NASA GSFC ModelWeb *Earth Global Reference Atmospheric Model (Earth-GRAM 2010) Category:Atmospheric sciences

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A camper stranded in snowy weather loses heat by wind convection. The camper is packing emergency rations consisting of $58 \%$ sucrose, $31 \%$ fat, and $11 \%$ protein by weight. Using the data provided in Problem P4.32 and assuming the fat content of the rations can be treated with palmitic acid data and the protein content similarly by the protein data in Problem P4.32, how much emergency rations must the camper consume in order to compensate for a reduction in body temperature of $3.5 \mathrm{~K}$ ? Assume the heat capacity of the body equals that of water. Assume the camper weighs $67 \mathrm{~kg}$.

Another formula used is: citing :M=VO_2\left(\frac{RQ-0.7}{0.3}e_c+\frac{1-RQ}{0.3}e_f\right) where RQ is the respiratory quotient (ratio of volume CO2 produced to volume of O2 consumed), e_c is , the heat released per litre of oxygen by the oxidation of carbohydrate, and e_f is , the value for fat. The ration was two-thirds of a pound (302 g) of bread and two-thirds of a pound of meat. fourpence (4d) was deducted daily from the soldiers' pay. Vegetables and boiled starchy foods should be cooked without added salt ==== Daily Messing Rate ==== The Daily Messing Rate (DMR) is used to provide the following daily calorific intake; Daily Messing Rate Type Calorific Intake Basic DMR 3000 Kcal Exercise (Field) DMR. 4000 Kcal Overseas Exercise (Field) DMR. 4000 Kcal Operational DMR. 4000 Kcal Nijmegen Marches. 4000 Kcal Norway DMR. 5000 Kcal The current Daily Messing Rate is; * £2.73 in the United Kingdom * £3.60 outside the United Kingdom ==== Catering for diversity ==== In accordance with current UK legislation and Government guidelines it is incumbent on the Armed Forces to cater for all personnel irrespective of gender, race, religious belief, medical requirements and committed lifestyle choices. ==United States== During the American Revolution, the Continental Congress regulated garrison rations, stipulating in the Militia Law of 1775 that they should consist of: :One pound of beef, or 3/4 of a pound of pork or one pound of fish, per day. Rations in camp. The theoretical bases of indirect calorimetry: a review." The daily ration scale in September 1941 was as follows; ==== Food ==== Meat Bacon and Ham Butter and margarine Cheese Cooking fats Sugar Tea Preserves Army rations Home Service Scale (Men) 12 oz (340 g) 1.14 oz (32 g) 1.89 oz (53 g) 0.57 oz (16 g) 0.28 oz (7 g) 4.28 oz (121 g) 0.57 oz (16 g) 1.14 oz (32 g) Army rations Home Service Scale (Women) 6 oz (170 g) 1.28 oz (36 g) 1.5 oz (42 g) (margarine only) 0.57 oz (16 g) - 2 oz (56 g) 0.28 oz (7 g) 1 oz (28 g) === Modern === ==== UK MOD Nutrition Policy Statement ==== Joint Service Publication (JSP) 456 Part 2 Volume 1 of December 2014, the Ministry of Defence policy on nutrition is as follows; The UK Ministry of Defence (MOD) undertakes to provide military personnel with a basic knowledge of nutrition, with the aim of optimising physical and mental function, long-term health, and morale. After a volume is met, Resting Energy Expenditure is calculated by the Weir formula and results are displayed in software attached to the system. This gives the same result as the Weir formula at RQ = 1 (burning only carbohydrates), and almost the same value at RQ = 0.7 (burning only fat). ⊅Σ′ ==History== Antoine Lavoisier noted in 1780 that heat production, in some cases, can be predicted from oxygen consumption, using multiple regression. Data on per capita food supplies are expressed in terms of quantity and by applying appropriate food composition factors for all primary and processed products also in terms of dietary energy value, protein and fat content. Ambient and diluted fractions of O2 and CO2 are measured for a known ventilation rate, and O2 consumption and CO2 production are determined and converted into Resting Energy Expenditure.Academy of Nutrition and Dietetics "Measuring RMR with Indirect Calorimetry (IC)." By World War I, the American garrison ration had improved dramatically, including 137 grams of protein, 129 grams of fat, and 539 grams of carbohydrate every day, with a total of roughly 4,000 calories. The dietary energy supply is the food available for human consumption, usually expressed in kilocalories or kilojoules per person per day. In the West Indies troops were issued with salt beef on five days with fresh meat being issued for two days a week. === Crimean War === Following initial disasters in the supply system, reforms were made and British troops were issued the following; 24 oz (680 g) of bread, 16 oz (453 g) meat, 2 oz (56 g) Rice, 2 oz (56 g) Sugar, 3 oz (85 g) Coffee, 1 Gill (0.118l) spirits and ½ oz (14 g) salt. === First World War === During the First World War British troops were issued the following daily ration; 1¼ pound (567 g) of meat, 1 pound (453 g) preserved meat, 1¼ (567 g) pound of bread, (or 1 pound (453 g) of biscuit and 4 oz (113 g) of bacon), 4 oz (113 g) Jam, 3 oz (85 g) sugar, ⅝ oz (17 g) tea, 8 oz (226 g) vegetables and 2 oz (56 g) of butter (weekly) ==== Horse Rations ==== As horses were a principal form of transport for the British Army, horses also had a scale of rations issued. A garrison ration (or mess ration for food rations of this type) is a type of military ration. thumb|Indirect calorimetry metabolic cart measuring oxygen uptake (O2) and carbon dioxide production (CO2) of a spontaneously breathing subject (dilution method with canopy hood). Indirect calorimetry calculates heat that living organisms produce by measuring either their production of carbon dioxide and nitrogen waste (frequently ammonia in aquatic organisms, or urea in terrestrial ones), or from their consumption of oxygen. Indirect calorimetry estimates the type and rate of substrate utilization and energy metabolism in vivo starting from gas exchange measurements (oxygen consumption and carbon dioxide production during rest and steady-state exercise). The per capita supply of each such food item available for human consumption is then obtained by dividing the respective quantity by the related data on the population actually partaking in it. Further advances in nutrition led to the replacement of the garrison ration in 1933 with the New Army ration, which ultimately developed into the rations system described at United States military ration. *Canopy (dilution): The dilution technique is considered the gold standard technology for Resting Energy Expenditure measurement in clinical nutrition. From 1815 to 1854 the daily ration for a British soldier in the United Kingdom was 1 pound of bread (453 g) and ¾ of a pound of meat (340 g). Food Item Ration I Ration II Ration III Ration IV Rye bread 700g (1.54 lb) 700g (1.54 lb) 700g (1.54 lb) 600g (1.32 lb) Fresh meat with bones 136g (4.8 oz) 107g (3.7 oz) 90g (3.17 oz) 56g (2 oz) Soy bean flour 7g (0.24 oz) 7g (0.24 oz) 7g (0.24 oz) 7g (0.24 oz) Headless fish 30g (1 oz) 30g (1 oz) 30g (1 oz) 30g (1 oz) Fresh vegetables and fruits 250g (8.8 oz) 250g (8.8 oz) 250g (8.8 oz) 250g (8.8 oz) Potatoes 320g (11.29 oz) 320g (11.29 oz) 320g (11.29 oz) 320g (11.29 oz) Legumes 80g (2.8 oz) 80g (2.8 oz) 80g (2.8 oz) 80g (2.8 oz) Pudding powder 20g (0.70 oz) 20g (0.70 oz) 20g (0.70 oz) 20g (0.70 oz) Sweetened condensed skim milk 25g (0.88 oz) 25g (0.88 oz) 25g (0.88 oz) 25g (0.88 oz) Salt 15g (0.5 oz) 15g (0.5 oz) 15g (0.5 oz) 15g (0.5 oz) Other seasonings 3g (0.1 oz) 3g (0.1 oz) 3g (0.1 oz) 3g (0.1 oz) Spices 1g (0.03 oz) 1g (0.03 oz) 1g (0.03 oz) 1g (0.03 oz) Fats and bread spreads 60g (2.11 oz) 50g (1.76 oz) 40g (1.41 oz) 35g (1.23 oz) Coffee 9g (0.32 oz) 9g (0.32 oz) 9g (0.32 oz) 9g (0.32 oz) Sugar 40g (1.4 oz) 35g (1.23 oz) 30g (1.05 oz) 30g (1.05 oz) Supplementary allowances 2g (0.07 oz) 2g (0.07 oz) 2g (0.07 oz) 2g (0.07 oz) Total Maximum Ration in grams 1698 1654 1622 1483 Total Maximum Ration in Pounds 3.74 3.64 3.57 3.26 == United Kingdom == In 1689 the first Royal warrant was published concerning the messing provisions for troops.

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At 303 . K, the vapor pressure of benzene is 120 . Torr and that of hexane is 189 Torr. Calculate the vapor pressure of a solution for which $x_{\text {benzene }}=0.28$ assuming ideal behavior.

Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. The second solution is switching to another vapor pressure equation with more than three parameters. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds", Hydrocarbon Processing, 68(10), Pages 65–68, 1989 ==See also== *Vapour pressure of water *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation *Raoult's law *Thermodynamic activity ==References== ==External links== * Gallica, scanned original paper * NIST Chemistry Web Book * Calculation of vapor pressures with the Antoine equation Category:Equations Category:Thermodynamic equations The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. The vapor phase is also assumed to behave like an ideal gas, so :v_v = \frac{k T}{P}, where k is the Boltzmann constant. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). (The small differences in the results are only caused by the used limited precision of the coefficients). ==Extension of the Antoine equations== To overcome the limits of the Antoine equation some simple extension by additional terms are used: : \begin{align} P &= \exp{\left( A + \frac{B}{C+T} + D \cdot T + E \cdot T^2 + F \cdot \ln \left( T \right) \right)} \\\ P &= \exp\left( A + \frac{B}{C+T} + D \cdot \ln \left( T \right) + E \cdot T^F\right). \end{align} The additional parameters increase the flexibility of the equation and allow the description of the entire vapor pressure curve. The equation was presented in 1888 by the French engineer (1825–1897). ==Equation== The Antoine equation is :\log_{10} p = A-\frac{B}{C+T}. where p is the vapor pressure, is temperature (in °C or in K according to the value of C) and , and are component-specific constants. Let v_l and v_v be the volume occupied by one molecule in the liquid phase and vapor phase respectively. The Antoine equation can also be transformed in a temperature-explicit form with simple algebraic manipulations: :T = \frac{B}{A-\log_{10}\, p} - C ==Validity range== Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the triple point to the critical point, because it is not flexible enough. "A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The Antoine equation is a class of semi-empirical correlations describing the relation between vapor pressure and temperature for pure substances. Equilibrium vapor pressure depends on droplet size. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. This doesn't affect the equation form. ==Sources for Antoine equation parameters== * NIST Chemistry WebBook * Dortmund Data Bank * Directory of reference books and data banks containing Antoine constants * Several reference books and publications, e. g. ** Lange's Handbook of Chemistry, McGraw-Hill Professional ** Wichterle I., Linek J., "Antoine Vapor Pressure Constants of Pure Compounds" ** Yaws C. L., Yang H.-C., "To Estimate Vapor Pressure Easily. This may be written in the following form, known as the Ostwald–Freundlich equation: \ln \frac{p}{p_{\rm sat}} = \frac{2 \gamma V_\text{m}}{rRT}, where p is the actual vapour pressure, p_{\rm sat} is the saturated vapour pressure when the surface is flat, \gamma is the liquid/vapor surface tension, V_\text{m} is the molar volume of the liquid, R is the universal gas constant, r is the radius of the droplet, and T is temperature. Image:VaporPressureFitAugust.png | Deviations of an August equation fit (2 parameters) Image:VaporPressureFitAntoine.png | Deviations of an Antoine equation fit (3 parameters) Image:VaporPressureFitDIPPR101.png | Deviations of a DIPPR 105 equation fit (4 parameters) ==Example parameters== Parameterisation for T in °C and P in mmHg A B C T min. (°C) T max. (°C) Water 8.07131 1730.63 233.426 1 100 Water 8.14019 1810.94 244.485 99 374 Ethanol 8.20417 1642.89 230.300 −57 80 Ethanol 7.68117 1332.04 199.200 77 243 ===Example calculation=== The normal boiling point of ethanol is TB = 78.32 °C. :\begin{align} P &= 10^{\left(8.20417 - \frac{1642.89}{78.32 + 230.300}\right)} = 760.0\ \text{mmHg} \\\ P &= 10^{\left(7.68117 - \frac{1332.04}{78.32 + 199.200}\right)} = 761.0\ \text{mmHg} \end{align} (760mmHg = 101.325kPa = 1.000atm = normal pressure) This example shows a severe problem caused by using two different sets of coefficients. Some of these are (in approximate order of increasing accuracy): Name Formula Description "Eq. 1" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Thus, the change in the Gibbs free energy for one molecule is :\Delta g = - k T \int\limits_{P_{sat}}^{P} \frac {dP}{P}, where P_{sat} is the saturated vapor pressure of x over a flat surface and P is the actual vapor pressure over the liquid.

0.4

0.69

170.0

24

6

C

Determine the molar standard Gibbs energy for ${ }^{35} \mathrm{Cl}^{35} \mathrm{Cl}$ where $\widetilde{\nu}=560 . \mathrm{cm}^{-1}, B=0.244 \mathrm{~cm}^{-1}$, and the ground electronic state is nondegenerate.

The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. These tables list values of molar ionization energies, measured in kJ⋅mol−1. Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. CL-35 or CL35 may refer to: * (CL-35), a United States Navy heavy cruiser * Chlorine-35 (Cl-35 or 35Cl), an isotope of chlorine The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. Unlike standard enthalpies of formation, the value of is absolute. LNG: Values refer to 300 K. * Ref. WEL: Values refer to 25 °C. ==References== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. The first molar ionization energy applies to the neutral atoms. That is, an element in its standard state has a definite, nonzero value of at room temperature. Under identical conditions, it is greater for a heavier gas. ==See also== *Entropy *Heat *Gibbs free energy *Helmholtz free energy *Standard state *Third law of thermodynamics ==References== ==External links== *Standard Thermodynamic Properties of Chemical Substances Table Category:Chemical properties Category:Thermodynamic entropy The notation (P=0) denotes low pressure limiting values. This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. All data from rutherfordium onwards is predicted. == All Ionization Energies == Number Symbol Name 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 1 H hydrogen 1312.0 2 He helium 2372.3 5250.5 3 Li lithium 520.2 7298.1 11,815.0 4 Be beryllium 899.5 1757.1 14,848.7 21,006.6 5 B boron 800.6 2427.1 3659.7 25,025.8 32,826.7 6 C carbon 1086.5 2352.6 4620.5 6222.7 37,831 47,277.0 7 N nitrogen 1402.3 2856 4578.1 7475.0 9444.9 53,266.6 64,360 8 O oxygen 1313.9 3388.3 5300.5 7469.2 10,989.5 13,326.5 71,330 84,078.0 9 F fluorine 1681.0 3374.2 6050.4 8407.7 11,022.7 15,164.1 17,868 92,038.1 106,434.3 10 Ne neon 2080.7 3952.3 6122 9371 12,177 15,238.90 19,999.0 23,069.5 115,379.5 131,432 11 Na sodium 495.8 4562 6910.3 9543 13,354 16,613 20,117 25,496 28,932 141,362 12 Mg magnesium 737.7 1450.7 7732.7 10,542.5 13,630 18,020 21,711 25,661 31,653 35,458 13 Al aluminium 577.5 1816.7 2744.8 11,577 14,842 18,379 23,326 27,465 31,853 38,473 14 Si silicon 786.5 1577.1 3231.6 4355.5 16,091 19,805 23,780 29,287 33,878 38,726 15 P phosphorus 1011.8 1907 2914.1 4963.6 6273.9 21,267 25,431 29,872 35,905 40,950 16 S sulfur 999.6 2252 3357 4556 7004.3 8495.8 27,107 31,719 36,621 43,177 17 Cl chlorine 1251.2 2298 3822 5158.6 6542 9362 11,018 33,604 38,600 43,961 18 Ar argon 1520.6 2665.8 3931 5771 7238 8781 11,995 13,842 40,760 46,186 19 K potassium 418.8 3052 4420 5877 7975 9590 11,343 14,944 16,963.7 48,610 20 Ca calcium 589.8 1145.4 4912.4 6491 8153 10,496 12,270 14,206 18,191 20,385 21 Sc scandium 633.1 1235.0 2388.6 7090.6 8843 10,679 13,310 15,250 17,370 21,726 22 Ti titanium 658.8 1309.8 2652.5 4174.6 9581 11,533 13,590 16,440 18,530 20,833 23 V vanadium 650.9 1414 2830 4507 6298.7 12,363 14,530 16,730 19,860 22,240 24 Cr chromium 652.9 1590.6 2987 4743 6702 8744.9 15,455 17,820 20,190 23,580 25 Mn manganese 717.3 1509.0 3248 4940 6990 9220 11,500 18,770 21,400 23,960 26 Fe iron 762.5 1561.9 2957 5290 7240 9560 12,060 14,580 22,540 25,290 27 Co cobalt 760.4 1648 3232 4950 7670 9840 12,440 15,230 17,959 26,570 28 Ni nickel 737.1 1753.0 3395 5300 7339 10,400 12,800 15,600 18,600 21,670 29 Cu copper 745.5 1957.9 3555 5536 7700 9900 13,400 16,000 19,200 22,400 30 Zn zinc 906.4 1733.3 3833 5731 7970 10,400 12,900 16,800 19,600 23,000 31 Ga gallium 578.8 1979.3 2963 6180 32 Ge germanium 762 1537.5 3302.1 4411 9020 33 As arsenic 947.0 1798 2735 4837 6043 12,310 34 Se selenium 941.0 2045 2973.7 4144 6590 7880 14,990 35 Br bromine 1139.9 2103 3470 4560 5760 8550 9940 18,600 36 Kr krypton 1350.8 2350.4 3565 5070 6240 7570 10,710 12,138 22,274 25,880 37 Rb rubidium 403.0 2633 3860 5080 6850 8140 9570 13,120 14,500 26,740 38 Sr strontium 549.5 1064.2 4138 5500 6910 8760 10,230 11,800 15,600 17,100 39 Y yttrium 600 1180 1980 5847 7430 8970 11,190 12,450 14,110 18,400 40 Zr zirconium 640.1 1270 2218 3313 7752 9500 41 Nb niobium 652.1 1380 2416 3700 4877 9847 12,100 42 Mo molybdenum 684.3 1560 2618 4480 5257 6640.8 12,125 13,860 15,835 17,980 43 Tc technetium 702 1470 2850 44 Ru ruthenium 710.2 1620 2747 45 Rh rhodium 719.7 1740 2997 46 Pd palladium 804.4 1870 3177 47 Ag silver 731.0 2070 3361 48 Cd cadmium 867.8 1631.4 3616 49 In indium 558.3 1820.7 2704 5210 50 Sn tin 708.6 1411.8 2943.0 3930.3 7456 51 Sb antimony 834 1594.9 2440 4260 5400 10,400 52 Te tellurium 869.3 1790 2698 3610 5668 6820 13,200 53 I iodine 1008.4 1845.9 3180 54 Xe xenon 1170.4 2046.4 3099.4 55 Cs caesium 375.7 2234.3 3400 56 Ba barium 502.9 965.2 3600 57 La lanthanum 538.1 1067 1850.3 4819 5940 58 Ce cerium 534.4 1050 1949 3547 6325 7490 59 Pr praseodymium 527 1020 2086 3761 5551 60 Nd neodymium 533.1 1040 2130 3900 61 Pm promethium 540 1050 2150 3970 62 Sm samarium 544.5 1070 2260 3990 63 Eu europium 547.1 1085 2404 4120 64 Gd gadolinium 593.4 1170 1990 4250 65 Tb terbium 565.8 1110 2114 3839 66 Dy dysprosium 573.0 1130 2200 3990 67 Ho holmium 581.0 1140 2204 4100 68 Er erbium 589.3 1150 2194 4120 69 Tm thulium 596.7 1160 2285 4120 70 Yb ytterbium 603.4 1174.8 2417 4203 71 Lu lutetium 523.5 1340 2022.3 4370 6445 72 Hf hafnium 658.5 1440 2250 3216 73 Ta tantalum 761 1500 74 W tungsten 770 1700 75 Re rhenium 760 1260 2510 3640 76 Os osmium 840 1600 77 Ir iridium 880 1600 78 Pt platinum 870 1791 79 Au gold 890.1 1980 80 Hg mercury 1007.1 1810 3300 81 Tl thallium 589.4 1971 2878 82 Pb lead 715.6 1450.5 3081.5 4083 6640 83 Bi bismuth 703 1610 2466 4370 5400 8520 84 Po polonium 812.1 85 At astatine 899.003 86 Rn radon 1037 87 Fr francium 393 88 Ra radium 509.3 979.0 89 Ac actinium 499 1170 1900 4700 90 Th thorium 587 1110 1978 2780 91 Pa protactinium 568 1128 1814 2991 92 U uranium 597.6 1420 1900 3145 93 Np neptunium 604.5 1128 1997 3242 94 Pu plutonium 584.7 1128 2084 3338 95 Am americium 578 1158 2132 3493 96 Cm curium 581 1196 2026 3550 97 Bk berkelium 601 1186 2152 3434 98 Cf californium 608 1206 2267 3599 99 Es einsteinium 619 1216 2334 3734 100 Fm fermium 629 1225 2363 3792 101 Md mendelevium 636 1235 2470 3840 102 No nobelium 639 1254 2643 3956 103 Lr lawrencium 479 1428 2228 4910 104 Rf rutherfordium 580 1390 2300 3080 105 Db dubnium 665 1547 2378 3299 4305 106 Sg seaborgium 757 1733 2484 3416 4562 5716 107 Bh bohrium 740 1690 2570 3600 4730 5990 7230 108 Hs hassium 730 1760 2830 3640 4940 6180 7540 8860 109 Mt meitnerium 800 1820 2900 3900 4900 110 Ds darmstadtium 960 1890 3030 4000 5100 111 Rg roentgenium 1020 2070 3080 4100 5300 112 Cn copernicium 1155 2170 3160 4200 5500 113 Nh nihonium 707.2 2309 3226 4382 5638 114 Fl flerovium 832.2 1600 3370 4400 5850 115 Mc moscovium 538.3 1760 2650 4680 5720 116 Lv livermorium 663.9 1330 2850 3810 6080 117 Ts tennessine 736.9 1435.4 2161.9 4012.9 5076.4 118 Og oganesson 860.1 1560 119 Uue ununennium 463.1 1700 120 Ubn unbinilium 563.3 121 Ubu unbiunium 429.4 1110 1710 4270 122 Ubb unbibium 545 1090 1848 2520 == 11th-20th ionisation energies == number symbol name 11th 12th 13th 14th 15th 16th 17th 18th 19th 20th 11 Na sodium 159,076 12 Mg magnesium 169,988 189,368 13 Al aluminium 42,647 201,266 222,316 14 Si silicon 45,962 50,502 235,196 257,923 15 P phosphorus 46,261 54,110 59,024 271,791 296,195 16 S sulfur 48,710 54,460 62,930 68,216 311,048 337,138 17 Cl chlorine 51,068 57,119 63,363 72,341 78,095 352,994 380,760 18 Ar argon 52,002 59,653 66,199 72,918 82,473 88,576 397,605 427,066 19 K potassium 54,490 60,730 68,950 75,900 83,080 93,400 99,710 444,880 476,063 20 Ca calcium 57,110 63,410 70,110 78,890 86,310 94,000 104,900 111,711 494,850 527,762 21 Sc scandium 24,102 66,320 73,010 80,160 89,490 97,400 105,600 117,000 124,270 547,530 22 Ti titanium 25,575 28,125 76,015 83,280 90,880 100,700 109,100 117,800 129,900 137,530 23 V vanadium 24,670 29,730 32,446 86,450 94,170 102,300 112,700 121,600 130,700 143,400 24 Cr chromium 26,130 28,750 34,230 37,066 97,510 105,800 114,300 125,300 134,700 144,300 25 Mn manganese 27,590 30,330 33,150 38,880 41,987 109,480 118,100 127,100 138,600 148,500 26 Fe iron 28,000 31,920 34,830 37,840 44,100 47,206 122,200 131,000 140,500 152,600 27 Co cobalt 29,400 32,400 36,600 39,700 42,800 49,396 52,737 134,810 145,170 154,700 28 Ni nickel 30,970 34,000 37,100 41,500 44,800 48,100 55,101 58,570 148,700 159,000 29 Cu copper 25,600 35,600 38,700 42,000 46,700 50,200 53,700 61,100 64,702 163,700 30 Zn zinc 26,400 29,990 40,490 43,800 47,300 52,300 55,900 59,700 67,300 71,200 36 Kr krypton 29,700 33,800 37,700 43,100 47,500 52,200 57,100 61,800 75,800 80,400 38 Sr strontium 31,270 39 Y yttrium 19,900 36,090 42 Mo molybdenum 20,190 22,219 26,930 29,196 52,490 55,000 61,400 67,700 74,000 80,400 == 21st-30th ionisation energies == number symbol name 21st 22nd 23rd 24th 25th 26th 27th 28th 29th 30th 21 Sc scandium 582,163 22 Ti titanium 602,930 639,294 23 V vanadium 151,440 661,050 699,144 24 Cr chromium 157,700 166,090 721,870 761,733 25 Mn manganese 158,600 172,500 181,380 785,450 827,067 26 Fe iron 163,000 173,600 188,100 195,200 851,800 895,161 27 Co cobalt 167,400 178,100 189,300 204,500 214,100 920,870 966,023 28 Ni nickel 169,400 182,700 194,000 205,600 221,400 231,490 992,718 1,039,668 29 Cu copper 174,100 184,900 198,800 210,500 222,700 239,100 249,660 1,067,358 1,116,105 30 Zn zinc 179,100 36 Kr krypton 85,300 90,400 96,300 101,400 111,100 116,290 282,500 296,200 311,400 326,200 42 Mo molybdenum 87,000 93,400 98,420 104,400 121,900 127,700 133,800 139,800 148,100 154,500 == References == * Ionization energies of the elements (data page) * (for predictions) * * (for predictions) Category:Properties of chemical elements

-57.2

226

12.0

7.27

-0.55

A

For the reaction $\mathrm{C}($ graphite $)+\mathrm{H}_2 \mathrm{O}(g) \rightleftharpoons$ $\mathrm{CO}(g)+\mathrm{H}_2(g), \Delta H_R^{\circ}=131.28 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298.15 \mathrm{~K}$. Use the values of $C_{P, m}^{\circ}$ at $298.15 \mathrm{~K}$ in the data tables to calculate $\Delta H_R^{\circ}$ at $125.0^{\circ} \mathrm{C}$.

== Heat of fusion == kJ/mol 1 H hydrogen (H2) use (H2) 0.1 CRC (H2) 0.12 LNG (H2) 0.117 WEL (per mol H atoms) 0.0585 2 He helium use 0.0138 LNG 0.0138 WEL 0.02 3 Li lithium use 3.00 CRC 3.00 LNG 3.00 WEL 3.0 4 Be beryllium use 7.895 CRC 7.895 LNG 7.895 WEL 7.95 5 B boron use 50.2 CRC 50.2 LNG 50.2 WEL 50 6 C carbon (graphite) use (graphite) 117 CRC (graphite) 117 LNG (graphite) 117 7 N nitrogen (N2) use (N2) 0.720 CRC (N2) 0.71 LNG (N2) 0.720 WEL (per mol N atoms) 0.36 8 O oxygen (O2) use (O2) 0.444 CRC (O2) 0.44 LNG (O2) 0.444 WEL (per mol O atoms) 0.222 9 F fluorine (F2) use (F2) 0.510 CRC (F2) 0.51 LNG (F2) 0.510 WEL (per mol F atoms) 0.26 10 Ne neon use 0.335 CRC 0.328 LNG 0.335 WEL 0.34 11 Na sodium use 2.60 CRC 2.60 LNG 2.60 WEL 2.60 12 Mg magnesium use 8.48 CRC 8.48 LNG 8.48 WEL 8.7 13 Al aluminium use 10.71 CRC 10.789 LNG 10.71 WEL 10.7 14 Si silicon use 50.21 CRC 50.21 LNG 50.21 WEL 50.2 15 P phosphorus use 0.66 CRC (white) 0.66 LNG 0.66 WEL 0.64 16 S sulfur use (mono) 1.727 CRC (mono) 1.72 LNG (mono) 1.727 WEL 1.73 17 Cl chlorine (Cl2) use (Cl2) 6.406 CRC (Cl2) 6.40 LNG (Cl2) 6.406 WEL (per mol Cl atoms) 3.2 18 Ar argon use 1.18 CRC 1.18 LNG 1.12 WEL 1.18 19 K potassium use 2.321 CRC 2.33 LNG 2.321 WEL 2.33 20 Ca calcium use 8.54 CRC 8.54 LNG 8.54 WEL 8.54 21 Sc scandium use 14.1 CRC 14.1 LNG 14.1 WEL 16 22 Ti titanium use 14.15 CRC 14.15 LNG 14.15 WEL 18.7 23 V vanadium use 21.5 CRC 21.5 LNG 21.5 WEL 22.8 24 Cr chromium use 21.0 CRC 21.0 LNG 21.0 WEL 20.5 25 Mn manganese use 12.91 CRC 12.91 LNG 12.9 WEL 13.2 26 Fe iron use 13.81 CRC 13.81 LNG 13.81 WEL 13.8 27 Co cobalt use 16.06 CRC 16.06 LNG 16.2 WEL 16.2 28 Ni nickel use 17.48 CRC 17.04 LNG 17.48 WEL 17.2 29 Cu copper use 13.26 CRC 12.93 LNG 13.26 WEL 13.1 30 Zn zinc use 7.32 CRC 7.068 LNG 7.32 WEL 7.35 31 Ga gallium use 5.59 CRC 5.576 LNG 5.59 WEL 5.59 32 Ge germanium use 36.94 CRC 36.94 LNG 36.94 WEL 31.8 33 As arsenic use (gray) 24.44 CRC (gray) 24.44 LNG 24.44 WEL 27.7 34 Se selenium use (gray) 6.69 CRC (gray) 6.69 LNG 6.69 WEL 5.4 35 Br bromine (Br2) use (Br2) 10.57 CRC (Br2) 10.57 LNG (Br2) 10.57 WEL (per mol Br atoms) 5.8 36 Kr krypton use 1.64 CRC 1.64 LNG 1.37 WEL 1.64 37 Rb rubidium use 2.19 CRC 2.19 LNG 2.19 WEL 2.19 38 Sr strontium use 7.43 CRC 7.43 LNG 7.43 WEL 8 39 Y yttrium use 11.42 CRC 11.4 LNG 11.42 WEL 11.4 40 Zr zirconium use 14 13.96 ± 0.36 CRC 21.00 LNG 21.00 WEL 21 41 Nb niobium use 30 CRC 30 LNG 30 WEL 26.8 42 Mo molybdenum use 37.48 CRC 37.48 LNG 37.48 WEL 36 43 Tc technetium use 33.29 CRC 33.29 LNG 33.29 WEL 23 44 Ru ruthenium use 38.59 CRC 38.59 LNG 38.59 WEL 25.7 45 Rh rhodium use 26.59 CRC 26.59 LNG 26.59 WEL 21.7 46 Pd palladium use 16.74 CRC 16.74 LNG 16.74 WEL 16.7 47 Ag silver use 11.28 CRC 11.28 LNG 11.95 WEL 11.3 48 Cd cadmium use 6.21 CRC 6.21 LNG 6.19 WEL 6.3 49 In indium use 3.281 CRC 3.281 LNG 3.28 WEL 3.26 50 Sn tin use (white) 7.03 CRC (white) 7.173 LNG (white) 7.03 WEL 7.0 51 Sb antimony use 19.79 CRC 19.79 LNG 19.87 WEL 19.7 52 Te tellurium use 17.49 CRC 17.49 LNG 17.49 WEL 17.5 53 I iodine (I2) use (I2) 15.52 CRC (I2) 15.52 LNG (I2) 150.66 [sic] WEL (per mol I atoms) 7.76 54 Xe xenon use 2.27 CRC 2.27 LNG 1.81 WEL 2.30 55 Cs caesium use 2.09 CRC 2.09 LNG 2.09 WEL 2.09 56 Ba barium use 7.12 CRC 7.12 LNG 7.12 WEL 8.0 57 La lanthanum use 6.20 CRC 6.20 LNG 6.20 WEL 6.2 58 Ce cerium use 5.46 CRC 5.46 LNG 5.46 WEL 5.5 59 Pr praseodymium use 6.89 CRC 6.89 LNG 6.89 WEL 6.9 60 Nd neodymium use 7.14 CRC 7.14 LNG 7.14 WEL 7.1 61 Pm promethium use 7.13 LNG 7.13 WEL about 7.7 62 Sm samarium use 8.62 CRC 8.62 LNG 8.62 WEL 8.6 63 Eu europium use 9.21 CRC 9.21 LNG 9.21 WEL 9.2 64 Gd gadolinium use 10.05 CRC 10.0 LNG 10.05 WEL 10.0 65 Tb terbium use 10.15 CRC 10.15 LNG 10.15 WEL 10.8 66 Dy dysprosium use 11.06 CRC 11.06 LNG 11.06 WEL 11.1 67 Ho holmium use 17.0 CRC 17.0 LNG 16.8 WEL 17.0 68 Er erbium use 19.90 CRC 19.9 LNG 19.90 WEL 19.9 69 Tm thulium use 16.84 CRC 16.84 LNG 16.84 WEL 16.8 70 Yb ytterbium use 7.66 CRC 7.66 LNG 7.66 WEL 7.7 71 Lu lutetium use ca. 22 CRC 22 LNG (22) WEL about 22 72 Hf hafnium use 27.2 CRC 27.2 LNG 27.2 WEL 25.5 73 Ta tantalum use 36.57 CRC 36.57 LNG 36.57 WEL 36 74 W tungsten use 52.31 CRC 52.31 LNG 52.31 WEL 35 75 Re rhenium use 60.43 CRC 60.43 LNG 60.43 WEL 33 76 Os osmium use 57.85 CRC 57.85 LNG 57.85 WEL 31 77 Ir iridium use 41.12 CRC 41.12 LNG 41.12 WEL 26 78 Pt platinum use 22.17 CRC 22.17 LNG 22.17 WEL 20 79 Au gold use 12.55 CRC 12.72 LNG 12.55 WEL 12.5 80 Hg mercury use 2.29 CRC 2.29 LNG 2.29 WEL 2.29 81 Tl thallium use 4.14 CRC 4.14 LNG 4.14 WEL 4.2 82 Pb lead use 4.77 CRC 4.782 LNG 4.77 WEL 4.77 83 Bi bismuth use 11.30 CRC 11.145 LNG 11.30 WEL 10.9 84 Po polonium use ca. 13 WEL about 13 85 At astatine use WEL (per mol At atoms) about 6 86 Rn radon use 3.247 LNG 3.247 WEL 3 87 Fr francium use ca. 2 WEL about 2 88 Ra radium use 8.5 LNG 8.5 WEL about 8 89 Ac actinium use 14 WEL 14 90 Th thorium use 13.81 CRC 13.81 LNG 13.81 WEL 16 91 Pa protactinium use 12.34 CRC 12.34 LNG 12.34 WEL 15 92 U uranium use 9.14 CRC 9.14 LNG 9.14 WEL 14 93 Np neptunium use 3.20 CRC 3.20 LNG 3.20 WEL 10 94 Pu plutonium use 2.82 CRC 2.82 LNG 2.82 95 Am americium use 14.39 CRC 14.39 LNG 14.39 == Notes == * Values refer to the enthalpy change between the liquid phase and the most stable solid phase at the melting point (normal, 101.325 kPa). == References == === CRC === As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. ==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup Graphane is a two-dimensional polymer of carbon and hydrogen with the formula unit (CH)n where n is large. P-doped graphane is proposed to be a high-temperature BCS theory superconductor with a Tc above 90 K. ==Variants== Partial hydrogenation leads to hydrogenated graphene rather than (fully hydrogenated) graphane. The structure was found, using a cluster expansion method, to be the most stable of all the possible hydrogenation ratios of graphene. * Values from CRC refer to "100 kPa (1 bar or 0.987 standard atmospheres)". :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). LNG: Values refer to 300 K. * Ref. WEL: Values refer to 25 °C. ==References== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Hydrogenation decreases the dependence of the lattice constant on temperature, which indicates a possible application in precision instruments. ==References== ==External links== * Sep 14, 2010 Hydrogen vacancies induce stable ferromagnetism in graphane * May 25, 2010 Graphane yields new potential * May 02 2010 Doped Graphane Should Superconduct at 90K Category:Two-dimensional nanomaterials Category:Polymers Category:Superconductors Category:Hydrocarbons Density functional theory calculations suggested that hydrogenated and fluorinated forms of other group IV (Si, Ge and Sn) nanosheets present properties similar to graphane. ==Potential applications== p-Doped graphane is postulated to be a high-temperature BCS theory superconductor with a Tc above 90 K. Graphane has been proposed for hydrogen storage. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). Partial hydrogenation results in hydrogenated graphene, which was reported by Elias et al in 2009 by a TEM study to be "direct evidence for a new graphene-based derivative". * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Enthalpy of Fusion === LNG === As quoted from various sources in: * J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds === WEL === As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Above 750 K Tc values may be in error by 10 K or more. Table 259, Critical Temperatures, Pressures, and Densities of Gases == See also == Category:Phase transitions Category:Units of pressure Category:Temperature Category:Properties of chemical elements Category:Chemical element data pages In the last case mechanical exfoliation of hydrogenated top layers can be used. ==Structure== The first theoretical description of graphane was reported in 2003. CR2: Values refer to 300 K and a pressure of "100 kPa (1 bar)", or to the saturation vapor pressure if that is less than 100 kPa.

30

8.8

1855.0

0.000216

132.9

E

Calculate the mean ionic activity of a $0.0350 \mathrm{~m} \mathrm{Na}_3 \mathrm{PO}_4$ solution for which the mean activity coefficient is 0.685 .

It is important to note that because the ions in the solution act together, the activity coefficient obtained from this equation is actually a mean activity coefficient. Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. : Aqua ions in seawater (salinity = 35) Ion Concentration (mol kg−1) 0.469 0.0102 0.0528 0.0103 Many other aqua ions are present in seawater in concentrations ranging from ppm to ppt. The Debye–Hückel limiting law enables one to determine the activity coefficient of an ion in a dilute solution of known ionic strength. Journal of Solution Chemistry, 26, 791–815. ==External links== * For easy calculation of activity coefficients in (non- micellar) solutions, check out the IUPAC open project Aq-solutions (freeware). D&H; say that, due to the "mutual electrostatic forces between the ions", it is necessary to modify the Guldberg–Waage equation by replacing K with \gamma K, where \gamma is an overall activity coefficient, not a "special" activity coefficient (a separate activity coefficient associated with each species)—which is what is used in modern chemistry . Typical values are 3Å for ions such as H+, Cl−, CN−, and HCOO−. right|thumb|The Madelung constant being calculated for the NaCl ion labeled 0 in the expanding spheres method. The electrostatic energy of the ion at site then is the product of its charge with the potential acting at its site :E_{el,i} = z_ieV_i = \frac{e^2}{4 \pi \varepsilon_0 r_0 } z_i M_i. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. == Formal expression == The Madelung constant allows for the calculation of the electric potential of all ions of the lattice felt by the ion at position :V_i = \frac{e}{4 \pi \varepsilon_0 } \sum_{j eq i} \frac{z_j}{r_{ij}}\,\\! where r_{ij} = |r_i-r_j| is the distance between the th and the th ion. In these solutions the activity coefficient may actually increase with ionic strength. center|The Debye–Hückel plot with different values for ion charge Z and ion diameter a The Debye–Hückel equation cannot be used in the solutions of surfactants where the presence of micelles influences on the electrochemical properties of the system (even rough judgement overestimates γ for ~50%). ==See also== * Strong electrolyte * Weak electrolyte * Ionic atmosphere * Debye–Hückel theory * Poisson–Boltzmann equation ==Notes== ==References== * Alt URL * * * * Malatesta, F., and Zamboni, R. (1997). Ionic potential is also a measure of the polarising power of a cation. Ions fall into four groups. This factor takes into account the interaction energy of ions in solution. == Debye–Hückel limiting law == In order to calculate the activity a_C of an ion C in a solution, one must know the concentration and the activity coefficient: a_C = \gamma \frac\mathrm{[C]}\mathrm{[C^\ominus]}, where * \gamma is the activity coefficient of C, * \mathrm{[C^\ominus]} is the concentration of the chosen standard state, e.g. 1 mol/kg if molality is used, * \mathrm{[C]} is a measure of the concentration of C. Dividing \mathrm{[C]} with \mathrm{[C^\ominus]} gives a dimensionless quantity. thumb|250px|Distribution of ions in a solution The chemists Peter Debye and Erich Hückel noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. A metal ion in aqueous solution or aqua ion is a cation, dissolved in water, of chemical formula [M(H2O)n]z+. Zn2+ −0.751 Ga3+ −0.53 Ge2+ +0.1 Rb −2.98 Sr2+ −2.899 Y3+ −2.37 ... Hg2+ −0.854 Tl3+ +0.73 Pb2+ −0.126 Bi3+ +0.16 Po4+ +0.76 Fr −2.9 Ra2+ −2.8 Lr3+ −1.96 La3+ −2.52 Ce3+ −2.32 Pr3+ −2.34 Nd3+ −2.32 Pm3+ −2.30 Sm3+ −2.28 Eu3+ −1.98 Gd3+ −2.27 Tb3+ −2.27 Dy3+ −2.32 Ho3+ −2.37 Er3+ −2.33 Tm3+ −2.30 Yb3+ −2.23 Ac3+ −2.18 Th4+ −1.83 Pa4+ −1.46 U4+ −1.51 Np4+ −1.33 Pu4+ −1.80 Am3+ −2.06 Cm3+ −2.07 Bk3+ −2.03 Cf3+ −2.01 Es3+ −1.99 Fm3+ −1.97 Md3+ −1.65 No3+ −1.20 : Standard electrode potentials /V for 1st. row transition metal ions Couple Ti V Cr Mn Fe Co Ni Cu M2+ / M −1.63 −1.18 −0.91 −1.18 −0.473 −0.28 −0.228 +0.345 M3+ / M −1.37 −0.87 −0.74 −0.28 −0.06 +0.41 : Miscellaneous standard electrode potentials /V Ag+ / Ag Pd2+ / Pd Pt2+ / Pt Zr4+ / Zr Hf4+ / Hf Au3+ / Au Ce4+ / Ce +0.799 +0.915 +1.18 −1.53 −1.70 +1.50 −1.32 As the standard electrode potential is more negative the aqua ion is more difficult to reduce. For ions in solution Shannon's "effective ionic radius" is the measure most often used.. Ions are 6-coordinate unless indicated differently in parentheses (e.g. 146 (4) for 4-coordinate N3−). Ions are 6-coordinate unless indicated differently in parentheses (e.g. 146 (4) for 4-coordinate N3−). Li+ -118.8 Na+ -87.4 Mg2+ -267.8 Al3+ -464.4 K+ -51.9 Ca2+ -209.2 ...

0.0547

399

34.0

0.2115

7.136

A

Consider the transition between two forms of solid tin, $\mathrm{Sn}(s$, gray $) \rightarrow \mathrm{Sn}(s$, white $)$. The two phases are in equilibrium at 1 bar and $18^{\circ} \mathrm{C}$. The densities for gray and white tin are 5750 and $7280 \mathrm{~kg} \mathrm{~m}^{-3}$, respectively, and the molar entropies for gray and white tin are 44.14 and $51.18 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, respectively. Calculate the temperature at which the two phases are in equilibrium at 350. bar.

J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? Transition temperature is the temperature at which a material changes from one crystal state (allotrope) to another. Tin(II) chloride, also known as stannous chloride, is a white crystalline solid with the formula . K (? °C), ? K (? °C), ? * Handbook of Chemistry and Physics, 71st edition, CRC Press, Ann Arbor, Michigan, 1990. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? It was first discovered in 1822. == Preparation == To obtain tin(II) acetate, tin(II) oxide is dissolved in glacial acetic acid and refluxed to obtain yellow Sn(CH3COO)2·2CH3COOH when cooled. More formally, it is the temperature at which two crystalline forms of a substance can co-exist in equilibrium. White tin may also refer specifically to β-tin, the metallic allotrope of the pure element, as opposed to the nonmetallic allotrope α-tin (also known as gray tin), which occurs at temperatures below , a transformation known as tin pest). Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? At 95.6 °C the two forms can co-exist. Tin(II) acetate is the acetate salt of tin(II), with the chemical formula of Sn(CH3COO)2. Another example is tin, which transitions from a cubic crystal below 13.2 °C to a tetragonal crystal above that temperature. White tin is refined metallic tin. Tin(II) chloride should not be confused with the other chloride of tin; tin(IV) chloride or stannic chloride (SnCl4). ==Chemical structure== SnCl2 has a lone pair of electrons, such that the molecule in the gas phase is bent. The main part of the molecule stacks into double layers in the crystal lattice, with the "second" water sandwiched between the layers. thumb|460px|left|Structures of tin(II) chloride and related compounds ==Chemical properties== Tin(II) chloride can dissolve in less than its own mass of water without apparent decomposition, but as the solution is diluted, hydrolysis occurs to form an insoluble basic salt: :SnCl2 (aq) + H2O (l) Sn(OH)Cl (s) + HCl (aq) Therefore, if clear solutions of tin(II) chloride are to be used, it must be dissolved in hydrochloric acid (typically of the same or greater molarity as the stannous chloride) to maintain the equilibrium towards the left-hand side (using Le Chatelier's principle). J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? Solutions of tin(II) chloride can also serve simply as a source of Sn2+ ions, which can form other tin(II) compounds via precipitation reactions. Complex tin (II) acetates. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? In the case of ferroelectric or ferromagnetic crystals, a transition temperature may be known as the Curie temperature. == See also == * Crystal system Category:Crystallography Category:Threshold temperatures

479

2.24

'-242.6'

7.82

-3.5

E

The densities of pure water and ethanol are 997 and $789 \mathrm{~kg} \mathrm{~m}^{-3}$, respectively. For $x_{\text {ethanol }}=0.35$, the partial molar volumes of ethanol and water are 55.2 and $17.8 \times 10^{-3} \mathrm{~L} \mathrm{~mol}^{-1}$, respectively. Calculate the change in volume relative to the pure components when $2.50 \mathrm{~L}$ of a solution with $x_{\text {ethanol }}=0.35$ is prepared.

It can be calculated from this table that at 25 °C, 45 g of ethanol has volume 57.3 ml, 55 g of water has volume 55.2 ml; these sum to 112.5 ml. The volume of alcohol in the solution can then be estimated. Whereas to make a 50% v/v ethanol solution, 50 ml of ethanol and 50 ml of water could be mixed but the resulting volume of solution will measure less than 100 ml due to the change of volume on mixing, and will contain a higher concentration of ethanol. Mixing two solutions of alcohol of different strengths usually causes a change in volume. For example, to make 100 ml of 50% ABV ethanol solution, water would be added to 50 ml of ethanol to make up exactly 100 ml. This source gives density data for ethanol:water mixes by %weight ethanol in 5% increments and against temperature including at 25 °C, used here. Therefore, one can use the following equation to convert between ABV and ABW: \text{ABV} = \text{ABW} \times \frac{\text{density of beverage}}{\text{density of alcohol}} At relatively low ABV, the alcohol percentage by weight is about 4/5 of the ABV (e.g., 3.2% ABW is about 4% ABV). The following formulas can be used to calculate the volumes of solute (Vsolute) and solvent (Vsolvent) to be used: *Vsolute = Vtotal / F *Vsolvent = Vtotal \- Vsolute , where: *Vtotal is the desired total volume *F is the desired dilution factor number (the number in the position of F if expressed as "1:F dilution factor" or "xF dilution") However, some solutions and mixtures take up slightly less volume than their components. The International Organization of Legal Metrology has tables of density of water–ethanol mixtures at different concentrations and temperatures. At 0% and 100% ABV is equal to ABW, but at values in between ABV is always higher, up to ~13% higher around 60% ABV. ==See also== *Apparent molar property *Excess molar quantity *Standard drink *Unit of alcohol *Volume fraction == Notes == == References == ==Bibliography== * * * * ==External links== * * Category:Alcohol measurement The number of millilitres of pure ethanol is the mass of the ethanol divided by its density at , which is .Haynes, William M., ed. (2011). Mixing pure water with a solution less than 24% by mass causes a slight increase in total volume, whereas the mixing of two solutions above 24% causes a decrease in volume. Alcohol by volume (abbreviated as ABV, abv, or alc/vol) is a standard measure of how much alcohol (ethanol) is contained in a given volume of an alcoholic beverage (expressed as a volume percent). The volume of the cup is irrelevant, as is any stirring of the mixtures. == Solution == Conservation of substance implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine. The amount by which the liquid rises in the cylinder (∆V) is equal to the volume of the object. However, because of the miscibility of alcohol and water, the conversion factor is not constant but rather depends upon the concentration of alcohol. It is defined as the number of millilitres (mL) of pure ethanol present in of solution at . The density of sugar in water is greater than the density of alcohol in water. The volume was rounded up to 750 mL and then was used as the base size for French wine containers, with all subdivisions and multiples figured from it. In some countries, e.g. France, alcohol by volume is often referred to as degrees Gay-Lussac (after the French chemist Joseph Louis Gay-Lussac), although there is a slight difference since the Gay- Lussac convention uses the International Standard Atmosphere value for temperature, . ==Volume change== thumb|upright=1.48|Change in volume with increasing ABV. In the wine/water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. Originally there were different standard gallons depending on the type of alcohol.

0.8561

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-36.5

8.7

B

For $\mathrm{N}_2$ at $298 \mathrm{~K}$, what fraction of molecules has a speed between 200. and $300 . \mathrm{m} / \mathrm{s}$ ?

To help compare different orders of magnitude, the following list describes various speed levels between approximately 2.2 m/s and 3.0 m/s (the speed of light). These atoms effuse out of a hole in the oven with average speeds on the order of hundreds of m/s and large velocity distributions (due to their high temperature). The inch per second is a unit of speed or velocity. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between entities of species A and species B is:chem.libretexts.org: Collision Frequency : Z = N_\text{A} N_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}}, which has units of [volume][time]−1. Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the "effective area" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. Current fastest macroscopic man-made system. 450,000 1,600,000 1,000,000 0.0015 Typical speed of a particle of the solar wind, relative to the Sun. 552,000 1,990,000 1,230,000 0.0018 Speed of the Milky Way, relative to the cosmic microwave background. 617,700 2,224,000 1,382,000 0.0021 Escape velocity from the surface of the Sun. 106 1,000,000 3,600,000 2,200,000 0.0030 Typical speed of a Moreton wave across the surface of the Sun. 1,610,000 5,800,000 3,600,000 0.0054 Speed of hypervelocity star PSR B2224+65, which currently seems to be leaving the Milky Way. 5,000,000 18,000,000 11,000,000 0.017 Estimated minimum speed of star S2 at its closest approach to Sagittarius A*. 107 14,000,000 50,000,000 31,000,000 0.047 Typical speed of a fast neutron. 30,000,000 100,000,000 70,000,000 0.1 Typical speed of an electron in a cathode ray tube. 108 100,000,000 360,000,000 220,000,000 0.3 The escape velocity of a neutron star. 100,000,000 360,000,000 220,000,000 0.3 Typical speed of the return stroke of lightning (cf. stepped leader above). 124,000,000 447,000,000 277,000,000 0.4 Speed of light in a diamond (Refractive index 2.417). 200,000,000 720,000,000 440,000,000 0.7 Speed of a signal in an optical fiber. 299,792,456 1,079,252,840 670,615,282 1 − 9 Speed of the 7 TeV protons in the Large Hadron Collider at full power. 299,792,457.996 1,079,252,848.786 670,616,629.38 1 − 1 Maximal speed of an electron in LEP (104.5 GeV). 299,792,458 − 1.5×10−15 1,079,252,848.8 − 5.4×10−15 670,616,629.4 1 − 4.9×10−24 Speed of the Oh-My-God particle ultra-high-energy cosmic ray. 299,792,458 1,079,252,848.8 670,616,629.4 1 Speed of light or other electromagnetic radiation in a vacuum or massless particles. >299,792,458 >1,079,252,848.8 >670,616,629.4 >1 Expansion rate of the universe between objects farther apart than the Hubble radius ==See also== *Typical projectile speeds - also showing the corresponding kinetic energy per unit mass *Neutron temperature ==References== Category:Units of velocity Category:Physical quantities Speed The newton-second (also newton second; symbol: N⋅s or N s) is the unit of impulse in the International System of Units (SI). In the constant deceleration approach we get: ::v\left(z\right)=\sqrt{v_{i}^{2}-2az} ::B\left(z\right)=\frac{\hbar k}{\mu'}v+\frac{\hbar \delta}{\mu'}=\frac{\hbar kv_{i}}{\mu'}\sqrt{1-\frac{2a}{v_{i}^{2}}z}+\frac{\hbar \delta}{\mu'} where v_{i} is the maximum velocity class that will be slowed; all the atoms in the velocity distribution that have velocities v will be slowed, and those with velocities v>v_{i} will not be slowed at all. Mass (kg) Speed (m/s) Momentum (N⋅s) Explanation 0.42 2.4 1 A football (FIFA specified weight for outdoor size 5) kicked to a speed of . 0.42 38 16 The momentum of the famous football kick of the Brazilian player Roberto Carlos in the match against France in 1997. Kinetic diameter is a measure applied to atoms and molecules that expresses the likelihood that a molecule in a gas will collide with another molecule. Here, the Maxwell–Boltzmann distribution of energies must be considered, which leads to the modified expression,Freude, p. 4 :d^2 = {1 \over \sqrt 2 \pi l n} ==List of diameters== The following table lists the kinetic diameters of some common molecules; Molecule Molecule Molecular mass Kinetic diameter (pm) ref Name Formula Molecular mass Kinetic diameter (pm) ref Hydrogen H2 2 289 Helium He 4 260 Matteucci et al., p. 6 Methane CH4 16 380 Ammonia NH3 17 260 Breck Water H2O 18 265 Neon Ne 20 275 Acetylene C2H2 26 330 Nitrogen N2 28 364 Carbon monoxide CO 28 376 Ethylene C2H4 28 390 Nitric oxide NO 30 317 Oxygen O2 32 346 Hydrogen sulfide H2S 34 360 Hydrogen chloride HCl 36 320 Argon Ar 40 340 Propylene C3H6 42 450 Carbon dioxide CO2 44 330 Nitrous oxide N2O 44 330 Propane C3H8 44 430 Sulfur dioxide SO2 64 360 Chlorine Cl2 70 320 Benzene C6H6 78 585 Li & Talu, p. 373 Hydrogen bromide HBr 81 350 Krypton Kr 84 360 Xenon Xe 131 396 Sulfur hexafluoride SF6 146 550 Carbon tetrachloride CCl4 154 590 Bromine Br2 160 350 ==Dissimilar particles== Collisions between two dissimilar particles occur when a beam of fast particles is fired into a gas consisting of another type of particle, or two dissimilar molecules randomly collide in a gas mixture. For particles of different size, more elaborate expressions can be derived for estimating u. ==References== Category:Chemical kinetics thumb|A Zeeman slower before its incorporation into a larger cold atom experiment. The average acceleration (due to many photon absorption events over time) of an atom with mass, M, a cycling transition with frequency, \omega=ck+\delta, and linewidth, \gamma, that is in the presence of a laser beam that has wavenumber, k, and intensity I=s_{0}I_{s} (where I_s=\hbar c \gamma k^{3}/12\pi is the saturation intensity of the laser) is :: \vec{a}=\frac{\hbar\vec{k}\gamma}{2M}\frac{s_{0}}{1+s_{0}+\left(2\delta'/\gamma\right)^2} In the rest frame of the atoms with velocity, v, in the atomic beam, the frequency of the laser beam is shifted by k_{L}v. The SSERVI - Impact Dust Accelerator Facility at the University of Colorado, 47th Lunar and Planetary Science Conference (2016), accessed May 30, 2017 140,000 540,000 313,170 0.00047 Approaching velocity of Messier 98 to our galaxy. 192,000 690,000 430,000 0.00064 Predicted top speed of the Parker Solar Probe at its closest perihelion in 2024. 200,000 700,000 450,000 0.00070 Orbital speed of the Solar System in the Milky Way galaxy. 308,571 1,080,000 694,288 0.001 Approaching velocity of Andromeda Galaxy to our galaxy. 440,000 1,600,000 980,000 0.0015 Typical speed of the stepped leader of lightning (cf. return stroke below). 445,000 1,600,000 995,000 0.0015 Max velocity of the remaining shell (mass about 0.1 mg) of an inertial confinement fusion capsule driven by the National Ignition Facility for the 'Bigfoot' capsule campaign. Thus it aims at a final velocity of about 10 m/s (depending on the atom used), starting with a beam of atoms with a velocity of a few hundred meters per second. For a fast moving particle (that is, one moving much faster than the particles it is moving through) the kinetic diameter is given by,Ismail et al., p. 14 :d^2 = {1 \over \pi l n} :where, :d is the kinetic diameter, :r is the kinetic radius, r = d/2, :l is the mean free path, and :n is the number density of particles However, a more usual situation is that the colliding particle being considered is indistinguishable from the population of particles in general. It is dimensionally equivalent to the momentum unit kilogram-metre per second (kg⋅m/s). One newton-second corresponds to a one-newton force applied for one second. :\vec F \cdot t = \Delta m \vec v It can be used to identify the resultant velocity of a mass if a force accelerates the mass for a specific time interval. ==Definition== Momentum is given by the formula: :\mathbf{p} = m \mathbf{v}, * \mathbf{p} is the momentum in newton-seconds (N⋅s) or "kilogram-metres per second" (kg⋅m/s) * m is the mass in kilograms (kg) * \mathbf{v} is the velocity in metres per second (m/s) ==Examples== This table gives the magnitudes of some momenta for various masses and speeds. The kinetic diameter is not the same as atomic diameter defined in terms of the size of the atom's electron shell, which is generally a lot smaller, depending on the exact definition used. Abbreviations include in/s, in/sec, ips, and less frequently in s−1. ==Conversions== 1 inch per second is equivalent to: : = 0.0254 metres per second (exactly) : = or 0.083 feet per second (exactly) : = or 0.05681 miles per hour (exactly) : = 0.09144 km·h−1 (exactly) 1 metre per second ≈ 39.370079 inches per second (approximately) 1 foot per second = 12 inches per second (exactly) 1 mile per hour = 17.6 inches per second (exactly) 1 kilometre per hour ≈ 10.936133 inches per second (approximately) ==Uses== In magnetic tape sound recording, magnetic tape speed is often quoted in inches per second (abbreviated "ips").

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E

Calculate the pressure exerted by Ar for a molar volume of $1.31 \mathrm{~L} \mathrm{~mol}^{-1}$ at $426 \mathrm{~K}$ using the van der Waals equation of state. The van der Waals parameters $a$ and $b$ for Ar are 1.355 bar dm${ }^6 \mathrm{~mol}^{-2}$ and $0.0320 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, respectively. Is the attractive or repulsive portion of the potential dominant under these conditions?

The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. The first form of the Van der Waals equation of state given above can be recast in the following reduced form: :\left(p_R + \frac{3}{v_R^2}\right)(3v_R - 1) = (8 T_R) This equation is invariant for all fluids; that is, the same reduced form equation of state applies, no matter what a and b may be for the particular fluid. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). In Van der Waals' original derivation, given below, b' is four times the proper volume of the particle. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. The polarizability is related the van der Waals volume by the relation V_{\rm w} = {1\over{4\pi\varepsilon_0}}\alpha , so the van der Waals volume of helium V = = 0.2073 Å by this method, corresponding to r = 0.37 Å. Van der Waals forces are responsible for certain cases of pressure broadening (van der Waals broadening) of spectral lines and the formation of van der Waals molecules. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). For example, the pairwise attractive van der Waals interaction energy between H atoms in different H2 molecules equals 0.06 kJ/mol (0.6 meV) and the pairwise attractive interaction energy between O atoms in different O2 molecules equals 0.44 kJ/mol (4.6 meV). Accordingly, van der Waals forces can range from weak to strong interactions, and support integral structural loads when multitudes of such interactions are present. The main characteristics of van der Waals forces are: * They are weaker than normal covalent and ionic bonds. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The molar van der Waals volume should not be confused with the molar volume of the substance. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. * Van der Waals forces are additive and cannot be saturated. The excluded volume b is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules.

26.9

144

4.68

14

840

A

For water, $\Delta H_{\text {vaporization }}$ is $40.656 \mathrm{~kJ} \mathrm{~mol}^{-1}$, and the normal boiling point is $373.12 \mathrm{~K}$. Calculate the boiling point for water on the top of Mt. Everest (elevation $8848 \mathrm{~m}$ ), where the barometric pressure is 253 Torr.

For comparison, on top of Mount Everest, at elevation, the pressure is about and the boiling point of water is . At higher elevations, where the atmospheric pressure is much lower, the boiling point is also lower. For every increase in elevation, water's boiling point is lowered by approximately 0.5 °C. Because of this, water boils at under standard pressure at sea level, but at at altitude. At in elevation, water boils at just . Charles Darwin commented on this phenomenon in The Voyage of the Beagle:Journal and remarks, Chapter XV, March 21, 1835 by Charles Darwin. ==Boiling point of pure water at elevated altitudes== Based on standard sea- level atmospheric pressure (courtesy, NOAA): Altitude, ft (m) Boiling point of water, °F (°C) 0 (0 m) 212°F (100°C) 500 (150 m) 211.1°F (99.5°C) 1,000 (305 m) 210.2°F (99°C) 2,000 (610 m) 208.4°F (98°C) 5,000 (1,524 m) 203°F (95°C) 6,000 (1,829 m) 201.1°F (94°C) 8,000 (2,438 m) 197.4°F (91.9°C) 10,000 (3,048 m) 193.6°F (89.8°C) 12,000 (3,658 m) 189.8°F (87.6°C) 14,000 (4,267 m) 185.9°F (85.5°C) 15,000 (4,572 m) 184.1°F (84.5°C) Source: NASA. ==References== ==External links== *Is it true that you can't make a decent cup of tea up a mountain? physics.org, accessed 2012-11-02 Category:Cooking techniques The normal boiling point (also called the atmospheric boiling point or the atmospheric pressure boiling point) of a liquid is the special case in which the vapor pressure of the liquid equals the defined atmospheric pressure at sea level, one atmosphere.General Chemistry Glossary Purdue University website page At that temperature, the vapor pressure of the liquid becomes sufficient to overcome atmospheric pressure and allow bubbles of vapor to form inside the bulk of the liquid. If the heat of vaporization and the vapor pressure of a liquid at a certain temperature are known, the boiling point can be calculated by using the Clausius–Clapeyron equation, thus: :T_\text{B} = \left(\frac{1}{T_0} - \frac{R\,\ln \frac{P}{P_0}}{\Delta H_\text{vap}}\right)^{-1} where: :T_\text{B} is the boiling point at the pressure of interest, :R is the ideal gas constant, :P is the vapor pressure of the liquid, :P_0 is some pressure where the corresponding T_0 is known (usually data available at 1 atm or 100 kPa), :\Delta H_\text{vap} is the heat of vaporization of the liquid, :T_0 is the boiling temperature, :\ln is the natural logarithm. At elevated altitudes, any cooking that involves boiling or steaming generally requires compensation for lower temperatures because the boiling point of water is lower at higher altitudes due to the decreased atmospheric pressure. Mount Everest is the world's highest mountain, with a peak at 8,849 metres (29,031.7 ft) above sea level. The boiling point corresponds to the temperature at which the vapor pressure of the liquid equals the surrounding environmental pressure. There are two conventions regarding the standard boiling point of water: The normal boiling point is at a pressure of 1 atm (i.e., 101.325 kPa). By comparison, reasonable base elevations for Everest range from on the south side to on the Tibetan Plateau, yielding a height above base in the range of .Mount Everest (1:50,000 scale map), prepared under the direction of Bradford Washburn for the Boston Museum of Science, the Swiss Foundation for Alpine Research, and the National Geographic Society, 1991, . The air pressure at the summit is generally about one-third what it is at sea level. The primary peak of Mount Everest is elevation above sea level. == Overview == The peak is a dome-shaped peak of snow and ice, and is connected to the summit of Mount Everest by the Cornice Traverse and Hillary Step, approximately from the higher peak. It also has the lowest normal boiling point (−24.2 °C), which is where the vapor pressure curve of methyl chloride (the blue line) intersects the horizontal pressure line of one atmosphere (atm) of absolute vapor pressure. In the expedition, the summit's altitude was measured as 8848.13 metres. * 1975 British Mount Everest Southwest Face expedition - On September 24, a British expedition led by Chris Bonington achieved the first ascent of the Southwest Face. The atmospheric pressure at the top of Everest is about a third of sea level pressure or , resulting in the availability of only about a third as much oxygen to breathe. The standard boiling point has been defined by IUPAC since 1982 as the temperature at which boiling occurs under a pressure of one bar. The boiling point of a liquid varies depending upon the surrounding environmental pressure. Boiling points may be published with respect to the NIST, USA standard pressure of 101.325 kPa (or 1 atm), or the IUPAC standard pressure of 100.000 kPa. Towards the end of the season, due to a stalled high-pressure system, conditions on Everest were better than usual, being warmer, drier, and less windy, facilitating a higher-than-usual summitting success rate of 70%.

7.0

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273.0

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E

An ideal solution is formed by mixing liquids $\mathrm{A}$ and $B$ at $298 \mathrm{~K}$. The vapor pressure of pure A is 151 Torr and that of pure B is 84.3 Torr. If the mole fraction of $\mathrm{A}$ in the vapor is 0.610 , what is the mole fraction of $\mathrm{A}$ in the solution?

The boiling temperature of water for atmospheric pressures can be approximated by the Antoine equation: :\log_{10}\left(\frac{P}{1\text{ Torr}}\right) = 8.07131 - \frac{1730.63\ {}^\circ\text{C}}{233.426\ {}^\circ\text{C} + T_b} or transformed into this temperature-explicit form: :T_b = \frac{1730.63\ {}^\circ\text{C}}{8.07131 - \log_{10}\left(\frac{P}{1\text{ Torr}}\right)} - 233.426\ {}^\circ\text{C} where the temperature T_b is the boiling point in degrees Celsius and the pressure P is in torr. ==Dühring's rule== Dühring's rule states that a linear relationship exists between the temperatures at which two solutions exert the same vapor pressure. ==Examples== The following table is a list of a variety of substances ordered by increasing vapor pressure (in absolute units). The mole fraction is one way of expressing the composition of a mixture with a dimensionless quantity; mass fraction (percentage by weight, wt%) and volume fraction (percentage by volume, vol%) are others. ==Properties== Mole fraction is used very frequently in the construction of phase diagrams. When Raoult's law and Dalton's law hold for the mixture, the K factor is defined as the ratio of the vapor pressure to the total pressure of the system: :K_i = \frac{P'_i}{P} Given either of x_i or y_i and either the temperature or pressure of a two-component system, calculations can be performed to determine the unknown information. ==References== ==See also== * Phase diagram * Azeotrope * Dew point Category:Temperature Category:Phase transitions Category:Gases The mole fraction is also called the amount fraction. It states that the activity (pressure or fugacity) of a single-phase mixture is equal to the mole-fraction-weighted sum of the components' vapor pressures: : P_{\rm tot} =\sum_i P y_i = \sum_i P_i^{\rm sat} x_i \, where P_{\rm tot} is the mixture's vapor pressure, x_i is the mole fraction of component i in the liquid phase and y_i is the mole fraction of component i in the vapor phase respectively. At the normal boiling point of a liquid, the vapor pressure is equal to the standard atmospheric pressure defined as 1 atmosphere, 760Torr, 101.325kPa, or 14.69595psi. If the heat of vaporization and the vapor pressure of a liquid at a certain temperature are known, the boiling point can be calculated by using the Clausius–Clapeyron equation, thus: :T_\text{B} = \left(\frac{1}{T_0} - \frac{R\,\ln \frac{P}{P_0}}{\Delta H_\text{vap}}\right)^{-1} where: :T_\text{B} is the boiling point at the pressure of interest, :R is the ideal gas constant, :P is the vapor pressure of the liquid, :P_0 is some pressure where the corresponding T_0 is known (usually data available at 1 atm or 100 kPa), :\Delta H_\text{vap} is the heat of vaporization of the liquid, :T_0 is the boiling temperature, :\ln is the natural logarithm. This is important for volatile inhalational anesthetics, most of which are liquids at body temperature, but with a relatively high vapor pressure. ==Estimating vapor pressures with Antoine equation== The Antoine equationWhat is the Antoine Equation? Whereas mole fraction is a ratio of moles to moles, molar concentration is a quotient of moles to volume. In chemistry, the mole fraction or molar fraction (xi or ) is defined as unit of the amount of a constituent (expressed in moles), ni, divided by the total amount of all constituents in a mixture (also expressed in moles), ntot. The vapor pressure of a liquid at its boiling point equals the pressure of its surrounding environment. ==Liquid mixtures: Raoult's law== Raoult's law gives an approximation to the vapor pressure of mixtures of liquids. The basic form of the equation is: :\log P = A-\frac{B}{C+T} and it can be transformed into this temperature-explicit form: :T = \frac{B}{A-\log P} - C where: * P is the absolute vapor pressure of a substance * T is the temperature of the substance * A, B and C are substance-specific coefficients (i.e., constants or parameters) * \log is typically either \log_{10} or \log_e A simpler form of the equation with only two coefficients is sometimes used: :\log P = A- \frac{B}{T} which can be transformed to: :T = \frac{B}{A-\log P} Sublimations and vaporizations of the same substance have separate sets of Antoine coefficients, as do components in mixtures. When the volatilities of both key components are equal, \alpha = 1 and separation of the two by distillation would be impossible under the given conditions because the compositions of the liquid and the vapor phase are the same (azeotrope). This doesn't affect the equation form. ==Sources for Antoine equation parameters== * NIST Chemistry WebBook * Dortmund Data Bank * Directory of reference books and data banks containing Antoine constants * Several reference books and publications, e. g. ** Lange's Handbook of Chemistry, McGraw-Hill Professional ** Wichterle I., Linek J., "Antoine Vapor Pressure Constants of Pure Compounds" ** Yaws C. L., Yang H.-C., "To Estimate Vapor Pressure Easily. The second solution is switching to another vapor pressure equation with more than three parameters. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds", Hydrocarbon Processing, 68(10), Pages 65–68, 1989 ==See also== *Vapour pressure of water *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation *Raoult's law *Thermodynamic activity ==References== ==External links== * Gallica, scanned original paper * NIST Chemistry Web Book * Calculation of vapor pressures with the Antoine equation Category:Equations Category:Thermodynamic equations The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. The Antoine equation is a class of semi-empirical correlations describing the relation between vapor pressure and temperature for pure substances. Experimental measurement of vapor pressure is a simple procedure for common pressures between 1 and 200 kPa. Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. Actually, as stated by Dalton's law (known since 1802), the partial pressure of water vapor or any substance does not depend on air at all, and the relevant temperature is that of the liquid. thumb|300px|Boiling waterThe boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor.

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The mean solar flux at Earth's surface is $\sim 2.00 \mathrm{~J}$ $\mathrm{cm}^{-2} \mathrm{~min}^{-1}$. In a nonfocusing solar collector, the temperature reaches a value of $79.5^{\circ} \mathrm{C}$. A heat engine is operated using the collector as the hot reservoir and a cold reservoir at $298 \mathrm{~K}$. Calculate the area of the collector needed to produce 1000. W. Assume that the engine operates at the maximum Carnot efficiency.

Solar energy – Solar thermal collectors – Test methods International Organization for Standardization, Geneva, Switzerland states that the efficiency of solar thermal collectors should be measured in terms of gross area and this might favour flat plates in respect to evacuated tube collectors in direct comparisons. thumb|An array of evacuated flat plate collectors next to compact solar concentrators thumb|A comparison of the energy output (kW.h/day) of a flat plate collector (blue lines; Thermodynamics S42-P; absorber 2.8 m2) and an evacuated tube collector (green lines; SunMaxx 20EVT; absorber 3.1 m2. In non-concentrating collectors, the aperture area (i.e., the area that receives the solar radiation) is roughly the same as the absorber area (i.e., the area absorbing the radiation). A solar thermal collector collects heat by absorbing sunlight. Transpired solar collectors are usually wall- mounted to capture the lower sun angle in the winter heating months as well as sun reflection off the snow and achieve their optimum performance and return on investment when operating at flow rates of between 4 and 8 CFM per square foot (72 to 144 m3/h.m2) of collector area. The extensive monitoring by Natural Resources Canada and NREL has shown that transpired solar collector systems reduce between 10-50% of the conventional heating load and that RETScreen is an accurate predictor of system performance. Its value is about 86%, which is the Chambadal-Novikov efficiency, an approximation related to the Carnot limit, based on the temperature of the photons emitted by the Sun's surface. == Effect of band gap energy == Solar cells operate as quantum energy conversion devices, and are therefore subject to the thermodynamic efficiency limit. This value depends on the size of the storage unit (hot water tank or storage battery), the size of the harvesting surface (sun collection surface or surface area of photovoltaic modules), and on the amount of energy required. The collector absorbs the incoming solar radiation, converting it into thermal energy. The solar energy flux (irradiance) incident on the Earth's surface has a variable and relatively low surface density, usually not exceeding 1100 W/m² without concentration systems. For a solar cell powered by the Sun's unconcentrated black-body radiation, the theoretical maximum efficiency is 43% whereas for a solar cell powered by the Sun's full concentrated radiation, the efficiency limit is up to 85%. Solar collector may refer to: * Solar thermal collector, a solar collector that collects heat by absorbing sunlight * Solar Collector (sculpture), a 2008 interactive light art installation in Cambridge, Ontario, Canada ==See also== *Concentrating solar power *Renewable heat *Solar air heating *Solar water heating *Solar panel They offer the highest energy conversion efficiency of any non-concentrating solar thermal collector, but require sophisticated technology for manufacturing. The exterior surface of a transpired solar collector consists of thousands of tiny micro-perforations that allow the boundary layer of heat to be captured and uniformly drawn into an air cavity behind the exterior panels. The internal combustion engine efficiency is determined by its two temperature reservoirs, the temperatures of ambient air and its upper limit operating temperature. Engine efficiency of thermal engines is the relationship between the total energy contained in the fuel, and the amount of energy used to perform useful work. In locations with average available solar energy, flat plate collectors are sized approximately 1.2 to 2.4 square decimeter per liter of one day's hot water use. === Applications === The main use of this technology is in residential buildings where the demand for hot water has a large impact on energy bills. In 1954 the solar constant was evaluated as 2.00 cal/min/cm2 ± 2%. The term "solar collector" commonly refers to a device for solar hot water heating, but may refer to large power generating installations such as solar parabolic troughs and solar towers or non water heating devices such as solar cooker, solar air heaters. With the increasing drive to install renewable energy systems on buildings, transpired solar collectors are now used across the entire building stock because of high energy production (up to 750 peak thermal Watts/square metre), high solar conversion (up to 90%) and lower capital costs when compared against solar photovoltaic and solar water heating. thumb|Solar air heating is a solar thermal technology in which the energy from the sun, solar insolation, is captured by an absorbing medium and used to heat air. Thermodynamic efficiency limit is the absolute maximum theoretically possible conversion efficiency of sunlight to electricity. Solar thermal collectors are either non-concentrating or concentrating. The solar coverage rate is the percentage of an amount of energy that is provided by the sun.

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A hiker caught in a thunderstorm loses heat when her clothing becomes wet. She is packing emergency rations that if completely metabolized will release $35 \mathrm{~kJ}$ of heat per gram of rations consumed. How much rations must the hiker consume to avoid a reduction in body temperature of $2.5 \mathrm{~K}$ as a result of heat loss? Assume the heat capacity of the body equals that of water and that the hiker weighs $51 \mathrm{~kg}$.

The dietary energy supply is the food available for human consumption, usually expressed in kilocalories or kilojoules per person per day. kJ/kg may refer to: * kilojoules per kilogram * The SI derived units of specific energy * Specific Internal energy * Specific kinetic energy * Heat of fusion * Heat of combustion International Journal of Heat and Mass Transfer is a peer-reviewed scientific journal in the field of heat transfer and mass transfer, published by Elsevier. Heat and Mass Transfer is a peer-reviewed scientific journal published by Springer. Heatwork is the combined effect of temperature and time. Dietary energy supply is correlated with the rate of obesity. ==Regions== Daily dietary energy supply per capita: Region 1964-1966 1974-1976 1984-1986 1997-1999 World Sub-Saharan Africa ==Countries== Country 1979-1981 1989-1991 2001-2003 Canada China Eswatini United Kingdom United States ==See also== *Diet and obesity thumb|300x300px|Average dietary energy supply by region ==References== ==External links== * Category:Obesity Category:Diets *Temperature equivalents table & description of Orton pyrometric cones. As of 1995 the title Wärme- und Stoffübertragung was changed to Heat and Mass Transfer. == Indexing == Among others, the journal is indexed in Google Scholar, INIS Atomindex, Journal Citation Reports/Science Edition, OCLC, PASCAL, Science Citation Index, Science Citation Index Expanded (SciSearch) and Scopus. == External links == *Heat and Mass Transfer Category:Energy and fuel journals Category:Engineering journals Category:English-language journals Category:Monthly journals Category:Springer Science+Business Media academic journals Heatwork is taught in material science courses, but is not a precise measurement or a valid scientific concept. == External links == *Temperature equivalents table & description of Bullers Rings. *Temperature Equivalents, °F & °C for Bullers Ring. It gives an overestimate of the total amount of food consumed as it reflects both food consumed and food wasted. It serves the circulation of new developments in the field of basic research of heat and mass transfer phenomena, as well as related material properties and their measurements. *Temperature equivalents table & description of Nimra Cerglass pyrometric cones. Within tolerances, firing can be undertaken at lower temperatures for a longer period to achieve comparable results. *Temperature equivalents table of Seger pyrometric cones. When the amount of heatwork of two firings is the same, the pieces may look identical, but there may be differences not visible, such as mechanical strength and microstructure. It is important to several industries: *Ceramics *Glass and metal annealing *Metal heat treating Pyrometric devices can be used to gauge heat work as they deform or contract due to heatwork to produce temperature equivalents. Category:Glass physics Category:Pottery Category:Metallurgy Category:Ceramic engineering The editor-in-chief is T. S. Zhao (Hong Kong University of Science and Technology). ==Abstracting and indexing== The journal is abstracted and indexed in: According to the Journal Citation Reports, the journal has a 2020 impact factor of 5.584. ==References== ==External links== * Category:Physics journals Category:English-language journals Category:Engineering journals Category:Elsevier academic journals Category:Academic journals established in 1960 Category:Monthly journals Thereby applications to engineering problems are promoted. The journal publishes original research reports. It varies markedly between different regions and countries of the world.

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Calculate the degree of dissociation of $\mathrm{N}_2 \mathrm{O}_4$ in the reaction $\mathrm{N}_2 \mathrm{O}_4(g) \rightleftharpoons 2 \mathrm{NO}_2(g)$ at 300 . $\mathrm{K}$ and a total pressure of 1.50 bar. Do you expect the degree of dissociation to increase or decrease as the temperature is increased to 550 . K? Assume that $\Delta H_R^{\circ}$ is independent of temperature.

Figure 8 shows the corresponding enthalpy drop for the reaction = 0 case. none|310px|thumb|Figure 8. Stage enthalpy diagram for degree of reaction = 1⁄2 in a turbine and pump. left|thumb|Figure 6. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. Hence from Tds = dh - \frac{dp}{\rho}, 400px|alt=enthalpy diagram|thumb|Figure 1. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. An algorithm for computing a 4/3-approximation of the dissociation number in bipartite graphs was published in 2022. The dissociation constant is the inverse of the association constant. From the relation for degree of reaction, || α2 > β3. right|380px|thumb|Figure 7. The dissociation constant for a particular ligand-protein interaction can change significantly with solution conditions (e.g., temperature, pH and salt concentration). Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity. In chemistry, biochemistry, and pharmacology, a dissociation constant (K_D) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex falls apart into its component molecules, or when a salt splits up into its component ions. thumb|Examples for the definition of the dissociation number In the mathematical discipline of graph theory, a subset of vertices in a graph G is called dissociation if it induces a subgraph with maximum degree 1. [B^{-3}] \over [H B^{-2}]} & \mathrm{p}K_3 &= -\log K_3 \end{align} ==Dissociation constant of water== The dissociation constant of water is denoted Kw: :K_\mathrm{w} = [\ce{H}^+] [\ce{OH}^-] The concentration of water, [H2O], is omitted by convention, which means that the value of Kw differs from the value of Keq that would be computed using that concentration. The degree of reaction now depends only on ϕ and \tan{\beta_m} which again depend on geometrical parameters β3 and β2 i.e. the vane angles of stator outlet and rotor outlet. (The symbol K_a, used for the acid dissociation constant, can lead to confusion with the association constant and it may be necessary to see the reaction or the equilibrium expression to know which is meant.) The value of Kw varies with temperature, as shown in the table below. From the relation for degree of reaction,|| α2 < β3 which is also shown in corresponding Figure 7. === Reaction = zero === This is special case used for impulse turbine which suggest that entire pressure drop in the turbine is obtained in the stator. The problem of computing diss(G) (dissociation number problem) was firstly studied by Yannakakis. : Water temperature Kw pKw 0 °C 0.112 14.95 25 °C 1.023 13.99 50 °C 5.495 13.26 75 °C 19.95 12.70 100 °C 56.23 12.25 == See also == * Acid * Equilibrium constant * Ki Database * Competitive inhibition * pH * Scatchard plot * Ligand binding * Avidity ==References== Category:Equilibrium chemistry Category:Enzyme kinetics A molecule can have several acid dissociation constants. Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. The number of vertices in a maximum cardinality dissociation set in G is called the dissociation number of G, denoted by diss(G).

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Calculate $\Delta G$ for the isothermal expansion of $2.25 \mathrm{~mol}$ of an ideal gas at $325 \mathrm{~K}$ from an initial pressure of 12.0 bar to a final pressure of 2.5 bar.

For an ideal gas, the change in entropyTipler, P., and Mosca, G. Physics for Scientists and Engineers (with modern physics), 6th edition, 2008. pages 602 and 647. is the same as for isothermal expansion where all heat is converted to work: \Delta S = \int_\text{i}^\text{f} dS = \int_{V_\text{i}}^{V_\text{f}} \frac{P\,dV}{T} = \int_{V_\text{i}}^{V_\text{f}} \frac{n R\,dV}{V} = n R \ln \frac{V_\text{f}}{V_\text{i}} = N k_\text{B} \ln \frac{V_\text{f}}{V_\text{i}}. During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. The best method (i.e. the method involving the least work) is that of a reversible isothermal compression, which would take work given by W = -\int_{2V_0}^{V_0} P\,\mathrm{d}V = - \int_{2V_0}^{V_0} \frac{nRT}{V} \mathrm{d}V = nRT\ln 2 = T \Delta S_\text{gas}. This type of expansion is named after James Prescott Joule who used this expansion, in 1845, in his study for the mechanical equivalent of heat, but this expansion was known long before Joule e.g. by John Leslie, in the beginning of the 19th century, and studied by Joseph-Louis Gay-Lussac in 1807 with similar results as obtained by Joule.D.S.L. Cardwell, From Watt to Clausius, Heinemann, London (1957)M.J. Klein, Principles of the theory of heat, D. Reidel Pub.Cy., Dordrecht (1986) The Joule expansion should not be confused with the Joule–Thomson expansion or throttling process which refers to the steady flow of a gas from a region of higher pressure to one of lower pressure via a valve or porous plug. ==Description== The process begins with gas under some pressure, P_{\mathrm{i}}, at temperature T_{\mathrm{i}}, confined to one half of a thermally isolated container (see the top part of the drawing at the beginning of this article). We might ask what the work would be if, once the Joule expansion has occurred, the gas is put back into the left-hand side by compressing it. Heating the gas up to the initial temperature increases the entropy by the amount \Delta S = n \int_{T}^{T_i} C_\mathrm{V} \frac{\mathrm{d}T'}{T'} = nR \ln 2. "A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Thus, at low temperatures the Joule expansion process provides information on intermolecular forces. ===Ideal gases=== If the gas is ideal, both the initial (T_{\mathrm{i}}, P_{\mathrm{i}}, V_{\mathrm{i}}) and final (T_{\mathrm{f}}, P_{\mathrm{f}}, V_{\mathrm{f}}) conditions follow the Ideal Gas Law, so that initially P_{\mathrm{i}} V_{\mathrm{i}} = n R T_{\mathrm{i}} and then, after the tap is opened, P_{\mathrm{f}} V_{\mathrm{f}} = n R T_{\mathrm{f}}. For the above defined path we have that and thus , and hence the increase in entropy for the Joule expansion is \Delta S=\int_i^f\mathrm{d}S=\int_{V_0}^{2V_0} \frac{P\,\mathrm{d}V}{T}=\int_{V_0}^{2V_0} \frac{n R\,\mathrm{d}V}{V}=n R\ln 2. For a monatomic ideal gas , with the molar heat capacity at constant volume. thumb|240px|The Joule expansion, in which a volume is expanded to a volume in a thermally isolated chamber. thumb|180px|A free expansion of a gas can be achieved by moving the piston out faster than the fastest molecules in the gas. The Joule expansion, treated as a thought experiment involving ideal gases, is a useful exercise in classical thermodynamics. Here n is the number of moles of gas and R is the molar ideal gas constant. For pressures above 1 bar, that means, above the normal boiling point, the typical errors are below 2%. For other gases this "Joule inversion temperature" appears to be extremely high. ==Entropy production== Entropy is a function of state, and therefore the entropy change can be computed directly from the knowledge of the final and initial equilibrium states. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. Pressure gain combustion is a combustion process whereby gas expansion by heat release is constrained, causing a rise in stagnation pressure and allowing work extraction by expansion to the initial pressure. thumb|400px|Diagram showing pressure difference induced by a temperature difference. After thermal equilibrium is reached, we then let the gas undergo another free expansion by and wait until thermal equilibrium is reached. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup The Joule expansion (also called free expansion) is an irreversible process in thermodynamics in which a volume of gas is kept in one side of a thermally isolated container (via a small partition), with the other side of the container being evacuated. The fact that the temperature does not change makes it easy to compute the change in entropy of the universe for this process. ===Real gases=== Unlike ideal gases, the temperature of a real gas will change during a Joule expansion.

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Determine the total collisional frequency for $\mathrm{CO}_2$ at $1 \mathrm{~atm}$ and $298 \mathrm{~K}$.

Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the "effective area" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between entities of species A and species B is:chem.libretexts.org: Collision Frequency : Z = N_\text{A} N_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}}, which has units of [volume][time]−1. It can be more than 1 if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity. \text{Coefficient of restitution } (e) = \frac{\left | \text{Relative velocity after collision} \right |}{\left | \text{Relative velocity before collision}\right |} The mathematics were developed by Sir Isaac Newton in 1687. The coefficient of restitution (COR, also denoted by e), is the ratio of the final to initial relative speed between two objects after they collide. Setting the above estimate for \Delta m_e v_\perp equal to mv, we find the lower cut-off to the impact parameter to be about :b_0 = \frac{Ze^2}{4\pi\epsilon_0} \, \frac{1}{m_e v^2} We can also use \pi b_0^2 as an estimate of the cross section for large-angle collisions. If a given object collides with two different objects, each collision would have its own COR. Escande DF, Elskens Y, Doveil F (2015) Uniform derivation of Coulomb collisional transport thanks to Debye shielding. For this reason, determining the COR of a material when there is not identical material for collision is best done by using a much harder material. Collision-induced emission and absorption by simultaneous collisions of three or more particles generally do involve pairwise-additive dipole components, as well as important irreducible dipole contributions and their spectra. == Historical sketch == Collision-induced absorption was first reported in compressed oxygen gas in 1949 by Harry Welsch and associates at frequencies of the fundamental band of the O2 molecule. In this case, the change in Ek is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus: C_R = \sqrt{\frac{E_\text{p, (at bounce height)}}{E_\text{p, (at drop height)}}} = \sqrt{\frac{mgh}{mgH}} = \sqrt{\frac{h}{H}} == Speeds after impact == The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between. v_\text{a} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b}-u_\text{a})}{m_\text{a}+m_\text{b}} and v_\text{b} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{a} C_R(u_\text{a}-u_\text{b})}{m_\text{a}+m_\text{b}} where *v_\text{a} is the final velocity of the first object after impact *v_\text{b} is the final velocity of the second object after impact *u_\text{a} is the initial velocity of the first object before impact *u_\text{b} is the initial velocity of the second object before impact *m_\text{a} is the mass of the first object *m_\text{b} is the mass of the second object === Derivation === The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). Finally, with certain modifications (replacement of Z by Z − 1, and use of the integers 1 and 2 for the ns to give a numerical value of for the difference of their inverse squares), the Rydberg formula provides correct values in the special case of K-alpha lines, since the transition in question is the K-alpha transition of the electron from the 1s orbital to the 2p orbital. Collision-induced spectra appear at the frequencies of the rotovibrational and electronic transition bands of the unperturbed molecules, and also at sums and differences of such transition frequencies: simultaneous transitions in two (or more) interacting molecules are well known to generate optical transitions of molecular complexes. == Virial expansions of spectral intensities == Intensities of spectra of individual atoms or molecules typically vary linearly with the numerical gas density. It is possible that e = \infty for a perfect explosion of a rigid system. === Paired objects === The COR is a property of a pair of objects in a collision, not a single object. thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. (Note that an unperturbed O2 molecule, like all other diatomic homonuclear molecules, is infrared inactive on account of the inversion symmetry and does thus not possess a "dipole allowed" rotovibrational spectrum at any frequency). == Collision-induced spectra == Molecular fly-by collisions take little time, something like 10−13 s. The upper cut-off to the impact parameter should thus be approximately equal to the Debye length: : \lambda_D = \sqrt{\frac{\epsilon_0 kT_e}{n_e e^2}} == Coulomb logarithm == The integral of 1/b thus yields the logarithm of the ratio of the upper and lower cut-offs. Within the same approximations, a more elegant derivation of the collisional transport coefficients was provided, by using the Balescu–Lenard equation (see Sec. 8.4 of Balescu, R. 1997 Statistical Dynamics: Matter Out of Equilibrium. These are the collision-induced spectra of two-body (and quite possibly three-body,...) collisional complexes. The collision-induced spectra have sometimes been separated from the continua of individual atoms and molecules, based on the characteristic density dependences. Also, some recent articles have described superelastic collisions in which it is argued that the COR can take a value greater than one in a special case of oblique collisions.

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The volatile liquids $A$ and $\mathrm{B}$, for which $P_A^*=165$ Torr and $P_B^*=85.1$ Torr are confined to a piston and cylinder assembly. Initially, only the liquid phase is present. As the pressure is reduced, the first vapor is observed at a total pressure of 110 . Torr. Calculate $x_{\mathrm{A}}$

If the heat of vaporization and the vapor pressure of a liquid at a certain temperature are known, the boiling point can be calculated by using the Clausius–Clapeyron equation, thus: :T_\text{B} = \left(\frac{1}{T_0} - \frac{R\,\ln \frac{P}{P_0}}{\Delta H_\text{vap}}\right)^{-1} where: :T_\text{B} is the boiling point at the pressure of interest, :R is the ideal gas constant, :P is the vapor pressure of the liquid, :P_0 is some pressure where the corresponding T_0 is known (usually data available at 1 atm or 100 kPa), :\Delta H_\text{vap} is the heat of vaporization of the liquid, :T_0 is the boiling temperature, :\ln is the natural logarithm. thumb|300px|A representative pressure–volume diagram for a refrigeration cycle Vapour-compression refrigeration or vapor-compression refrigeration system (VCRS), in which the refrigerant undergoes phase changes, is one of the many refrigeration cycles and is the most widely used method for air conditioning of buildings and automobiles. Between point 3 and point 4, the vapor travels through the remainder of the condenser and is condensed into a high temperature, high pressure subcooled liquid. When the volatilities of both key components are equal, \alpha = 1 and separation of the two by distillation would be impossible under the given conditions because the compositions of the liquid and the vapor phase are the same (azeotrope). thumb|right|300px|Otto cycle pressure–volume diagram An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark ignition piston engine. The presence of other volatile components in a mixture affects the vapor pressures and thus boiling points and dew points of all the components in the mixture. Relative volatilities are not used in separation or absorption processes that involve components reacting with each other (for example, the absorption of gaseous carbon dioxide in aqueous solutions of sodium hydroxide). ==Definition== For a liquid mixture of two components (called a binary mixture) at a given temperature and pressure, the relative volatility is defined as :\alpha=\frac {(y_i/x_i)}{(y_j/x_j)} = K_i/K_j where: \alpha = the relative volatility of the more volatile component i to the less volatile component j y_i = the vapor–liquid equilibrium mole fraction of component i in the vapor phase x_i = the vapor–liquid equilibrium mole fraction of component i in the liquid phase y_j = the vapor–liquid equilibrium concentration of component j in the vapor phase x_j = the vapor–liquid equilibrium concentration of component j in the liquid phase (y/x) = Henry's law constant (also called the K value or vapor-liquid distribution ratio) of a component When their liquid concentrations are equal, more volatile components have higher vapor pressures than less volatile components. As the piston is capable of moving along the cylinder, the volume of the air changes with its position in the cylinder. thumb|250px|Diagram of cylinder and piston valve. If the pressure in a system remains constant (isobaric), a vapor at saturation temperature will begin to condense into its liquid phase as thermal energy (heat) is removed. That pressure reduction results in the adiabatic flash evaporation of a part of the liquid refrigerant. thumb|300px|Boiling waterThe boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. Relative volatility is a measure comparing the vapor pressures of the components in a liquid mixture of chemicals. Similarly, a liquid at saturation pressure and temperature will tend to flash into its vapor phase as system pressure is decreased. Similarly, a liquid at saturation temperature and pressure will boil into its vapor phase as additional thermal energy is applied. For a given pressure, different liquids will boil at different temperatures. The system, in this case, is defined to be the fluid (gas) within the cylinder. Furthermore, at any given temperature, the composition of the vapor is different from the composition of the liquid in most such cases. The condensed liquid refrigerant, in the thermodynamic state known as a saturated liquid, is next routed through an expansion valve where it undergoes an abrupt reduction in pressure. The cold refrigerant liquid and vapor mixture is then routed through the coil or tubes in the evaporator. thumb|right|400px|An illustration of fluid simulation using VOF method. The heat of vaporization is the energy required to transform a given quantity (a mol, kg, pound, etc.) of a substance from a liquid into a gas at a given pressure (often atmospheric pressure).

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The osmotic pressure of an unknown substance is measured at $298 \mathrm{~K}$. Determine the molecular weight if the concentration of this substance is $31.2 \mathrm{~kg} \mathrm{~m}^{-3}$ and the osmotic pressure is $5.30 \times 10^4 \mathrm{~Pa}$. The density of the solution is $997 \mathrm{~kg} \mathrm{~m}^{-3}$.

Both sodium and chloride ions affect the osmotic pressure of the solution. The osmotic coefficient based on molality m is defined by: \phi = \frac{\mu_A^* - \mu_A}{RTM_A \sum_i m_i} and on a mole fraction basis by: \phi = -\frac{\mu_A^* - \mu_A}{RT \ln x_A} where \mu_A^* is the chemical potential of the pure solvent and \mu_A is the chemical potential of the solvent in a solution, MA is its molar mass, xA its mole fraction, R the gas constant and T the temperature in Kelvin. The transfer of solvent molecules will continue until equilibrium is attained. ==Theory and measurement== Jacobus van 't Hoff found a quantitative relationship between osmotic pressure and solute concentration, expressed in the following equation: :\Pi = icRT where \Pi is osmotic pressure, i is the dimensionless van 't Hoff index, c is the molar concentration of solute, R is the ideal gas constant, and T is the absolute temperature (usually in kelvins). This value allows the measurement of the osmotic pressure of a solution and the determination of how the solvent will diffuse across a semipermeable membrane (osmosis) separating two solutions of different osmotic concentration. == Unit == The unit of osmotic concentration is the osmole. Osmotic concentration, formerly known as osmolarity, is the measure of solute concentration, defined as the number of osmoles (Osm) of solute per litre (L) of solution (osmol/L or Osm/L). This is a non-SI unit of measurement that defines the number of moles of solute that contribute to the osmotic pressure of a solution. The Pfeffer cell was developed for the measurement of osmotic pressure. == Applications == thumb|upright=1.15|Osmotic pressure on red blood cells Osmotic pressure measurement may be used for the determination of molecular weights. The proportionality to concentration means that osmotic pressure is a colligative property. In order to find \Pi, the osmotic pressure, we consider equilibrium between a solution containing solute and pure water. :\mu_v(x_v,p+\Pi) = \mu_v^0(p). Potential osmotic pressure is the maximum osmotic pressure that could develop in a solution if it were separated from its pure solvent by a semipermeable membrane. For aqueous solutions, the osmotic coefficients can be calculated theoretically by Pitzer equationsI. The osmotic pressure of ocean water is approximately 27 atm. Reverse osmosis desalinates fresh water from ocean salt water. == Derivation of the van 't Hoff formula == Consider the system at the point when it has reached equilibrium. For example, a 3 Osm solution might consist of: 3 moles glucose, or 1.5 moles NaCl, or 1 mole glucose + 1 mole NaCl, or 2 moles glucose + 0.5 mole NaCl, or any other such combination. ==Definition== The osmolarity of a solution, given in osmoles per liter (osmol/L) is calculated from the following expression: \mathrm{osmolarity} = \sum_i \varphi_i \, n_i C_i where * is the osmotic coefficient, which accounts for the degree of non-ideality of the solution. Temperature(°F) Specific weight (lbf/ft3) 32 62.42 40 62.43 50 62.41 60 62.37 70 62.30 80 62.22 90 62.11 100 62.00 110 61.86 120 61.71 130 61.55 140 61.38 150 61.20 160 61.00 170 60.80 180 60.58 190 60.36 200 60.12 212 59.83 ==Specific weight of air== Specific weight of air at standard sea-level atmospheric pressure (Metric units) Temperature(°C) Specific weight (N/m3) −40 14.86 −20 13.86 0 12.68 10 12.24 20 11.82 30 11.43 40 11.06 60 10.4 80 9.81 100 9.28 200 7.33 Specific weight of air at standard sea-level atmospheric pressure (English units) Temperature(°F) Specific Weight (lbf/ft3) −40 −20 0.0903 0 0.08637 10 0.08453 20 0.08277 30 0.08108 40 0.07945 50 0.0779 60 0.0764 70 0.07495 80 0.07357 90 0.07223 100 0.07094 120 0.06849 140 0.0662 160 0.06407 180 0.06206 200 0.06018 250 0.05594 ==References== ==External links== * Submerged weight calculator * Specific weight calculator * http://www.engineeringtoolbox.com/density-specific-weight-gravity-d_290.html * http://www.themeter.net/pesi-spec_e.htm Category:Soil mechanics Category:Fluid mechanics Category:Physical chemistry Category:Physical quantities Category:Density For example, the intracellular fluid and extracellular can be hyperosmotic, but isotonic – if the total concentration of solutes in one compartment is different from that of the other, but one of the ions can cross the membrane (in other words, a penetrating solute), drawing water with it, thus causing no net change in solution volume. ==Plasma osmolarity vs. osmolality== Plasma osmolarity can be calculated from plasma osmolality by the following equation: where: * is the density of the solution in g/ml, which is 1.025 g/ml for blood plasma. * is the (anhydrous) solute concentration in g/ml – not to be confused with the density of dried plasma According to IUPAC, osmolality is the quotient of the negative natural logarithm of the rational activity of water and the molar mass of water, whereas osmolarity is the product of the osmolality and the mass density of water (also known as osmotic concentration). Here, the difference in pressure of the two compartments \Pi \equiv p' - p is defined as the osmotic pressure exerted by the solutes. Also, the molar volume V_m may be written as volume per mole, V_m = V/n_v. In soil mechanics, specific weight may refer to: ===Civil and mechanical engineering=== Specific weight can be used in civil engineering and mechanical engineering to determine the weight of a structure designed to carry certain loads while remaining intact and remaining within limits regarding deformation. ==Specific weight of water== Specific weight of water at standard sea-level atmospheric pressure (Metric units) Temperature(°C) Specific weight (kN/m3) 0 9.805 5 9.807 10 9.804 15 9.798 20 9.789 25 9.777 30 9.765 40 9.731 50 9.690 60 9.642 70 9.589 80 9.530 90 9.467 100 9.399 Specific weight of water at standard sea-level atmospheric pressure (English units) Finnemore, J. E. (2002). The compartment containing the pure solvent has a chemical potential of \mu^0(p), where p is the pressure. The molecular formula C27H34O3 (molar mass: 406.56 g/mol, exact mass: 406.2508 u) may refer to: * Nandrolone phenylpropionate (NPP), or nandrolone phenpropionate * Testosterone phenylacetate The specific weight, also known as the unit weight, is the weight per unit volume of a material. Osmotic pressure is the basis of filtering ("reverse osmosis"), a process commonly used in water purification.

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One mole of Ar initially at 310 . K undergoes an adiabatic expansion against a pressure $P_{\text {external }}=0$ from a volume of $8.5 \mathrm{~L}$ to a volume of $82.0 \mathrm{~L}$. Calculate the final temperature using the ideal gas

We assume the expansion occurs without exchange of heat (adiabatic expansion). The constant volume adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any work, heat transfer or changes in kinetic or potential energy. The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). For an ideal gas, the change in entropyTipler, P., and Mosca, G. Physics for Scientists and Engineers (with modern physics), 6th edition, 2008. pages 602 and 647. is the same as for isothermal expansion where all heat is converted to work: \Delta S = \int_\text{i}^\text{f} dS = \int_{V_\text{i}}^{V_\text{f}} \frac{P\,dV}{T} = \int_{V_\text{i}}^{V_\text{f}} \frac{n R\,dV}{V} = n R \ln \frac{V_\text{f}}{V_\text{i}} = N k_\text{B} \ln \frac{V_\text{f}}{V_\text{i}}. In the case of the constant pressure adiabatic flame temperature, the pressure of the system is held constant, which results in the following equation for the work: : {}_RW_P = \int\limits_R^P {pdV} = p\left( {V_P - V_R } \right) Again there is no heat transfer occurring because the process is defined to be adiabatic: {}_RQ_P = 0 . The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). With our present knowledge of the thermodynamic properties of air Refprop, software package developed by National Institute of Standards and Technology (NIST) we can calculate that the temperature of the air should drop by about 3 degrees Celsius when the volume is doubled under adiabatic conditions. There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. In the constant volume adiabatic flame temperature case, the volume of the system is held constant and hence there is no work occurring: : {}_RW_P = \int\limits_R^P {pdV} = 0 There is also no heat transfer because the process is defined to be adiabatic: {}_RQ_P = 0 . The potential temperature of a parcel of fluid at pressure P is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure P_{0}, usually . During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to: : dh = v \, dp. For a monatomic ideal gas , with the molar heat capacity at constant volume. As a result, these substances will burn at a constant pressure, which allows the gas to expand during the process. == Common flame temperatures == Assuming initial atmospheric conditions (1bar and 20 °C), the following tableSee under "Tables" in the external references below. lists the flame temperature for various fuels under constant pressure conditions. For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state, pv = RT can be substituted into the 1st law yielding, after some rearrangement: : \frac{dp}{p} = {\frac{c_p}{R}\frac{dT}{T}}, where the dh = c_{p}dT was used and both terms were divided by the product pv Integrating yields: : \left(\frac{p_1}{p_0}\right)^{R/c_p} = \frac{T_1}{T_0}, and solving for T_{0}, the temperature a parcel would acquire if moved adiabatically to the pressure level p_{0}, you get: : T_0 = T_1 \left(\frac{p_0}{p_1}\right)^{R/c_p} \equiv \theta. == Potential virtual temperature == The potential virtual temperature \theta_{v}, defined by : \theta_v = \theta \left( 1 + 0.61 r - r_L \right), is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pressure P0. In atmospheric science, equivalent temperature is the temperature of air in a parcel from which all the water vapor has been extracted by an adiabatic process. For an imperfect or non-ideal gas, Chandrasekhar defined three different adiabatic indices so that the adiabatic relations can be written in the same form as above; these are used in the theory of stellar structure: : \begin{align} \Gamma_1 &= \left.\frac{\partial \ln P}{\partial \ln \rho}\right|_{S}, \\\\[2pt] \frac{\Gamma_2 - 1}{\Gamma_2} &= \left.\frac{\partial \ln T}{\partial \ln P}\right|_{S}, \\\\[2pt] \Gamma_3 - 1 &= \left.\frac{\partial \ln T}{\partial \ln \rho}\right|_{S}. \end{align} All of these are equal to \gamma in the case of an ideal gas. == See also == * Relations between heat capacities * Heat capacity * Specific heat capacity * Speed of sound * Thermodynamic equations * Thermodynamics * Volumetric heat capacity == References == == Notes == Category:Thermodynamic properties Category:Physical quantities Category:Ratios Category:Thought experiments in physics There are two types adiabatic flame temperature: constant volume and constant pressure, depending on how the process is completed. From the classical expression for the entropy it can be derived that the temperature after the doubling of the volume at constant entropy is given as: T = T_i 2^{-R/C_V} = T_i2^{-2/3} for the monoatomic ideal gas. The potential temperature is denoted \theta and, for a gas well-approximated as ideal, is given by : \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, where T is the current absolute temperature (in K) of the parcel, R is the gas constant of air, and c_p is the specific heat capacity at a constant pressure.

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A refrigerator is operated by a $0.25-\mathrm{hp}(1 \mathrm{hp}=$ 746 watts) motor. If the interior is to be maintained at $4.50^{\circ} \mathrm{C}$ and the room temperature on a hot day is $38^{\circ} \mathrm{C}$, what is the maximum heat leak (in watts) that can be tolerated? Assume that the coefficient of performance is $50 . \%$ of the maximum theoretical value. What happens if the leak is greater than your calculated maximum value?

The basic SI units equation for deriving cooling capacity is of the form: :\dot{Q}=\dot{m}C_p\Delta T Where :\dot{Q} is the cooling capacity [kW] :\dot{m} is the mass rate [kg/s] :C_p is the specific heat capacity [kJ/kg K] :\Delta T is the temperature change [K] ==References== Category:Heating, ventilation, and air conditioning As the target temperature of the refrigerator approaches ambient temperature, without exceeding it, the refrigeration capacity increases thus increasing the refrigerator's COP. This is a table of specific heat capacities by magnitude. Cooling capacity is the measure of a cooling system's ability to remove heat. Modern cogeneration plants have power loss ratios of about 1/5 to 1/9 when delivering heat in the range of 80 °C-120 °C.Danny Harvey: Clean building - contribution from cogeneration, trigeneration and district energy, Cogeneration and On-Site Power Production, september–october 2006, pp. 107-115 (Fig. 1) That means in exchange of one kWh of electrical energy ca. 5 up to 9 kWh of useful heat are obtained. The power loss factor β describes the loss of electrical power in CHP systems with a variable power-to-heat ratio when an increasing heat flow is extracted from the main thermodynamic electricity generating process in order to provide useful heat. right|thumb|150px|Thermal mass refrigerator A thermal mass refrigerator is a refrigerator that is foreseen with thermal mass as well as insulation to decrease the energy use of the refrigerator. A refrigerator death is death by suffocation in a refrigerator or other air- tight appliance. A particularly popular thermal mass refrigerator was conceived by Michael Reynolds and detailed in the book "Earthship Volume 3". On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . This refrigerator was a DIY refrigerator designed around a (Sun-Frost) DC refrigeration unit run on P.V. panels.Earthship Volume 3 by Michael Reynolds ==Design details== The thermal mass used in Michael Reynolds' design is a combination of a liquid (i.e. water or beer) together with concrete mass. The partial steam flow, which goes into the heating condenser at high temperature can no longer work in the low-pressure section and is responsible for the loss of power. It is equivalent to the heat supplied to the evaporator/boiler part of the refrigeration cycle and may be called the "rate of refrigeration" or "refrigeration capacity". The Refrigerator Safety Act in 1956 was a U.S. law that required a change in the way refrigerator doors stay shut. The Refrigerator Safety Act was a factor in the decline, in combination with other factors such as "reduced exposure and increased parental supervision". ==Entrapment hazards== Hazardous items for refrigerator deaths are "places with a poor air supply, a heavy lid or a self-latching door". The corresponding expression for the ratio of specific heat capacities remains the same since the thermodynamic system size- dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. Parents or caregivers can lessen the risks of refrigerator deaths. Another unit common in non-metric regions or sectors is the ton of refrigeration, which describes the amount of water at freezing temperature that can be frozen in 24 hours, equivalent to or . In thermodynamics, the heat capacity at constant volume, C_{V}, and the heat capacity at constant pressure, C_{P}, are extensive properties that have the magnitude of energy divided by temperature. == Relations == The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): :C_{P} - C_{V}= V T\frac{\alpha^{2}}{\beta_{T}}\, :\frac{C_{P}}{C_{V}}=\frac{\beta_{T}}{\beta_{S}}\, Here \alpha is the thermal expansion coefficient: :\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\, \beta_{T} is the isothermal compressibility (the inverse of the bulk modulus): :\beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\, and \beta_{S} is the isentropic compressibility: :\beta_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S}\, A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is: : c_p - c_v = \frac{T \alpha^2}{\rho \beta_T} where ρ is the density of the substance under the applicable conditions. Usually, the power loss factor refers to extraction steam turbines in thermal power stations, which conduct a part of the steam in a heating condenser for the production of useful heat, instead of the low pressure part of the steam turbine where is could perform mechanical work. thumb|Power loss within an extraction steam turbine: CHP plant section (left) and T-s-diagram (right) \beta = \frac{\Delta P_\text{el}}{\dot Q_\text{utile}} The picture on the right shows in the left part the principle of steam extraction. Based on the equivalence of power loss and gain of heat, the power loss method assigns CO2 emissions and primary energy from the fuel to the useful heat and the electrical energy. == References == Category:Cogeneration Category:Energy conversion The ratio relation allows one to express the isentropic compressibility in terms of the heat capacity ratio. ==Derivation== If an infinitesimally small amount of heat \delta Q is supplied to a system in a reversible way then, according to the second law of thermodynamics, the entropy change of the system is given by: :dS = \frac{\delta Q}{T}\, Since :\delta Q = C dT\, where C is the heat capacity, it follows that: :T dS = C dT\, The heat capacity depends on how the external variables of the system are changed when the heat is supplied.

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In order to get in shape for mountain climbing, an avid hiker with a mass of $60 . \mathrm{kg}$ ascends the stairs in the world's tallest structure, the $828 \mathrm{~m}$ tall Burj Khalifa in Dubai, United Arab Emirates. Assume that she eats energy bars on the way up and that her body is $25 \%$ efficient in converting the energy content of the bars into the work of climbing. How many energy bars does she have to eat if a single bar produces $1.08 \times 10^3 \mathrm{~kJ}$ of energy upon metabolizing?

Manufacturing of energy bars may supply nutrients in sufficient quantity to be used as meal replacements. ==Nutrition== A typical energy bar weighs between 30 and 50 g and is likely to supply about 200–300 Cal (840–1,300 kJ), 3–9 g of fat, 7–15 g of protein, and 20–40 g of carbohydrates — the three sources of energy in food. Energy bars are supplemental bars containing cereals, micronutrients, and flavor ingredients intended to supply quick food energy. Energy bars may be used as an energy source during athletic events such as marathons, triathlons and other activities which require a high energy expenditure for long periods of time. Because most energy bars contain added protein, carbohydrates, dietary fiber, and other nutrients, they may be marketed as functional foods. YouBar is an online nutrition bar company that makes customized energy bars. The dietary energy supply is the food available for human consumption, usually expressed in kilocalories or kilojoules per person per day. thumb|Energy bars vary in size, ingredients, and nutritional benefits. CalorieMate (カロリーメイト karorīmeito) is a brand of nutritional energy bar and energy gel foods produced by Otsuka Pharmaceutical Co., in Japan. Physical Activity MET Light Intensity Activities < 3 sleeping 0.9 watching television 1.0 writing, desk work, typing 1.8 walking, 1.7 mph (2.7 km/h), level ground, strolling, very slow 2.3 walking, 2.5 mph (4 km/h) 2.9 Moderate Intensity Activities 3 to 6 bicycling, stationary, 50 watts, very light effort 3.0 walking 3.0 mph (4.8 km/h) 3.3 calisthenics, home exercise, light or moderate effort, general 3.5 walking 3.4 mph (5.5 km/h) 3.6 bicycling, <10 mph (16 km/h), leisure, to work or for pleasure 4.0 bicycling, stationary, 100 watts, light effort 5.5 Vigorous Intensity Activities > 6 jogging, general 7.0 calisthenics (e.g. pushups, situps, pullups, jumping jacks), heavy, vigorous effort 8.0 running jogging, in place 8.0 rope jumping 10.0 ==Fuel Used== The body uses different amounts of energy substrates (carbohydrates or fats) depending on the intensity of the exercise and the heart rate of the exerciser. One MET, which is equal to 3.5 mL/kg per minute, is considered to be the average resting energy expenditure of a typical human being. High intensity activity also yields a higher total caloric expenditure. For those who are malnourished, energy bars, such as Plumpy'nut, are an effective tool for treating malnutrition. == See also == * Candy bar * Protein bar * Energy gel * Sports drink * High energy biscuits * Flapjack (oat bar) * D ration ==References== Category:Dietary supplements Category:Energy food products Category:Snack foods Dietary energy supply is correlated with the rate of obesity. ==Regions== Daily dietary energy supply per capita: Region 1964-1966 1974-1976 1984-1986 1997-1999 World Sub-Saharan Africa ==Countries== Country 1979-1981 1989-1991 2001-2003 Canada China Eswatini United Kingdom United States ==See also== *Diet and obesity thumb|300x300px|Average dietary energy supply by region ==References== ==External links== * Category:Obesity Category:Diets An intensity of exercise equivalent to 6 METs means that the energy expenditure of the exercise is six times the resting energy expenditure.Vehrs, P., Ph.D. (2011). Intensity of exercise can be expressed as multiples of resting energy expenditure. On the other hand, high intensity activity utilizes a larger percentage of carbohydrates in the calories expended because its quick production of energy makes it the preferred energy substrate for high intensity exercise. Intensity (%MHR) Heart Rate (bpm) % Carbohydrate % Fat 65-70 130-140 15 85 70-75 140-150 35 65 75-80 150-160 65 35 80-85 160-170 80 20 85-90 170-180 90 10 90-95 180-190 95 5 100 190-200 100 - These estimates are valid only when glycogen reserves are able to cover the energy needs. Fats sources are often cocoa butter and dark chocolate. == Usage == Energy bars are used in a variety of contexts. right|300px Exercise intensity refers to how much energy is expended when exercising. Protein is a third energy substrate, but it contributes minimally and is therefore discounted in the percent contribution graphs reflecting different intensities of exercise. This table outlines the estimated distribution of energy consumption at different intensity levels for a healthy 20-year-old with a Max Heart Rate (MHR) of 200. thumb|A Lärabar bar Lärabar is a brand of energy bars produced by General Mills.

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The half-life of ${ }^{238} \mathrm{U}$ is $4.5 \times 10^9$ years. How many disintegrations occur in $1 \mathrm{~min}$ for a $10 \mathrm{mg}$ sample of this element?

The short half-life of 87.7 years of 238Pu means that a large amount of it decayed during its time inside his body, especially when compared to the 24,100 year half-life of 239Pu. The half-life of 242Pu is about 15 times that of 239Pu; so it is one-fifteenth as radioactive, and not one of the larger contributors to nuclear waste radioactivity. 242Pu's gamma ray emissions are also weaker than those of the other isotopes. Plutonium-242 decays via spontaneous fission in about 5.5 × 10−4% of casesChart of all nuclei which includes half life and mode of decay ==References== Category:Actinides Category:Nuclear materials Category:Isotopes of plutonium Plutonium-244 (244Pu) is an isotope of plutonium that has a half-life of 80 million years. Plutonium-238 (238Pu or Pu-238) is a fissile, radioactive isotope of plutonium that has a half-life of 87.7 years. Modern calculations of his lifetime absorbed dose give an incredible 64 Sv (6400 rem) total. ===Weapons=== The first application of 238Pu was its use in nuclear weapon components made at Mound Laboratories for Lawrence Radiation Laboratory (now Lawrence Livermore National Laboratory). Presence of 244Pu fission tracks can be established by using the initial ratio of 244Pu to 238U (Pu/U)0 at a time T0 = years, when Xe formation first began in meteorites, and by considering how the ratio of Pu/U fission tracks varies over time. Plutonium-242 (242Pu or Pu-242) is one of the isotopes of plutonium, the second longest-lived, with a half-life of 375,000 years. This is longer than any of the other isotopes of plutonium and longer than any other actinide isotope except for the three naturally abundant ones: uranium-235 (704 million years), uranium-238 (4.468 billion years), and thorium-232 (14.05 billion years). The amount of 244Pu in the pre-Solar nebula (4.57×109 years ago) was estimated as 0.8% the amount of 238U..As the age of the Earth is about 57 half-lives of 244Pu, the amount of plutonium-244 left should be very small; Hoffman et al. estimated its content in the rare-earth mineral bastnasite as = 1.0×10−18 g/g, which corresponded to the content in the Earth crust as low as 3×10−25 g/g (i.e. the total mass of plutonium-244 in Earth's crust is about 9 g). This gives a density for 238Pu of (1.66053906660×10−24g/dalton×238.0495599 daltons/atom×16 atoms/unit cell)/(319.96 Å3/unit cell × 10−24cc/Å3) or 19.8 g/cc. It decays by electron capture to stable cadmium-111 with a half-life of 2.8 days. However, 238Pu is far more dangerous than 239Pu due to its short half-life and being a strong alpha-emitter. Historically, most plutonium-238 has been produced by Savannah River in their weapons reactor, by irradiating with neutrons neptunium-237 (half life ). \+ → Neptunium-237 is a by-product of the production of plutonium-239 weapons-grade material, and when the site was shut down in 1988, 238Pu was mixed with about 16% 239Pu. ===Human radiation experiments=== Plutonium was first synthesized in 1940 and isolated in 1941 by chemists at the University of California, Berkeley. They also reported an even longer half-life for alpha decay of bismuth-209 to the first excited state of thallium-205 (at 204 keV), was estimated to be 1.66 years. In February 2013, a small amount of 238Pu was successfully produced by Oak Ridge's High Flux Isotope Reactor, and on December 22, 2015, they reported the production of of 238Pu. Although 209Bi holds the half-life record for alpha decay, bismuth does not have the longest half-life of any radionuclide to be found experimentally--this distinction belongs to tellurium-128 (128Te) with a half-life estimated at 7.7 × 1024 years by double β-decay (double beta decay). Bismuth-209 (209Bi) is the isotope of bismuth with the longest known half-life of any radioisotope that undergoes α-decay (alpha decay). Significant amounts of pure 238Pu could also be produced in a thorium fuel cycle. The density of plutonium-238 at room temperature is about 19.8 g/cc.Calculated from the atomic weight and the atomic volume. However, 242Pu's low cross section means that relatively little of it will be transmuted during one cycle in a thermal reactor. ==Decay== Plutonium-242 mainly decays into uranium-238 via alpha decay, before continuing along the uranium series. The material will generate about 0.57 watts per gram of 238Pu.

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Calculate the ionic strength in a solution that is 0.0750 $m$ in $\mathrm{K}_2 \mathrm{SO}_4, 0.0085 \mathrm{~m}$ in $\mathrm{Na}_3 \mathrm{PO}_4$, and $0.0150 \mathrm{~m}$ in $\mathrm{MgCl}_2$.

For the electrolyte MgSO4, however, each ion is doubly- charged, leading to an ionic strength that is four times higher than an equivalent concentration of sodium chloride: :I = \frac{1}{2}[c(+2)^2+c(-2)^2] = \frac{1}{2}[4c + 4c] = 4c Generally multivalent ions contribute strongly to the ionic strength. ===Calculation example=== As a more complex example, the ionic strength of a mixed solution 0.050 M in Na2SO4 and 0.020 M in KCl is: : \begin{align} I & = \tfrac 1 2 \times \left[\begin{array}{l} \\{(\text{concentration of }\ce{Na2SO4}\text{ in M}) \times (\text{number of }\ce{Na+}) \times (\text{charge of }\ce{Na+})^2\\}\ + \\\ \\{(\text{concentration of }\ce{Na2SO4}\text{ in M}) \times (\text{number of }\ce{SO4^2-}) \times (\text{charge of }\ce{SO4^2-})^2\\} \ + \\\ \\{(\text{concentration of }\ce{KCl}\text{ in M}) \times (\text{number of }\ce{K+}) \times (\text{charge of }\ce{K+})^2\\}\ + \\\ \\{(\text{concentration of }\ce{KCl}\text{ in M}) \times (\text{number of }\ce{Cl-}) \times (\text{charge of }\ce{Cl-})^2\\} \end{array}\right] \\\ & = \tfrac 1 2 \times [\\{0.050 M \times 2 \times (+1)^2\\} + \\{0.050 M \times 1 \times (-2)^2\\} + \\{0.020 M \times 1 \times (+1)^2\\} + \\{0.020 M \times 1 \times (-1)^2\\}] \\\ & = 0.17 M \end{align} ==Non-ideal solutions== Because in non-ideal solutions volumes are no longer strictly additive it is often preferable to work with molality b (mol/kg of H2O) rather than molarity c (mol/L). Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. The ionic strength of a solution is a measure of the concentration of ions in that solution. Ionic strength can be molar (mol/L solution) or molal (mol/kg solvent) and to avoid confusion the units should be stated explicitly. For a 1:1 electrolyte such as sodium chloride, where each ion is singly-charged, the ionic strength is equal to the concentration. One of the main characteristics of a solution with dissolved ions is the ionic strength. The concept of ionic strength was first introduced by Lewis and Randall in 1921 while describing the activity coefficients of strong electrolytes. ==Quantifying ionic strength== The molar ionic strength, I, of a solution is a function of the concentration of all ions present in that solution. In that case, molal ionic strength is defined as: : I = \frac{1}{2}\sum_{{i}=1}^{n} b_{i}z_{i}^{2} in which :i = ion identification number :z = charge of ion :b = molality (mol solute per Kg solvent)Standard definition of molality ==Importance== The ionic strength plays a central role in the Debye–Hückel theory that describes the strong deviations from ideality typically encountered in ionic solutions. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. Ionic radius, rion, is the radius of a monatomic ion in an ionic crystal structure. That is, the Debye length, which is the inverse of the Debye parameter (κ), is inversely proportional to the square root of the ionic strength. In condensed matter physics and inorganic chemistry, the cation-anion radius ratio can be used to predict the crystal structure of an ionic compound based on the relative size of its atoms. Comparison between observed and calculated ion separations (in pm) MX Observed Soft-sphere model LiCl 257.0 257.2 LiBr 275.1 274.4 NaCl 282.0 281.9 NaBr 298.7 298.2 In the soft-sphere model, k has a value between 1 and 2. Magnesium orthosilicate is a chemical compound with the formula Mg2SiO4.Magnesium orthosilicate at Chemister It is the orthosilicate salt of magnesium. _Effective_ ionic radii in pm of elements as a function of ionic charge and spin (ls = low spin, hs = high spin). Curiously, no theoretical justification for the equation containing k has been given. == Non-spherical ions == The concept of ionic radii is based on the assumption of a spherical ion shape. To be consistent with Pauling's radii, Shannon has used a value of rion(O2−) = 140 pm; data using that value are referred to as "effective" ionic radii. One approach to improving the calculated accuracy is to model ions as "soft spheres" that overlap in the crystal. Natural waters such as mineral water and seawater have often a non-negligible ionic strength due to the presence of dissolved salts which significantly affects their properties. ==See also== * Activity (chemistry) * Activity coefficient * Bromley equation * Davies equation * Debye–Hückel equation * Debye–Hückel theory * Double layer (interfacial) * Double layer (electrode) * Double layer forces * Electrical double layer * Gouy-Chapman model * Flocculation * Peptization (the inverse of flocculation) * DLVO theory (from Derjaguin, Landau, Verwey and Overbeek) * Interface and colloid science * Osmotic coefficient * Pitzer equations * Poisson–Boltzmann equation * Specific ion Interaction Theory * Salting in * Salting out ==External links== * Ionic strength * Ionic strength introduction at the EPA web site == References == Category:Analytical chemistry Category:Colloidal chemistry Category:Electrochemical equations Category:Electrochemical concepts Category:Equilibrium chemistry Category:Physical chemistry Category:Physical quantities Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides. Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides. All compounds crystallize in the NaCl structure. thumb|300 px|Relative radii of atoms and ions.

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The interior of a refrigerator is typically held at $36^{\circ} \mathrm{F}$ and the interior of a freezer is typically held at $0.00^{\circ} \mathrm{F}$. If the room temperature is $65^{\circ} \mathrm{F}$, by what factor is it more expensive to extract the same amount of heat from the freezer than from the refrigerator? Assume that the theoretical limit for the performance of a reversible refrigerator is valid in this case.

Direct Cool Vs Frost Free Refrigerators – Know the Differences Direct cool is less expensive in production and in operation, as it consumes less energy when compared to frost free refrigerators ==References== Category:Refrigerants 2\. While having the same total pressure throughout the system, the refrigerator maintains a low partial pressure of the refrigerant (therefore high evaporation rate) in the part of the system that draws heat out of the low-temperature interior of the refrigerator, but maintains the refrigerant at high partial pressure (therefore low evaporation rate) in the part of the system that expels heat to the ambient-temperature air outside the refrigerator. A single-pressure absorption refrigerator takes advantage of the fact that a liquid's evaporation rate depends upon the partial pressure of the vapor above the liquid and goes up with lower partial pressure. An absorption refrigerator is a refrigerator that uses a heat source (e.g., solar energy, a fossil-fueled flame, waste heat from factories, or district heating systems) to provide the energy needed to drive the cooling process. Heat flows from the hotter interior of the refrigerator to the colder liquid, promoting further evaporation. 3\. A refrigerator designed to reach cryogenic temperatures (below ) is often called a cryocooler. The refrigerator was less efficient than existing appliances, although having no moving parts made it more reliable; the introduction of non-toxic Freon — later found to be responsible for serious depletion of the Earth's ozone layer — to replace toxic refrigerant gases made it even less attractive commercially. Progress in the cryocooler field in recent decades is in large part due to development of new materials having high heat capacity below 10 K.T. Kuriyama, R. Hakamada, H. Nakagome, Y. Tokai, M. Sahashi, R. Li, O. Yoshida, K. Matsumoto, and T. Hashimoto, Advances in Cryogenic Engineering 35B, 1261 (1990) == Stirling refrigerators == ===Components=== 300px|thumb| Fig.1 Schematic diagram of a Stirling cooler. right|thumb|150px|Thermal mass refrigerator A thermal mass refrigerator is a refrigerator that is foreseen with thermal mass as well as insulation to decrease the energy use of the refrigerator. In the 1960s, absorption refrigeration saw a renaissance due to the substantial demand for refrigerators for caravans (travel trailers). Unlike more common vapor-compression refrigeration systems, an absorption refrigerator has no moving parts. ==History== In the early years of the 20th century, the vapor absorption cycle using water- ammonia systems was popular and widely used, but after the development of the vapor compression cycle it lost much of its importance because of its low coefficient of performance (about one fifth of that of the vapor compression cycle). An absorption refrigerator changes the gas back into a liquid using a method that needs only heat, and has no moving parts other than the fluids. 300px|right|Absorption cooling process The absorption cooling cycle can be described in three phases: #Evaporation: A liquid refrigerant evaporates in a low partial pressure environment, thus extracting heat from its surroundings (e.g. the refrigerator's compartment). Direct-cool refrigerators produce the cooling effect by a natural convection process from cooled surfaces in the insulated compartment that is being cooled. Compression refrigerators typically use an HCFC or HFC, while absorption refrigerators typically use ammonia or water and need at least a second fluid able to absorb the coolant, the absorbent, respectively water (for ammonia) or brine (for water). The water evaporated from the salt solution is re- condensed, and rerouted back to the evaporative cooler. ===Single pressure absorption refrigeration=== thumb|right|300px|Domestic absorption refrigerator. 1\. The refrigerator is a small unit placed over a campfire, that can later be used to cool of water to just above freezing for 24 hours in a environment. The main difference between the two systems is the way the refrigerant is changed from a gas back into a liquid so that the cycle can repeat. thumb|200px|Einstein's and Szilárd's patent application thumb|200px|Annotated patent drawing The Einstein–Szilard or Einstein refrigerator is an absorption refrigerator which has no moving parts, operates at constant pressure, and requires only a heat source to operate. This refrigerator was a DIY refrigerator designed around a (Sun-Frost) DC refrigeration unit run on P.V. panels.Earthship Volume 3 by Michael Reynolds ==Design details== The thermal mass used in Michael Reynolds' design is a combination of a liquid (i.e. water or beer) together with concrete mass. Direct cool is one of the two major types of techniques used in domestic refrigerators, the other being the "frost-free" type. The "Tang-Dresselhaus Theory" (Shuang Tang and Mildred Dresselhaus) has pointed out that anisotropic transport behaviors of quantum confined BiSb alloys nanostructures can optimize the pertinent thermoelectric cooling performance below 77 K for applications in satellites and space stations. ==See also== * Cryogenic processor * Adiabatic demagnetization refrigerator * Dilution refrigerator * Hampson-Linde cycle * Pulse tube refrigerator * Stirling engine (Stirling cryocooler) * Entropy production ==References== Category:Cooling technology Category:Cryogenics Category:Industrial gases

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Calculate the rotational partition function for $\mathrm{SO}_2$ at $298 \mathrm{~K}$ where $B_A=2.03 \mathrm{~cm}^{-1}, B_B=0.344 \mathrm{~cm}^{-1}$, and $B_C=0.293 \mathrm{~cm}^{-1}$

For each value of J, we have rotational degeneracy, g_j = (2J+1), so the rotational partition function is therefore \zeta^\text{rot} = \sum_{J=0}^\infty g_j e^{-E_J/k_\text{B} T} = \sum_{J=0}^\infty (2J+1) e^{-J(J+1) B / k_\text{B} T}. In terms of these constants, the rotational partition function can be written in the high temperature limit as G. Herzberg, ibid, Equation (V,29) \zeta^\text{rot} \approx \frac{\sqrt{\pi} }{\sigma} \sqrt{ \frac{ (k_\text{B} T)^3 }{ A B C }} with \sigma again known as the rotational symmetry number G. Herzberg, ibid; see Table 140 for values for common molecular point groups which in general equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms. Another convenient expression for the rotational partition function for symmetric and asymmetric tops is provided by Gordy and Cook: \zeta^\text{rot} \approx \frac{5.34 \times 10^6}{\sigma} \sqrt{ \frac{T^3}{A B C}} where the prefactor comes from \sqrt{\frac{(\pi k_\text{B}) ^3}{h^3}} = 5.34 \times 10^6 when A, B, and C are expressed in units of MHz. The expressions for \zeta^\text{rot} works for asymmetric, symmetric and spherical top rotors. ==References== ==See also== * Translational partition function * Vibrational partition function * Partition function (mathematics) Category:Equations of physics Category:Partition functions In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. == Definition == The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions \zeta:Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 Z = \frac{ \zeta^N }{ N! } with: \zeta = \sum_j g_j e^{ -E_j / k_\text{B} T} , where g_j is the degeneracy of the j-th quantum level of an individual particle, k_\text{B} is the Boltzmann constant, and T is the absolute temperature of system. Atkins and J. de Paula "Physical Chemistry", 9th edition (W.H. Freeman 2010), p.597 :\theta_{\mathrm{R}} = \frac{hc \overline{B}}{k_{\mathrm{B}}} = \frac{\hbar^2}{2k_{\mathrm{B}}I}, where \overline{B} = B/hc is the rotational constant, is a molecular moment of inertia, is the Planck constant, is the speed of light, is the reduced Planck constant and is the Boltzmann constant. For a diatomic molecule like CO or HCl, or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are E_J^\text{rot} = \frac{\mathbf{J}^2}{2I} = \frac{J(J+1)\hbar^2}{2I} = J(J+1)B. J is the quantum number for total rotational angular momentum and takes all integer values starting at zero, i.e., J = 0,1,2, \ldots, B = \frac{\hbar^2}{2I} is the rotational constant, and I is the moment of inertia. thumb|350px|Dini's Surface with constants a = 1, b = 0.5 and 0 ≤ u ≤ 4 and 0 It is named after Ulisse Dini and described by the following parametric equations: : \begin{align} x&=a \cos u \sin v \\\ y&=a \sin u \sin v \\\ z&=a \left(\cos v +\ln \tan \frac{v}{2} \right) + bu \end{align} thumb|350px|right|Dini's surface with 0 ≤ u ≤ 4 and 0.01 ≤ v ≤ 1 and constants a = 1.0 and b = 0.2. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945, Equation (V,21) \zeta^\text{rot} = \frac{ k_\text{B} T}{B} + \frac{1}{3} + \frac{1}{15} \left( \frac{B}{ k_\text{B} T} \right) + \frac{4}{315} \left( \frac{B}{k_\text{B} T} \right)^2 + \frac{1}{315} \left( \frac{B}{k_\text{B} T} \right)^3 + \cdots . In the high temperature limit, it is traditional to correct for the missing nuclear spin states by dividing the rotational partition function by a factor \sigma = 2 with \sigma known as the rotational symmetry number which is 2 for linear molecules with a center of symmetry and 1 for linear molecules without. ==Nonlinear molecules== A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written A, B, and C , which can often be determined by rotational spectroscopy. This suggests we can approximate the sum by replacing the sum over J by an integral of J treated as a continuous variable. \zeta^\text{rot} \approx \int_0^{\infty} (2J+1)e^{-J(J+1) B /k_\text{B} T} dJ = \frac{ k_\text{B} T}{B} . In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is k_\text{B} T . == Quantum symmetry effects == For a diatomic molecule with a center of symmetry, such as \rm H_2, N_2, CO_2, or \mathrm{ H_2 C_2} (i.e. D_{\infty h} point group), rotation of a molecule by \pi radian about an axis perpendicular to the molecule axis and going through the center of mass will interchange pairs of equivalent atoms. The mean thermal rotational energy per molecule can now be computed by taking the derivative of \zeta^\text{rot} with respect to temperature T. The characteristic rotational temperature ( or ) is commonly used in statistical thermodynamics to simplify the expression of the rotational partition function and the rotational contribution to molecular thermodynamic properties. ASM Handbook; Alloy Phase Diagrams; v. 3; ASM International, USA; 1992, pp. 491–492. For molecules, under the assumption that total energy levels E_j can be partitioned into its contributions from different degrees of freedom (weakly coupled degrees of freedom)Donald A. McQuarrie, ibid E_j = \sum_i E_j^i = E_j^\text{trans} + E_j^\text{ns} + E_j^\text{rot} + E_j^\text{vib} + E_j^\text{e} and the number of degenerate states are given as products of the single contributions g_j = \prod_i g_j^i = g_j^\text{trans} g_j^\text{ns} g_j^\text{rot} g_j^\text{vib} g_j^\text{e}, where "trans", "ns", "rot", "vib" and "e" denotes translational, nuclear spin, rotational and vibrational contributions as well as electron excitation, the molecular partition functions \zeta = \sum_j g_j e^{-E_j/k_\text{B} T} can be written as a product itself \zeta = \prod_i \zeta^i = \zeta^\text{trans} \zeta^\text{ns} \zeta^\text{rot} \zeta^\text{vib}\zeta^\text{e}. == Linear molecules == Rotational energies are quantized. This alloy presents a eutectic temperature of 382 K (109 °C; 228.2 °F). On 19 February 1772, the agreement of partition was signed in Vienna. thumb|right|300px|Picture of Europe for July 1772, satirical British plate The Partition Sejm () was a Sejm lasting from 1773 to 1775 in the Polish–Lithuanian Commonwealth, convened by its three neighbours (the Russian Empire, Prussia and Austria) in order to legalize their First Partition of Poland. *BiIn2 (from 52.5 to 53.5 wt% of In), having a hexagonal structure with 2 atoms per unit cell. The Sejm on 30 September 1773 accepted the partition treaty. Atkins and J. de Paula "Physical Chemistry", 10th edition, Table 12D.1, p.987 ==References== ==See also== *Rotational spectroscopy *Vibrational temperature *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics

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For a two-level system where $v=1.50 \times 10^{13} \mathrm{~s}^{-1}$, determine the temperature at which the internal energy is equal to $0.25 \mathrm{Nhv}$, or $1 / 2$ the limiting value of $0.50 \mathrm{Nhv}$.

The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39 Therefore, a convenient null reference point may be chosen for the internal energy. The internal energy of an ideal gas is proportional to its mass (number of moles) n and to its temperature T : U = C_V n T, where C_V is the isochoric (at constant volume) molar heat capacity of the gas. Knowing temperature and pressure to be the derivatives T = \frac{\partial U}{\partial S}, P = -\frac{\partial U}{\partial V}, the ideal gas law PV = nRT immediately follows as below: : T = \frac{\partial U}{\partial S} = \frac{U}{C_V n} : P = -\frac{\partial U}{\partial V} = U \frac{R}{C_V V} : \frac{P}{T} = \frac{\frac{U R}{C_V V}}{\frac{U}{C_V n}} = \frac{n R}{V} : PV = nRT ==Internal energy of a closed thermodynamic system== The above summation of all components of change in internal energy assumes that a positive energy denotes heat added to the system or the negative of work done by the system on its surroundings. The differential internal energy may be written as :\mathrm{d} U = \frac{\partial U}{\partial S} \mathrm{d} S + \frac{\partial U}{\partial V} \mathrm{d} V + \sum_i\ \frac{\partial U}{\partial N_i} \mathrm{d} N_i\ = T \,\mathrm{d} S - P \,\mathrm{d} V + \sum_i\mu_i \mathrm{d} N_i, which shows (or defines) temperature T to be the partial derivative of U with respect to entropy S and pressure P to be the negative of the similar derivative with respect to volume V, : T = \frac{\partial U}{\partial S}, : P = -\frac{\partial U}{\partial V}, and where the coefficients \mu_{i} are the chemical potentials for the components of type i in the system. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The expression relating changes in internal energy to changes in temperature and volume is : \mathrm{d}U =C_{V} \, \mathrm{d}T +\left[T\left(\frac{\partial P}{\partial T}\right)_{V} - P\right] \mathrm{d}V. Appendix D. ISBN 978-1305079113 Na sodium 107 K potassium 89 Rb rubidium 81 Cs caesium 76 Mg magnesium 148 Ca calcium 178 Sr strontium 164 Ba barium 180 Fe iron 416 Ni nickel 430 Cu copper 338 Zn zinc 131 Ag silver 285 W tungsten 849 Au gold 366 C graphite 717 C diamond 715 Si silicon 456 Sn tin 302 Pb lead 195 I2 iodine 62.4 C10H8 naphthalene 72.9 CO2 carbon dioxide 25 ==See also== * Heat * Sublimation (chemistry) * Phase transition * Clausius-Clapeyron equation == References == Category:Enthalpy For a closed system, with transfers only as heat and work, the change in the internal energy is : \mathrm{d} U = \delta Q - \delta W, expressing the first law of thermodynamics. Although the heat capacity has a peak, it does not tend towards infinity (contrary to what the graph may suggest), but has finite limiting values when approaching the transition from above and below. The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization.Crawford, F. H. (1963), pp. 106–107.Haase, R. (1971), pp. 24–28. The internal energy is an extensive function of the extensive variables S, V, and the amounts N_j, the internal energy may be written as a linearly homogeneous function of first degree: : U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots ) = \alpha U(S,V,N_{1},N_{2},\ldots), where \alpha is a factor describing the growth of the system. For those phase transitions specific heat does tend to infinity. Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): U = U(n,T). If a real gas can be described by the van der Waals equation of state p = \frac{nRT}{V-nb} - a \frac{n^2}{V^2} it follows from the thermodynamic equation of state that \pi_T = a \frac{n^2}{V^2} Since the parameter a is always positive, so is its internal pressure: internal energy of a van der Waals gas always increases when it expands isothermally. A Bordwell thermodynamic cycle use experimentally determined and reasonable estimates of Gibbs free energy (ΔG˚) values to determine unknown and experimentally inaccessible values. == Overview == Analogous to Hess's Law which deal with the summation of enthalpy (ΔH) values, Bordwell thermodynamic cycles deal with the summation of Gibbs free energy (ΔG) values. On the energetics of maximum-entropy temperature profiles, Q. J. R. Meteorol. The equation of state is the ideal gas law :P V = n R T. Solve for pressure: :P = \frac{n R T}{V}. The change in internal energy becomes : \mathrm{d}U = T \, \mathrm{d}S - P \, \mathrm{d}V. ===Changes due to temperature and volume=== The expression relating changes in internal energy to changes in temperature and volume is This is useful if the equation of state is known. Generalized Thermodynamics, M.I.T. Press, Cambridge MA. Equilibrium Thermodynamics, second edition, McGraw-Hill, London, . In thermodynamics, the enthalpy of sublimation, or heat of sublimation, is the heat required to sublimate (change from solid to gas) one mole of a substance at a given combination of temperature and pressure, usually standard temperature and pressure (STP).

2.2

655

0.000216

46.7

0.000226

B

Calculate $K_P$ at $600 . \mathrm{K}$ for the reaction $\mathrm{N}_2 \mathrm{O}_4(l) \rightleftharpoons 2 \mathrm{NO}_2(g)$ assuming that $\Delta H_R^{\circ}$ is constant over the interval 298-725 K.

The molecular formula C18H12O4 (molar mass: 292.28 g/mol, exact mass: 292.0736 u) may refer to: * Karanjin * Polyporic acid Category:Molecular formulas The molecular formula C6H6N4O4 (molar mass: 198.14 g/mol, exact mass: 198.0389 u) may refer to: * 2,4-Dinitrophenylhydrazine * Nitrofurazone The molecular formula C6H7KO6 (molar mass: 214.21 g/mol, exact mass: 213.9880 u) may refer to: * Potassium ascorbate * Potassium erythorbate The molecular formula C12H14O4 (molar mass: 222.23 g/mol, exact mass: 222.0892 u) may refer to: * Apiole * Blattellaquinone * Diethyl phthalate * Dillapiole * Monobutyl phthalate The molecular formula C27H33NO4 (molar mass: 435.56 g/mol, exact mass: 435.2410 u) may refer to: * Paxilline, a potassium channel blocker * BU-48 The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas

0.33333333

1.7

2.25

0.00017

4.76

E

Count Rumford observed that using cannon boring machinery a single horse could heat $11.6 \mathrm{~kg}$ of ice water $(T=273 \mathrm{~K})$ to $T=355 \mathrm{~K}$ in 2.5 hours. Assuming the same rate of work, how high could a horse raise a $225 \mathrm{~kg}$ weight in 2.5 minutes? Assume the heat capacity of water is $4.18 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~g}^{-1}$.

Based on differences in the definition of what constitutes the "power of a horse", a horsepower-hour differs slightly from the German "Pferdestärkenstunde" (PSh): :1.014 PSh = 1 hp⋅h = 1,980,000 lbf⋅ft = 0.7457 kW⋅h. :1 The unit represents an amount of work a horse is supposed capable of delivering during an hour (1 horsepower integrated over a time interval of an hour). Pound per hour is a mass flow unit. For example, if Railroad A borrows a 2,500 horsepower locomotive from Railroad B and operates it for twelve hours, Railroad A owes a debt of (2,500 hp × 12 h) = 30,000 hp⋅h. In the US utility industry, steam and water flows throughout turbine cycles are typically expressed in PPH, while in Europe these mass flows are usually expressed in metric tonnes per hour: :1 lb/h = 0.4535927 kg/h = 126.00 mg/s Minimum fuel intake on a jumbo jet can be as low as 150 lb/h when idling; however, this is not enough to sustain flight. PSh = 0.73549875 kW⋅h = 2647.7955 kJ (exactly by definition) The horsepower-hour is still used in the railroad industry when sharing motive power (locomotives). The steam to oil ratio is a measure of the water and energy consumption related to oil production in cyclic steam stimulation and steam assisted gravity drainage oil production. Humber Fifteen 15 horsepower cars were medium to large cars, classified as medium weight, with a less powerful than usual engine which attracted less annual taxation and provided more stately progress. Their equivalent prewar car with an engine of 3.3 Litres had twin overhead camshafts. ===Bodies=== The 15.9 was available as a saloon or a 5-seater tourer. Ice Water (foaled 1963 in Ontario) was a Canadian Thoroughbred racehorse. ==Background== Ice Water was a bay mare owned and bred by George Gardiner. This means two to eight barrels of water converted into steam is used to produce one barrel of oil. == References == * Glossary at Schlumberger. Railroad A may repay the debt by loaning Railroad B a 3,000 horsepower locomotive for ten hours. ==References== Category:Imperial units Category:Units of energy The GWR 2021 Class was a class of 140 steam locomotives. Ice Water raced and won at age four and five, notably winning her second and third consecutive runnings of the Belle Mahone Stakes. ==Breeding record== She was retired to broodmare duty for the 1969 season at her owner's breeding farm where she had limited success. ==References== * Ice Water's pedigree and partial racing stats * Article on Gardiner Farms and Ice Water at the Jockey Club of Canada Category:1963 racehorse births Category:Thoroughbred family 13-c Category:Racehorses bred in Ontario Category:Racehorses trained in Canada A horsepower-hour (symbol: hp⋅h) is an outdated unit of energy, not used in the International System of Units. Ice Water's sire was Nearctic who also sired the most influential sire of the 20th Century, Northern Dancer. The car's steering was delightful but its brakes and suspension were only satisfactory. ==Fifteen 40== The 15-40-hp, a lightly revised 15.9, was displayed at the Olympia Motor Show in October 1924. ===Engine=== The Times noted some trouble had been taken to dampen engine vibration. The experiment was unpopular with engine crews, and the bodywork removed in 1911. ==See also== * GWR 0-6-0PT – list of classes of GWR 0-6-0 pannier tank, including table of preserved locomotives ==References== ==Sources== * Ian Allan ABC of British Railways Locomotives, 1948 edition, part 1, pp 16,51 * * 2021 Category:0-6-0ST locomotives Category:Railway locomotives introduced in 1897 Category:Standard gauge steam locomotives of Great Britain Category:Scrapped locomotives Against females, Ice Water won the Wonder Where Stakes and the Belle Mahone Stakes. They were superseded by the short-lived GWR 1600 Class, nominally a Hawksworth design, but in reality a straightforward update of the then 75-year-old design, with new boiler, bigger cab and bunker. ==Coachwork== When autotrains were introduced on the GWR, a trial was made of enclosing the engine in coachwork to resemble the coaches. The typical values are three to eight and two to five respectively. The newspaper noted that a few drops of oil two or three times a week ensures tappets run for a long time without shake otherwise they soon become noisy.

30

0.3085

2.0

-0.0301

-2

A

The vibrational frequency of $I_2$ is $208 \mathrm{~cm}^{-1}$. At what temperature will the population in the first excited state be half that of the ground state?

The vibrational temperature is used commonly when finding the vibrational partition function. 2MASS J03480772−6022270 (abbreviated to 2MASS J0348−6022) is a brown dwarf of spectral class T7, located in the constellation Reticulum approximately 27.2 light-years from the Sun. The high estimated age of 2MASS J0348−6022 is due to its late T-type spectral class, which is generally expected to describe the later evolutionary stages of brown dwarfs as they cool. == Rotation == === Photometric variability and periodicity === 2MASS J0348−6022 is the fastest-rotating brown dwarf confirmed , with a photometric periodicity of hours. The near-infrared spectrum of 2MASS J0348−6022 also displays a pair of narrow absorption lines at 1.243 and 1.252 μm, which are attributed to the presence of neutral potassium (K I) in the brown dwarf's atmosphere. Photometric variability in 2MASS J0348−6022 was first reported in 2008 by Fraser Clarke and collaborators using the New Technology Telescope's (NTT) near-infrared spectrograph. 2MASS J09373487+2931409, or 2MASSI J0937347+293142 (abbreviated to 2MASS 0937+2931) is a brown dwarf of spectral class T6, located in the constellation Leo about 19.96 light-years from Earth. ==Discovery== 2MASS 0937+2931 was discovered in 2002 by Adam J. Burgasser et al. from Two Micron All-Sky Survey (2MASS), conducted from 1997 to 2001. In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. 2MASS J12195156+3128497 (abbreviated to 2MASS J1219+3128) is a rapidly- rotating brown dwarf of spectral class L8, located in the constellation Coma Berenices about 66 light-years from Earth. Absorption bands of iron(I) hydride (FeH) have also been found in 2MASS J0348−6022's spectrum between 1.72–1.78 μm. The mass, radius, and age of 2MASS J0348−6022 are estimated by interpolation of brown dwarf evolutionary models based on effective temperature and surface gravity. 2MASS J02431371−2453298 (abbreviated to 2MASS 0243−2453) is a brown dwarf of spectral class T6, located in the constellation Fornax about 34.84 light-years from Earth. ==Discovery== 2MASS 0243−2453 was discovered in 2002 by Adam J. Burgasser et al. from Two Micron All-Sky Survey (2MASS), conducted from 1997 to 2001. The high spin rate and oblateness of 2MASS J0348−6022 places it at about 45% of its rotational stability limit, assuming a smoothly varying fluid interior. A less precise parallax of this object, measured under U.S. Naval Observatory Infrared Astrometry Program, was published in 2004 by Vrba et al. ==Properties== 2MASS 0937+2931 has an unusual spectrum, indicating a metal- poor atmosphere and/or a high surface gravity (high pressure at the surface). Its effective temperature is estimated at about 800 Kelvin. The inclination of 2MASS J0348−6022's spin axis to Earth is , derived from its v sin i value. A previous estimate by Burgasser and collaborators from the spectrophotometric relation of spectral type and near-infrared absolute magnitude resulted in a value of , based on 2MASS JHK-band photometry. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics In a 2021 study, Megan Tannock and collaborators compared the near-infrared spectrum of 2MASS J0348−6022 to various published photospheric models and derived multiple best-fit solutions for its effective temperature and surface gravity. The trigonometric parallax of 2MASS J0348−6022 has been measured to be milliarcseconds, from 16 observations by the New Technology Telescope (NTT) collected over 6.4 years. Given the distance estimate from trigonometric parallax, the corresponding tangential velocity is , consistent with the kinematics of the stars of the Galactic disk. == Spectral class == 2MASS J0348−6022 is classified as a late T-type brown dwarf with the spectral class T7, distinguished by the presence of strong methane (CH4) and water (H2O) absorption bands in its near-infrared spectrum between wavelengths 1.2 and 2.35 μm. This can be explained by the presence of CH4 in its atmosphere, which is opaque to wavelengths around 3.3 μm. === Physical effects === The spectral lines in 2MASS J0348−6022's spectrum are Doppler-broadened due to the brown dwarf's rapid rotation, consistent with its short photometric periodicity. The vibrational temperature is commonly used in thermodynamics, to simplify certain equations.

432

3

7.0

21

-6.04697

A

One mole of $\mathrm{H}_2 \mathrm{O}(l)$ is compressed from a state described by $P=1.00$ bar and $T=350$. K to a state described by $P=590$. bar and $T=750$. K. In addition, $\beta=2.07 \times 10^{-4} \mathrm{~K}^{-1}$ and the density can be assumed to be constant at the value $997 \mathrm{~kg} \mathrm{~m}^{-3}$. Calculate $\Delta S$ for this transformation, assuming that $\kappa=0$.

J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). The limit of this sum as N \rightarrow \infty becomes an integral: :S^\circ = \sum_{k=1}^N \Delta S_k = \sum_{k=1}^N \frac{dQ_k}{T} \rightarrow \int _0 ^{T_2} \frac{dS}{dT} dT = \int _0 ^{T_2} \frac {C_{p_k}}{T} dT In this example, T_2 =298.15 K and C_{p_k} is the molar heat capacity at a constant pressure of the substance in the reversible process . Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. From these sources: :*a - K.K. Kelley, Bur. Mines Bull. 383, (1935). :*b - :*c - Brewer, The thermodynamic and physical properties of the elements, Report for the Manhattan Project, (1946). :*d - :*e - :*f - :*g - ; :*h - :*i - . :*j - :*k - Int. National Critical Tables, vol. 3, p. 306, (1928). :*l - :*m - :*n - :*o - :*p - :*q - :*r - :*s - H.A. Jones, I. Langmuir, Gen. Electric Rev., vol. 30, p. 354, (1927). == See also == Category:Properties of chemical elements Category:Chemical element data pages The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. With :T \sim t_ u and location-scale family transformation :X = \mu + \tau T we get :X \sim lst(\mu, \tau^2, u) The resulting distribution is also called the non- standardized Student's t-distribution. ===Density and first two moments=== The location-scale t distribution has a density defined by: :p(x\mid u,\mu,\tau) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u}\tau\,} \left(1+\frac{1}{ u} \left( \frac{ x-\mu } {\tau } \right)^2\right)^{-( u+1)/2} Equivalently, the density can be written in terms of \tau^2: :p(x\mid u, \mu, \tau^2) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u\tau^2}} \left(1+\frac{1}{ u}\frac{(x-\mu)^2}{{\tau}^2}\right)^{-( u+1)/2} Other properties of this version of the distribution are: :\begin{align} \operatorname{E}(X) &= \mu & \text{ for } u > 1 \\\ \operatorname{var}(X) &= \tau^2\frac{ u}{ u-2} & \text{ for } u > 2 \\\ \operatorname{mode}(X) &= \mu \end{align} ===Special cases=== * If X follows a location-scale t-distribution X \sim \mathrm{lst}\left(\mu, \tau^2, u\right) then for u \rightarrow \infty X is normally distributed X \sim \mathrm{N}\left(\mu, \tau^2\right) with mean \mu and variance \tau^2. The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. The following images show the density of the t-distribution for increasing values of u. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b).

524

7.27

0.195

-233

57.2

E

A mass of $34.05 \mathrm{~g}$ of $\mathrm{H}_2 \mathrm{O}(s)$ at $273 \mathrm{~K}$ is dropped into $185 \mathrm{~g}$ of $\mathrm{H}_2 \mathrm{O}(l)$ at $310 . \mathrm{K}$ in an insulated container at 1 bar of pressure. Calculate the temperature of the system once equilibrium has been reached. Assume that $C_{P, m}$ for $\mathrm{H}_2 \mathrm{O}(l)$ is constant at its values for $298 \mathrm{~K}$ throughout the temperature range of interest.

The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. The potential temperature is denoted \theta and, for a gas well-approximated as ideal, is given by : \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, where T is the current absolute temperature (in K) of the parcel, R is the gas constant of air, and c_p is the specific heat capacity at a constant pressure. The analysis here shows the simple unattributed formula and the Antoine equation are reasonably accurate at 100 °C, but quite poor for lower temperatures above freezing. For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state, pv = RT can be substituted into the 1st law yielding, after some rearrangement: : \frac{dp}{p} = {\frac{c_p}{R}\frac{dT}{T}}, where the dh = c_{p}dT was used and both terms were divided by the product pv Integrating yields: : \left(\frac{p_1}{p_0}\right)^{R/c_p} = \frac{T_1}{T_0}, and solving for T_{0}, the temperature a parcel would acquire if moved adiabatically to the pressure level p_{0}, you get: : T_0 = T_1 \left(\frac{p_0}{p_1}\right)^{R/c_p} \equiv \theta. == Potential virtual temperature == The potential virtual temperature \theta_{v}, defined by : \theta_v = \theta \left( 1 + 0.61 r - r_L \right), is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pressure P0. "A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The equilibrium constants may be derived by best-fitting of the experimental data with a chemical model of the equilibrium system. == Experimental methods == There are four main experimental methods. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Some of these are (in approximate order of increasing accuracy): Name Formula Description "Eq. 1" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. The Boyle temperature is formally defined as the temperature for which the second virial coefficient, B_{2}(T), becomes zero. The equilibrium constant value can be determined if any one of these concentrations can be measured. The (unattributed) constants are given as {| class="wikitable" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Equilibrium constants are determined in order to quantify chemical equilibria. A large number of general-purpose computer programs for equilibrium constant calculation have been published. The former is an extremely simple Antoine equation, while the latter is a polynomial. ==Graphical pressure dependency on temperature== ==See also== *Dew point *Gas laws *Lee–Kesler method *Molar mass ==References== ==Further reading== * * * * ==External links== * * Category:Thermodynamic properties Category:Atmospheric thermodynamics The potential temperature of a parcel of fluid at pressure P is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure P_{0}, usually . It is at this temperature that the attractive forces and the repulsive forces acting on the gas particles balance out P = RT \left(\frac{1}{V_m} + \frac{B_{2}(T)}{V_m^2} + \cdots \right) This is the virial equation of state and describes a real gas. :H2O <=> H+ + OH-: K_\mathrm{W}^' = \frac{[H^+][OH^-]}{[H_2O]} With dilute solutions the concentration of water is assumed constant, so the equilibrium expression is written in the form of the ionic product of water. As expected, Buck's equation for > 0 °C is significantly more accurate than Tetens, and its superiority increases markedly above 50 °C, though it is more complicated to use. For this assumption to be valid, equilibrium constants must be determined in a medium of relatively high ionic strength. One or more equilibrium constants may be parameters of the refinement.

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Calculate $\Delta H_f^{\circ}$ for $N O(g)$ at $975 \mathrm{~K}$, assuming that the heat capacities of reactants and products are constant over the temperature interval at their values at $298.15 \mathrm{~K}$.

Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. Integration of this equation permits the evaluation of the heat of reaction at one temperature from measurements at another temperature.Laidler K.J. and Meiser J.H., "Physical Chemistry" (Benjamin/Cummings 1982), p.62Atkins P. and de Paula J., "Atkins' Physical Chemistry" (8th edn, W.H. Freeman 2006), p.56 :\Delta H^\circ \\! \left( T \right) = \Delta H^\circ \\! \left( T^\circ \right) + \int_{T^\circ}^{T} \Delta C_P^\circ \, \mathrm{d} T Pressure variation effects and corrections due to mixing are generally minimal unless a reaction involves non-ideal gases and/or solutes, or is carried out at extremely high pressures. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Therefore, the heat capacity ratio in this example is 1.4. The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? See relations between specific heats for the derivation of the thermodynamic relations between the heat capacities. Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. \, Then :1-e^{-\hbar\omega_\alpha/k_{\rm B}T} \approx \hbar\omega_\alpha/k_{\rm B}T \, and we have :F=N\varepsilon_0+Nk_{\rm B}T\sum_{\alpha}\log\left(\frac{\hbar\omega_{\alpha}}{k_{\rm B}T}\right). 'At, Astatine', system no. 8a, Springer-Verlag, Berlin, , pp. 116–117 ===LNG=== As quoted from various sources in: * J.A. Dean (ed.), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T. H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity.

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A two-level system is characterized by an energy separation of $1.30 \times 10^{-18} \mathrm{~J}$. At what temperature will the population of the ground state be 5 times greater than that of the excited state?

Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. An excited state is any state with energy greater than the ground state. The temperature of a group of particles is indicative of the level of excitation (with the notable exception of systems that exhibit negative temperature). The excitation temperature can even be negative for a system with inverted levels (such as a maser). It satisfies : \frac{n_{\rm u}}{n_{\rm l}} = \frac{g_{\rm u}}{g_{\rm l}} \exp{ \left(-\frac{\Delta E}{k T_{\rm ex}} \right) }, where * is the number of particles in an upper (e.g. excited) state; * is the statistical weight of those upper-state particles; * is the number of particles in a lower (e.g. ground) state; * is the statistical weight of those lower-state particles; * is the exponential function; * is the Boltzmann constant; * is the difference in energy between the upper and lower states. Excitation refers to an increase in energy level above a chosen starting point, usually the ground state, but sometimes an already excited state. However, it is not easy to measure them compared to ground-state absorption, and in some cases complete bleaching of the ground state is required to measure excited-state absorption. ==Reaction== A further consequence of excited-state formation may be reaction of the atom or molecule in its excited state, as in photochemistry. == See also == * Rydberg formula * Stationary state * Repulsive state == References == == External links == * NASA background information on ground and excited states Category:Quantum mechanics In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit negative temperature. == Absence of nodes in one dimension == In one dimension, the ground state of the Schrödinger equation can be proven to have no nodes. Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. thumb|250px|Schematic picture of energy levels and examples of different states. In observations of the 21 cm line of hydrogen, the apparent value of the excitation temperature is often called the "spin temperature". ==References== Category:Temperature 300 px|thumb|A Jablonski diagram showing the excitation of molecule A to its singlet excited state (1A*) followed by intersystem crossing to the triplet state (3A) that relaxes to the ground state by phosphorescence. This phenomenon has been studied in the case of a two-dimensional gas in some detail, analyzing the time taken to relax to equilibrium. == Calculation of excited states == Excited states are often calculated using coupled cluster, Møller–Plesset perturbation theory, multi-configurational self-consistent field, configuration interaction, and time-dependent density functional theory. ==Excited-state absorption== The excitation of a system (an atom or molecule) from one excited state to a higher-energy excited state with the absorption of a photon is called excited-state absorption (ESA). Fortunately, it is often possible to define a quasi-Fermi level and quasi-temperature for a given location, that accurately describe the occupation of states in terms of a thermal distribution. The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. The ground state (blue) is x2–y2 orbitals; the excited orbitals are in green; the arrows illustrate inelastic x-ray spectroscopy. Relaxation of the excited state to its lowest vibrational level is called vibrational relaxation. The vibrational ground states of each electronic state are indicated with thick lines, the higher vibrational states with thinner lines. The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. Excited-state absorption measurements are done using pump–probe techniques such as flash photolysis.

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