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SciBench-TruthfulQA-RAG

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An electric field $\vec{E}$ with an average magnitude of about $150 \mathrm{~N} / \mathrm{C}$ points downward in the atmosphere near Earth's surface. We wish to "float" a sulfur sphere weighing $4.4 \mathrm{~N}$ in this field by charging the sphere. What charge (both sign and magnitude) must be used?

This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. The Pauthenier equation states K. Adamiak, "Rate of charging of spherical particles by monopolar ions in electric fields", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. A surface charge is an electric charge present on a two-dimensional surface. Charges are arbitrarily labeled as positive(+) or negative(-). At a certain pH, the average surface charge will be equal to zero; this is known as the point of zero charge (PZC). It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). Which leads to the simple expression: \sigma=\frac{\varepsilon \varepsilon_0 \psi_0}{\lambda_D} thumb|right|upright=1.25|alt=A bulk solid, containing positive charge, borders a bulk liquid, containing negative charge. These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge distribution on the surface. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. Surface charge practically always appears on the particle surface when it is placed into a fluid. Overall, two layers of charge and a potential drop from the electrode to the edge of the outer layer (outer Helmholtz Plane) are observed. If the number of adsorbed cations exceeds the number of adsorbed anions, the surface would have a net positive electric charge. When the cross does not reach the edges of the field, it becomes a mobile charge. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. Thereby is the electrical charge enclosed by the Gaussian surface. The term charge can also be used as a verb; for example, if an escutcheon depicts three lions, it is said to be charged with three lions; similarly, a crest or even a charge itself may be "charged", such as a pair of eagle wings charged with trefoils (as on the coat of arms of Brandenburg). By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e. In chemistry, there are many different processes which can lead to a surface being charged, including adsorption of ions, protonation or deprotonation, and, as discussed above, the application of an external electric field.

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Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of $0.108 \mathrm{~N}$ when their center-to-center separation is $50.0 \mathrm{~cm}$. The spheres are then connected by a thin conducting wire. When the wire is removed, the spheres repel each other with an electrostatic force of $0.0360 \mathrm{~N}$. Of the initial charges on the spheres, with a positive net charge, what was (a) the negative charge on one of them?

thumb|300px|right|visualized induced-charge electrokinetic flow pattern around a carbon-steel sphere (diameter=1.2mm). For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. Surface Charging and Points of Zero Charge. Double layer forces occur between charged objects across liquids, typically water. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. An electrical double layer develops near charged surfaces (or another charged objects) in aqueous solutions. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. The electrodipping force is a force proposed to explain the observed attraction that arises among small colloidal particles attached to an interface between immiscible liquids. Chenhui Peng et al.C. Peng, I. Lazo, S. V. Shiyanovskii, O. D. Lavrentovich , Induced-charge electro-osmosis around metal and Janus spheres in water: Patterns of flow and breaking symmetries, arXiv preprint , (2014) also experimentally showed the patterns of electro-osmotic flow around an Au sphere when alternating current (AC) is involved (E=10mV/μm, f=1 kHz). For unequally charged objects and eventually at shorted distances, these forces may also be attractive. Within this double layer, the first layer corresponds to the charged surface. As a result of this migration, the negative charges move to the side which is close to the positive (or higher) voltage while the positive charges move to the opposite side of the particle. :*If abla = 0, then exactly one solution exists, i.e. the line just touches the sphere in one point (case 2). Aspects, 73, (1993) 29-48.Electrokinetics and Electrohydrodynamics in Microsystems CISM Courses and Lectures Volume 530, 2011, pp 221-297 Induced-Charge Electrokinetic Phenomena Martin Z. BazantY. At larger distances, oppositely charged surfaces repel and equally charged ones attract. ==Charge regulating surfaces== While the superposition approximation is actually exact at larger distances, it is no longer accurate at smaller separations. In this case, the surface charge density decreases upon approach. Even if the large spheres are not in a close-packed arrangement, it is always possible to insert some smaller spheres of up to 0.29099 of the radius of the larger sphere. Induced-charge electrokinetics in physics is the electrically driven fluid flow and particle motion in a liquid electrolyte.V. G. Levich, Physicochemical Hydrodynamics. Englewood Cliffs, N.J., Prentice-Hall, (1962) Consider a metal particle (which is neutrally charged but electrically conducting) in contact with an aqueous solution in a chamber/channel. The problem of finding the arrangement of n identical spheres that maximizes the number of contact points between the spheres is known as the "sticky-sphere problem".

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A uniform electric field exists in a region between two oppositely charged plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, $2.0 \mathrm{~cm}$ away, in a time $1.5 \times 10^{-8} \mathrm{~s}$. What is the speed of the electron as it strikes the second plate?

This velocity is the speed with which electromagnetic waves penetrate into the conductor and is not the drift velocity of the conduction electrons. Without the presence of an electric field, the electrons have no net velocity. * "Velocity of Propagation of Electric Field", Theory and Calculation of Transient Electric Phenomena and Oscillations by Charles Proteus Steinmetz, Chapter VIII, p. 394-, McGraw-Hill, 1920. Relativistic electron beams are streams of electrons moving at relativistic speeds. When a DC voltage is applied, the electron drift velocity will increase in speed proportionally to the strength of the electric field. The inch per second is a unit of speed or velocity. That is, the velocity of propagation has no appreciable effect unless the return conductor is very distant, or entirely absent, or the frequency is so high that the distance to the return conductor is an appreciable portion of the wavelength.Theory and calculation of transient electric phenomena and oscillations By Charles Proteus Steinmetz ==Electric drift== The drift velocity deals with the average velocity of a particle, such as an electron, due to an electric field. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. AC voltages cause no net movement; the electrons oscillate back and forth in response to the alternating electric field (over a distance of a few micrometers – see example calculation). ==See also== *Speed of light *Speed of gravity *Speed of sound *Telegrapher's equations *Reflections of signals on conducting lines ==References== ==Further reading== * Alfvén, H. (1950). The important part of the electric field of a conductor extends to the return conductor, which usually is only a few feet distant. The word electricity refers generally to the movement of electrons (or other charge carriers) through a conductor in the presence of a potential difference or an electric field. In everyday electrical and electronic devices, the signals travel as electromagnetic waves typically at 50%–99% of the speed of light in vacuum, while the electrons themselves move much more slowly; see drift velocity and electron mobility. ==Electromagnetic waves== The speed at which energy or signals travel down a cable is actually the speed of the electromagnetic wave traveling along (guided by) the cable. As a consequence of Snell's Law and the extremely low speed, electromagnetic waves always enter good conductors in a direction that is within a milliradian of normal to the surface, regardless of the angle of incidence. ===Electromagnetic waves in circuits=== In the theoretical investigation of electric circuits, the velocity of propagation of the electromagnetic field through space is usually not considered; the field is assumed, as a precondition, to be present throughout space. It has been suggested that relativistic electron beams could be used to heat and accelerate the reaction mass in electrical rocket engines that Dr. Robert W. Bussard called quiet electric-discharge engines (QEDs). ==References== ==External links== *PEARL Lab @ UHawaii *Applying REBs for the development of high-powered microwaves (HPM) Category:Electron beam Category:Quantum mechanics Category:Special relativity In copper at 60Hz, v \approx 3.2m/s. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. In other words, the greater the distance from the conductor, the more the electric field lags. In general, an electron will propagate randomly in a conductor at the Fermi velocity.Academic Press dictionary of science and technology By Christopher G. Morris, Academic Press. The electric field starts at the conductor, and propagates through space at the velocity of light (which depends on the material it is traveling through). 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. The drift velocity in a 2 mm diameter copper wire in 1 ampere current is approximately 8 cm per hour.

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Two point charges of $30 \mathrm{nC}$ and $-40 \mathrm{nC}$ are held fixed on an $x$ axis, at the origin and at $x=72 \mathrm{~cm}$, respectively. A particle with a charge of $42 \mu \mathrm{C}$ is released from rest at $x=28 \mathrm{~cm}$. If the initial acceleration of the particle has a magnitude of $100 \mathrm{~km} / \mathrm{s}^2$, what is the particle's mass?

The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. An ion with a mass of 100 Da (daltons) () carrying two charges () will be observed at . thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. This differential equation is the classic equation of motion for charged particles. The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. This differential equation is the classic equation of motion of a charged particle in a vacuum. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. In this case, one must solve the PB equation together with an appropriate model of the surface charging process. The CODATA recommended value for an electron is ==Origin== When charged particles move in electric and magnetic fields the following two laws apply: *Lorentz force law: \mathbf{F} = Q (\mathbf{E} + \mathbf{v} \times \mathbf{B}), *Newton's second law of motion:\mathbf{F}=m\mathbf{a} = m \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} where F is the force applied to the ion, m is the mass of the particle, a is the acceleration, Q is the electric charge, E is the electric field, and v × B is the cross product of the ion's velocity and the magnetic flux density. Thus, the m/z of an ion alone neither infers mass nor the number of charges. Often, the charge can be inferred from theoretical considerations, so the charge-to-mass ratio provides a way to calculate the mass of a particle. A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. At very low energies, space charge has a large effect on a particle beam and thus becomes hard to calculate. Often, the charge-to-mass ratio can be determined by observing the deflection of a charged particle in an external magnetic field. A charged particle accelerator is a complex machine that takes elementary charged particles and accelerates them to very high energies. Assuming a normal distribution of particle positions and impulses, a charged particle beam (or a bunch of the beam) is characterized by * the species of particle, e.g. electrons, protons, or atomic nuclei * the mean energy of the particles, often expressed in electronvolts (typically keV to GeV) * the (average) particle current, often expressed in amperes * the particle beam size, often using the so-called β-function * the beam emittance, a measure of the area occupied by the beam in one of several phase spaces. When performing a modeling task for any accelerator operation, the results of charged particle beam dynamics simulations must feed into the associated application. This equation can be extended to more highly charged particles by reinterpreting the charge Q as an effective charge. Codes for this computation include *ABCI ABCI home page at kek.jp *ACE3P ACE3P at slac.stanford.gov *CST Studio Suite CST, Computer Simulation Technology at cst.com *GdfidLGdfidL, Gitter drueber, fertig ist die Laube at gdfidl.de *TBCI T. Weiland, DESY * VSim ==Magnet and other hardware-modeling codes== To control the charged particle beam, appropriate electric and magnetic fields must be created.

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In Fig. 21-26, particle 1 of charge $-5.00 q$ and particle 2 of charge $+2.00 q$ are held at separation $L$ on an $x$ axis. If particle 3 of unknown charge $q_3$ is to be located such that the net electrostatic force on it from particles 1 and 2 is zero, what must be the $x$ coordinate of particle 3?

The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and \hat\mathbf r is the unit vector in the direction of . In physics, a charged particle is a particle with an electric charge. Surface Charging and Points of Zero Charge. A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order {v^2}/{c^2} between two charged particles in a vacuum and is given by L = L_\text{f} + L_\text{int}, where the free particle Lagrangian is L_\text{f} = \frac{1}{2} m_1 v_1^2 + \frac{1}{8c^2} m_1 v_1^4 + \frac{1}{2} m_2 v_2^2 + \frac{1}{8c^2} m_2 v_2^4, and the interaction Lagrangian is L_\text{int} = L_\text{C} + L_\text{D}, where the Coulomb interaction is L_\text{C} = -\frac{q_1 q_2}{r}, and the Darwin interaction is L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf{r} \hat\mathbf{r}\right] \cdot \mathbf v_2. Related values associated with the soil characteristics exist along with the pzc value, including zero point of charge (zpc), point of zero net charge (pznc), etc. == Term definition of point of zero charge == The point of zero charge is the pH for which the net surface charge of adsorbent is equal to zero. With the electrostatic force being proportional to r^{-2}, individual particle-particle interactions are long- range in nature, presenting a challenging computational problem in the simulation of particulate systems. IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. thumb|Electrical double layer (EDL) around a negatively charged particle in suspension in water. For example, the surface charge of adsorbent is described by the ion that lies on the surface of the particle (adsorbent) structure like image. Of interest, this represents the first three-dimensional solution. Charges are arbitrarily labeled as positive(+) or negative(-). A related concept in electrochemistry is the electrode potential at the point of zero charge. The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. Also, this represents the first two-dimensional solution. Simply applying this cutoff method introduces a discontinuity in the force at r_c that results in particles experiencing sudden impulses when other particles cross the boundary of their respective interaction spheres. Assuming a normal distribution of particle positions and impulses, a charged particle beam (or a bunch of the beam) is characterized by * the species of particle, e.g. electrons, protons, or atomic nuclei * the mean energy of the particles, often expressed in electronvolts (typically keV to GeV) * the (average) particle current, often expressed in amperes * the particle beam size, often using the so-called β-function * the beam emittance, a measure of the area occupied by the beam in one of several phase spaces. The single electron may reside at any point on the surface of the unit sphere. Only the existence of two 'types' of charges is known, there isn't anything inherent about positive charges that makes them positive, and the same goes for the negative charge. == Examples == === Positively charged particles === * protons and atomic nuclei * positrons (antielectrons) * alpha particles * positive charged pions * cations === Negatively charged particles === * electrons * antiprotons * muons * tauons * negative charged pions * anions === Particles without an electric charge=== * neutrons * photons * neutrinos * neutral pions *z boson *higgs boson *atoms ==References== * * * * * ==External links== * Charged particle motion in E/B Field Category:Charge carriers Category:Particle physics

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An isolated conductor has net charge $+10 \times 10^{-6} \mathrm{C}$ and a cavity with a particle of charge $q=+3.0 \times 10^{-6} \mathrm{C}$. What is the charge on the cavity wall?

Cavity quantum electrodynamics (cavity QED) is the study of the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of photons is significant. In modern cavity wall construction, cavity insulation is typically added. thumb|Circuit diagram of a charge qubit circuit. A cavity wall is a type of wall that has a hollow center. He shares half of the prize for developing a new field called cavity quantum electrodynamics (CQED) – whereby the properties of an atom are controlled by placing it in an optical or microwave cavity. This results in two overlapping layers of the superconducting metal, in between which a thin layer of insulator (normally aluminum oxide) is deposited. == Hamiltonian == If the Josephson junction has a junction capacitance C_{\rm J}, and the gate capacitor C_{\rm g}, then the charging (Coulomb) energy of one Cooper pair is: :E_{\rm C}=(2e)^2/2(C_{\rm g}+C_{\rm J}). In superconducting quantum computing, a charge qubit is formed by a tiny superconducting island coupled by a Josephson junction (or practically, superconducting tunnel junction) to a superconducting reservoir (see figure). In quantum computing, a charge qubit (also known as Cooper-pair box) is a qubit whose basis states are charge states (i.e. states which represent the presence or absence of excess Cooper pairs in the island). QIT may refer to: * QIT-Fer et Titane, a Canadian mining company * Quadrupole ion trap * Quantum information theory * Queensland University of Technology * Q = It, the formula describing charge in terms of current and time The conductance quantum, denoted by the symbol , is the quantized unit of electrical conductance. The resonance frequencies are given by \lambda/2: \quad u_n=\frac{c}{\sqrt{\varepsilon_{\text{eff}}}}\frac{n}{2 \ell} \quad (n=1,2,3,\ldots) \qquad \lambda/4:\quad u_n=\frac{c}{\sqrt{\varepsilon_{\text{eff}}}}\frac{2n+1}{4 \ell} \quad (n=0,1,2,\ldots) with \varepsilon_{\text{eff}} being the effective dielectric permittivity of the device. == Artificial atoms, Qubits == The first realized artificial atom in circuit QED was the so-called Cooper-pair box, also known as the charge qubit. Circuit quantum electrodynamics (circuit QED) provides a means of studying the fundamental interaction between light and matter (quantum optics). The voltage is: V = -\frac{(\mu_1 - \mu_2)}{e} , where e is the electron charge. If the detuning is significantly larger than the combined cavity and atomic linewidth the cavity states are merely shifted by \pm g^2/\Delta (with the detuning \Delta=\omega_a-\omega_r) depending on the atomic state. *Cavity wall, Energy Saving Trust *Cavity wall insulation (CWI): consumer guide to issues arising from installations, 14 October 2019, Department for Business, Energy & Industrial Strategy *Cavity Wall Insulation Victims Alliance Category:Masonry Category:Construction Category:Types of wall Structural properties of a concrete-block cavity- wall construction sponsored by the National Concrete Masonry Association. This immobilises the air within the cavity (air is still the actual insulator), preventing convection, and can substantially reduce space heating costs. thumb|A wall that has had cavity wall insulation installed (after construction), with refilled holes highlighted with arrows During construction of new buildings, cavities are often filled with glass fiber wool or mineral wool panels placed between the two leaves (sides) of the wall, but many other building insulation materials offer various advantages and many others are also widely used. Recent work has shown T2 times approaching 100 μs using a type of charge qubit known as a transmon inside a three-dimensional superconducting cavity.C. Rigetti et al., "Superconducting qubit in waveguide cavity with coherence time approaching 0.1 ms," arXiv:1202.5533 (2012) Understanding the limits of T2 is an active area of research in the field of superconducting quantum computing. == Fabrication == Charge qubits are fabricated using techniques similar to those used for microelectronics. Cavity wall insulation also helps to prevent convection and can keep a house warm by making sure that less heat is lost through walls; this can also thus be a more cost-efficient way of heating a house. One function of the cavity is to drain water through weep holes at the base of the wall system or above windows. In 1999, coherent oscillations in the charge Qubit were first observed by Nakamura et al. Manipulation of the quantum states and full realization of the charge qubit was observed 2 years later. Thermal mass cavity walls are thick walls.

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Point charges of $+6.0 \mu \mathrm{C}$ and $-4.0 \mu \mathrm{C}$ are placed on an $x$ axis, at $x=8.0 \mathrm{~m}$ and $x=16 \mathrm{~m}$, respectively. What charge must be placed at $x=24 \mathrm{~m}$ so that any charge placed at the origin would experience no electrostatic force?

Surface Charging and Points of Zero Charge. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. Electric charge can be positive or negative (commonly carried by protons and electrons respectively, by convention). IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. The sign of the space charge can be either negative or positive. The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). ChargePoint (formerly Coulomb Technologies) is an American electric vehicle infrastructure company based in Campbell, California. By convention, the charge of an electron is negative, −e, while that of a proton is positive, +e. Space charge is an interpretation of a collection of electric charges in which excess electric charge is treated as a continuum of charge distributed over a region of space (either a volume or an area) rather than distinct point-like charges. The proton has a charge of +e, and the electron has a charge of −e. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. No force, either of attraction or of repulsion, can be observed between an electrified body and a body not electrified.James Clerk Maxwell (1891) A Treatise on Electricity and Magnetism, pp. 32–33, Dover Publications ==The role of charge in electric current== Electric current is the flow of electric charge through an object. Electric charges produce electric fields. In ordinary matter, negative charge is carried by electrons, and positive charge is carried by the protons in the nuclei of atoms. Rear..JPG|Ford Focus being charged on a roadside ChargePoint station. In June 2017, ChargePoint took over 9,800 electric vehicle charging spots from GE. In contemporary understanding, positive charge is now defined as the charge of a glass rod after being rubbed with a silk cloth, but it is arbitrary which type of charge is called positive and which is called negative. A related concept in electrochemistry is the electrode potential at the point of zero charge. In this case, the electrical potential profile ψ(z) near a charged interface will only depend on the position z.

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The electric field in a certain region of Earth's atmosphere is directed vertically down. At an altitude of $300 \mathrm{~m}$ the field has magnitude $60.0 \mathrm{~N} / \mathrm{C}$; at an altitude of $200 \mathrm{~m}$, the magnitude is $100 \mathrm{~N} / \mathrm{C}$. Find the net amount of charge contained in a cube $100 \mathrm{~m}$ on edge, with horizontal faces at altitudes of 200 and $300 \mathrm{~m}$.

220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. The Pauthenier equation states K. Adamiak, "Rate of charging of spherical particles by monopolar ions in electric fields", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. thumb|400px|A snapshot of the variation of the Earth's magnetic field from its intrinsic field at 400 km altitude, due to the ionospheric current systems. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Thus, v_i is the number of vertices where the given number of edges meet, e is the total number of edges, f_3 is the number of triangular faces, f_4 is the number of quadrilateral faces, and \theta_1 is the smallest angle subtended by vectors associated with the nearest charge pair. * For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. The horizontal intensity of magnetic field peaks at ~12 LT. For conductors, p=3. * For N = 4, electrons reside at the vertices of a regular tetrahedron. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. The mechanism that produced the variation in the magnetic field was proposed as a band of current about 300 km in width flowing over the dip equator. Int.,159, 521-547. == External links == * A movie of the magnetic fields generated by the equatorial electrojet, . This electric field gives a primary eastwards Pedersen current. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. Global-scale ionospheric circulation establishes a Sq (solar quiet) current system in the E region of the Earth's ionosphere (100-130 km altitude), and a primary eastwards electric field near day-side magnetic equator, where the magnetic field is horizontal and northwards. * For N = 3, electrons reside at the vertices of an equilateral triangle about any great circle.. Electr. 52, 449 – 451 *Chapman, S. 1951, The equatorial electrojet as detected from the abnormal electric current distribution above Huancayo, Peru, and elsewhere. The equatorial electrojet (EEJ) is a narrow ribbon of current flowing eastward in the day time equatorial region of the Earth's ionosphere. We can similarly describe the electric field E so that . Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell.

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What would be the magnitude of the electrostatic force between two 1.00 C point charges separated by a distance of $1.00 \mathrm{~m}$ if such point charges existed (they do not) and this configuration could be set up?

If is the distance between the charges, the magnitude of the force is |\mathbf{F}|=\frac{|q_1q_2|}{4\pi\varepsilon_0 r^2}, where is the electric constant. The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules. ==Electrostatic potential energy of one point charge== ===One point charge q in the presence of another point charge Q=== right|A point charge q in the electric field of another charge Q.|thumb|434px The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). A common question arises concerning the interaction of a point charge with its own electrostatic potential. The force is along the straight line joining the two charges. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. The resulting force vector is parallel to the electric field vector at that point, with that point charge removed. If the product is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive. == Vector form == thumb|right|350px|In the image, the vector is the force experienced by , and the vector is the force experienced by . The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. The scalar form gives the magnitude of the vector of the electrostatic force between two point charges and , but not its direction. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). Coulomb's law in vector form states that the electrostatic force \mathbf{F}_1 experienced by a charge, q_1 at position \mathbf{r}_1, in the vicinity of another charge, q_2 at position \mathbf{r}_2, in a vacuum is equal to \mathbf{F}_1 = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{r}_1-\mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}_{12}}{|\mathbf{r}_{12}|^2} where \boldsymbol{r}_{12} = \boldsymbol{r}_1 - \boldsymbol{r}_2 is the vectorial distance between the charges, \widehat{\mathbf{r}}_{12}=\frac{\mathbf{r}_{12}}{|\mathbf{r}_{12}|} a unit vector pointing from q_2 to and \varepsilon_0 the electric constant. The electrostatic force \mathbf{F}_2 experienced by q_2, according to Newton's third law, is === System of discrete charges === The law of superposition allows Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply the vector addition of the individual forces acting alone on that point charge due to each one of the charges. Using Coulomb's law, it is known that the electrostatic force F and the electric field E created by a discrete point charge Q are radially directed from Q. In 1767, he conjectured that the force between charges varied as the inverse square of the distance. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

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E

An electric dipole consisting of charges of magnitude $1.50 \mathrm{nC}$ separated by $6.20 \mu \mathrm{m}$ is in an electric field of strength 1100 $\mathrm{N} / \mathrm{C}$. What is the magnitude of the electric dipole moment?

Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The magnitude of the torque is proportional to both the magnitude of the charges and the separation between them, and varies with the relative angles of the field and the dipole: \left|\mathbf{\tau}\right| = \left|q\mathbf{r}\right| \left|\mathbf{E}\right|\sin\theta. (See electron electric dipole moment). For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: :U = \mathbf d_{\rm e} \cdot \mathbf E. It can be shown that this net force is generally parallel to the dipole moment. ==Expression (general case)== More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is: \mathbf{p}(\mathbf{r}) = \int_{V} \rho(\mathbf{r}') \left(\mathbf{r}' - \mathbf{r}\right) d^3 \mathbf{r}', where r locates the point of observation and d3r′ denotes an elementary volume in V. An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. The nucleon magnetic moments are the intrinsic magnetic dipole moments of the proton and neutron, symbols μp and μn. The dipole moment is particularly useful in the context of an overall neutral system of charges, for example a pair of opposite charges, or a neutral conductor in a uniform electric field. The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . Some authors may split in half and use since this quantity is the distance between either charge and the center of the dipole, leading to a factor of two in the definition. In the case of two classical point charges, +q and -q, with a displacement vector, \mathbf{r}, pointing from the negative charge to the positive charge, the electric dipole moment is given by \mathbf{p} = q\mathbf{r}. The electric dipole moment vector also points from the negative charge to the positive charge. It is possible to calculate dipole moments from electronic structure theory, either as a response to constant electric fields or from the density matrix. If the charge, e, is omitted from the electric dipole operator during this calculation, one obtains \mathbf{R}_\alpha as used in oscillator strength. == Applications == The transition dipole moment is useful for determining if transitions are allowed under the electric dipole interaction. This is the vector sum of the individual dipole moments of the neutral charge pairs. The transition dipole moment or transition moment, usually denoted \mathbf{d}_{nm} for a transition between an initial state, m, and a final state, n, is the electric dipole moment associated with the transition between the two states. This quantity is used in the definition of polarization density. ==Energy and torque== thumb|187x187px|Electric dipole p and its torque τ in a uniform E field. To the accuracy of this dipole approximation, as shown in the previous section, the dipole moment density p(r) (which includes not only p but the location of p) serves as P(r).

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B

What equal positive charges would have to be placed on Earth and on the Moon to neutralize their gravitational attraction?

The acceleration due to gravity on the surface of the Moon is approximately 1.625 m/s2, about 16.6% that on Earth's surface or 0.166 . The lunar GM is 1/81.30057 of the Earth's GM. == Theory == For the lunar gravity field, it is conventional to use an equatorial radius of R = 1738.0 km. There is growing evidence that fine particles of moondust might actually float, ejected from the lunar surface by electrostatic repulsion. The charged particle lunar environment experiment was deployed approximately 3 meters northeast of the central station. Volta Latitude Longitude Diameter B 54.6° N 83.5° W 9 km D 52.5° N 83.3° W 20 km ==References== * * * * * * * * * * * * Category:Impact craters on the Moon Category:Alessandro Volta Volta is a lunar impact crater near the northwest limb of the Moon. Lunar Gravity Coefficients nm Jn Cnm Snm 20 203.3 × 10−6 — — 21 — 0 0 22 — 22.4 × 10−6 0 30 8.46 × 10−6 — — 31 — 28.48 × 10−6 5.89 × 10−6 32 — 4.84 × 10−6 1.67 × 10−6 33 — 1.71 × 10−6 −0.25 × 10−6 The J2 coefficient for an oblate shape to the gravity field is affected by rotation and solid-body tides whereas C22 is affected by solid-body tides. Because weight is directly dependent upon gravitational acceleration, things on the Moon will weigh only 16.6% (= 1/6) of what they weigh on the Earth. ==Gravitational field== The gravitational field of the Moon has been measured by tracking the radio signals emitted by orbiting spacecraft. Consequently, it is conventional to express the lunar mass M multiplied by the gravitational constant G. The gravitational potential V at an external point is conventionally expressed as positive in astronomy and geophysics, but negative in physics. Lunar Gravity Fields Designation Degree Mission IDs Citation LP165P 165 LO A15 A16 Cl LP GLGM3 150 LO A15 A16 Cl LP CEGM01 50 Ch 1 SGM100h 100 LO A15 A16 Cl LP K/S SGM150J 150 LO A15 A16 Cl LP K/S CEGM02 100 LO A15 A16 Cl LP K/S Ch1 GL0420A 420 G GL0660B 660 G GRGM660PRIM 660 G GL0900D 900 G GRGM900C 900 G GRGM1200A 1200 G CEGM03 100 LO A15 A16 Cl LP Ch1 K/S Ch5T1 A major feature of the Moon's gravitational field is the presence of mascons, which are large positive gravity anomalies associated with some of the giant impact basins. Detailed data collected has shown that for low lunar orbit the only "stable" orbits are at inclinations near 27°, 50°, 76°, and 86°. The mass of the Moon is M = 7.3458 × 1022 kg and the mean density is 3346 kg/m3. Much of these details are still speculative, but the Lunar Prospector spacecraft detected changes in the lunar nightside voltage during magnetotail crossings, jumping from -200 V to -1000 V. In physics, a neutral particle is a particle with no electric charge, such as a neutron. One of these (analyzer A) pointed toward local lunar vertical, and the other (analyzer B) to a point 60 deg from vertical toward lunar west. The C31 coefficient is large. ==Simulating lunar gravity== In January 2022 China was reported by the South China Morning Post to have built a small (60 centimeters in diameter) research facility to simulate low lunar gravity with the help of magnets. thumb|right|300px|Total magnetic field strength at the surface of the Moon as derived from the Lunar Prospector electron reflectometer experiment. Voltaire is an impact crater on Mars's moon Deimos and is approximately across. thumb|upright=1.2|A close-up view of the CPLEE on the Moon's surface thumb|upright|The CPLEE with the ALSEP central station in the background The Charged Particle Lunar Environment Experiment (CPLEE), placed on the lunar surface by the Apollo 14 mission as part of the Apollo Lunar Surface Experiments Package (ALSEP), was designed to measure the energy spectra of low-energy charged particles striking the lunar surface. The center of gravity of the Moon does not coincide exactly with its geometric center, but is displaced toward the Earth by about 2 kilometers.Nine Planets == Mass of Moon == The gravitational constant G is less accurate than the product of G and masses for Earth and Moon. The most notable of these are Volta D in the southeast and Volta B to the northeast. ==Satellite craters== By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Volta.

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D

The initial charges on the three identical metal spheres in Fig. 21-24 are the following: sphere $A, Q$; sphere $B,-Q / 4$; and sphere $C, Q / 2$, where $Q=2.00 \times 10^{-14}$ C. Spheres $A$ and $B$ are fixed in place, with a center-to-center separation of $d=1.20 \mathrm{~m}$, which is much larger than the spheres. Sphere $C$ is touched first to sphere $A$ and then to sphere $B$ and is then removed. What then is the magnitude of the electrostatic force between spheres $A$ and $B$ ?

The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. thumb|right|280px|Scheme of the colloidal probe technique for direct force measurements in the sphere-plane and sphere-sphere geometries. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). For unequally charged objects and eventually at shorted distances, these forces may also be attractive. Simply applying this cutoff method introduces a discontinuity in the force at r_c that results in particles experiencing sudden impulses when other particles cross the boundary of their respective interaction spheres. The Bjerrum length (after Danish chemist Niels Bjerrum 1879–1958 ) is the separation at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy scale, k_\text{B} T, where k_\text{B} is the Boltzmann constant and T is the absolute temperature in kelvins. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. The electrodipping force is a force proposed to explain the observed attraction that arises among small colloidal particles attached to an interface between immiscible liquids. It is also possible to study forces between colloidal particles by attaching another particle to the substrate and perform the measurement in the sphere-sphere geometry, see figure above. thumb|left|350px|Principle of the force measurements by the colloidal probe technique. According to Nikolaides, the electrostatic force engenders a long range capillary attraction. thumb|Computed electrostatic equipotentials (black contours) between two electrically charged spheres In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential.Weisstein, Eric W. "Equipotential Curve." A way to correct this problem is to shift the force to zero at r_c, thus removing the discontinuity. This model treats the electrostatic and hard-core interactions between all individual ions explicitly. In the particular case of electrostatic forces, as the force magnitude is large at the boundary, this unphysical feature can compromise simulation accuracy. The exponential nature of these repulsive forces and the fact that its range is given by the Debye length was confirmed experimentally by direct force measurements, including surface forces apparatus, colloidal probe technique, or optical tweezers.

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A $10.0 \mathrm{~g}$ block with a charge of $+8.00 \times 10^{-5} \mathrm{C}$ is placed in an electric field $\vec{E}=(3000 \hat{\mathrm{i}}-600 \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C}$. What is the magnitude of the electrostatic force on the block?

The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules. ==Electrostatic potential energy of one point charge== ===One point charge q in the presence of another point charge Q=== right|A point charge q in the electric field of another charge Q.|thumb|434px The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Alternatively, the electric potential energy of any given charge or system of charges is termed as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration. {{block indent|em=1.2|text=The electrostatic potential energy, UE, of one point charge q at position r in the presence of an electric field E is defined as the negative of the work W done by the electrostatic force to bring it from the reference position rrefThe reference zero is usually taken to be a state in which the individual point charges are very well separated ("are at infinite separation") and are at rest. to that position r.Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 {{Equation box 1 |indent=: |equation=U_\mathrm{E}(\mathbf r) = -W_{r_{\rm ref} \rightarrow r } = -\int_{{\mathbf{r}}_{\rm ref}}^\mathbf{r} q\mathbf{E}(\mathbf{r'}) \cdot \mathrm{d} \mathbf{r'}, |cellpadding |border |border colour = #50C878 |background colour = #ECFCF4}} where E is the electrostatic field and dr' is the displacement vector in a curve from the reference position rref to the final position r.}} Using Coulomb's law, it is known that the electrostatic force F and the electric field E created by a discrete point charge Q are radially directed from Q. Electromotive force in electrostatic units is the statvolt (in the centimeter gram second system of units equal in amount to an erg per electrostatic unit of charge). ==Formal definitions== Inside a source of emf (such as a battery) that is open-circuited, a charge separation occurs between the negative terminal N and the positive terminal P. In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal{E} or {\xi}) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. (A virtual experiment based on the energy transfert between capacitor plates reveals that an additional term must be taken into account when the electrostatic energy is expressed in terms of the electric field and displacement vectors . The electrostatic field does not contribute to the net emf around a circuit because the electrostatic portion of the electric field is conservative (i.e., the work done against the field around a closed path is zero, see Kirchhoff's voltage law, which is valid, as long as the circuit elements remain at rest and radiation is ignored ). This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The magnitude of the emf for the battery (or other source) is the value of this open-circuit voltage. In the particular case of electrostatic forces, as the force magnitude is large at the boundary, this unphysical feature can compromise simulation accuracy. The following outline of proof states the derivation from the definition of electric potential energy and Coulomb's law to this formula. {{math proof |title=Outline of proof |proof= The electrostatic force F acting on a charge q can be written in terms of the electric field E as \mathbf{F} = q\mathbf{E} , By definition, the change in electrostatic potential energy, UE, of a point charge q that has moved from the reference position rref to position r in the presence of an electric field E is the negative of the work done by the electrostatic force to bring it from the reference position rref to that position r. The total electrostatic potential energy stored in a capacitor is given by U_E = \frac{1}{2}QV = \frac{1}{2} CV^2 = \frac{Q^2}{2C} where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. Years earlier, Alessandro Volta, who had measured a contact potential difference at the metal–metal (electrode–electrode) interface of his cells, held the incorrect opinion that contact alone (without taking into account a chemical reaction) was the origin of the emf. == Notation and units of measurement == Electromotive force is often denoted by \mathcal{E} or ℰ. Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. This emf is the work done on a unit charge by the source's nonelectrostatic field \boldsymbol{E}' when the charge moves from N to P. So, E and ds must be parallel: \mathbf{E} \cdot \mathrm{d} \mathbf{s} = |\mathbf{E}| \cdot |\mathrm{d}\mathbf{s}|\cos(0) = E \mathrm{d}s Using Coulomb's law, the electric field is given by |\mathbf{E}| = E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{s^2} and the integral can be easily evaluated: U_E(r) = -\int_\infty^r q\mathbf{E} \cdot \mathrm{d} \mathbf{s} = -\int_\infty^r \frac{1}{4\pi\varepsilon_0}\frac{qQ}{s^2}{\rm d}s = \frac{1}{4\pi\varepsilon_0}\frac{qQ}{r} = k_e\frac{qQ}{r} }} ===One point charge q in the presence of n point charges Qi=== thumb|Electrostatic potential energy of q due to Q1 and Q2 charge system:U_E = q\frac{1}{4 \pi \varepsilon_0} \left(\frac{Q_1}{r_1} + \frac{Q_2}{r_2} \right) The electrostatic potential energy, UE, of one point charge q in the presence of n point charges Qi, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, ri is the distance between the point charges q and Qi, and q and Qi are the assigned values of the charges. ==Electrostatic potential energy stored in a system of point charges== The electrostatic potential energy UE stored in a system of N charges q1, q2, …, qN at positions r1, r2, …, rN respectively, is: |}} where, for each i value, Φ(ri) is the electrostatic potential due to all point charges except the one at ri,The factor of one half accounts for the 'double counting' of charge pairs. * Both a 1 volt emf and a 1 volt potential difference correspond to 1 joule per coulomb of charge.

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E

Two long, charged, thin-walled, concentric cylindrical shells have radii of 3.0 and $6.0 \mathrm{~cm}$. The charge per unit length is $5.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the inner shell and $-7.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the outer shell. What is the magnitude $E$ of the electric field at radial distance $r=4.0 \mathrm{~cm}$?

Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Let f(\mathbf{r}^{\prime}) be the second charge density, and define \lambda(\rho, \theta) as its integral over z \lambda(\rho, \theta) = \int dz \, f(\rho, \theta, z) The electrostatic energy is given by the integral of the charge multiplied by the potential due to the cylindrical multipoles U = \int d\theta \, d\rho \, \rho \, \lambda(\rho, \theta) \Phi(\rho, \theta) If the cylindrical multipoles are exterior, this equation becomes U = \frac{-Q_1}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \ln \rho \+ \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} \int d\theta \, d\rho \left[ C_{1k} \frac{\cos k\theta}{\rho^{k-1}} + S_{1k} \frac{\sin k\theta}{\rho^{k-1}}\right] \lambda(\rho, \theta) where Q_{1}, C_{1k} and S_{1k} are the cylindrical multipole moments of charge distribution 1. By assumption, the line charges are infinitely long and aligned with the z axis. ==Cylindrical multipole moments of a line charge== frame|right|Figure 1: Definitions for cylindrical multipoles; looking down the z' axis The electric potential of a line charge \lambda located at (\rho', \theta') is given by \Phi(\rho, \theta) = \frac{-\lambda}{2\pi\epsilon} \ln R = \frac{-\lambda}{4\pi\epsilon} \ln \left| \rho^{2} + \left( \rho' \right)^{2} - 2\rho\rho'\cos (\theta - \theta' ) \right| where R is the shortest distance between the line charge and the observation point. Here, R is the distance from the origin while r is the distance from the central axis of a cylinder as in the (r,\phi,z) cylindrical coordinate system. It is the region of a ball between two concentric spheres of differing radii. ==Volume== The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac{4}{3}\pi R^3- \frac{4}{3}\pi r^3 : V=\frac{4}{3}\pi \left(R^3-r^3\right) : V=\frac{4}{3}\pi (R-r)\left(R^2+Rr+r^2\right) where is the radius of the inner sphere and is the radius of the outer sphere. ==Approximation== An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). The same equation shows that multipole moments higher than the first non-zero moment do depend on the choice of origin (in general). ==Interior axial multipole moments== Conversely, if the radius r is smaller than the smallest \left| \zeta \right| for which \lambda(\zeta) is significant (denoted \zeta_\text{min}), the electric potential may be written \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k} r^{k} P_{k}(\cos \theta ) where the interior axial multipole moments I_{k} are defined I_{k} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{k+1}} Special cases include the interior axial monopole moment ( eq the total charge) M_{0} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta}, the interior axial dipole moment M_{1} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{2}}, etc. Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as \ln \ R. However, the axial multipole expansion can also be applied to any potential or field that varies inversely with the distance to the source, i.e., as \frac{1}{R}. We can similarly describe the electric field E so that . If the radius r of the observation point P is greater than the largest \left| \zeta \right| for which \lambda(\zeta) is significant (denoted \zeta_\text{max}), the electric potential may be written \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} M_{k} \left( \frac{1}{r^{k+1}} \right) P_{k}(\cos \theta ) where the axial multipole moments M_{k} are defined M_{k} \equiv \int d\zeta \ \lambda(\zeta) \zeta^{k} Special cases include the axial monopole moment (=total charge) M_{0} \equiv \int d\zeta \ \lambda(\zeta), the axial dipole moment M_{1} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta, and the axial quadrupole moment M_{2} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta^{2}. This energy formula can be reduced to a remarkably simple form U = \frac{-Q_{1}}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \ln \rho \+ \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} k \left( C_{1k} I_{2k} + S_{1k} J_{2k} \right) where I_{2k} and J_{2k} are the interior cylindrical multipoles of the second charge density. Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the z-axis. In alternative notation, the Sommerfeld identity can be more easily seen as an expansion of a spherical wave in terms of cylindrically-symmetric waves: : \frac{{e^{ik_0 r} }} {r} = i\int\limits_0^\infty {dk_\rho \frac J_0 (k_\rho \rho )e^{ik_z \left| z \right|} } Where : k_z=(k_0^2-k_\rho^2)^{1/2} The notation used here is different form that above: r is now the distance from the origin and \rho is the radial distance in a cylindrical coordinate system defined as (\rho,\phi,z). The analogous formula holds if charge density 1 is composed of interior cylindrical multipoles U = \frac{-Q_1\ln \rho'}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \+ \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} k \left( C_{2k} I_{1k} + S_{2k} J_{1k} \right) where I_{1k} and J_{1k} are the interior cylindrical multipole moments of charge distribution 1, and C_{2k} and S_{2k} are the exterior cylindrical multipoles of the second charge density. For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density \lambda(z) localized to the z-axis. frame|right|Figure 1: Point charge on the z axis; Definitions for axial multipole expansion ==Axial multipole moments of a point charge== The electric potential of a point charge q located on the z-axis at z=a (Fig. 1) equals \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon} \frac{1}{R} = \frac{q}{4\pi\varepsilon} \frac{1}{\sqrt{r^{2} + a^{2} - 2 a r \cos \theta}}. At short distances (\frac{r}{\zeta_\text{min}} \ll 1), the potential is well- approximated by the leading nonzero interior multipole term. ==See also== * Potential theory * Multipole expansion * Spherical multipole moments * Cylindrical multipole moments * Solid harmonics * Laplace expansion ==References== Category:Electromagnetism Category:Potential theory Category:Moment (physics) If the radius r of the observation point is greater than a, we may factor out \frac{1}{r} and expand the square root in powers of (a/r)<1 using Legendre polynomials \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon r} \sum_{k=0}^{\infty} \left( \frac{a}{r} \right)^{k} P_{k}(\cos \theta ) \equiv \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} M_{k} \left( \frac{1}{r^{k+1}} \right) P_{k}(\cos \theta ) where the axial multipole moments M_{k} \equiv q a^{k} contain everything specific to a given charge distribution; the other parts of the electric potential depend only on the coordinates of the observation point P. Special cases include the axial monopole moment M_{0}=q, the axial dipole moment M_{1}=q a and the axial quadrupole moment M_{2} \equiv q a^{2}. 300px|thumb|spherical shell, right: two halves In geometry, a spherical shell is a generalization of an annulus to three dimensions. Thus, at large distances (\frac{\zeta_\text{max}}{r} \ll 1), the potential is well-approximated by the leading nonzero multipole term. Conversely, if the radius r is less than a, we may factor out \frac{1}{a} and expand in powers of (r/a)<1, once again using Legendre polynomials \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon a} \sum_{k=0}^{\infty} \left( \frac{r}{a} \right)^{k} P_{k}(\cos \theta ) \equiv \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k} r^{k} P_{k}(\cos \theta ) where the interior axial multipole moments I_{k} \equiv \frac{q}{a^{k+1}} contain everything specific to a given charge distribution; the other parts depend only on the coordinates of the observation point P. ==General axial multipole moments== To get the general axial multipole moments, we replace the point charge of the previous section with an infinitesimal charge element \lambda(\zeta)\ d\zeta, where \lambda(\zeta) represents the charge density at position z=\zeta on the z-axis. thumb|An annular solar eclipse has a magnitude of less than 1.0 The magnitude of eclipse is the fraction of the angular diameter of a celestial body being eclipsed. The total surface area of the spherical shell is 4 \pi r^2. == See also == * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere == References == Category:Elementary geometry Category:Geometric shapes Category:Spherical geometry

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A particle of charge $1.8 \mu \mathrm{C}$ is at the center of a Gaussian cube $55 \mathrm{~cm}$ on edge. What is the net electric flux through the surface?

It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). It is defined as the closed surface in three dimensional space by which the flux of vector field be calculated. == Common Gaussian surfaces == Most calculations using Gaussian surfaces begin by implementing Gauss's law (for electricity):Introduction to electrodynamics (4th Edition), D. J. Griffiths, 2012, :}{\varepsilon_0}.}} For a closed Gaussian surface, electric flux is given by: where * is the electric field, * is any closed surface, * is the total electric charge inside the surface , * is the electric constant (a universal constant, also called the "permittivity of free space") () This relation is known as Gauss' law for electric fields in its integral form and it is one of Maxwell's equations. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. Since the flux is defined as an integral of the electric field, this expression of Gauss's law is called the integral form. thumb|A tiny Gauss's box whose sides are perpendicular to a conductor's surface is used to find the local surface charge once the electric potential and the electric field are calculated by solving Laplace's equation. Thereby is the electrical charge enclosed by the Gaussian surface. Electric flux through its surface is zero. If the Gaussian surface is chosen such that for every point on the surface the component of the electric field along the normal vector is constant, then the calculation will not require difficult integration as the constants which arise can be taken out of the integral. Gauss's law may be expressed as: \Phi_E = \frac{Q}{\varepsilon_0} where is the electric flux through a closed surface enclosing any volume , is the total charge enclosed within , and is the electric constant. A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field.Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, It is an arbitrary closed surface (the boundary of a 3-dimensional region ) used in conjunction with Gauss's law for the corresponding field (Gauss's law, Gauss's law for magnetism, or Gauss's law for gravity) by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution. The flux passing consists of the three contributions: : For surfaces a and b, and will be perpendicular. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. In electromagnetism, electric flux is the measure of the electric field through a given surface,Purcell, pp. 22–26 although an electric field in itself cannot flow. The electric flux over a surface is therefore given by the surface integral: \Phi_E = \iint_S \mathbf{E} \cdot \textrm{d}\mathbf{S} where is the electric field and is a differential area on the closed surface with an outward facing surface normal defining its direction. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. While the electric flux is not affected by charges that are not within the closed surface, the net electric field, can be affected by charges that lie outside the closed surface. *MISN-0-133 Gauss's Law Applied to Cylindrical and Planar Charge Distributions (PDF file) by Peter Signell for Project PHYSNET. If the electric field is uniform, the electric flux passing through a surface of vector area is \Phi_E = \mathbf{E} \cdot \mathbf{S} = ES \cos \theta, where is the electric field (having units of ), is its magnitude, is the area of the surface, and is the angle between the electric field lines and the normal (perpendicular) to . The electric field is perpendicular, locally, to the equipotential surface of the conductor, and zero inside; its flux πa2·E, by Gauss's law equals πa2·σ/ε0. Under these circumstances, Gauss's law modifies to \Phi_E = \frac{Q_\mathrm{free}}{\varepsilon} for the integral form, and abla \cdot \mathbf{E} = \frac{\rho_\mathrm{free}}{\varepsilon} for the differential form. ==Interpretations== ===In terms of fields of force=== Gauss's theorem can be interpreted in terms of the lines of force of the field as follows: The flux through a closed surface is dependent upon both the magnitude and direction of the electric field lines penetrating the surface. The electric flux is defined as a surface integral of the electric field: : where is the electric field, is a vector representing an infinitesimal element of area of the surface, and represents the dot product of two vectors. In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface.

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The drum of a photocopying machine has a length of $42 \mathrm{~cm}$ and a diameter of $12 \mathrm{~cm}$. The electric field just above the drum's surface is $2.3 \times 10^5 \mathrm{~N} / \mathrm{C}$. What is the total charge on the drum?

The total charge divided by the length, surface area, or volume will be the average charge densities: \langle\lambda_q \rangle = \frac{Q}{\ell}\,,\quad \langle\sigma_q\rangle = \frac{Q}{S}\,,\quad\langle\rho_q\rangle = \frac{Q}{V}\,. == Free, bound and total charge == In dielectric materials, the total charge of an object can be separated into "free" and "bound" charges. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. Unlike the case of the metal, the image charge q' is not exactly opposite to the real charge: q'=\frac{\varepsilon_1 - \varepsilon_2}{\varepsilon_1 + \varepsilon_2}q. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge distribution on the surface. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. Electrostatic charge on an object can be measured by placing it into the Faraday Cup. If we assume for simplicity (without loss of generality) that the inner charge lies on the z-axis, then the induced charge density will be simply a function of the polar angle θ and is given by: : \sigma(\theta) = \varepsilon_0 \left.\frac{\partial V}{\partial r} \right|_{r=R} =\frac{-q\left(R^2-p^2\right)}{4\pi R\left(R^2+p^2-2pR\cos\theta\right)^{3/2}} The total charge on the sphere may be found by integrating over all angles: : Q_t=\int_0^\pi d\theta \int_0^{2\pi} d\phi\,\,\sigma(\theta) R^2\sin\theta = -q Note that the reciprocal problem is also solved by this method. It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. A surface charge is an electric charge present on a two-dimensional surface. Just as in the first case, the image charge will have charge −qR/p and will be located at vector position \left(R^2 / p^2\right) \mathbf{p}. Free charges are the excess charges which can move into electrostatic equilibrium, i.e. when the charges are not moving and the resultant electric field is independent of time, or constitute electric currents. === Total charge densities === In terms of volume charge densities, the total charge density is: \rho = \rho_\text{f} + \rho_\text{b}\,. as for surface charge densities: \sigma = \sigma_\text{f} + \sigma_\text{b}\,. where subscripts "f" and "b" denote "free" and "bound" respectively. === Bound charge === The bound surface charge is the charge piled up at the surface of the dielectric, given by the dipole moment perpendicular to the surface: q_b = \frac{\mathbf{d} \cdot\mathbf{\hat{n}}}{|\mathbf{s}|} where s is the separation between the point charges constituting the dipole, \mathbf{d} is the electric dipole moment, \mathbf{\hat{n}} is the unit normal vector to the surface. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. Overall, two layers of charge and a potential drop from the electrode to the edge of the outer layer (outer Helmholtz Plane) are observed.

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A spherical water drop $1.20 \mu \mathrm{m}$ in diameter is suspended in calm air due to a downward-directed atmospheric electric field of magnitude $E=462 \mathrm{~N} / \mathrm{C}$. What is the magnitude of the gravitational force on the drop?

If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). From Stokes' solution, the drag force acting on the sphere of diameter d can be obtained as where the Reynolds number, Re = \frac{\rho d}{\mu} V. thumb|upright=0.7|The downward force of gravity (Fg) equals the restraining force of drag (Fd) plus the buoyancy. Fluid Dynamics Research 12.2 (1993): 61-93 In the case of floating, a drop will float on the surface for several seconds. A new model for the equilibrium shape of raindrops. The resulting outcome depends on the properties of the drop, the surface, and the surrounding fluid, which is most commonly a gas. == On a dry solid surface == When a liquid drop strikes a dry solid surface, it generally spreads on the surface, and then will retract if the impact is energetic enough to cause the drop to spread out more than it would generally spread due to its static receding contact angle. The Beard and Chuang model is a well known and leading theoretical force balance model used to derive the rotational cross-sections of raindrops in their equilibrium state by employing Chebyshev polynomials in series. thumb|Beard and Chuang model of raindrop The radius-vector of the raindrop's surface r(\theta) in vertical angular direction \theta is equal to r(\theta)=a [ 1 + \sum c_n cos(n \theta) ] , where shape coefficients c_n \cdot 10^4 are defined for the raindrops with different equivolumetric diameter as in following table d(mm) n = 0 1 2 3 4 5 6 7 8 9 10 2.0 -131 -120 -376 -96 -4 15 5 0 -2 0 1 2.5 -201 -172 -567 -137 3 29 8 -2 -4 0 1 3.0 -282 -230 -779 -175 21 46 11 -6 -7 0 3 3.5 -369 -285 -998 -207 48 68 13 -13 -10 0 5 4.0 -458 -335 -1211 -227 83 89 12 -21 -13 1 8 4.5 -549 -377 -1421 -240 126 110 9 -31 -16 4 11 5.0 -644 -416 -1629 -246 176 131 2 -44 -18 9 14 5.5 -742 -454 -1837 -244 234 150 -7 -58 -19 15 19 6.0 -840 -480 -2034 -237 297 166 -21 -72 -19 24 23 == Applications == The description of raindrop shape has some rather practical uses. To find a relationship between drop size and contact time for low Weber number impacts (We << 1) on superhydrophobic surfaces (which experience little deformation), a simple balance between inertia (\rho R / \tau^2) and capillarity (\sigma/R^2) can be used,Richard, Denis, Christophe Clanet, and David Quéré. alt=|thumb|300x300px|Low Force Waterfalls Low Force is an 18-foot (5.5m) high set of falls on the River Tees, England, UK. This outcome is representative of impact of small, low-velocity drops onto smooth wetting surfaces. In fluid dynamics, drop impact occurs when a drop of liquid strikes a solid or liquid surface. Hydrometeor loading is the induced drag effects on the atmosphere from a falling hydrometeor. thumb|A drop striking a liquid surface; in this case, both the drop and the surface are water. After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. "Phenomena of liquid drop impact on solid and liquid surfaces." When falling at terminal velocity, the value of this drag is equal to grh, where g is the acceleration due to gravity and rh is the mixing ratio of the hydrometeors. If the droplet is split into multiple droplets, the contact time is reduced. thumb|Breakup of a water drop impacting a superhydrophobic surface at a Weber number of approximately 214. Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass m has to be reduced by the displaced fluid mass \rho V, with V the volume of the object. "Surface phenomena: Contact time of a bouncing drop." Hydrometeor loading has a net-negative effect on the atmospheric buoyancy equations. * For large We (for which the magnitude depends on the specific surface structure), many satellite drops break off during spreading and/or retraction of the drop. == On a wet solid surface == When a liquid drop strikes a wet solid surface (a surface covered with a thin layer of liquid that exceeds the height of surface roughness), either spreading or splashing will occur. The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object.

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A

How many electrons would have to be removed from a coin to leave it with a charge of $+1.0 \times 10^{-7} \mathrm{C}$ ?

For an electron, it has a value of . Ten years later, he switched to electron to describe these elementary charges, writing in 1894: "... an estimate was made of the actual amount of this most remarkable fundamental unit of electricity, for which I have since ventured to suggest the name electron". For example, in one instance a Penning trap was used to contain a single electron for a period of 10 months. When there is an excess of electrons, the object is said to be negatively charged. The electron ( or ) is a subatomic particle with a negative one elementary electric charge. The word electron is a combination of the words _electr_ ic and i _on_."electron, n.2". The electron, on the other hand, is thought to be stable on theoretical grounds: the electron is the least massive particle with non-zero electric charge, so its decay would violate charge conservation. # Use the previous solution for making r, # excluding coin elif coin > r: m[c][r] = m[c - 1][r] # coin can be used. The version of this problem assumed that the people making change will use the minimum number of coins (from the denominations available). thumb|upright=1.7|Contrasting differences between discrete and continuous electron multipliers. Using the previous solution for making r (without using coin). thumb|upright=1.35|Coin of Tennes. The electron's mass is approximately 1/1836 that of the proton. The change-making problem addresses the question of finding the minimum number of coins (of certain denominations) that add up to a given amount of money. The upper bound of the electron radius of 10−18 meters can be derived using the uncertainty relation in energy. Within the limits of experimental accuracy, the electron charge is identical to the charge of a proton, but with the opposite sign. In turn, he divided the shells into a number of cells each of which contained one pair of electrons. Hence, about one electron for every billion electron-positron pairs survived. When there are fewer electrons than the number of protons in nuclei, the object is said to be positively charged. On the other hand, a point-like electron (zero radius) generates serious mathematical difficulties due to the self-energy of the electron tending to infinity.Eduard Shpolsky, Atomic physics (Atomnaia fizika), second edition, 1951 Observation of a single electron in a Penning trap suggests the upper limit of the particle's radius to be 10−22 meters. thumb|upright=1.5|Coin of Epander. Using the previous solution for making r - coin (without # using coin) plus this 1 extra coin. else: m[c][r] = min(m[c - 1][r], 1 + m[c][r - coin]) return m[-1][-1] ===Dynamic programming with the probabilistic convolution tree=== The probabilistic convolution tree can also be used as a more efficient dynamic programming approach.

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A

An unknown charge sits on a conducting solid sphere of radius $10 \mathrm{~cm}$. If the electric field $15 \mathrm{~cm}$ from the center of the sphere has the magnitude $3.0 \times 10^3 \mathrm{~N} / \mathrm{C}$ and is directed radially inward, what is the net charge on the sphere?

It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. thumb|300px|right|visualized induced-charge electrokinetic flow pattern around a carbon-steel sphere (diameter=1.2mm). We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. The 15th parallel north is a circle of latitude that is 15 degrees north of the Earth's equatorial plane. The great circle is often considered to define an equator about the sphere and the two points perpendicular to the plane are often considered poles to aid in discussions about the electrostatic configurations of many-N electron solutions. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. Thereby is the electrical charge enclosed by the Gaussian surface. This may also be written as V = \frac{2\pi r^3}{3} (1-\cos\varphi)\,, where is half the cone angle, i.e., is the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center. 400px|thumb|A spherical sector (blue) thumb|A spherical sector In geometry, a spherical sector, also known as a spherical cone, is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. It is the three-dimensional analogue of the sector of a circle. ==Volume== If the radius of the sphere is denoted by and the height of the cap by , the volume of the spherical sector is V = \frac{2\pi r^2 h}{3}\,. The volume of the sector is related to the area of the cap by: V = \frac{rA}{3}\,. ==Area== The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is A = 2\pi rh\,. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. * For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness.

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Space vehicles traveling through Earth's radiation belts can intercept a significant number of electrons. The resulting charge buildup can damage electronic components and disrupt operations. Suppose a spherical metal satellite $1.3 \mathrm{~m}$ in diameter accumulates $2.4 \mu \mathrm{C}$ of charge in one orbital revolution. Find the resulting surface charge density.

A satellite shielded by 3 mm of aluminium in an elliptic orbit () passing the radiation belts will receive about 2,500 rem (25 Sv) per year. Radiation belt electrons are also constantly removed by collisions with Earth's atmosphere, losses to the magnetopause, and their outward radial diffusion. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. Charge carrier densities involve equations concerning the electrical conductivity and related phenomena like the thermal conductivity. ==Calculation== The carrier density is usually obtained theoretically by integrating the density of states over the energy range of charge carriers in the material (e.g. integrating over the conduction band for electrons, integrating over the valence band for holes). This charge is equal to d q = \rho \, v\, dt \, dA, where is the charge density at . To show this mathematically, charge carrier density is a particle density, so integrating it over a volume V gives the number of charge carriers N in that volume N=\int_V n(\mathbf r) \,dV. where n(\mathbf r) is the position-dependent charge carrier density. As always, the integral of the charge density over a region of space is the charge contained in that region. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. The outer electron radiation belt is mostly produced by inward radial diffusion and local acceleration due to transfer of energy from whistler-mode plasma waves to radiation belt electrons. If the total number of charge carriers is known, the carrier density can be found by simply dividing by the volume. At position at time , the distribution of charge flowing is described by the current density: \mathbf{j}(\mathbf{r}, t) = \rho(\mathbf{r},t) \; \mathbf{v}_\text{d} (\mathbf{r},t) where * is the current density vector; * is the particles' average drift velocity (SI unit: m∙s−1); *\rho(\mathbf{r}, t) = q \, n(\mathbf{r},t) is the charge density (SI unit: coulombs per cubic metre), in which ** is the number of particles per unit volume ("number density") (SI unit: m−3); ** is the charge of the individual particles with density (SI unit: coulombs). The outer Van Allen belt consists mainly of electrons. Spacecraft charging is what happens when charged particles from the surrounding energetic environment stop on either the exterior of a spacecraft or the interior, such as in conductors. ==References== Category:Spaceflight The charge density of the system at a point r is a sum of the charge densities for each charge qi at position ri, where : \rho_q(\mathbf{r})=\sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) The delta function for each charge qi in the sum, δ(r − ri), ensures the integral of charge density over R returns the total charge in R: Q = \int_R d^3 \mathbf{r} \sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i \int_R d^3 \mathbf{r} \delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i If all charge carriers have the same charge q (for electrons q = −e, the electron charge) the charge density can be expressed through the number of charge carriers per unit volume, n(r), by \rho_q(\mathbf{r}) = q n(\mathbf{r})\,. Material Number of valence electrons Carrier density (1/cm3) at 300K Copper 1 Silver 1 Gold 1 Beryllium 2 Magnesium 2 Calcium 2 Strontium 2 Barium 2 Niobium 1 Iron 2 Manganese 2 Zinc 2 Cadmium 2 Aluminum 3 Gallium 3 Indium 3 Thallium 3 Tin 4 Lead 4 Bismuth 5 Antimony 5 The values for n among metals inferred for example by the Hall effect are often on the same orders of magnitude, but this simple model cannot predict carrier density to very high accuracy. ==Measurement== The density of charge carriers can be determined in many cases using the Hall effect, the voltage of which depends inversely on the carrier density. ==References== Category:Density Category:Charge carriers thumb|320px|A cross section of Van Allen radiation belts A Van Allen radiation belt is a zone of energetic charged particles, most of which originate from the solar wind, that are captured by and held around a planet by that planet's magnetosphere. As of 2014, it remains uncertain if there are any negative unintended consequences to removing these radiation belts. ==See also== * Dipole model of the Earth's magnetic field * L-shell * List of artificial radiation belts * List of plasma (physics) articles * Space weather == Explanatory notes == == Citations == ==Additional sources== * * * Part I: Radial transport, pp. 1679–1693, ; Part II: Local acceleration and loss, pp. 1694–1713, . == External links == * An explanation of the belts by David P. Stern and Mauricio Peredo * Background: Trapped particle radiation models—Introduction to the trapped radiation belts by SPENVIS * SPENVIS—Space Environment, Effects, and Education System—Gateway to the SPENVIS orbital dose calculation software *The Van Allen Probes Web Site Johns Hopkins University Applied Physics Laboratory Category:1958 in science Category:Articles containing video clips Category:Geomagnetism Category:Space physics Category:Space plasmas The equivalent proofs for linear charge density and surface charge density follow the same arguments as above. == Discrete charges == For a single point charge q at position r0 inside a region of 3d space R, like an electron, the volume charge density can be expressed by the Dirac delta function: \rho_q(\mathbf{r}) = q \delta(\mathbf{r} - \mathbf{r}_0) where r is the position to calculate the charge. Charge carrier density, also known as carrier concentration, denotes the number of charge carriers in per volume. Miniaturization and digitization of electronics and logic circuits have made satellites more vulnerable to radiation, as the total electric charge in these circuits is now small enough so as to be comparable with the charge of incoming ions.

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A charge of $20 \mathrm{nC}$ is uniformly distributed along a straight rod of length $4.0 \mathrm{~m}$ that is bent into a circular arc with a radius of $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the center of curvature of the arc?

The radius of such a curve is 5729.57795. Where degree of curvature is based on 100 units of arc length, the conversion between degree of curvature and radius is , where is degree and is radius. The distance from the vertex to the center of curvature is the radius of curvature of the surface. Bend radius, which is measured to the inside curvature, is the minimum radius one can bend a pipe, tube, sheet, cable or hose without kinking it, damaging it, or shortening its life. thumb|A concave mirror with light rays thumb|400px|Center of curvature In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. Degree of curve or degree of curvature is a measure of curvature of a circular arc used in civil engineering for its easy use in layout surveying. ==Definition== The degree of curvature is defined as the central angle to the ends of an agreed length of either an arc or a chord; various lengths are commonly used in different areas of practice. thumb|upright=1.0|The inscribed angle θ is half of the central angle 2θ that subtends the same arc on the circle. * If the vertex lies to the right of the center of curvature, the radius of curvature is negative. The is called a station, used to define length along a road or other alignment, annotated as stations plus feet 1+00, 2+00, etc. Metric work may use similar notation, such as kilometers plus meters 1+000. ==Formulas for radius of curvature== alt=Degree of Curvature Formula Explanation|thumb|Diagram showing different parts of the curve used in the formula Degree of curvature can be converted to radius of curvature by the following formulae: ===Formula from arc length=== r = \frac{180^\circ A}{\pi D_\text{C}} where A is arc length, r is radius of curvature, and D_\text{C} is degree of curvature, arc definition Substitute deflection angle for degree of curvature or make arc length equal to 100 feet. ===Formula from chord length=== r = \frac{C}{2 \sin \left( \frac{D_\text{C}}{2} \right) } where C is chord length, r is radius of curvature and D_\text{C} is degree of curvature, chord definition ===Formula from radius=== D_\text{C} = 5729.58/r === Example === As an example, a curve with an arc length of 600 units that has an overall sweep of 6 degrees is a 1-degree curve: For every 100 feet of arc, the bearing changes by 1 degree. Cauchy defined the center of curvature C as the intersection point of two infinitely close normal lines to the curve.* The osculating circle to the curve is centered at the centre of curvature. By using degrees of curvature, curve setting can be easily done with the help of a transit or theodolite and a chain, tape, or rope of a prescribed length. === Length selection === The usual distance used to compute degree of curvature in North American road work is of arc. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. For metal tubing, bend radius is to the centerline of tubing, not the exterior. ==References== Category:Cables Category:Fiber optics Category:Plumbing vi:Bán kính cong The sign convention for the optical radius of curvature is as follows: * If the vertex lies to the left of the center of curvature, the radius of curvature is positive. Suppose this arc includes point E within it. The minimum bending radius will vary with different cable designs. The locus of centers of curvature for each point on the curve comprise the evolute of the curve. In an n-degree curve, the forward bearing changes by n degrees over the standard length of arc or chord. ==Usage== Curvature is usually measured in radius of curvature. thumb|upright=1.3|Radius of curvature sign convention for optical design Radius of curvature (ROC) has specific meaning and sign convention in optical design. The smaller the bend radius, the greater the material flexibility (as the radius of curvature decreases, the curvature increases). If the chord definition is used, each 100-unit chord length will sweep 1 degree with a radius of 5729.651 units, and the chord of the whole curve will be slightly shorter than 600 units. == See also == * Geometric design of roads * Highway engineering * Lateral motion device * Minimum railway curve radius * Radius of curvature (applications) * Railway systems engineering * Track geometry * Track transition curve * Transition curve * Turning radius == References == == External links == * * http://www.tpub.com/content/engineering/14071/css/14071_242.htm * * * * * * * Category:Surveying Category:Transportation engineering Category:Track geometry

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Calculate the number of coulombs of positive charge in 250 $\mathrm{cm}^3$ of (neutral) water. (Hint: A hydrogen atom contains one proton; an oxygen atom contains eight protons.)

One coulomb is the charge of approximately , where the number is the reciprocal of This is also 160.2176634 zC of charge. In this case, the effective nuclear charge can be calculated by Coulomb's law. For example, a neutral chlorine atom has 17 protons and 17 electrons, whereas a Cl− anion has 17 protons and 18 electrons for a total charge of −1. The internationally accepted value of a proton's charge radius is . In the SI system of units, the value of the elementary charge is exactly defined as e = coulombs, or 160.2176634 zeptocoulombs (zC). It is positive (repulsive) to a radial distance of about 0.6 fm, negative (attractive) at greater distances, and very weak beyond about 2 fm. === Charge radius in solvated proton, hydronium === The radius of the hydrated proton appears in the Born equation for calculating the hydration enthalpy of hydronium. == Interaction of free protons with ordinary matter == Although protons have affinity for oppositely charged electrons, this is a relatively low-energy interaction and so free protons must lose sufficient velocity (and kinetic energy) in order to become closely associated and bound to electrons. It is impossible to realize exactly 1 C of charge, since the number of elementary charges is not an integer. In this case, one says that the "elementary charge" is three times as large as the "quantum of charge". Because protons are not fundamental particles, they possess a measurable size; the root mean square charge radius of a proton is about 0.84–0.87 fm ( = ). The 4s electrons in iron, which are furthest from the nucleus, feel an effective atomic number of only 5.43 because of the 25 electrons in between it and the nucleus screening the charge. In January 2013, an updated value for the charge radius of a proton——was published. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. Protons are composed of two up quarks of charge +e and one down quark of charge −e. The effective nuclear charge on such an electron is given by the following equation: Z_\mathrm{eff} = Z - S where *Z is the number of protons in the nucleus (atomic number), and *S is the shielding constant. A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 e (elementary charge). By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e. It is possible to determine the strength of the nuclear charge by the oxidation number of the atom. In atomic physics, the effective nuclear charge is the actual amount of positive (nuclear) charge experienced by an electron in a multi-electron atom. The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). Their measurement of the root-mean-square charge radius of a proton is ", which differs by 5.0 standard deviations from the CODATA value of ". The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. The chemical properties of each atom are determined by the number of (negatively charged) electrons, which for neutral atoms is equal to the number of (positive) protons so that the total charge is zero.

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A charged cloud system produces an electric field in the air near Earth's surface. A particle of charge $-2.0 \times 10^{-9} \mathrm{C}$ is acted on by a downward electrostatic force of $3.0 \times 10^{-6} \mathrm{~N}$ when placed in this field. What is the magnitude of the electric field?

Near the surface of the Earth, the magnitude of the field is on average around 100 V/m. Because the atmospheric electric field is negatively directed in fair weather, the convention is to refer to the potential gradient, which has the opposite sign and is about 100 V/m at the surface, away from thunderstorms. From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with the inverse square of the distance from the charge. This Carnegie curve variation has been described as "the fundamental electrical heartbeat of the planet".Liz Kalaugher, Atmospheric electricity affects cloud height 3 March 2013, physicsworld.com accessed 15 April 2021 Even away from thunderstorms, atmospheric electricity can be highly variable, but, generally, the electric field is enhanced in fogs and dust whereas the atmospheric electrical conductivity is diminished. === Links with biology === The atmospheric potential gradient leads to an ion flow from the positively charged atmosphere to the negatively charged earth surface. The electric field is a vector field... and has a magnitude and direction." This aura is the electric field due to the charge. The electric field is defined as a vector field that associates to each point in space the electrostatic (Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Atmospheric electricity is an interdisciplinary topic with a long history, involving concepts from electrostatics, atmospheric physics, meteorology and Earth science. Atmospheric electricity is the study of electrical charges in the Earth's atmosphere (or that of another planet). Thunderstorms act as a giant battery in the atmosphere, charging up the electrosphere to about 400,000 volts with respect to the surface. The movement of charge between the Earth's surface, the atmosphere, and the ionosphere is known as the global atmospheric electrical circuit. thumb|upright=1.25|Convective cloud's thickness, between its base and top, shown on the background scale at different stages of its life The cloud height, more commonly known as cloud thickness or depth, is the distance between the cloud base and the cloud top. The Coulomb force on a charge of magnitude q at any point in space is equal to the product of the charge and the electric field at that point \mathbf{F} = q\mathbf{E} The SI unit of the electric field is the newton per coulomb (N/C), or volt per meter (V/m); in terms of the SI base units it is kg⋅m⋅s−3⋅A−1. ===Superposition principle=== Due to the linearity of Maxwell's equations, electric fields satisfy the superposition principle, which states that the total electric field, at a point, due to a collection of charges is equal to the vector sum of the electric fields at that point due to the individual charges. Over a flat field on a day with clear skies, the atmospheric potential gradient is approximately 120 V/m. The derived SI unit for the electric field is the volt per meter (V/m), which is equal to the newton per coulomb (N/C)., p. 23 ==Description== The electric field is defined at each point in space as the force per unit charge that would be experienced by a vanishingly small positive test charge if held stationary at that point. This is the electric field at point \mathbf{x}_0 due to the point charge q_1; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position \mathbf{x}_0. There is a weak conduction current of atmospheric ions moving in the atmospheric electric field, about 2 picoamperes per square meter, and the air is weakly conductive due to the presence of these atmospheric ions. ===Variations=== Global daily cycles in the atmospheric electric field, with a minimum around 03 UT and peaking roughly 16 hours later, were researched by the Carnegie Institution of Washington in the 20th century. Discoveries about the electrification of the atmosphere via sensitive electrical instruments and ideas on how the Earth's negative charge is maintained were developed mainly in the 20th century, with CTR Wilson playing an important part.Encyclopedia of Geomagnetism and Paleomagnetism - Page 359 Current research on atmospheric electricity focuses mainly on lightning, particularly high-energy particles and transient luminous events, and the role of non-thunderstorm electrical processes in weather and climate. ==Description== Atmospheric electricity is always present, and during fine weather away from thunderstorms, the air above the surface of Earth is positively charged, while the Earth's surface charge is negative. As the electric field is defined in terms of force, and force is a vector (i.e. having both magnitude and direction), it follows that an electric field is a vector field. To make it easy to calculate the Coulomb force on any charge at position \mathbf{x}_0 this expression can be divided by q_0 leaving an expression that only depends on the other charge (the source charge) \mathbf{E}_{1} (\mathbf{x}_0) = \frac{ \mathbf{F}_{01} } {q_0} = \frac{q_1 }{4\pi\varepsilon_0} \frac{\hat \mathbf{r}_{01}}{r_{01}^2} Where *\mathbf{E}_{1} (\mathbf{x}_0) is the component of the electric field at q_0 due to q_1 . An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them.Browne, p 225: "... around every charge there is an aura that fills all space.

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An electric dipole with dipole moment $$ \vec{p}=(3.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{j}})\left(1.24 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}\right) $$ is in an electric field $\vec{E}=(4000 \mathrm{~N} / \mathrm{C}) \hat{\mathrm{i}}$. What is the potential energy of the electric dipole? (b) What is the torque acting on it?

An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . For a spatially uniform electric field across the small region occupied by the dipole, the energy U and the torque \boldsymbol{\tau} are given by U = - \mathbf{p} \cdot \mathbf{E},\qquad\ \boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}, The scalar dot "" product and the negative sign shows the potential energy minimises when the dipole is parallel with field and is maximum when antiparallel while zero when perpendicular. This quantity is used in the definition of polarization density. ==Energy and torque== thumb|187x187px|Electric dipole p and its torque τ in a uniform E field. The E-field vector and the dipole vector define a plane, and the torque is directed normal to that plane with the direction given by the right-hand rule. The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: :U = \mathbf d_{\rm e} \cdot \mathbf E. The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The magnitude of the torque is proportional to both the magnitude of the charges and the separation between them, and varies with the relative angles of the field and the dipole: \left|\mathbf{\tau}\right| = \left|q\mathbf{r}\right| \left|\mathbf{E}\right|\sin\theta. A dipole aligned parallel to an electric field has lower potential energy than a dipole making some angle with it. The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. In the case of two classical point charges, +q and -q, with a displacement vector, \mathbf{r}, pointing from the negative charge to the positive charge, the electric dipole moment is given by \mathbf{p} = q\mathbf{r}. The torque tends to align the dipole with the field. The potential energy of a magnet or magnetic moment \mathbf{m} in a magnetic field \mathbf{B} is defined as the mechanical work of the magnetic force (actually magnetic torque) on the re-alignment of the vector of the magnetic dipole moment and is equal to: E_\text{p,m} = -\mathbf{m} \cdot \mathbf{B} while the energy stored in an inductor (of inductance L) when a current I flows through it is given by: E_\text{p,m} = \frac{1}{2} LI^2. To the accuracy of this dipole approximation, as shown in the previous section, the dipole moment density p(r) (which includes not only p but the location of p) serves as P(r). In the presence of an electric field, such as that due to an electromagnetic wave, the two charges will experience a force in opposite directions, leading to a net torque on the dipole. The potential at a position r is: \phi (\mathbf{r}) = \frac{1}{4 \pi \varepsilon_0} \int \frac{\rho \left(\mathbf{r}_0\right)}{\left|\mathbf{r} - \mathbf{r}_0\right|} d^3 \mathbf{r}_0 \ + \frac {1}{4 \pi \varepsilon_0}\int \frac{\mathbf{p} \left(\mathbf{r}_0\right) \cdot \left(\mathbf{r} - \mathbf{r}_0\right)} {| \mathbf{r} - \mathbf{r}_0 |^3 } d^3 \mathbf{ r}_0 , where ρ(r) is the unpaired charge density, and p(r) is the dipole moment density.For example, for a system of ideal dipoles with dipole moment p confined within some closed surface, the dipole density p(r) is equal to p inside the surface, but is zero outside. It can be shown that this net force is generally parallel to the dipole moment. ==Expression (general case)== More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is: \mathbf{p}(\mathbf{r}) = \int_{V} \rho(\mathbf{r}') \left(\mathbf{r}' - \mathbf{r}\right) d^3 \mathbf{r}', where r locates the point of observation and d3r′ denotes an elementary volume in V. (See electron electric dipole moment). The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The electric dipole moment vector also points from the negative charge to the positive charge.

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D

What is the total charge in coulombs of $75.0 \mathrm{~kg}$ of electrons?

One coulomb is the charge of approximately , where the number is the reciprocal of This is also 160.2176634 zC of charge. For an electron, it has a value of . (In other words, the charge of one mole of electrons, divided by the number of electrons in a mole, equals the charge of a single electron.) Electrons have an electric charge of coulombs,The original source for CODATA is :Individual physical constants from the CODATA are available at: which is used as a standard unit of charge for subatomic particles, and is also called the elementary charge. Ten years later, he switched to electron to describe these elementary charges, writing in 1894: "... an estimate was made of the actual amount of this most remarkable fundamental unit of electricity, for which I have since ventured to suggest the name electron". In the SI system of units, the value of the elementary charge is exactly defined as e = coulombs, or 160.2176634 zeptocoulombs (zC). The formal charge of any atom in a molecule can be calculated by the following equation: q^{*} = V - L - \frac{B}{2} where is the number of valence electrons of the neutral atom in isolation (in its ground state); is the number of non-bonding valence electrons assigned to this atom in the Lewis structure of the molecule; and is the total number of electrons shared in bonds with other atoms in the molecule. By carefully analyzing the noise of a current, the charge of an electron can be calculated. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. * Subtract the number of electrons in the circle from the number of valence electrons of the neutral atom in isolation (in its ground state) to determine the formal charge. :center|350px * The formal charges computed for the remaining atoms in this Lewis structure of carbon dioxide are shown below. :center|450px It is important to keep in mind that formal charges are just that – formal, in the sense that this system is a formalism. Somewhat confusingly, in atomic physics, e sometimes denotes the electron charge, i.e. the negative of the elementary charge. Together with the mass-to-charge ratio, the electron mass was determined with reasonable precision. In atomic physics, a partial charge (or net atomic charge) is a non-integer charge value when measured in elementary charge units. Later, the name electron was assigned to the particle and the unit of charge e lost its name. There are different ways to draw the Lewis structure **Carbon single bonded to both oxygen atoms (carbon = +2, oxygens = −1 each, total formal charge = 0) **Carbon single bonded to one oxygen and double bonded to another (carbon = +1, oxygendouble = 0, oxygensingle = −1, total formal charge = 0) **Carbon double bonded to both oxygen atoms (carbon = 0, oxygens = 0, total formal charge = 0) Even though all three structures gave us a total charge of zero, the final structure is the superior one because there are no charges in the molecule at all. === Pictorial method === The following is equivalent: *Draw a circle around the atom for which the formal charge is requested (as with carbon dioxide, below) :center|150px * Count up the number of electrons in the atom's "circle." It is also impossible to realize charge at the yoctocoulomb scale. ==SI prefixes== Like other SI units, the coulomb can be modified by adding a prefix that multiplies it by a power of 10. ==Conversions== *The magnitude of the electrical charge of one mole of elementary charges (approximately , the Avogadro number) is known as a faraday unit of charge (closely related to the Faraday constant). Therefore, the "quantum of charge" is e. However, the unit of energy electronvolt (eV) is a remnant of the fact that the elementary charge was once called electron. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). The electron (symbol e) is on the left. In simple terms, formal charge is the difference between the number of valence electrons of an atom in a neutral free state and the number assigned to that atom in a Lewis structure.

-0.38

0.15

'-1.32'

-0.10

+80

C

A uniformly charged conducting sphere of $1.2 \mathrm{~m}$ diameter has surface charge density $8.1 \mu \mathrm{C} / \mathrm{m}^2$. Find the net charge on the sphere.

It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge distribution on the surface. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. A surface charge is an electric charge present on a two-dimensional surface. In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential (electrical potential), which is purely geometric: : \begin{matrix} \phi_1 = p_{11}Q_1 + \cdots + p_{1n}Q_n \\\ \phi_2 = p_{21}Q_1 + \cdots + p_{2n}Q_n \\\ \vdots \\\ \phi_n = p_{n1}Q_1 + \cdots + p_{nn}Q_n \end{matrix}. where is the surface charge on conductor . thumb|Viviani's curve: intersection of a sphere with a tangent cylinder. thumb|upright=0.75|The light blue part of the half sphere can be squared. In this case, the surface charge density decreases upon approach. Within this double layer, the first layer corresponds to the charged surface. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. Let us introduce the factor that describes how the actual charge density differs from the average and itself on a position on the surface of the -th conductor: :\frac{\sigma_j}{\langle\sigma_j\rangle} = f_j, or : \sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j. In this case, one must solve the PB equation together with an appropriate model of the surface charging process. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. The ability of the surface to regulate its charge can be quantified by the regulation parameter : p = \frac{C_{\rm D}}{C_{\rm I}+C_{\rm D}} where CD = ε0 ε κ is the diffuse layer capacitance and CI the inner (or regulation) capacitance. Thereby is the electrical charge enclosed by the Gaussian surface. Overall, two layers of charge and a potential drop from the electrode to the edge of the outer layer (outer Helmholtz Plane) are observed. Instead, the entirety of the charge of the conductor resides on the surface, and can be expressed by the equation: \sigma = E\varepsilon_0 where E is the electric field caused by the charge on the conductor and \varepsilon_0 is the permittivity of the free space. In the plane \textstyle \textbf{R}^2, this expression simplifies to : H_s(r)=1-e^{-\lambda \pi r^2}. ==Relationship to other functions== ===Nearest neighbour function=== In general, the spherical contact distribution function and the corresponding nearest neighbour function are not equal. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law.

76

3.52

4.979

9

37

E

The magnitude of the electrostatic force between two identical ions that are separated by a distance of $5.0 \times 10^{-10} \mathrm{~m}$ is $3.7 \times 10^{-9}$ N. What is the charge of each ion?

The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. An ion with a mass of 100 Da (daltons) () carrying two charges () will be observed at . The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. At large impact parameters, the charge of the ion is shielded by the tendency of electrons to cluster in the neighborhood of the ion and other ions to avoid it. The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. (The size of an atom, measured in angstroms (Å, or 10−10 m), is five orders of magnitude larger). This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. This model treats the electrostatic and hard-core interactions between all individual ions explicitly. In passing through a field of ions with density n, an electron will have many such encounters simultaneously, with various impact parameters (distance to the ion) and directions. The CODATA recommended value for an electron is ==Origin== When charged particles move in electric and magnetic fields the following two laws apply: *Lorentz force law: \mathbf{F} = Q (\mathbf{E} + \mathbf{v} \times \mathbf{B}), *Newton's second law of motion:\mathbf{F}=m\mathbf{a} = m \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} where F is the force applied to the ion, m is the mass of the particle, a is the acceleration, Q is the electric charge, E is the electric field, and v × B is the cross product of the ion's velocity and the magnetic flux density. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). Thus, the m/z of an ion alone neither infers mass nor the number of charges. thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. thumb|540x540px|Oppositely charged particles interact as they are moved through a column. A Coulomb collision is a binary elastic collision between two charged particles interacting through their own electric field. This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. Such forces between atoms are much weaker than the attractive electrical forces that hold the atoms themselves together (i.e., that bind electrons to the nucleus), and their range between atoms is shorter, because they arise from small separation of charges inside the neutral atom. List of orders of magnitude for electric charge Factor [Coulomb] SI prefix8th edition of the official brochure of the BIPM (SI units and prefixes). Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation.

0.33333333

35.2

3.2

0.123

0.88

C

How many megacoulombs of positive charge are in $1.00 \mathrm{~mol}$ of neutral molecular-hydrogen gas $\left(\mathrm{H}_2\right)$ ?

The ionization energy of the hydrogen molecule is 15.603 eV. The charge transfer gives 0.011 electron charge units to each helium atom. Because the two 1s electrons screen the protons to give an effective atomic number for the 2s electron close to 1, we can treat this 2s valence electron with a hydrogenic model. In this case, the effective nuclear charge can be calculated by Coulomb's law. Figure 2 - A molecular orbital diagram for open and closed hydrogen bridged cations with carbon is shown above. A hydrogen molecular ion cluster or hydrogen cluster ion is a positively charged cluster of hydrogen molecules. A free energy change of dissociation of −360 kJ/mol is equivalent to a pKa of −63 at 298 K. ===Other helium-hydrogen ions=== Additional helium atoms can attach to HeH+ to form larger clusters such as He2H+, He3H+, He4H+, He5H+ and He6H+. The dihydrogen cation or hydrogen molecular ion is a cation (positive ion) with formula . The helium hydride ion or hydridohelium(1+) ion or helonium is a cation (positively charged ion) with chemical formula HeH+. He22+ is the smallest possible molecule with a double positive charge. The calculated dipole moment of HeH+ is 2.26 or 2.84 D. The length of the covalent bond in the ion is 0.772 Å. ===Isotopologues=== The helium hydride ion has six relatively stable isotopologues, that differ in the isotopes of the two elements, and hence in the total atomic mass number (A) and the total number of neutrons (N) in the two nuclei: * or (A = 4, N = 1) * or (A = 5, N = 2) * or (A = 6, N = 3; radioactive) * or (A = 5, N = 2) * or (A = 6, N = 3) * or (A = 7, N = 4; radioactive) They all have three protons and two electrons. The amount of the dimer formed in the gas beam is of the order of one percent. ==Molecular ions== He2+ is a related ion bonded by a half covalent bond. It consists of two hydrogen nuclei (protons) sharing a single electron. Negative hydrogen clusters have not been found to exist. is theoretically unstable, but in theory is bound at 0.003 eV. ==Decay== in the free gas state decays by giving off H atoms and molecules. However hydrogen also forms singly charged clusters () with n up to 120. ==Experiments== Hydrogen ion clusters can be formed in liquid helium or with lesser cluster size in pure hydrogen. is far more common than higher even numbered clusters. is stable in solid hydrogen. High speed electrons also cause ionization of hydrogen molecules with a peak cross section around 50 eV. The ion can be formed from the ionization of a neutral hydrogen molecule . The He2 molecule has a large separation distance between the atoms of about 5200 pm (= 52 ångström). Due to the increased number of bonds that the hydrogen atom is part of, the higher electron density around hydrogen shields the 1H nucleus, causing its chemical shift to appear at negative ppm. The molecular helium anion is also found in liquid helium that has been excited by electrons with an energy level higher than 22 eV. Hydrogen-bridged cations are a type of charged species in which a hydrogen atom is simultaneously bonded to two atoms through partial sigma bonds.

0.19

0.68

0.69

0.132

15.1

A

A charge (uniform linear density $=9.0 \mathrm{nC} / \mathrm{m}$ ) lies on a string that is stretched along an $x$ axis from $x=0$ to $x=3.0 \mathrm{~m}$. Determine the magnitude of the electric field at $x=4.0 \mathrm{~m}$ on the $x$ axis.

We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. The electric field is a vector field... and has a magnitude and direction." Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Assuming infinite planes, the magnitude of the electric field E is: E = - \frac{\Delta V}{d} where ΔV is the potential difference between the plates and d is the distance separating the plates. To make it easy to calculate the Coulomb force on any charge at position \mathbf{x}_0 this expression can be divided by q_0 leaving an expression that only depends on the other charge (the source charge) \mathbf{E}_{1} (\mathbf{x}_0) = \frac{ \mathbf{F}_{01} } {q_0} = \frac{q_1 }{4\pi\varepsilon_0} \frac{\hat \mathbf{r}_{01}}{r_{01}^2} Where *\mathbf{E}_{1} (\mathbf{x}_0) is the component of the electric field at q_0 due to q_1 . From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with the inverse square of the distance from the charge. The electric field is defined as a vector field that associates to each point in space the electrostatic (Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. Analysis of the physics of string bending suggests that the resultant pitch of a string bent at its midpoint is given by u = \frac{1}{2L} \sqrt{\frac{T + \cos\theta (T - EA)}{\mu_{o}}} where L is the length of the vibrating element, T is the tension of the string prior to bending, \theta is the bend angle, E is the Young's Modulus of the string material, A is the string cross sectional area and \mu_{o} is the linear density of the string material. This is the electric field at point \mathbf{x}_0 due to the point charge q_1; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position \mathbf{x}_0. The Coulomb force on a charge of magnitude q at any point in space is equal to the product of the charge and the electric field at that point \mathbf{F} = q\mathbf{E} The SI unit of the electric field is the newton per coulomb (N/C), or volt per meter (V/m); in terms of the SI base units it is kg⋅m⋅s−3⋅A−1. ===Superposition principle=== Due to the linearity of Maxwell's equations, electric fields satisfy the superposition principle, which states that the total electric field, at a point, due to a collection of charges is equal to the vector sum of the electric fields at that point due to the individual charges. The electric field of such a uniformly moving point charge is hence given by: \mathbf{E} = \frac q {4 \pi \epsilon_0 r^3} \frac {1- \beta^2} {(1- \beta^2 \sin^2 \theta)^{3/2}} \mathbf{r} where q is the charge of the point source, \mathbf{r} is the position vector from the point source to the point in space, \beta is the ratio of observed speed of the charge particle to the speed of light and \theta is the angle between \mathbf{r} and the observed velocity of the charged particle. 300px|thumb|right|A string half the length (1/2), four times the tension (4), or one-quarter the mass per length (1/4) is an octave higher (2/1). By considering the charge \rho(\mathbf{x}')dV in each small volume of space dV at point \mathbf{x}' as a point charge, the resulting electric field, d\mathbf{E}(\mathbf{x}), at point \mathbf{x} can be calculated as d\mathbf{E}(\mathbf{x}) = \frac{\rho(\mathbf{x}')}{4\pi\varepsilon_0}\frac{\hat \mathbf{r}'}{{r'}^2} dV where *\hat \mathbf{r}' is the unit vector pointing from \mathbf{x}' to \mathbf{x}. *r' is the distance from \mathbf{x}' to \mathbf{x}. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. The force required to bend a string at its midpoint to a given angle \theta is given by F_{B} = 2\left(T + EA\left(\frac{1 - \cos\theta}{\cos\theta} \right) \right)\sin\theta . thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. As the electric field is defined in terms of force, and force is a vector (i.e. having both magnitude and direction), it follows that an electric field is a vector field. In that case, Coulomb's law fully describes the field.Purcell, pp. 5-7. ===Parallels between electrostatic and gravitational fields=== Coulomb's law, which describes the interaction of electric charges: \mathbf{F} = q \left(\frac{Q}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right) = q \mathbf{E} is similar to Newton's law of universal gravitation: \mathbf{F} = m\left(-GM\frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right) = m\mathbf{g} (where \mathbf{\hat{r}} = \mathbf{\frac{r}{|r|}}). Alternatively the electric field of uniformly moving point charges can be derived from the Lorentz transformation of four-force experienced by test charges in the source's rest frame given by Coulomb's law and assigning electric field and magnetic field by their definition given by the form of Lorentz force. The above equation, although consistent with that of uniformly moving point charges as well as its non-relativistic limit, are not corrected for quantum- mechanical effects. == Some common electric field values == * Infinite wire having uniform charge density \lambda has electric field at a distance x from it as \frac{\lambda}{2\pi\epsilon_0x} \hat{x} * Infinitely large surface having charge density \sigma has electric field at a distance x from it as \frac{\sigma}{2\epsilon_0} \hat{x} * Infinitely long cylinder having Uniform charge density \lambda that is charge contained along unit length of the cylinder has electric field at a distance x from it as \frac{\lambda}{2\pi\epsilon_0x} \hat{x} while it is 0 everywhere inside the cylinder * Uniformly charged non-conducting sphere of radius R, volume charge density \rho and total charge Q has electric field at a distance x from it as \frac{Q}{4\pi\epsilon_0x^2} \hat{x} while the electric field at a point \vec{r} inside sphere from its center is given by \frac{Q}{4\pi\epsilon_0R^3}\vec{r} * Uniformly charged conducting sphere of radius R, surface charge density \sigma and total charge Q has electric field at a distance x from it as \frac{Q}{4\pi\epsilon_0x^2} \hat{x} while the electric field inside is 0 * Electric field infinitely close to a conducting surface in electrostatic equilibrium having charge density \sigma at that point is \frac{\sigma}{\epsilon_0} \hat{x} * Uniformly charged ring having total charge Q has electric field at a distance x along its axis as \frac{Qx}{4\pi\epsilon_0(R^2+x^2)^{3/2}} \hat{x}' * Uniformly charged disc of radius R and charge density \sigma has electric field at a distance x along its axis from it as \frac{\sigma}{2\epsilon_0} \left[1-\left(\frac{R^2}{x^2}-1\right)^{-1/2}\right] \hat{x} * Electric field due to dipole of dipole moment \vec{p} at a distance x from their center along equatorial plane is given as -\frac{\vec{p}}{4\pi\epsilon_0x^3} and the same along the axial line is approximated to \frac{\vec{p}}{2\pi\epsilon_0x^3} for x much bigger than the distance between dipoles. 275px|thumb|Stick figure of 1.75 meters standing next to a violin string of .33 meters and a long string instrument string of 10 meters.

13

5840

61.0

+10

131

C

A long, straight wire has fixed negative charge with a linear charge density of magnitude $3.6 \mathrm{nC} / \mathrm{m}$. The wire is to be enclosed by a coaxial, thin-walled nonconducting cylindrical shell of radius $1.5 \mathrm{~cm}$. The shell is to have positive charge on its outside surface with a surface charge density $\sigma$ that makes the net external electric field zero. Calculate $\sigma$.

It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. With the same example, using a larger Gaussian surface outside the shell where , Gauss's law will produce a non-zero electric field. And, as mentioned, any exterior charges do not count. === Cylindrical surface === A cylindrical Gaussian surface is used when finding the electric field or the flux produced by any of the following: * an infinitely long line of uniform charge * an infinite plane of uniform charge * an infinitely long cylinder of uniform charge As example "field near infinite line charge" is given below; Consider a point P at a distance from an infinite line charge having charge density (charge per unit length) λ. Let us introduce the factor that describes how the actual charge density differs from the average and itself on a position on the surface of the -th conductor: :\frac{\sigma_j}{\langle\sigma_j\rangle} = f_j, or : \sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. Given the electrical potential on a conductor surface (the equipotential surface or the point chosen on surface ) contained in a system of conductors : :\phi_i = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{ji}} \mbox{ (i=1, 2..., n)}, where , i.e. the distance from the area- element to a particular point on conductor . is not, in general, uniformly distributed across the surface. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential (electrical potential), which is purely geometric: : \begin{matrix} \phi_1 = p_{11}Q_1 + \cdots + p_{1n}Q_n \\\ \phi_2 = p_{21}Q_1 + \cdots + p_{2n}Q_n \\\ \vdots \\\ \phi_n = p_{n1}Q_1 + \cdots + p_{nn}Q_n \end{matrix}. where is the surface charge on conductor . Thereby is the electrical charge enclosed by the Gaussian surface. Surface Charging and Points of Zero Charge. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. If the Gaussian surface is chosen such that for every point on the surface the component of the electric field along the normal vector is constant, then the calculation will not require difficult integration as the constants which arise can be taken out of the integral. If the electrode is polarizable, then its surface charge depends on the electrode potential. On a capacitor, the charge on the two conductors is equal and opposite: . Hence, with :p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}, we have :\phi_i=\sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1, 2, ..., n)}. ==Example== In this example, we employ the method of coefficients of potential to determine the capacitance on a two-conductor system. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The electrostatic potential at point is \phi_P = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{j}}. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h.

4

3.8

'-1270.0'

38

8.7

B

Beams of high-speed protons can be produced in "guns" using electric fields to accelerate the protons. (a) What acceleration would a proton experience if the gun's electric field were $2.00 \times 10^4 \mathrm{~N} / \mathrm{C}$ ? (b) What speed would the proton attain if the field accelerated the proton through a distance of $1.00 \mathrm{~cm}$ ?

An electrostatic particle accelerator is a particle accelerator in which charged particles are accelerated to a high energy by a static high voltage potential. A special application of electrostatic particle accelerator are dust accelerators in which nanometer to micrometer sized electrically charged dust particles are accelerated to speeds up to 100 km/s. For the special case of an electrostatic field that is surpassed by a particle, the acceleration voltage is directly given by integrating the electric field along its path. The advantages of electrostatic accelerators over oscillating field machines include lower cost, the ability to produce continuous beams, and higher beam currents that make them useful to industry. The accelerating voltage on electrostatic machines is in the range 0.1 to 25 MV and the charge on particles is a few elementary charges, so the particle energy is in the low MeV range. When performing a modeling task for any accelerator operation, the results of charged particle beam dynamics simulations must feed into the associated application. High energy oscillating field accelerators usually incorporate an electrostatic machine as their first stage, to accelerate particles to a high enough velocity to inject into the main accelerator. See particle- beam weapon for more information on this type of weapon. ==See also== * Ion source * Ion thruster * Ion wind ==References== ==External links== * Stopping parameters of ion beams in solids calculated by MELF-GOS model * ISOLDE – Facility dedicated to the production of a large variety of radioactive ion beams located at CERN Category:Plasma physics Category:Semiconductor device fabrication Category:Semiconductor analysis Category:Thin film deposition Category:Ions Category:Accelerator physics In particle accelerators, a common mechanism for accelerating a charged particle beam is via copper resonant cavities in which electric and magnetic fields form a standing wave, the mode of which is designed so that the E field points along the axis of the accelerator, producing forward acceleration of the particles when in the correct phase. A charged particle accelerator is a complex machine that takes elementary charged particles and accelerates them to very high energies. Accelerator physics is a field of physics encompassing all the aspects required to design and operate the equipment and to understand the resulting dynamics of the charged particles. Oscillating accelerators do not have this limitation, so they can achieve higher particle energies than electrostatic machines. While all linacs accelerate particles in a straight line, electrostatic accelerators use a fixed accelerating field from a single high voltage source, while radiofrequency linacs use oscillating electric fields across a series of accelerating gaps. == Applications == Electrostatic accelerators have a wide array of applications in science and industry. This innovative propulsion technique named Ion Beam Shepherd has been shown to be effective in the area of active space debris removal as well as asteroid deflection. ===High-energy ion beams=== High-energy ion beams produced by particle accelerators are used in atomic physics, nuclear physics and particle physics. ===Weaponry=== The use of ion beams as a particle-beam weapon is theoretically possible, but has not been demonstrated. This contrasts with the other major category of particle accelerator, oscillating field particle accelerators, in which the particles are accelerated by oscillating electric fields. The maximum particle energy produced by electrostatic accelerators is limited by the maximum voltage which can be achieved the machine. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. Omitting practical problems, if the platform is positively charged, it will repel the ions of the same electric polarity, accelerating them. The following considerations are generalized for time-dependent fields. == Longitudinal voltage == The longitudinal effective acceleration voltage is given by the kinetic energy gain experienced by a particle with velocity \beta c along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge, V_\parallel(\beta) = \frac 1 q \vec e_s \cdot \int \vec F_L(s,t) \,\mathrm{d}s = \frac 1 q \vec e_s \cdot \int \vec F_L(s, t = \frac{s}{\beta c}) \,\mathrm d s . The ion current density j that can be accelerated using a gridded ion source is limited by the space charge effect, which is described by Child's law: j \approx \frac{4\epsilon_0}{9} \sqrt{\frac{2e}{m}} \frac{(\Delta V)^{\frac{3}{2}}}{d^2}, where \Delta V is the voltage between the grids, d is the distance between the grids, and m is the ion mass. Electrostatic accelerators are a subset of linear accelerators (linacs). Many universities worldwide have electrostatic accelerators for research purposes.

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An infinite line of charge produces a field of magnitude $4.5 \times$ $10^4 \mathrm{~N} / \mathrm{C}$ at distance $2.0 \mathrm{~m}$. Find the linear charge density.

Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m−1), at any point on a line charge distribution. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. Like mass density, charge density can vary with position. The linear charge density is the ratio of an infinitesimal electric charge dQ (SI unit: C) to an infinitesimal line element, \lambda_q = \frac{d Q}{d \ell}\,, similarly the surface charge density uses a surface area element dS \sigma_q = \frac{d Q}{d S}\,, and the volume charge density uses a volume element dV \rho_q =\frac{d Q}{d V} \, , Integrating the definitions gives the total charge Q of a region according to line integral of the linear charge density λq(r) over a line or 1d curve C, Q = \int_L \lambda_q(\mathbf{r}) \, d\ell similarly a surface integral of the surface charge density σq(r) over a surface S, Q = \int_S \sigma_q(\mathbf{r}) \, dS and a volume integral of the volume charge density ρq(r) over a volume V, Q = \int_V \rho_q(\mathbf{r}) \, dV where the subscript q is to clarify that the density is for electric charge, not other densities like mass density, number density, probability density, and prevent conflict with the many other uses of λ, σ, ρ in electromagnetism for wavelength, electrical resistivity and conductivity. As always, the integral of the charge density over a region of space is the charge contained in that region. Similar equations are used for the linear and surface charge densities. == Charge density in special relativity == In special relativity, the length of a segment of wire depends on velocity of observer because of length contraction, so charge density will also depend on velocity. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). The equivalent proofs for linear charge density and surface charge density follow the same arguments as above. == Discrete charges == For a single point charge q at position r0 inside a region of 3d space R, like an electron, the volume charge density can be expressed by the Dirac delta function: \rho_q(\mathbf{r}) = q \delta(\mathbf{r} - \mathbf{r}_0) where r is the position to calculate the charge. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. Derivation: qvB = mv\frac{v}{r} or Since F_\text{electric} = F_\text{magnetic}, E q = B q v or Equations () and () yield \frac{q}{m}=\frac{E}{B^2r} === Significance === In some experiments, the charge-to-mass ratio is the only quantity that can be measured directly. The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. Since all charge is carried by subatomic particles, which can be idealized as points, the concept of a continuous charge distribution is an approximation, which becomes inaccurate at small length scales. It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. Charge density can be either positive or negative, since electric charge can be either positive or negative. And, as mentioned, any exterior charges do not count. === Cylindrical surface === A cylindrical Gaussian surface is used when finding the electric field or the flux produced by any of the following: * an infinitely long line of uniform charge * an infinite plane of uniform charge * an infinitely long cylinder of uniform charge As example "field near infinite line charge" is given below; Consider a point P at a distance from an infinite line charge having charge density (charge per unit length) λ. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. The value of nuclear magneton System of units Value Unit SI J·T CGS Erg·G eV eV·T MHz/T (per h) MHz/T The nuclear magneton (symbol μ) is a physical constant of magnetic moment, defined in SI units by: :\mu_\text{N} = {{e \hbar} \over {2 m_\text{p}}} and in Gaussian CGS units by: :\mu_\text{N} = {{e \hbar} \over{2 m_\text{p}c}} where: :e is the elementary charge, :ħ is the reduced Planck constant, :m is the proton rest mass, and :c is the speed of light In SI units, its value is approximately: :μ = In Gaussian CGS units, its value can be given in convenient units as :μ = The nuclear magneton is the natural unit for expressing magnetic dipole moments of heavy particles such as nucleons and atomic nuclei. An ion with a mass of 100 Da (daltons) () carrying two charges () will be observed at .

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B

A charged nonconducting rod, with a length of $2.00 \mathrm{~m}$ and a cross-sectional area of $4.00 \mathrm{~cm}^2$, lies along the positive side of an $x$ axis with one end at the origin. The volume charge density $\rho$ is charge per unit volume in coulombs per cubic meter. How many excess electrons are on the rod if $\rho$ is uniform, with a value of $-4.00 \mu \mathrm{C} / \mathrm{m}^3$?

Since the field is uniform in the direction of wire length, for a conductor having uniformly consistent resistivity ρ, the current density will also be uniform in any cross-sectional area and oriented in the direction of wire length, so we may write:Lerner L, Physics for scientists and engineers, Jones & Bartlett, 1997, pp. 732–733 J = \frac{I}{a}. In these situations, the density simplifies to :\rho(\mathbf{r})=\sum_{k=1}^N n_{k}|\varphi_k(\mathbf{r})|^2. == General properties == From its definition, the electron density is a non-negative function integrating to the total number of electrons. Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. The charge density of the system at a point r is a sum of the charge densities for each charge qi at position ri, where : \rho_q(\mathbf{r})=\sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) The delta function for each charge qi in the sum, δ(r − ri), ensures the integral of charge density over R returns the total charge in R: Q = \int_R d^3 \mathbf{r} \sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i \int_R d^3 \mathbf{r} \delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i If all charge carriers have the same charge q (for electrons q = −e, the electron charge) the charge density can be expressed through the number of charge carriers per unit volume, n(r), by \rho_q(\mathbf{r}) = q n(\mathbf{r})\,. The total charge divided by the length, surface area, or volume will be the average charge densities: \langle\lambda_q \rangle = \frac{Q}{\ell}\,,\quad \langle\sigma_q\rangle = \frac{Q}{S}\,,\quad\langle\rho_q\rangle = \frac{Q}{V}\,. == Free, bound and total charge == In dielectric materials, the total charge of an object can be separated into "free" and "bound" charges. At position at time , the distribution of charge flowing is described by the current density: \mathbf{j}(\mathbf{r}, t) = \rho(\mathbf{r},t) \; \mathbf{v}_\text{d} (\mathbf{r},t) where * is the current density vector; * is the particles' average drift velocity (SI unit: m∙s−1); *\rho(\mathbf{r}, t) = q \, n(\mathbf{r},t) is the charge density (SI unit: coulombs per cubic metre), in which ** is the number of particles per unit volume ("number density") (SI unit: m−3); ** is the charge of the individual particles with density (SI unit: coulombs). This charge is equal to d q = \rho \, v\, dt \, dA, where is the charge density at . It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Charge density can be either positive or negative, since electric charge can be either positive or negative. In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. As always, the integral of the charge density over a region of space is the charge contained in that region. In quantum chemical calculations, the electron density, ρ(r), is a function of the coordinates r, defined so ρ(r)dr is the number of electrons in a small volume dr. The linear charge density is the ratio of an infinitesimal electric charge dQ (SI unit: C) to an infinitesimal line element, \lambda_q = \frac{d Q}{d \ell}\,, similarly the surface charge density uses a surface area element dS \sigma_q = \frac{d Q}{d S}\,, and the volume charge density uses a volume element dV \rho_q =\frac{d Q}{d V} \, , Integrating the definitions gives the total charge Q of a region according to line integral of the linear charge density λq(r) over a line or 1d curve C, Q = \int_L \lambda_q(\mathbf{r}) \, d\ell similarly a surface integral of the surface charge density σq(r) over a surface S, Q = \int_S \sigma_q(\mathbf{r}) \, dS and a volume integral of the volume charge density ρq(r) over a volume V, Q = \int_V \rho_q(\mathbf{r}) \, dV where the subscript q is to clarify that the density is for electric charge, not other densities like mass density, number density, probability density, and prevent conflict with the many other uses of λ, σ, ρ in electromagnetism for wavelength, electrical resistivity and conductivity. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. The electric current is dI = dq/dt = \rho v dA, it follows that the current density vector is the vector normal dA (i.e. parallel to ) and of magnitude dI/dA = \rho v \mathbf{j} = \rho \mathbf{v}. The electrical resistance of a uniform conductor is given in terms of resistivity by: {R} = \rho \frac{\ell}{a} where ℓ is the length of the conductor in SI units of meters, is the cross-sectional area (for a round wire if is radius) in units of meters squared, and ρ is the resistivity in units of ohm·meters. If (SI unit: A) is the electric current flowing through , then electric current density at is given by the limit: j = \lim_{A \to 0} \frac{I_A}{A} = \left.\frac{\partial I}{\partial A} \right|_{A=0}, with surface remaining centered at and orthogonal to the motion of the charges during the limit process. The delta function has the sifting property for any function f: \int_R d^3 \mathbf{r} f(\mathbf{r})\delta(\mathbf{r} - \mathbf{r}_0) = f(\mathbf{r}_0) so the delta function ensures that when the charge density is integrated over R, the total charge in R is q: Q =\int_R d^3 \mathbf{r} \, \rho_q =\int_R d^3 \mathbf{r} \, q \delta(\mathbf{r} - \mathbf{r}_0) = q \int_R d^3 \mathbf{r} \, \delta(\mathbf{r} - \mathbf{r}_0) = q This can be extended to N discrete point-like charge carriers. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m−1), at any point on a line charge distribution.

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-31.95

A

A one-particle, one-dimensional system has $\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \mathrm{~nm}$. At $t=0$, the particle's position is measured. (b) Find the probability that the measured value is between $x=0$ and $x=2 \mathrm{~nm}$.

Hence, at a given time , is the probability density function of the particle's position. If the oscillator spends an infinitesimal amount of time in the vicinity of a given -value, then the probability of being in that vicinity will be :P(x)\, dx \propto dt. If it corresponds to a non-degenerate eigenvalue of , then |\psi (x)|^2 gives the probability of the corresponding value of for the initial state . Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. Thus the probability that the particle is in the volume at is :\mathbf{P}(V)=\int_V \rho(\mathbf {x})\, \mathrm{d\mathbf {x}} = \int_V |\psi(\mathbf {x}, t_0)|^2\, \mathrm{d\mathbf {x}}. If the system is known to be in some eigenstate of (e.g. after an observation of the corresponding eigenvalue of ) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of (so long as no other important forces act between the measurements). We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. The probability of measuring |u\rangle is given by :P(|u\rangle) = \langle r|u\rangle\langle u|r\rangle = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} Which agrees with experiment. ==Normalization== In the example above, the measurement must give either or , so the total probability of measuring or must be 1. In other words the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. Plugging this into the expression for yields :P(x) = \frac{1}{\pi}\frac{1}{\sqrt{A^2-x^2}}. Suppose a wavefunction is a solution of the wave equation, giving a description of the particle (position , for time ). Note that if any solution to the wave equation is normalisable at some time , then the defined above is always normalised, so that :\rho_t(\mathbf x)=\left |\psi(\mathbf x, t)\right |^2 = \left|\frac{\psi_0(\mathbf x, t)}{a}\right|^2 is always a probability density function for all . Plugging this into our expression for yields :P(x) = \frac{1}{T} \sqrt{\frac{2m}{E-U(x)}}. If the set of eigenstates to which the system can jump upon measurement of is the same as the set of eigenstates for measurement of , then subsequent measurements of either or always produce the same values with probability of 1, no matter the order in which they are applied. 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. Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function is the wave function. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In this system, all momenta are equally probable. ==See also== *Probability density function *Correspondence principle *Classical limit *Wave function ==References== Category:Concepts in physics Category:Classical mechanics Category:Theoretical physics Then \psi (x) is the "probability amplitude" for the eigenstate . Once this is done, is readily obtained for any allowed energy . ==Examples== ===Simple harmonic oscillator=== thumb|300px|right|The probability density function of the state of the quantum harmonic oscillator. Therefore, if the system is known to be in some eigenstate of (all probability amplitudes zero except for one eigenstate), then when is observed the probability amplitudes are changed. Following a similar argument as above, the result is :P(p) = \frac{2}{T}\frac{1}{|F(x)|}, where is the force acting on the particle as a function of position.

7.42

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B

Calculate the ground-state energy of the hydrogen atom using SI units and convert the result to electronvolts.

To convert from "value of ionization energy" to the corresponding "value of molar ionization energy", the conversion is: *:1 eV = 96.48534 kJ/mol *:1 kJ/mol = 0.0103642688 eV == References == === WEL (Webelements) === As quoted at http://www.webelements.com/ from these sources: * J.E. Huheey, E.A. Keiter, and R.L. Keiter in Inorganic Chemistry : Principles of Structure and Reactivity, 4th edition, HarperCollins, New York, USA, 1993. == Numerical values == For each atom, the column marked 1 is the first ionization energy to ionize the neutral atom, the column marked 2 is the second ionization energy to remove a second electron from the +1 ion, the column marked 3 is the third ionization energy to remove a third electron from the +2 ion, and so on. "use" and "WEL" give ionization energy in the unit kJ/mol; "CRC" gives atomic ionization energy in the unit eV. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 H hydrogen use 1312.0 WEL 1312.0 CRC 13.59844 2 He helium use 2372.3 5250.5 WEL 2372.3 5250.5 CRC 24.58738 54.41776 3 Li lithium use 520.2 7298.1 11815.0 WEL 520.2 7298.1 11815.0 CRC 5.39171 75.64009 122.45435 4 Be beryllium use 899.5 1757.1 14848.7 21006.6 WEL 899.5 1757.1 14848.7 21006.6 CRC 9.32269 18.21115 153.89620 217.71858 5 B boron use 800.6 2427.1 3659.7 25025.8 32826.7 WEL 800.6 2427.1 3659.7 25025.8 32826.7 CRC 8.29803 25.15484 37.93064 259.37521 340.22580 6 C carbon use 1086.5 2352.6 4620.5 6222.7 37831 47277.0 WEL 1086.5 2352.6 4620.5 6222.7 37831 47277.0 CRC 11.26030 24.38332 47.8878 64.4939 392.087 489.99334 7 N nitrogen use 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 WEL 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 CRC 14.53414 29.6013 47.44924 77.4735 97.8902 552.0718 667.046 8 O oxygen use 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 WEL 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 CRC 13.61806 35.11730 54.9355 77.41353 113.8990 138.1197 739.29 871.4101 9 F fluorine use 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 WEL 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 CRC 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 953.9112 1103.1176 10 Ne neon use 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 WEL 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 CRC 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 1195.8286 1362.1995 11 Na sodium use 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 159076 WEL 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 CRC 5.13908 47.2864 71.6200 98.91 138.40 172.18 208.50 264.25 299.864 1465.121 1648.702 12 Mg magnesium use 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 169988 189368 WEL 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 CRC 7.64624 15.03528 80.1437 109.2655 141.27 186.76 225.02 265.96 328.06 367.50 1761.805 1962.6650 13 Al aluminium use 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 42647 201266 222316 WEL 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 CRC 5.98577 18.82856 28.44765 119.992 153.825 190.49 241.76 284.66 330.13 398.75 442.00 2085.98 2304.1410 14 Si silicon use 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 45962 50502 235196 257923 WEL 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 CRC 8.15169 16.34585 33.49302 45.14181 166.767 205.27 246.5 303.54 351.12 401.37 476.36 523.42 2437.63 2673.182 15 P phosphorus use 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 46261 54110 59024 271791 296195 WEL 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 CRC 10.48669 19.7694 30.2027 51.4439 65.0251 220.421 263.57 309.60 372.13 424.4 479.46 560.8 611.74 2816.91 3069.842 16 S sulfur use 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 48710 54460 62930 68216 311048 337138 WEL 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 CRC 10.36001 23.3379 34.79 47.222 72.5945 88.0530 280.948 328.75 379.55 447.5 504.8 564.44 652.2 707.01 3223.78 3494.1892 17 Cl chlorine use 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 51068 57119 63363 72341 78095 352994 380760 WEL 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 CRC 12.96764 23.814 39.61 53.4652 67.8 97.03 114.1958 348.28 400.06 455.63 529.28 591.99 656.71 749.76 809.40 3658.521 3946.2960 18 Ar argon use 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 52002 59653 66199 72918 82473 88576 397605 427066 WEL 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 CRC 15.75962 27.62967 40.74 59.81 75.02 91.009 124.323 143.460 422.45 478.69 538.96 618.26 686.10 755.74 854.77 918.03 4120.8857 4426.2296 19 K potassium use 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 54490 60730 68950 75900 83080 93400 99710 444880 476063 WEL 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 CRC 4.34066 31.63 45.806 60.91 82.66 99.4 117.56 154.88 175.8174 503.8 564.7 629.4 714.6 786.6 861.1 968 1033.4 4610.8 4934.046 20 Ca calcium use 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 63410 70110 78890 86310 94000 104900 111711 494850 527762 WEL 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 CRC 6.11316 11.87172 50.9131 67.27 84.50 108.78 127.2 147.24 188.54 211.275 591.9 657.2 726.6 817.6 894.5 974 1087 1157.8 5128.8 5469.864 21 Sc scandium use 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 WEL 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 CRC 6.5615 12.79967 24.75666 73.4894 91.65 110.68 138.0 158.1 180.03 225.18 249.798 687.36 756.7 830.8 927.5 1009 1094 1213 1287.97 5674.8 6033.712 22 Ti titanium use 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 639294 WEL 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 CRC 6.8281 13.5755 27.4917 43.2672 99.30 119.53 140.8 170.4 192.1 215.92 265.07 291.500 787.84 863.1 941.9 1044 1131 1221 1346 1425.4 6249.0 6625.82 23 V vanadium use 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 661050 699144 WEL 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 CRC 6.7462 14.66 29.311 46.709 65.2817 128.13 150.6 173.4 205.8 230.5 255.7 308.1 336.277 896.0 976 1060 1168 1260 1355 1486 1569.6 6851.3 7246.12 24 Cr chromium use 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 166090 721870 761733 WEL 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 CRC 6.7665 16.4857 30.96 49.16 69.46 90.6349 160.18 184.7 209.3 244.4 270.8 298.0 354.8 384.168 1010.6 1097 1185 1299 1396 1496 1634 1721.4 7481.7 7894.81 25 Mn manganese use 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 172500 181380 785450 827067 WEL 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 CRC 7.43402 15.63999 33.668 51.2 72.4 95.6 119.203 194.5 221.8 248.3 286.0 314.4 343.6 403.0 435.163 1134.7 1224 1317 1437 1539 1644 1788 1879.9 8140.6 8571.94 26 Fe iron use 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 173600 188100 195200 851800 895161 WEL 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 CRC 7.9024 16.1878 30.652 54.8 75.0 99.1 124.98 151.06 233.6 262.1 290.2 330.8 361.0 392.2 457 489.256 1266 1358 1456 1582 1689 1799 1950 2023 8828 9277.69 27 Co cobalt use 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 178100 189300 204500 214100 920870 966023 WEL 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 CRC 7.8810 17.083 33.50 51.3 79.5 102.0 128.9 157.8 186.13 275.4 305 336 379 411 444 511.96 546.58 1397.2 1504.6 1603 1735 1846 1962 2119 2219.0 9544.1 10012.12 28 Ni nickel use 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 182700 194000 205600 221400 231490 992718 1039668 WEL 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 CRC 7.6398 18.16884 35.19 54.9 76.06 108 133 162 193 224.6 321.0 352 384 430 464 499 571.08 607.06 1541 1648 1756 1894 2011 2131 2295 2399.2 10288.8 10775.40 29 Cu copper use 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 184900 198800 210500 222700 239100 249660 1067358 1116105 WEL 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 CRC 7.72638 20.29240 36.841 57.38 79.8 103 139 166 199 232 265.3 369 401 435 484 520 557 633 670.588 1697 1804 1916 2060 2182 2308 2478 2587.5 11062.38 11567.617 30 Zn zinc use 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 WEL 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 CRC 9.3942 17.96440 39.723 59.4 82.6 108 134 174 203 238 274 310.8 419.7 454 490 542 579 619 698 738 1856 31 Ga gallium use 578.8 1979.3 2963 6180 WEL 578.8 1979.3 2963 6180 CRC 5.99930 20.5142 30.71 64 32 Ge germanium use 762 1537.5 3302.1 4411 9020 WEL 762 1537.5 3302.1 4411 9020 CRC 7.8994 15.93462 34.2241 45.7131 93.5 33 As arsenic use 947.0 1798 2735 4837 6043 12310 WEL 947.0 1798 2735 4837 6043 12310 CRC 9.7886 18.633 28.351 50.13 62.63 127.6 34 Se selenium use 941.0 2045 2973.7 4144 6590 7880 14990 WEL 941.0 2045 2973.7 4144 6590 7880 14990 CRC 9.75238 21.19 30.8204 42.9450 68.3 81.7 155.4 35 Br bromine use 1139.9 2103 3470 4560 5760 8550 9940 18600 WEL 1139.9 2103 3470 4560 5760 8550 9940 18600 CRC 11.81381 21.8 36 47.3 59.7 88.6 103.0 192.8 36 Kr krypton use 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 90400 96300 101400 111100 116290 282500 296200 311400 326200 WEL 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 CRC 13.99961 24.35985 36.950 52.5 64.7 78.5 111.0 125.802 230.85 268.2 308 350 391 447 492 541 592 641 786 833 884 937 998 1051 1151 1205.3 2928 3070 3227 3381 37 Rb rubidium use 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 WEL 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 CRC 4.17713 27.285 40 52.6 71.0 84.4 99.2 136 150 277.1 38 Sr strontium use 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 31270 WEL 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 CRC 5.6949 11.03013 42.89 57 71.6 90.8 106 122.3 162 177 324.1 39 Y yttrium use 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 19900 36090 WEL 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 CRC 6.2171 12.24 20.52 60.597 77.0 93.0 116 129 146.2 191 206 374.0 40 Zr zirconium use 640.1 1270 2218 3313 7752 9500 WEL 640.1 1270 2218 3313 7752 9500 CRC 6.63390 13.13 22.99 34.34 80.348 41 Nb niobium use 652.1 1380 2416 3700 4877 9847 12100 WEL 652.1 1380 2416 3700 4877 9847 12100 CRC 6.75885 14.32 25.04 38.3 50.55 102.057 125 42 Mo molybdenum use 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 93400 98420 104400 121900 127700 133800 139800 148100 154500 WEL 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 CRC 7.09243 16.16 27.13 46.4 54.49 68.8276 125.664 143.6 164.12 186.4 209.3 230.28 279.1 302.60 544.0 570 636 702 767 833 902 968 1020 1082 1263 1323 1387 1449 1535 1601 43 Tc technetium use 702 1470 2850 WEL 702 1470 2850 CRC 7.28 15.26 29.54 44 Ru ruthenium use 710.2 1620 2747 WEL 710.2 1620 2747 CRC 7.36050 16.76 28.47 45 Rh rhodium use 719.7 1740 2997 WEL 719.7 1740 2997 CRC 7.45890 18.08 31.06 46 Pd palladium use 804.4 1870 3177 WEL 804.4 1870 3177 CRC 8.3369 19.43 32.93 47 Ag silver use 731.0 2070 3361 WEL 731.0 2070 3361 CRC 7.5762 21.49 34.83 48 Cd cadmium use 867.8 1631.4 3616 WEL 867.8 1631.4 3616 CRC 8.9938 16.90832 37.48 49 In indium use 558.3 1820.7 2704 5210 WEL 558.3 1820.7 2704 5210 CRC 5.78636 18.8698 28.03 54 50 Sn tin use 708.6 1411.8 2943.0 3930.3 7456 WEL 708.6 1411.8 2943.0 3930.3 7456 CRC 7.3439 14.63225 30.50260 40.73502 72.28 51 Sb antimony use 834 1594.9 2440 4260 5400 10400 WEL 834 1594.9 2440 4260 5400 10400 CRC 8.6084 16.53051 25.3 44.2 56 108 52 Te tellurium use 869.3 1790 2698 3610 5668 6820 13200 WEL 869.3 1790 2698 3610 5668 6820 13200 CRC 9.0096 18.6 27.96 37.41 58.75 70.7 137 53 I iodine use 1008.4 1845.9 3180 WEL 1008.4 1845.9 3180 CRC 10.45126 19.1313 33 54 Xe xenon use 1170.4 2046.4 3099.4 WEL 1170.4 2046.4 3099.4 CRC 12.1298 21.20979 32.1230 55 Cs caesium use 375.7 2234.3 3400 WEL 375.7 2234.3 3400 CRC 3.89390 23.15745 56 Ba barium use 502.9 965.2 3600 WEL 502.9 965.2 3600 CRC 5.21170 10.00390 57 La lanthanum use 538.1 1067 1850.3 4819 5940 WEL 538.1 1067 1850.3 4819 5940 CRC 5.5769 11.060 19.1773 49.95 61.6 58 Ce cerium use 534.4 1050 1949 3547 6325 7490 WEL 534.4 1050 1949 3547 6325 7490 CRC 5.5387 10.85 20.198 36.758 65.55 77.6 59 Pr praseodymium use 527 1020 2086 3761 5551 WEL 527 1020 2086 3761 5551 CRC 5.473 10.55 21.624 38.98 57.53 60 Nd neodymium use 533.1 1040 2130 3900 WEL 533.1 1040 2130 3900 CRC 5.5250 10.73 22.1 40.41 61 Pm promethium use 540 1050 2150 3970 WEL 540 1050 2150 3970 CRC 5.582 10.90 22.3 41.1 62 Sm samarium use 544.5 1070 2260 3990 WEL 544.5 1070 2260 3990 CRC 5.6436 11.07 23.4 41.4 63 Eu europium use 547.1 1085 2404 4120 WEL 547.1 1085 2404 4120 CRC 5.6704 11.241 24.92 42.7 64 Gd gadolinium use 593.4 1170 1990 4250 WEL 593.4 1170 1990 4250 CRC 6.1501 12.09 20.63 44.0 65 Tb terbium use 565.8 1110 2114 3839 WEL 565.8 1110 2114 3839 CRC 5.8638 11.52 21.91 39.79 66 Dy dysprosium use 573.0 1130 2200 3990 WEL 573.0 1130 2200 3990 CRC 5.9389 11.67 22.8 41.47 67 Ho holmium use 581.0 1140 2204 4100 WEL 581.0 1140 2204 4100 CRC 6.0215 11.80 22.84 42.5 68 Er erbium use 589.3 1150 2194 4120 WEL 589.3 1150 2194 4120 CRC 6.1077 11.93 22.74 42.7 69 Tm thulium use 596.7 1160 2285 4120 WEL 596.7 1160 2285 4120 CRC 6.18431 12.05 23.68 42.7 70 Yb ytterbium use 603.4 1174.8 2417 4203 WEL 603.4 1174.8 2417 4203 CRC 6.25416 12.1761 25.05 43.56 71 Lu lutetium use 523.5 1340 2022.3 4370 6445 WEL 523.5 1340 2022.3 4370 6445 CRC 5.4259 13.9 20.9594 45.25 66.8 72 Hf hafnium use 658.5 1440 2250 3216 WEL 658.5 1440 2250 3216 CRC 6.82507 14.9 23.3 33.33 73 Ta tantalum use 761 1500 WEL 761 1500 CRC 7.5496 74 W tungsten use 770 1700 WEL 770 1700 CRC 7.8640 75 Re rhenium use 760 1260 2510 3640 WEL 760 1260 2510 3640 CRC 7.8335 76 Os osmium use 840 1600 WEL 840 1600 CRC 8.4382 77 Ir iridium use 880 1600 WEL 880 1600 CRC 8.9670 78 Pt platinum use 870 1791 WEL 870 1791 CRC 8.9587 18.563 28https://dept.astro.lsa.umich.edu/~cowley/ionen.htm archived 79 Au gold use 890.1 1980 WEL 890.1 1980 CRC 9.2255 20.5 30 80 Hg mercury use 1007.1 1810 3300 WEL 1007.1 1810 3300 CRC 10.43750 18.756 34.2 81 Tl thallium use 589.4 1971 2878 WEL 589.4 1971 2878 CRC 6.1082 20.428 29.83 82 Pb lead use 715.6 1450.5 3081.5 4083 6640 WEL 715.6 1450.5 3081.5 4083 6640 CRC 7.41666 15.0322 31.9373 42.32 68.8 83 Bi bismuth use 703 1610 2466 4370 5400 8520 WEL 703 1610 2466 4370 5400 8520 CRC 7.2856 16.69 25.56 45.3 56.0 88.3 84 Po polonium use 812.1 WEL 812.1 CRC 8.417 85 At astatine use 899.003 WEL 920 CRC talk Originally quoted as 9.31751(8) eV. 86 Rn radon use 1037 WEL 1037 CRC 10.74850 87 Fr francium use 380 WEL 380 CRC 4.0727 talk give 4.0712±0.00004 eV (392.811(4) kJ/mol) 88 Ra radium use 509.3 979.0 WEL 509.3 979.0 CRC 5.2784 10.14716 89 Ac actinium use 499 1170 WEL 499 1170 CRC 5.17 12.1 90 Th thorium use 587 1110 1930 2780 WEL 587 1110 1930 2780 CRC 6.3067 11.5 20.0 28.8 91 Pa protactinium use 568 WEL 568 CRC 5.89 92 U uranium use 597.6 1420 WEL 597.6 1420 CRC 6.19405 14.72 93 Np neptunium use 604.5 WEL 604.5 CRC 6.2657 94 Pu plutonium use 584.7 WEL 584.7 CRC 6.0262 95 Am americium use 578 WEL 578 CRC 5.9738 96 Cm curium use 581 WEL 581 CRC 5.9915 97 Bk berkelium use 601 WEL 601 CRC 6.1979 98 Cf californium use 608 WEL 608 CRC 6.2817 99 Es einsteinium use 619 WEL 619 CRC 6.42 100 Fm fermium use 627 WEL 627 CRC 6.50 101 Md mendelevium use 635 WEL 635 CRC 6.58 102 No nobelium use 642 WEL CRC 6.65 103 Lr lawrencium use 470 WEL CRC 4.9 104 Rf rutherfordium use 580 WEL CRC 6.0 == Notes == * Values from CRC are ionization energies given in the unit eV; other values are molar ionization energies given in the unit kJ/mol. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. The Prout is an obsolete unit of energy, whose value is: 1 Prout = 2.9638 \times 10^{-14} J This is equal to one twelfth of the binding energy of the deuteron. This page shows the electron configurations of the neutral gaseous atoms in their ground states. This website is also cited in the CRC Handbook as source of Section 1, subsection Electron Configuration of Neutral Atoms in the Ground State. *91 Pa : [Rn] 5f2(3H4) 6d 7s2 *92 U : [Rn] 5f3(4Io9/2) 6d 7s2 *93 Np : [Rn] 5f4(5I4) 6d 7s2 *103 Lr : [Rn] 5f14 7s2 7p1 question-marked *104 Rf : [Rn] 5f14 6d2 7s2 question-marked ===CRC=== *David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition, online version. After ionization, the electron and proton recombine to form a new hydrogen atom. Boca Raton, Florida, 2003; Section 1, Basic Constants, Units, and Conversion Factors; Electron Configuration of Neutral Atoms in the Ground State. (elements 1-104) *Also subsection Periodic Table of the Elements, (elements 1-103) based on: **G. J. Leigh, Editor, Nomenclature of Inorganic Chemistry, Blackwell Scientific Publications, Oxford, 1990. **Atomic Weights of the Elements, 1999, Pure Appl. Chem., 73, 667, 2001. ===WebElements=== *http://www.webelements.com/ ; retrieved July 2005, electron configurations based on: **Atomic, Molecular, & Optical Physics Handbook, Ed. alt=|thumb|upright=1.2|Hemispherical electron energy analyzer. Therefore, the H-alpha line occurs where hydrogen is being ionized. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. These tables list values of molar ionization energies, measured in kJ⋅mol−1. (Elements 1-106) *58 Ce : [Xe] 4f2 6s2 *103 Lr : [Rn] 5f14 6d1 7s2 *104 Rf : [Rn] 5f14 6d2 7s2 (agrees with guess above) *105 Db : [Rn] 5f14 6d3 7s2 *106 Sg : [Rn] 5f14 6d4 7s2 ===Hoffman, Lee, and Pershina=== This book contains predicted electron configurations for the elements up to 172, as well as 184, based on relativistic Dirac–Fock calculations by B. Fricke in * Category:Chemical element data pages * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. == External links == * NIST Atomic Spectra Database Ionization Energies == See also == *Molar ionization energies of the elements Category:Properties of chemical elements Category:Chemical element data pages The first molar ionization energy applies to the neutral atoms. Since it takes nearly as much energy to excite the hydrogen atom's electron from n = 1 to n = 3 (12.1 eV, via the Rydberg formula) as it does to ionize the hydrogen atom (13.6 eV), ionization is far more probable than excitation to the n = 3 level. In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. Note that these electron configurations are given for neutral atoms in the gas phase, which are not the same as the electron configurations for the same atoms in chemical environments. The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion.

2500

0.18162

1.51

15.757

-13.598

E

Find the probability that the electron in the ground-state $\mathrm{H}$ atom is less than a distance $a$ from the nucleus.

The diameter of the nucleus is in the range of () for hydrogen (the diameter of a single proton) to about for uranium. 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. A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. In the simplest atom, hydrogen, a single electron orbits the nucleus, and its smallest possible orbit, with the lowest energy, has an orbital radius almost equal to the Bohr radius. 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. * 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. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the "reduced" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). 180px|thumb|right|Diagram of a helium atom, showing the electron probability density as shades of gray. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. The expected value of the radial distance of the electron, by contrast, . == Related constants == The Bohr radius is one of a trio of related units of length, the other two being the reduced Compton wavelength of the electron ( \lambda_{\mathrm{e}} / 2\pi ) and the classical electron radius ( r_{\mathrm{e}} ). H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. An atom is composed of a positively charged nucleus, with a cloud of negatively charged electrons surrounding it, bound together by electrostatic force. 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. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. Since it takes nearly as much energy to excite the hydrogen atom's electron from n = 1 to n = 3 (12.1 eV, via the Rydberg formula) as it does to ionize the hydrogen atom (13.6 eV), ionization is far more probable than excitation to the n = 3 level. This is because there are more energy levels and therefore a greater distance between protons and electrons. The value of the radius may depend on the atom's state and context. The CODATA value of the Bohr radius (in SI units) is ==History== In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction. The electric repulsion between each pair of protons in a nucleus contributes toward decreasing its binding energy. The Bohr model of the atom was superseded by an electron probability cloud obeying the Schrödinger equation as published in 1926. 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).

5300

0.23333333333

0.19

0.323

9

D

A one-particle, one-dimensional system has $\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \mathrm{~nm}$. At $t=0$, the particle's position is measured. (a) Find the probability that the measured value lies between $x=1.5000 \mathrm{~nm}$ and $x=1.5001 \mathrm{~nm}$.

Hence, at a given time , is the probability density function of the particle's position. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. If it corresponds to a non-degenerate eigenvalue of , then |\psi (x)|^2 gives the probability of the corresponding value of for the initial state . Thus the probability that the particle is in the volume at is :\mathbf{P}(V)=\int_V \rho(\mathbf {x})\, \mathrm{d\mathbf {x}} = \int_V |\psi(\mathbf {x}, t_0)|^2\, \mathrm{d\mathbf {x}}. Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. If the system is known to be in some eigenstate of (e.g. after an observation of the corresponding eigenvalue of ) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of (so long as no other important forces act between the measurements). The probability of measuring |u\rangle is given by :P(|u\rangle) = \langle r|u\rangle\langle u|r\rangle = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} Which agrees with experiment. ==Normalization== In the example above, the measurement must give either or , so the total probability of measuring or must be 1. In other words the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. Note that if any solution to the wave equation is normalisable at some time , then the defined above is always normalised, so that :\rho_t(\mathbf x)=\left |\psi(\mathbf x, t)\right |^2 = \left|\frac{\psi_0(\mathbf x, t)}{a}\right|^2 is always a probability density function for all . Suppose a wavefunction is a solution of the wave equation, giving a description of the particle (position , for time ). If the set of eigenstates to which the system can jump upon measurement of is the same as the set of eigenstates for measurement of , then subsequent measurements of either or always produce the same values with probability of 1, no matter the order in which they are applied. The distribution is expressed in the form: :p_i \propto \exp\left(- \frac{\varepsilon_i}{kT} \right) where is the probability of the system being in state , is the exponential function, is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . Then \psi (x) is the "probability amplitude" for the eigenstate . 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. Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function is the wave function. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In this case, if the vector has the norm 1, then is just the probability that the quantum system resides in the state . The difference of a density function from simply a numerical probability means that one should integrate this modulus-squared function over some (small) domains in to obtain probability values – as was stated above, the system can't be in some state with a positive probability. Therefore, if the system is known to be in some eigenstate of (all probability amplitudes zero except for one eigenstate), then when is observed the probability amplitudes are changed. upright=1.75|right|thumb|Boltzmann's distribution is an exponential distribution. upright=1.75|right|thumb|Boltzmann factor (vertical axis) as a function of temperature for several energy differences . If we have a system consisting of many particles, the probability of a particle being in state is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state .

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In this example, $2.50 \mathrm{~mol}$ of an ideal gas with $C_{V, m}=12.47 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ is expanded adiabatically against a constant external pressure of 1.00 bar. The initial temperature and pressure of the gas are $325 \mathrm{~K}$ and $2.50 \mathrm{bar}$, respectively. The final pressure is 1.25 bar. Calculate the final temperature, $q, w, \Delta U$.

The adiabatic constant remains the same, but with the resulting pressure unknown : P_2 V_2^\gamma = \mathrm{constant}_1 = 6.31~\text{Pa}\,\text{m}^{21/5} = P \times (0.0001~\text{m}^3)^\frac75, We can now solve for the final pressure : P_2 = P_1\left (\frac{V_1}{V_2}\right)^\gamma = 100\,000~\text{Pa} \times \text{10}^{7/5} = 2.51 \times 10^6~\text{Pa} or 25.1 bar. The model assumptions are: the uncompressed volume of the cylinder is one litre (1 L = 1000 cm3 = 0.001 m3); the gas within is the air consisting of molecular nitrogen and oxygen only (thus a diatomic gas with 5 degrees of freedom, and so ); the compression ratio of the engine is 10:1 (that is, the 1 L volume of uncompressed gas is reduced to 0.1 L by the piston); and the uncompressed gas is at approximately room temperature and pressure (a warm room temperature of ~27 °C, or 300 K, and a pressure of 1 bar = 100 kPa, i.e. typical sea-level atmospheric pressure). : \begin{align} & P_1 V_1^\gamma = \mathrm{constant}_1 = 100\,000~\text{Pa} \times (0.001~\text{m}^3)^\frac75 \\\ & = 10^5 \times 6.31 \times 10^{-5}~\text{Pa}\,\text{m}^{21/5} = 6.31~\text{Pa}\,\text{m}^{21/5}, \end{align} so the adiabatic constant for this example is about 6.31 Pa m4.2. Adiabatic heating occurs when the pressure of a gas is increased by work done on it by its surroundings, e.g., a piston compressing a gas contained within a cylinder and raising the temperature where in many practical situations heat conduction through walls can be slow compared with the compression time. At the same time, the work done by the pressure–volume changes as a result from this process, is equal to Since we require the process to be adiabatic, the following equation needs to be true By the previous derivation, Rearranging (b4) gives : P = P_1 \left(\frac{V_1}{V} \right)^\gamma. For such an adiabatic process, the modulus of elasticity (Young's modulus) can be expressed as , where is the ratio of specific heats at constant pressure and at constant volume () and is the pressure of the gas. === Various applications of the adiabatic assumption === For a closed system, one may write the first law of thermodynamics as , where denotes the change of the system's internal energy, the quantity of energy added to it as heat, and the work done by the system on its surroundings. 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. 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 . 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). We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. In addition, through the use of the Euler chain relation it can be shown that \left ( \frac{\partial U}{\partial V} \right )_T = - \left ( \frac{\partial U}{\partial T} \right )_V \left ( \frac{\partial T}{\partial V} \right )_U Defining \mu_J = \left ( \frac{\partial T}{\partial V} \right )_U as the "Joule coefficient" J. Westin, A Course in Thermodynamics, Volume 1, Taylor and Francis, New York (1979). and recognizing \left ( \frac{\partial U}{\partial T} \right )_V as the heat capacity at constant volume = C_V , we have \pi_T = - C_V \mu_J The coefficient \mu_J can be obtained by measuring the temperature change for a constant-U experiment, i.e., an adiabatic free expansion (see below). 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. Some chemical and physical processes occur too rapidly for energy to enter or leave the system as heat, allowing a convenient "adiabatic approximation".Bailyn, M. (1994), pp. 52–53. We know the compressed gas has = 0.1 L and = , so we can solve for temperature: : T = \frac{P V}{\mathrm{constant}_2} = \frac{2.51 \times 10^6~\text{Pa} \times 10^{-4}~\text{m}^3}{0.333~\text{Pa}\,\text{m}^3\text{K}^{-1}} = 753~\text{K}. Adiabatic expansion against pressure, or a spring, causes a drop in temperature. Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible. ===Derivation of P–V relation for adiabatic heating and cooling=== The definition of an adiabatic process is that heat transfer to the system is zero, . For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to: : dh = v \, dp. *If the system walls are adiabatic () but not rigid (), and, in a fictive idealized process, energy is added to the system in the form of frictionless, non-viscous pressure–volume work (), and there is no phase change, then the temperature of the system will rise. Adiabatic cooling occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand, thus causing it to do work on its surroundings. 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 . For example, the adiabatic flame temperature uses this approximation to calculate the upper limit of flame temperature by assuming combustion loses no heat to its surroundings. Simplifying, : T_2 - T_1 = T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right), : \frac{T_2}{T_1} - 1 = \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1, : T_2 = T_1 \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}}. ===Derivation of discrete formula and work expression=== The change in internal energy of a system, measured from state 1 to state 2, is equal to : At the same time, the work done by the pressure–volume changes as a result from this process, is equal to Since we require the process to be adiabatic, the following equation needs to be true By the previous derivation, Rearranging (c4) gives : P = P_1 \left(\frac{V_1}{V} \right)^\gamma. Our initial conditions being 100 kPa of pressure, 1 L volume, and 300 K of temperature, our experimental constant (nR) is: : \frac{PV}{T} = \mathrm{constant}_2 = \frac{10^5~\text{Pa} \times 10^{-3}~\text{m}^3}{300~\text{K}} = 0.333~\text{Pa}\,\text{m}^3\text{K}^{-1}.

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Find $Y_l^m(\theta, \phi)$ for $l=0$.

The solutions are usually written in terms of complex exponentials: Y_{\ell, m}(\theta, \phi) = \sqrt{\frac{(2\ell+1)(\ell-m)!}{4\pi(\ell+m)!}}\ Y_{\ell, m}^*(\theta, \phi) = (-1)^m Y_{\ell, -m}(\theta, \phi). The functions Y_{\ell, m}(\theta, \phi) are the spherical harmonics, and the quantity in the square root is a normalizing factor. P_\ell^{m}(\cos \theta)\ e^{im\phi}\qquad -\ell \le m \le \ell. When the partial differential equation \frac{\partial^2\psi}{\partial\theta^2} + \cot \theta \frac{\partial \psi}{\partial \theta} + \csc^2 \theta\frac{\partial^2\psi}{\partial\phi^2} + \lambda \psi = 0 is solved by the method of separation of variables, one gets a φ-dependent part \sin(m\phi) or \cos(m\phi) for integer m≥0, and an equation for the θ-dependent part \frac{d^{2}y}{d\theta^2} + \cot \theta \frac{dy}{d\theta} + \left[\lambda - \frac{m^2}{\sin^2\theta}\right]\,y = 0\, for which the solutions are P_\ell^{m}(\cos \theta) with \ell{\ge}m and \lambda = \ell(\ell+1). Therefore, the equation abla^2\psi + \lambda\psi = 0 has nonsingular separated solutions only when \lambda = \ell(\ell+1), and those solutions are proportional to P_\ell^{m}(\cos \theta)\ \cos (m\phi)\ \ \ \ 0 \le m \le \ell and P_\ell^{m}(\cos \theta)\ \sin (m\phi)\ \ \ \ 0 < m \le \ell. That the functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters ℓ and m follows by differentiating m times the Legendre equation for :. \left(1-x^2\right) \frac{d^2}{dx^2}P_\ell(x) -2x\frac{d}{dx}P_\ell(x)+ \ell(\ell+1)P_\ell(x) = 0. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is abla^2\psi = \frac{\partial^2\psi}{\partial\theta^2} + \cot \theta \frac{\partial \psi}{\partial \theta} + \csc^2 \theta\frac{\partial^2\psi}{\partial\phi^2}. The longitude angle, \phi, appears in a multiplying factor. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. P_\ell ^{m} (This followed from the Rodrigues' formula definition. L.M.L. is the second English album and fifth overall studio album by Nu Virgos. == Content == The title of the album comes from the song in the album titled "L.M.L.". == Release == The album was released in Russia on September 13, 2007, and in Asia on September 19, 2007. In mathematics and computing, the Levenberg–Marquardt algorithm (LMA or just LM), also known as the damped least-squares (DLS) method, is used to solve non-linear least squares problems. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials. It is also commonly denoted as zn(u,k) :\Theta(u)=\Theta_{4}\left(\frac{\pi u}{2K}\right) :Z(u)=\frac{\partial}{\partial u}\ln\Theta(u) =\frac{\Theta'(u)}{\Theta(u)} :Z(\phi|m)=E(\phi|m)-\frac{E(m)}{K(m)}F(\phi|m) :Where E, K, and F are generic Incomplete Elliptical Integrals of the first and second kind. Moreover, since by Rodrigues' formula, P_\ell(x) = \frac{1}{2^\ell\,\ell!} \ \frac{d^\ell}{dx^\ell}\left[(x^2-1)^\ell\right], the P can be expressed in the form P_\ell^{m}(x) = \frac{(-1)^m}{2^\ell \ell!} (1-x^2)^{m/2}\ \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell. This definition also makes the various recurrence formulas work for positive or negative .) \text{If}\quad |m| > \ell\,\quad\text{then}\quad P_\ell^{m} = 0.\, The differential equation is also invariant under a change from to , and the functions for negative are defined by P_{-\ell} ^{m} = P_{\ell-1} ^{m},\ (\ell=1,\,2,\, \dots). ==Parity== From their definition, one can verify that the Associated Legendre functions are either even or odd according to P_\ell ^{m} (-x) = (-1)^{\ell + m} P_\ell ^{m}(x) ==The first few associated Legendre functions== thumb|300px|Associated Legendre functions for m = 0 thumb|300px|Associated Legendre functions for m = 1 thumb|300px|Associated Legendre functions for m = 2 The first few associated Legendre functions, including those for negative values of m, are: P_{0}^{0}(x)=1 \begin{align} P_{1}^{-1}(x)&=-\tfrac{1}{2}P_{1}^{1}(x) \\\ P_{1}^{0}(x)&=x \\\ P_{1}^{1}(x)&=-(1-x^2)^{1/2} \end{align} \begin{align} P_{2}^{-2}(x)&=\tfrac{1}{24}P_{2}^{2}(x) \\\ P_{2}^{-1}(x)&=-\tfrac{1}{6}P_{2}^{1}(x) \\\ P_{2}^{0}(x)&=\tfrac{1}{2}(3x^{2}-1) \\\ P_{2}^{1}(x)&=-3x(1-x^2)^{1/2} \\\ P_{2}^{2}(x)&=3(1-x^2) \end{align} \begin{align} P_{3}^{-3}(x)&=-\tfrac{1}{720}P_{3}^{3}(x) \\\ P_{3}^{-2}(x)&=\tfrac{1}{120}P_{3}^{2}(x) \\\ P_{3}^{-1}(x)&=-\tfrac{1}{12}P_{3}^{1}(x) \\\ P_{3}^{0}(x)&=\tfrac{1}{2}(5x^3-3x) \\\ P_{3}^{1}(x)&=\tfrac{3}{2}(1-5x^{2})(1-x^2)^{1/2} \\\ P_{3}^{2}(x)&=15x(1-x^2) \\\ P_{3}^{3}(x)&=-15(1-x^2)^{3/2} \end{align} \begin{align} P_{4}^{-4}(x)&=\tfrac{1}{40320}P_{4}^{4}(x) \\\ P_{4}^{-3}(x)&=-\tfrac{1}{5040}P_{4}^{3}(x) \\\ P_{4}^{-2}(x)&=\tfrac{1}{360}P_{4}^{2}(x) \\\ P_{4}^{-1}(x)&=-\tfrac{1}{20}P_{4}^{1}(x) \\\ P_{4}^{0}(x)&=\tfrac{1}{8}(35x^{4}-30x^{2}+3) \\\ P_{4}^{1}(x)&=-\tfrac{5}{2}(7x^3-3x)(1-x^2)^{1/2} \\\ P_{4}^{2}(x)&=\tfrac{15}{2}(7x^2-1)(1-x^2) \\\ P_{4}^{3}(x)&= - 105x(1-x^2)^{3/2} \\\ P_{4}^{4}(x)&=105(1-x^2)^{2} \end{align} ==Recurrence formula== These functions have a number of recurrence properties: (\ell-m-1)(\ell-m)P_{\ell}^{m}(x) = -P_{\ell}^{m+2}(x) + P_{\ell-2}^{m+2}(x) + (\ell+m)(\ell+m-1)P_{\ell-2}^{m}(x) (\ell-m+1)P_{\ell+1}^{m}(x) = (2\ell+1)xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x) 2mxP_{\ell}^{m}(x)=-\sqrt{1-x^2}\left[P_{\ell}^{m+1}(x)+(\ell+m)(\ell-m+1)P_{\ell}^{m-1}(x)\right] \frac{1}{\sqrt{1-x^2}}P_\ell^m(x) = \frac{-1}{2m} \left[ P_{\ell-1}^{m+1}(x) + (\ell+m-1)(\ell+m)P_{\ell-1}^{m-1}(x) \right] \frac{1}{\sqrt{1-x^2}}P_\ell^m(x) = \frac{-1}{2m} \left[ P_{\ell+1}^{m+1}(x) + (\ell-m+1)(\ell-m+2)P_{\ell+1}^{m-1}(x) \right] \sqrt{1-x^2}P_\ell^m(x) = \frac1{2\ell+1} \left[ (\ell-m+1)(\ell-m+2) P_{\ell+1}^{m-1}(x) - (\ell+m-1)(\ell+m) P_{\ell-1}^{m-1}(x) \right] \sqrt{1-x^2}P_\ell^m(x) = \frac{-1}{2\ell+1} \left[ P_{\ell+1}^{m+1}(x) - P_{\ell-1}^{m+1}(x) \right] \sqrt{1-x^2}P_\ell^{m+1}(x) = (\ell-m)xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x) \sqrt{1-x^2}P_\ell^{m+1}(x) = (\ell-m+1)P_{\ell+1}^m(x) - (\ell+m+1)xP_\ell^m(x) \sqrt{1-x^2}\frac{d}{dx}{P_\ell^m}(x) = \frac12 \left[ (\ell+m)(\ell-m+1)P_\ell^{m-1}(x) - P_\ell^{m+1}(x) \right] (1-x^2)\frac{d}{dx}{P_\ell^m}(x) = \frac1{2\ell+1} \left[ (\ell+1)(\ell+m)P_{\ell-1}^m(x) - \ell(\ell-m+1)P_{\ell+1}^m(x) \right] (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = {\ell}xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x) (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = -(\ell+1)xP_{\ell}^{m}(x) + (\ell-m+1)P_{\ell+1}^{m}(x) (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = \sqrt{1-x^2}P_{\ell}^{m+1}(x) + mxP_{\ell}^{m}(x) (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = -(\ell+m)(\ell-m+1)\sqrt{1-x^2}P_{\ell}^{m-1}(x) - mxP_{\ell}^{m}(x) Helpful identities (initial values for the first recursion): P_{\ell +1}^{\ell +1}(x) = - (2\ell+1) \sqrt{1-x^2} P_{\ell}^{\ell}(x) P_{\ell}^{\ell}(x) = (-1)^\ell (2\ell-1)!! (1- x^2)^{(\ell/2)} P_{\ell +1}^{\ell}(x) = x (2\ell+1) P_{\ell}^{\ell}(x) with the double factorial. ==Gaunt's formula== The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. For each choice of ℓ, there are functions for the various values of m and choices of sine and cosine. Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m: \int_{-1}^{1} P_k ^{m} P_\ell ^{m} dx = \frac{2 (\ell+m)!}{(2\ell+1)(\ell-m)!}\ \delta _{k,\ell} Where is the Kronecker delta. P^{m}_\ell(x). ===Alternative notations=== The following alternative notations are also used in literature: P_{\ell m}(x) = (-1)^m P_\ell^{m}(x) ===Closed Form=== The Associated Legendre Polynomial can also be written as: P_l^m(x)=(-1)^{m} \cdot 2^{l} \cdot (1-x^2)^{m/2} \cdot \sum_{k=m}^l \frac{k!}{(k-m)!}\cdot x^{k-m} \cdot \binom{l}{k} \binom{\frac{l+k-1}{2}}{l} with simple monomials and the generalized form of the binomial coefficient. ==Orthogonality== The associated Legendre polynomials are not mutually orthogonal in general. Indeed, equate the coefficients of equal powers on the left and right hand side of \frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^m \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell}, then it follows that the proportionality constant is c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} , so that P^{-m}_\ell(x) = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}

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The lowest-frequency pure-rotational absorption line of ${ }^{12} \mathrm{C}^{32} \mathrm{~S}$ occurs at $48991.0 \mathrm{MHz}$. Find the bond distance in ${ }^{12} \mathrm{C}^{32} \mathrm{~S}$.

The C-S-C angles are 102° and C-S bond distance is 177 picometers. Since one atomic unit of length(i.e., a Bohr radius) is 52.9177 pm, the C–C bond length is 2.91 atomic units, or approximately three Bohr radii long. The bond lengths of these so- called "pancake bonds" are up to 305 pm. Shorter than average C–C bond distances are also possible: alkenes and alkynes have bond lengths of respectively 133 and 120 pm due to increased s-character of the sigma bond. The smallest theoretical C–C single bond obtained in this study is 131 pm for a hypothetical tetrahedrane derivative. In a bond between two identical atoms, half the bond distance is equal to the covalent radius. Stretching or squeezing the same bond by 15 pm required an estimated 21.9 or 37.7 kJ/mol. Bond lengths in organic compounds C–H Length (pm) C–C Length (pm) Multiple-bonds Length (pm) sp3–H 110 sp3–sp3 154 Benzene 140 sp2–H 109 sp3–sp2 150 Alkene 134 sp–H 108 sp2–sp2 147 Alkyne 120 sp3–sp 146 Allene 130 sp2–sp 143 sp–sp 137 ==References== == External links == * Bond length tutorial Length Category:Molecular geometry Category:Length In benzene all bonds have the same length: 139 pm. Carbon–carbon single bonds increased s-character is also notable in the central bond of diacetylene (137 pm) and that of a certain tetrahedrane dimer (144 pm). Bond distance of carbon to other elements Bonded element Bond length (pm) Group H 106–112 group 1 Be 193 group 2 Mg 207 group 2 B 156 group 13 Al 224 group 13 In 216 group 13 C 120–154 group 14 Si 186 group 14 Sn 214 group 14 Pb 229 group 14 N 147–210 group 15 P 187 group 15 As 198 group 15 Sb 220 group 15 Bi 230 group 15 O 143–215 group 16 S 181–255 group 16 Cr 192 group 6 Se 198–271 group 16 Te 205 group 16 Mo 208 group 6 W 206 group 6 F 134 group 17 Cl 176 group 17 Br 193 group 17 I 213 group 17 ==Bond lengths in organic compounds== The bond length between two atoms in a molecule depends not only on the atoms but also on such factors as the orbital hybridization and the electronic and steric nature of the substituents. Bond is located between carbons C1 and C2 as depicted in a picture below. thumb|center|150px| Hexaphenylethane skeleton based derivative containing longest known C-C bond between atoms C1 and C2 with a length of 186.2 pm Another notable compound with an extraordinary C-C bond length is tricyclobutabenzene, in which a bond length of 160 pm is reported. The carbon–carbon (C–C) bond length in diamond is 154 pm. By approximation the bond distance between two different atoms is the sum of the individual covalent radii (these are given in the chemical element articles for each element). In molecular geometry, bond length or bond distance is defined as the average distance between nuclei of two bonded atoms in a molecule. In an in silico experiment a bond distance of 136 pm was estimated for neopentane locked up in fullerene. Bond lengths are given in picometers. Longest C-C bond within the cyclobutabenzene category is 174 pm based on X-ray crystallography. By examining a large number of structures of molecular crystals, it is possible to find a minimum radius for each type of atom such that other non-bonded atoms do not encroach any closer. In this type of compound the cyclobutane ring would force 90° angles on the carbon atoms connected to the benzene ring where they ordinarily have angles of 120°. thumb|center|200px|Cyclobutabenzene with a bond length in red of 174 pm The existence of a very long C–C bond length of up to 290 pm is claimed in a dimer of two tetracyanoethylene dianions, although this concerns a 2-electron-4-center bond. Current record holder for the longest C-C bond with a length of 186.2 pm is 1,8-Bis(5-hydroxydibenzo[a,d]cycloheptatrien-5-yl)naphthalene, one of many molecules within a category of hexaaryl ethanes, which are derivatives based on hexaphenylethane skeleton. Bond lengths are measured in the solid phase by means of X-ray diffraction, or approximated in the gas phase by microwave spectroscopy. The 12.17×44mm RF is also known as "12×44RF Norwegian Remington Model 1871" and "12.7×44RF Norwegian". ==12.17×42mm RF and its subvariety the 12.17×44mm RF== thumb|250px|Model 1867 Remington rolling block chambered for the 12.17×42mm RF. The distance between both ends can also be evaluated by a plurality of segments according to a broken line passing through the successive inflection points (sinuosity of order 2).

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E

The strongest infrared band of ${ }^{12} \mathrm{C}^{16} \mathrm{O}$ occurs at $\widetilde{\nu}=2143 \mathrm{~cm}^{-1}$. Find the force constant of ${ }^{12} \mathrm{C}^{16} \mathrm{O}$.

The C-band is located between the short wavelengths (S) band (1460–1530 nm) and the long wavelengths (L) band (1565–1625 nm). The nuclear force is powerfully attractive between nucleons at distances of about 0.8 femtometre (fm, or 0.8×10−15 metre), but it rapidly decreases to insignificance at distances beyond about 2.5 fm. O16 or O-16 may refer to: * Curtiss O-16 Falcon, an observation aircraft of the United States Army Air Corps * Garberville Airport, in Humboldt County, California, United States * , a submarine of the Royal Netherlands Navy * Oxygen-16, an isotope of oxygen * , a submarine of the United States Navy The molecular formula for C10H12FNO (molar mass: 181.21 g/mol, exact mass: 181.0903 u) may refer to: * Flephedrone, also known as 4-fluoromethcathinone (4-FMC) * 3-Fluoromethcathinone In recent years, experimenters have concentrated on the subtleties of the nuclear force, such as its charge dependence, the precise value of the πNN coupling constant, improved phase-shift analysis, high-precision NN data, high-precision NN potentials, NN scattering at intermediate and high energies, and attempts to derive the nuclear force from QCD. ==The nuclear force as a residual of the strong force== The nuclear force is a residual effect of the more fundamental strong force, or strong interaction. thumb|Absorption in fiber in the range 900–1700 nm with a minimum at the C-band thumb|Transmittance of the atmosphere around the C-band In infrared optical communications, C-band (C for "conventional") refers to the wavelength range 1530–1565 nm, which corresponds to the amplification range of erbium doped fiber amplifiers (EDFAs). By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. ==Description== thumb|Comparison between the Nuclear Force and the Coulomb Force. a - residual strong force (nuclear force), rapidly decreases to insignificance at distances beyond about 2.5 fm, b - at distances less than ~ 0.7 fm between nucleons centers the nuclear force becomes repulsive, c - coulomb repulsion force between two protons (over 3 fm force becomes the main), d - equilibrium position for proton - proton, r - radius of a nucleon (a cloud composed of three quarks). The molecular formula C8H16 (molar mass: 112.21 g/mol, exact mass: 112.1252 u) may refer to: * Cyclooctane * Methylcycloheptane * Dimethylcyclohexanes * * Octenes ** 1-Octene Category:Molecular formulas thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. These nuclear forces are very weak compared to direct gluon forces ("color forces" or strong forces) inside nucleons, and the nuclear forces extend only over a few nuclear diameters, falling exponentially with distance. At distances less than 0.7 fm, the nuclear force becomes repulsive. The force depends on whether the spins of the nucleons are parallel or antiparallel, as it has a non-central or tensor component. After the verification of the quark model, strong interaction has come to mean QCD. ==Nucleon–nucleon potentials== Two- nucleon systems such as the deuteron, the nucleus of a deuterium atom, as well as proton–proton or neutron–proton scattering are ideal for studying the NN force. The model also gave good predictions for the binding energy of nuclei. (The size of an atom, measured in angstroms (Å, or 10−10 m), is five orders of magnitude larger). At small separations between nucleons (less than ~ 0.7 fm between their centers, depending upon spin alignment) the force becomes repulsive, which keeps the nucleons at a certain average separation. Nucleons have a radius of about 0.8 fm. A more recent approach is to develop effective field theories for a consistent description of nucleon–nucleon and three-nucleon forces. Sometimes, the nuclear force is called the residual strong force, in contrast to the strong interactions which arise from QCD. The constants are determined empirically. The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Note: 1 fm = 1E-15 m.

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Calculate the de Broglie wavelength for (a) an electron with a kinetic energy of $100 \mathrm{eV}$

In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. Thus, \textstyle \frac{1}{\lambda}=\frac{E}{hc} (in this formula the h represents Planck's constant). A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. If is the number of dimensions, and the relationship between energy () and momentum () is given by E=ap^s (with and being constants), then the thermal wavelength is defined as \lambda_{\mathrm{th}}=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{\mathrm B}T}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} , where is the Gamma function. For other spectral transitions in multi-electron atoms, the Rydberg formula generally provides incorrect results, since the magnitude of the screening of inner electrons for outer-electron transitions is variable and not possible to compensate for in the simple manner above. But the Rydberg formula also provides correct wavelengths for distant electrons, where the effective nuclear charge can be estimated as the same as that for hydrogen, since all but one of the nuclear charges have been screened by other electrons, and the core of the atom has an effective positive charge of +1. Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source. ==Massive particles== For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Rydberg was trying: \textstyle n=n_0 - \frac{C_0}{\left(m+m'\right)^2} when he became aware of Balmer's formula for the hydrogen spectrum \textstyle \lambda={hm^2 \over m^2-4} In this equation, m is an integer and h is a constant (not to be confused with the later Planck constant). 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. As stressed by Niels Bohr, expressing results in terms of wavenumber, not wavelength, was the key to Rydberg's discovery. Species Mass (kg) \lambda_{\mathrm{th}} (m) Electron Photon 0 H2 O2 ==References== * Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. One of the methods is to use the concept of dressed particle. == See also == * Energy level * Mode (electromagnetism) == References == Category:Electron The energy released is equal to the difference in energy levels between the electron energy states. The correction to the Rydberg formula for these atoms is known as the quantum defect. ==See also== * Balmer series * Hydrogen line * Rydberg–Ritz combination principle ==References== * * Category:Atomic physics Category:Foundational quantum physics Category:Hydrogen physics Category:History of physics Johannes (Janne) Robert Rydberg (; 8 November 1854 - 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom. == Biography == Rydberg was born 8 November 1854 in Halmstad in southern Sweden, the only child of Sven Rydberg and Maria Anderson Rydberg. Later models found that the values for n1 and n2 corresponded to the principal quantum numbers of the two orbitals. ==For hydrogen== \frac{1}{\lambda_{\mathrm{vac}}} = R_\text{H}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) , where *\lambda_{\mathrm{vac}} is the wavelength of electromagnetic radiation emitted in vacuum, *R_\text{H} is the Rydberg constant for hydrogen, approximately , *n_1 is the principal quantum number of an energy level, and *n_2 is the principal quantum number of an energy level for the atomic electron transition. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. When these deviations from the central trajectory are expressed in terms of the small parameters \varepsilon, \sigma defined as E_k=(1+\varepsilon)E_\textrm{P}, r_0=(1+\sigma)R_\textrm{P}, and having in mind that \alpha itself is small (of the order of 1°), the final radius of the electron's trajectory, r(\pi), can be expressed as :r_\pi\approx R_\textrm{P}(1+2\varepsilon-\sigma-2\alpha^2+2\varepsilon^2-6\alpha^2\varepsilon). 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.

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The threshold wavelength for potassium metal is $564 \mathrm{~nm}$. What is its work function?

alt=Potassium ferricyanide milled|thumb|Potassium ferricyanide when milled has lighter color Potassium ferricyanide is the chemical compound with the formula K3[Fe(CN)6]. Potassium is the chemical element with the symbol K (from Neo-Latin kalium) and atomic number19. In materials science, the threshold displacement energy () is the minimum kinetic energy that an atom in a solid needs to be permanently displaced from its site in the lattice to a defect position. Element K may refer to: * The chemical element Potassium given symbol K (Latin kalium) * An educational software package owned by Skillsoft Using the equations given above one can then translate the electron energy E into the threshold energy T. In infrared astronomy, the K band is an atmospheric transmission window centered on 2.2 μm (in the near-infrared 136 THz range). Metallic potassium is used in several types of magnetometers. ==Precautions== Potassium metal can react violently with water producing KOH and hydrogen gas. : thumb|left|alt=A piece of potassium metal is dropped into a clear container of water and skates around, burning with a bright pinkish or lilac flame for a short time until finishing with a pop and splash.|A reaction of potassium metal with water. Threshold displacement energies in typical solids are of the order of 10-50 eV. M. Nastasi, J. Mayer, and J. Hirvonen, Ion-Solid Interactions - Fundamentals and Applications, Cambridge University Press, Cambridge, Great Britain, 1996 P. Lucasson, The production of Frenkel defects in metals, in Fundamental Aspects of Radiation Damage in Metals, edited by M. T. Robinson and F. N. Young Jr., pages 42--65, Springfield, 1975, ORNL R. S. Averback and T. Diaz de la Rubia, Displacement damage in irradiated metals and semiconductors, in Solid State Physics, edited by H. Ehrenfest and F. Spaepen, volume 51, pages 281--402, Academic Press, New York, 1998.R. Smith (ed.), Atomic & ion collisions in solids and at surfaces: theory, simulation and applications, Cambridge University Press, Cambridge, UK, 1997 == Theory and simulation == The threshold displacement energy is a materials property relevant during high- energy particle radiation of materials. Potassium is silvery in appearance, but it begins to tarnish toward gray immediately on exposure to air.Greenwood, p. 76 In a flame test, potassium and its compounds emit a lilac color with a peak emission wavelength of 766.5 nanometers.Greenwood, p. 75 Neutral potassium atoms have 19 electrons, one more than the configuration of the noble gas argon. Potassium ferricyanide separates from the solution: :2 K4[Fe(CN)6] + Cl2 → 2 K3[Fe(CN)6] + 2 KCl ==Structure== Like other metal cyanides, solid potassium ferricyanide has a complicated polymeric structure. This filtering involves about 600g of sodium and 33g of potassium. Potassium peroxochromate, potassium tetraperoxochromate(V), or simply potassium perchromate, is an inorganic chemical having the chemical formula K3[Cr(O2)4]. Potassium ions are present in a wide variety of proteins and enzymes. ===Biochemical function=== Potassium levels influence multiple physiological processes, including *resting cellular-membrane potential and the propagation of action potentials in neuronal, muscular, and cardiac tissue. The stable isotopes of potassium can be laser cooled and used to probe fundamental and technological problems in quantum physics. This is the fundamental ("primary damage") threshold displacement energy, and also the one usually simulated by molecular dynamics computer simulations. It is not possible to write down a single analytical equation that would relate e.g. elastic material properties or defect formation energies to the threshold displacement energy. Pure potassium metal can be isolated by electrolysis of its hydroxide in a process that has changed little since it was first used by Humphry Davy in 1807. Even in the most studied materials such as Si and Fe, there are variations of more than a factor of two in the predicted threshold displacement energies. In contrast, the second ionization energy is very high (3052kJ/mol). ===Chemical=== Potassium reacts with oxygen, water, and carbon dioxide components in air. The K+\---NCFe linkages break when the solid is dissolved in water. ==Applications== The compound is also used to harden iron and steel, in electroplating, dyeing wool, as a laboratory reagent, and as a mild oxidizing agent in organic chemistry. ===Photography=== ==== Blueprint, cyanotype, toner ==== The compound has widespread use in blueprint drawing and in photography (Cyanotype process). Potassium ferricyanide is used to determine the ferric reducing power potential of a sample (extract, chemical compound, etc.).Nakajima, Y., Sato, Y., & Konishi, T. (2007). It is also used to bleach textiles and straw, and in the tanning of leathers. ===Industrial=== Major potassium chemicals are potassium hydroxide, potassium carbonate, potassium sulfate, and potassium chloride.

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Evaluate the series $$ S=\sum_{n=0}^{\infty} \frac{1}{3^n} $$

It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is :1\,+\,\frac{a}{1-r}\;=\;1\,+\,\frac{\frac{1}{3}}{1-\frac{4}{9}}\;=\;\frac{8}{5}. The series \sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots is known as the alternating harmonic series. Using the geometric series closed form as before :0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac{a}{1-r} \;=\; \frac{49/64}{1-1/64} \;=\; \frac{49/64}{63/64} \;=\; \frac{49}{63} \;=\; \frac{7}{9}. Explicitly, the asymptotic expansion of the series is \frac{1}{1} - \frac{1}{2} +\cdots + \frac{1}{2n-1} - \frac{1}{2n} = H_{2n} - H_n = \ln 2 - \frac{1}{2n} + O(n^{-2}) Using alternating signs with only odd unit fractions produces a related series, the Leibniz formula for \sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}. ===Riemann zeta function=== The Riemann zeta function is defined for real x>1 by the convergent series \zeta(x)=\sum_{n=1}^{\infty}\frac{1}{n^x}=\frac1{1^x}+\frac1{2^x}+\frac1{3^x}+\cdots, which for x=1 would be the harmonic series. That is, it is the sum : {\sideset{}{'}\sum_{n=1}^\infty} \frac{1}{n} where the prime indicates that n takes only values whose decimal expansion has no nines. The closed form geometric series 1 / (1 - r) is the black dashed line. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. One can derive that closed-form formula for the partial sum, sn, by subtracting out the many self-similar terms as follows: \begin{align} s_n &= ar^0 + ar^1 + \cdots + ar^{n-1},\\\ rs_n &= ar^1 + ar^2 + \cdots + ar^{n},\\\ s_n - rs_n &= ar^0 - ar^{n},\\\ s_n\left(1-r\right) &= a\left(1-r^{n}\right),\\\ s_n &= a\left(\frac{1-r^{n}}{1-r}\right), \text{ for } r eq 1. \end{align} As approaches infinity, the absolute value of must be less than one for the series to converge. The sum is :\frac{1}{1 -r}\;=\;\frac{1}{1 -\frac{1}{4}}\;=\;\frac{4}{3}. Its value can then be computed from the finite sum formula \sum_{k=0}^\infty ar^k = \lim_{n\to\infty}{\sum_{k=0}^{n} ar^k} = \lim_{n\to\infty}\frac{a(1-r^{n+1})}{1-r}= \frac{a}{1-r} - \lim_{n\to\infty}{\frac{ar^{n+1}}{1-r}} thumb|350px|Animation, showing convergence of partial sums of geometric progression \sum\limits_{k=0}^{n}q^k (red line) to its sum {1\over 1-q} (blue line) for |q|<1. thumb|350px|Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ⋯ which converges to 2. In addition to the expanded form of the geometric series, there is a generator form of the geometric series written as :\sum^{\infty}_{k=0} a r^k and a closed form of the geometric series written as :\frac{a}{1-r} \text{ for } |r|<1\. When r = 1, all of the terms of the series are the same and the series is infinite. Given that the last term is arn and the previous series remainder is s - sn-1 = arn / (1 - r)), this measure of the convergence rate of the geometric series is arn / (arn / (1 - r)) = 1 - r, if 0 ≤ r < 1\. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_{n=1}^\infty\frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots. The following table shows several geometric series: a r Example series 4 10 4 + 40 + 400 + 4000 + 40,000 + ··· 3 1 3 + 3 + 3 + 3 + 3 + ··· 1 2/3 1 + 2/3 + 4/9 + 8/27 + 16/81 + ··· 1/2 1/2 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ··· 9 1/3 9 + 3 + 1 + 1/3 + 1/9 + ··· 7 1/10 7 + 0.7 + 0.07 + 0.007 + 0.0007 + ··· 1 −1/2 1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ··· 3 −1 3 − 3 + 3 − 3 + 3 − ··· The convergence of the geometric series depends on the value of the common ratio r: :* If |r| < 1, the terms of the series approach zero in the limit (becoming smaller and smaller in magnitude), and the series converges to the sum a / (1 - r). The sum of the first n terms of a geometric series, up to and including the r n-1 term, is given by the closed-form formula: \begin{align} s_n &= ar^0 + ar^1 + \cdots + ar^{n-1}\\\ &= \sum_{k=0}^{n-1} ar^k = \sum_{k=1}^{n} ar^{k-1}\\\ &= \begin{cases} a\left(\frac{1-r^{n}}{1-r}\right), \text{ for } r eq 1\\\ an, \text{ for } r = 1 \end{cases} \end{align} where is the common ratio. The sum then becomes \begin{align} s &= a+ar+ar^2+ar^3+ar^4+\cdots\\\ &= \sum_{k=0}^\infty ar^{k} = \sum_{k=1}^\infty ar^{k-1}\\\ &= \frac{a}{1-r}, \text{ for } |r|<1\. \end{align} The formula also holds for complex , with the corresponding restriction that the modulus of is strictly less than one. The first dimension is horizontal, in the bottom row showing the geometric series S = 1/2 + 1/4 + 1/8 + 1/16 + ... , which is the geometric series with coefficient a = 1/2 and common ratio r = 1/2 that converges to S = a / (1-r) = (1/2) / (1-1/2) = 1.

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A

Evaluate the series $$ S=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2^n} $$

The Mercator series provides an analytic expression of the natural logarithm: \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n \;=\; \ln (1+x). It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. As another example, by Mercator series \ln(2) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots. In mathematics, an alternating series is an infinite series of the form \sum_{n=0}^\infty (-1)^n a_n or \sum_{n=0}^\infty (-1)^{n+1} a_n with for all . This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. For any \varepsilon \in (0,2), one thus finds ::\sum_{n=0}^\infty (-1+\varepsilon)^n=\frac{1}{1-(-1+\varepsilon)}=\frac{1}{2-\varepsilon}, and so the limit \varepsilon\to 0 of series evaluations is ::\lim_{\varepsilon\to 0}\lim_{N\to\infty}\sum_{n=0}^N (-1+\varepsilon)^n=\frac{1}{2}. In mathematics, the infinite series , also written : \sum_{n=0}^\infty (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. The series converges to the natural logarithm (shifted by 1) whenever -1 . ==History== The series was discovered independently by Johannes Hudde and Isaac Newton. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigor, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between mathematicians.Kline 1983 p.307Knopp p.457 == Relation to the geometric series == For any number r in the interval (-1,1), the sum to infinity of a geometric series can be evaluated via ::\lim_{N\to\infty}\sum_{n=0}^N r^n = \sum_{n=0}^\infty r^n=\frac{1}{1-r}. The same conclusion results from calculating −S, subtracting the result from S, and solving 2S = 1.Devlin p.77 The above manipulations do not consider what the sum of a series actually means and how said algebraic methods can be applied to divergent geometric series. In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots In summation notation, :\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n. The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. In contrast, as r approaches −1 the sum of the first several terms of the geometric series starts to converge to 1/2 but slightly flips up or down depending on whether the most recently added term has a power of r that is even or odd. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2. :\sum_{k=0}^{n} (-2)^k As a series of real numbers it diverges, so in the usual sense it has no sum. In the terms of complex analysis, \tfrac{1}{2} is thus seen to be the value at z=-1 of the analytic continuation of the series \sum_{n=0}^N z^n, which is only defined on the complex unit disk, |z|<1. ==Early ideas== ==Divergence== In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. Treating Grandi's series as a divergent geometric series and using the same algebraic methods that evaluate convergent geometric series to obtain a third value: :S = 1 − 1 + 1 − 1 + ..., so :1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S :1 − S = S :1 = 2S, resulting in S = . In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs. But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for \tfrac 1 2 \ln(2): \begin{align} & {} \quad \left(1-\frac{1}{2}\right)-\frac{1}{4} +\left(\frac{1}{3}-\frac{1}{6}\right) -\frac{1}{8}+\left(\frac{1}{5} -\frac{1}{10}\right)-\frac{1}{12}+\cdots \\\\[8pt] & = \frac{1}{2}-\frac{1}{4}+\frac{1}{6} -\frac{1}{8}+\frac{1}{10}-\frac{1}{12} +\cdots \\\\[8pt] & = \frac{1}{2}\left(1-\frac{1}{2} + \frac{1}{3} -\frac{1}{4}+\frac{1}{5}- \frac{1}{6}+ \cdots\right)= \frac{1}{2} \ln(2). \end{align} == Series acceleration == In practice, the numerical summation of an alternating series may be sped up using any one of a variety of series acceleration techniques. Therefore, an alternating series is also a unit series when -1 < r < 0 and a + r = 1 (for example, coefficient a = 1.7 and common ratio r = -0.7). In a much broader sense, the series is associated with another value besides ∞, namely , which is the limit of the series using the 2-adic metric. ==Historical arguments== Gottfried Leibniz considered the divergent alternating series as early as 1673.

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The relationship introduced in Problem $1-48$ has been interpreted to mean that a particle of mass $m\left(E=m c^2\right)$ can materialize from nothing provided that it returns to nothing within a time $\Delta t \leq h / m c^2$. Particles that last for time $\Delta t$ or more are called real particles; particles that last less than time $\Delta t$ are called virtual particles. The mass of the charged pion, a subatomic particle, is $2.5 \times 10^{-28} \mathrm{~kg}$. What is the minimum lifetime if the pion is to be considered a real particle?

It decays via the electromagnetic force, which explains why its mean lifetime is much smaller than that of the charged pion (which can only decay via the weak force). December 18, 2013 ===Neutral pion decays=== The meson has a mass of and a mean lifetime of . In particle physics, the pion decay constant is the square root of the coefficient in front of the kinetic term for the pion in the low-energy effective action. Each pion has isospin (I = 1) and third-component isospin equal to its charge (Iz = +1, 0 or −1). ===Charged pion decays=== The mesons have a mass of and a mean lifetime of . They are unstable, with the charged pions and decaying after a mean lifetime of 26.033 nanoseconds ( seconds), and the neutral pion decaying after a much shorter lifetime of 85 attoseconds ( seconds). Pions Particle name Particle symbol Antiparticle symbol Quark content Rest mass (MeV/c2) IG JPC S C B' Mean lifetime (s) Commonly decays to (>5% of decays) Pion 1− 0− 0 0 0 Pion Self \tfrac{\mathrm{u\bar{u}} - \mathrm{d\bar{d}}}{\sqrt 2} 1− 0−+ 0 0 0 [a] Make-up inexact due to non-zero quark masses. ==See also== *Pionium *Quark model *Static forces and virtual- particle exchange *Sanford-Wang parameterisation ==References== == Further reading == * Gerald Edward Brown and A. D. Jackson, The Nucleon-Nucleon Interaction (1976), North-Holland Publishing, Amsterdam ==External links== * * Mesons at the Particle Data Group Category:Mesons Also observed, for charged pions only, is the very rare "pion beta decay" (with branching fraction of about 10−8) into a neutral pion, an electron and an electron antineutrino (or for positive pions, a neutral pion, a positron, and electron neutrino). : → + + → + + The rate at which pions decay is a prominent quantity in many sub-fields of particle physics, such as chiral perturbation theory. In a series of articles published in Nature, they identified a cosmic particle having an average mass close to 200 times the mass of electron, today known as pions. This rate is parametrized by the pion decay constant (ƒπ), related to the wave function overlap of the quark and antiquark, which is about .Leptonic decays of charged pseudo- scalar mesons J. L. Rosner and S. Stone. Pions are not produced in radioactive decay, but commonly are in high-energy collisions between hadrons. The pion is one of the particles that mediate the residual strong interaction between a pair of nucleons. The existence of the neutral pion was inferred from observing its decay products from cosmic rays, a so-called "soft component" of slow electrons with photons. In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . The neutral pion has also been observed to decay into positronium with a branching fraction on the order of . According to Brown–Rho scaling, the masses of nucleons and most light mesons decrease at finite density as the ratio of the in-medium pion decay rate to the free-space pion decay constant. Giuseppe Paolo Stanislao "Beppo" Occhialini ForMemRS (; 5 December 1907 – 30 December 1993) was an Italian physicist who contributed to the discovery of the pion or pi-meson decay in 1947 with César Lattes and Cecil Frank Powell, the latter winning the Nobel Prize in Physics for this work. The pion mass is an exception to Brown-Rho scaling because the pion's mass is protected by its Goldstone boson nature. == References == # # # Particle Data Group: Decay constants of charged pseudoscalar mesons == External links == # Particle Data Group & WWW edition of Review of Particle Physics Category:Quantum chromodynamics Charged pions most often decay into muons and muon neutrinos, while neutral pions generally decay into gamma rays. The pion also plays a crucial role in cosmology, by imposing an upper limit on the energies of cosmic rays surviving collisions with the cosmic microwave background, through the Greisen–Zatsepin–Kuzmin limit. ==History== Theoretical work by Hideki Yukawa in 1935 had predicted the existence of mesons as the carrier particles of the strong nuclear force. Pions also result from some matter–antimatter annihilation events. If it does decay via a positron, the proton's half-life is constrained to be at least years. The exchange of virtual pions, along with vector, rho and omega mesons, provides an explanation for the residual strong force between nucleons.

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A household lightbulb is a blackbody radiator. Many lightbulbs use tungsten filaments that are heated by an electric current. What temperature is needed so that $\lambda_{\max }=550 \mathrm{~nm}$ ?

An incandescent lamp's light is thermal radiation, and the bulb approximates an ideal black-body radiator, so its color temperature is essentially the temperature of the filament. According to the Stefan–Boltzmann law, a black body at the Draper point emits 23 kW of radiation per square metre, almost exclusively infrared. ==See also== *Incandescence == References == Category:Heat transfer Category:Thermodynamics Category:Electromagnetic radiation Since the emissivity is a value between 0 and 1, the real temperature will be greater than or equal to the brightness temperature. The value of the Draper point can be calculated using Wien's displacement law: the peak frequency u_\text{peak} (in hertz) emitted by a blackbody relates to temperature as follows: u_\text{peak} = 2.821 \frac{kT}{h}, where * is Boltzmann's constant, * is Planck's constant, * is temperature (in kelvins). The specific heat capacity has a sharp peak as the temperature approaches the lambda point. The temperature of the ideal emitter that matches the color most closely is defined as the color temperature of the original visible light source. That makes the reciprocal of the brightness temperature: ::T_b^{-1} = \frac{k}{h u}\, \text{ln}\left[1 + \frac{e^{\frac{h u}{kT}}-1}{\epsilon}\right] At low frequency and high temperatures, when h u \ll kT, we can use the Rayleigh–Jeans law: ::I_{ u} = \frac{2 u^2k T}{c^2} so that the brightness temperature can be simply written as: ::T_b=\epsilon T\, In general, the brightness temperature is a function of u, and only in the case of blackbody radiation it is the same at all frequencies. 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). To the extent that a hot surface emits thermal radiation but is not an ideal black- body radiator, the color temperature of the light is not the actual temperature of the surface. The actual temperature will be higher than the brightness temperature if the emissivity of the object is greater than 1. Because such an approximation is not required for incandescent light, the CCT for an incandescent light is simply its unadjusted temperature, derived from comparison to a black-body radiator. ===The Sun=== The Sun closely approximates a black-body radiator. The brightness temperature can be used to calculate the spectral index of a body, in the case of non-thermal radiation. == Calculating by frequency == The brightness temperature of a source with known spectral radiance can be expressed as: : T_b=\frac{h u}{k} \ln^{-1}\left( 1 + \frac{2h u^3}{I_{ u}c^2} \right) When h u \ll kT we can use the Rayleigh–Jeans law: : T_b=\frac{I_{ u}c^2}{2k u^2} For narrowband radiation with very low relative spectral linewidth \Delta u \ll u and known radiance I we can calculate the brightness temperature as: : T_b=\frac{I c^2}{2k u^2\Delta u} == Calculating by wavelength == Spectral radiance of black-body radiation is expressed by wavelength as: : I_{\lambda}=\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{kT \lambda}} - 1} So, the brightness temperature can be calculated as: : T_b=\frac{hc}{k\lambda} \ln^{-1}\left(1 + \frac{2hc^2}{I_{\lambda}\lambda^5} \right) For long-wave radiation hc/\lambda \ll kT the brightness temperature is: : T_b=\frac{I_{\lambda}\lambda^4}{2kc} For almost monochromatic radiation, the brightness temperature can be expressed by the radiance I and the coherence length L_c: : T_b=\frac{\pi I \lambda^2 L_c}{4kc \ln{2} } ==In oceanography== In oceanography, the microwave brightness temperature, as measured by satellites looking at the ocean surface, depends on salinity as well as on the temperature of the water. ==References== Category:Temperature Category:Radio astronomy Category:Planetary science Thus a relatively low temperature emits a dull red and a high temperature emits the almost white of the traditional incandescent light bulb. The effective temperature, defined by the total radiative power per square unit, is 5772 K. The Draper point is the approximate temperature above which almost all solid materials visibly glow as a result of blackbody radiation. Wet-bulb potential temperature, sometimes referred to as pseudo wet-bulb potential temperature, is the temperature that a parcel of air at any level would have if, starting at the wet-bulb temperature, it were brought at the saturated adiabatic lapse rate to the standard pressure of 1,000 mbar. When the star's or planet's net emissivity in the relevant wavelength band is less than unity (less than that of a black body), the actual temperature of the body will be higher than the effective temperature. For radiation emitted by a non-thermal source such as a pulsar, synchrotron, maser, or a laser, the brightness temperature may be far higher than the actual temperature of the source. The fact that "warm" lighting in this sense actually has a "cooler" color temperature often leads to confusion.See the comments section of this LightNowBlog.com article on the recommendations of the American Medical Association to prefer LED- lighting with cooler color temperatures (i.e. warmer color). ==Categorizing different lighting== The color temperature of the electromagnetic radiation emitted from an ideal black body is defined as its surface temperature in kelvins, or alternatively in micro reciprocal degrees (mired). When the electromagnetic radiation observed is thermal radiation emitted by an object simply by virtue of its temperature, then the actual temperature of the object will always be equal to or higher than the brightness temperature. Color temperature is usually measured in kelvins. Color temperature is a parameter describing the color of a visible light source by comparing it to the color of light emitted by an idealized opaque, non-reflective body.

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Evaluate the series $$ S=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots $$

It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. Simplifying the fractions gives :1 \,+\, \frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots. It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. As it is a geometric series with first term and common ratio , its sum is :\sum_{n=1}^\infty \frac{1}{4^n}=\frac {\frac 1 4} {1 - \frac 1 4}=\frac 1 3. ==Visual demonstrations== left|thumb|3s = 1\. This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. Since the sum of an infinite series is defined as the limit of its partial sums, :1+\frac14+\frac{1}{4^2}+\frac{1}{4^3}+\cdots = \frac43. ==Notes== ==References== * * Page images at HTML with figures and commentary at * * * * * Category:Geometric series Category:Proof without words Since these three areas cover the unit square, the figure demonstrates that :3\left(\frac14+\frac{1}{4^2}+\frac{1}{4^3}+\frac{1}{4^4}+\cdots\right) = 1. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_{n=1}^\infty\frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots. Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series are: :1+\frac{1}{4}+\frac{1}{4^2}+\cdots+\frac{1}{4^n}=\frac{1-\left(\frac14\right)^{n+1}}{1-\frac14}. It is a geometric series whose first term is and whose common ratio is −, so its sum is :\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2^n}=\frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{\frac12}{1-(-\frac12)} = \frac13. ==Hackenbush and the surreals== frame|right|Demonstration of via a zero-value game A slight rearrangement of the series reads :1-\frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac13. Archimedes' own illustration, adapted at top,Heath p. 250 was slightly different, being closer to the equation right|thumb|3s = 1 again :\sum_{n=1}^\infty \frac{3}{4^n}=\frac34+\frac{3}{4^2}+\frac{3}{4^3}+\frac{3}{4^4}+\cdots = 1. The series \sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots is known as the alternating harmonic series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of , which is expressed by a famous formula. : 1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}, where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. Explicitly, the asymptotic expansion of the series is \frac{1}{1} - \frac{1}{2} +\cdots + \frac{1}{2n-1} - \frac{1}{2n} = H_{2n} - H_n = \ln 2 - \frac{1}{2n} + O(n^{-2}) Using alternating signs with only odd unit fractions produces a related series, the Leibniz formula for \sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}. ===Riemann zeta function=== The Riemann zeta function is defined for real x>1 by the convergent series \zeta(x)=\sum_{n=1}^{\infty}\frac{1}{n^x}=\frac1{1^x}+\frac1{2^x}+\frac1{3^x}+\cdots, which for x=1 would be the harmonic series. The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. The first dimension is horizontal, in the bottom row showing the geometric series S = 1/2 + 1/4 + 1/8 + 1/16 + ... , which is the geometric series with coefficient a = 1/2 and common ratio r = 1/2 that converges to S = a / (1-r) = (1/2) / (1-1/2) = 1. thumb|right|The geometric series 1/4 + 1/16 + 1/64 + 1/256 + ... shown as areas of purple squares. Using the geometric series closed form as before :0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac{a}{1-r} \;=\; \frac{49/64}{1-1/64} \;=\; \frac{49/64}{63/64} \;=\; \frac{49}{63} \;=\; \frac{7}{9}. Two of the best-known are listed below. ===Comparison test=== One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest power of two: \begin{alignat}{8} 1 & \+ \frac{1}{2} && \+ \frac{1}{3} && \+ \frac{1}{4} && \+ \frac{1}{5} && \+ \frac{1}{6} && \+ \frac{1}{7} && \+ \frac{1}{8} && \+ \frac{1}{9} && \+ \cdots \\\\[5pt] {} \geq 1 & \+ \frac{1}{2} && \+ \frac{1}{\color{red}{\mathbf{4}}} && \+ \frac{1}{4} && \+ \frac{1}{\color{red}{\mathbf{8}}} && \+ \frac{1}{\color{red}{\mathbf{8}}} && \+ \frac{1}{\color{red}{\mathbf{8}}} && \+ \frac{1}{8} && \+ \frac{1}{\color{red}{\mathbf{16}}} && \+ \cdots \\\\[5pt] \end{alignat} Grouping equal terms shows that the second series diverges (because every grouping of convergent series is only convergent): \begin{align} & 1 + \left(\frac{1}{2}\right) + \left(\frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) + \left(\frac{1}{16} + \cdots + \frac{1}{16}\right) + \cdots \\\\[5pt] {} = {} & 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots. \end{align} Because each term of the harmonic series is greater than or equal to the corresponding term of the second series (and the terms are all positive), and since the second series diverges, it follows (by the comparison test) that the harmonic series diverges as well. Among his insights into infinite series, in addition to his elegantly simple proof of the divergence of the harmonic series, Nicole Oresme proved that the series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. This list of mathematical series contains formulae for finite and infinite sums.

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Through what potential must a proton initially at rest fall so that its de Broglie wavelength is $1.0 \times 10^{-10} \mathrm{~m}$ ?

As a muon is 200 times heavier than an electron, its de Broglie wavelength is correspondingly shorter. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. This value is based on measurements involving a proton and an electron (namely, electron scattering measurements and complex calculation involving scattering cross section based on Rosenbluth equation for momentum-transfer cross section), and studies of the atomic energy levels of hydrogen and deuterium. When applied to the Planck relation, this gives: :\lambda = \frac {1}{ u} \cdot c = \frac {h}{E} \cdot c \approx \frac{\; 4.1357 \cdot 10^{-15} \ \mathrm{eV}\cdot\text{s} \;}{5.874\,33 \cdot 10^{-6}\ \mathrm{eV}}\, \cdot\, 2.9979 \cdot 10^8 \ \mathrm{m} \cdot \mathrm{s}^{-1} \approx 0.211\,06\ \mathrm{m} = 21.106\ \mathrm{cm}\; where is the wavelength of an emitted photon, is its frequency, is the photon energy, is the Planck constant, and is the speed of light. In kaonic hydrogen this strong contribution was found to be repulsive, shifting the ground state energy by 283 ± 36 (statistical) ± 6 (systematic) eV, thus making the system unstable with a resonance width of 541 ± 89 (stat) ± 22 (syst) eV (decay into Λπ and ΣπYiguang Yan, Kaonic hydrogen atom and kaon-proton scattering length, ). This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. Species Mass (kg) \lambda_{\mathrm{th}} (m) Electron Photon 0 H2 O2 ==References== * Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 e (elementary charge). Kaonic hydrogen is an exotic atom consisting of a negatively charged kaon orbiting a proton. A resolution came in 2019, when two different studies, using different techniques involving the Lamb shift of the electron in hydrogen, and electron–proton scattering, found the radius of the proton to be 0.833 fm, with an uncertainty of ±0.010 fm, and 0.831 fm. Consistent with the spectroscopy method, this produces a proton radius of about . ===2010 experiment=== In 2010, Pohl et al. published the results of an experiment relying on muonic hydrogen as opposed to normal hydrogen. In a laboratory setting, the hydrogen line parameters have been more precisely measured as: : λ = 21.106114054160(30) cm : ν = 1420405751.768(2) Hz in a vacuum. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. The result is again ~5% smaller than the previously-accepted proton radius. thumb|A hydrogen atom with proton and electron spins aligned (top) undergoes a flip of the electron spin, resulting in emission of a photon with a 21 cm wavelength (bottom) The hydrogen line, 21 centimeter line, or H I line is a spectral line that is created by a change in the energy state of solitary, electrically neutral hydrogen atoms. In the inertial frame, the accelerating proton should decay according to the formula above. In 2019, two different studies, using different techniques, found this radius to be 0.833 fm, with an uncertainty of ±0.010 fm. Free protons occur occasionally on Earth: thunderstorms can produce protons with energies of up to several tens of MeV. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. The internationally accepted value of a proton's charge radius is . Because protons are not fundamental particles, they possess a measurable size; the root mean square charge radius of a proton is about 0.84–0.87 fm ( = ). thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. (The size of an atom, measured in angstroms (Å, or 10−10 m), is five orders of magnitude larger).

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Example 5-3 shows that a Maclaurin expansion of a Morse potential leads to $$ V(x)=D \beta^2 x^2+\cdots $$ Given that $D=7.31 \times 10^{-19} \mathrm{~J} \cdot$ molecule ${ }^{-1}$ and $\beta=1.81 \times 10^{10} \mathrm{~m}^{-1}$ for $\mathrm{HCl}$, calculate the force constant of $\mathrm{HCl}$.

In fact, the real molecular spectra are generally fit to the form1 : E_n / hc = \omega_e (n+1/2) - \omega_e\chi_e (n+1/2)^2\, in which the constants \omega_e and \omega_e\chi_e can be directly related to the parameters for the Morse potential. More sophisticated versions are used for polyatomic molecules. ==See also== *Lennard-Jones potential *Molecular mechanics ==References== *1 CRC Handbook of chemistry and physics, Ed David R. Lide, 87th ed, Section 9, SPECTROSCOPIC CONSTANTS OF DIATOMIC MOLECULES pp. 9–82 * * * * * * * * I.G. Kaplan, in Handbook of Molecular Physics and Quantum Chemistry, Wiley, 2003, p207. The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. The Hamaker constant provides the means to determine the interaction parameter from the vdW-pair potential, :w(r) = \frac{-C}{r^6}. The molecular formula C12H12O4 (molar mass: 220.22 g/mol, exact mass: 220.0736 u) may refer to: * Eugenitin * Hispolon * Siderin The molecular formula C16H19BrN2 (molar mass: 319.24 g/mol, exact mass: 318.0732 u) may refer to: * Brompheniramine * Dexbrompheniramine Category:Molecular formulas Mathematically, the spacing of Morse levels is :E_{n+1} - E_n = h u_0 - (n+1) (h u_0)^2/2D_e.\, This trend matches the anharmonicity found in real molecules. The Morse potential energy function is of the form :V(r) = D_e ( 1-e^{-a(r-r_e)} )^2 Here r is the distance between the atoms, r_e is the equilibrium bond distance, D_e is the well depth (defined relative to the dissociated atoms), and a controls the 'width' of the potential (the smaller a is, the larger the well). Unlike the energy levels of the harmonic oscillator potential, which are evenly spaced by ħω, the Morse potential level spacing decreases as the energy approaches the dissociation energy. A further extension of the MLR potential referred to as the MLR3 potential was introduced in a 2010 study of Cs2, and this potential has since been used on HF, HCl, HBr and HI. == Function == The Morse/Long-range potential energy function is of the form V(r) = \mathfrak{D}_e \left( 1- \frac{u(r)}{u(r_e)} e^{-\beta(r) y_p^{r_{\rm{eq}}}(r)} \right)^2 where for large r, V(r) \simeq \mathfrak{D}_e - u(r) + \frac{u(r)^2}{4\mathfrak{D}_e}, so u(r) is defined according to the theoretically correct long-range behavior expected for the interatomic interaction. In molecular physics, the Hamaker constant (denoted ; named for H. C. Hamaker) is a physical constant that can be defined for a van der Waals (vdW) body–body interaction: :A=\pi^2C\rho_1\rho_2, where are the number densities of the two interacting kinds of particles, and is the London coefficient in the particle–particle pair interaction. The magnitude of this constant reflects the strength of the vdW-force between two particles, or between a particle and a substrate. The force constant (stiffness) of the bond can be found by Taylor expansion of V'(r) around r=r_e to the second derivative of the potential energy function, from which it can be shown that the parameter, a, is :a=\sqrt{k_e/2D_e}, where k_e is the force constant at the minimum of the well. right|thumb|illustrative example of C-C length molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C angle molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C-C torsion molecular energy dependence, numerical accuracy is not guaranteed A Molecular spring is a device or part of a biological system based on molecular mechanics and is associated with molecular vibration. The Morse/Long-range potential (MLR potential) is an interatomic interaction model for the potential energy of a diatomic molecule. Hamaker's method and the associated Hamaker constant ignores the influence of an intervening medium between the two particles of interaction. Since the zero of potential energy is arbitrary, the equation for the Morse potential can be rewritten any number of ways by adding or subtracting a constant value. When it is used to model the atom-surface interaction, the energy zero can be redefined so that the Morse potential becomes :V(r)= V'(r)-D_e = D_e ( 1-e^{-a(r-r_e)} )^2 -D_e which is usually written as :V(r) = D_e ( e^{-2a(r-r_e)}-2e^{-a(r-r_e)} ) where r is now the coordinate perpendicular to the surface. It clearly shows that the Morse potential is the combination of a short-range repulsion term (the former) and a long-range attractive term (the latter), analogous to the Lennard-Jones potential. ==Vibrational states and energies== 500px|thumb|Harmonic oscillator (grey) and Morse (black) potentials curves are shown along with their eigenfunctions (respectively green and blue for harmonic oscillator and morse) for the same vibrational levels for nitrogen. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. The latter is mathematically related to the particle mass, m, and the Morse constants via : u_0 = \frac{a}{2\pi} \sqrt{2D_e/m}. As is clear from dimensional analysis, for historical reasons the last equation uses spectroscopic notation in which \omega_e represents a wavenumber obeying E=hc\omega, and not an angular frequency given by E=\hbar\omega. == Morse/Long-range potential == An extension of the Morse potential that made the Morse form useful for modern (high-resolution) spectroscopy is the MLR (Morse/Long-range) potential.

9.8

1.45

'-194.0'

3.2

479

E

A line in the Lyman series of hydrogen has a wavelength of $1.03 \times 10^{-7} \mathrm{~m}$. Find the original energy level of the electron.

In physics and chemistry, the Lyman series is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1 (where n is the principal quantum number), the lowest energy level of the electron. Therefore, each wavelength of the emission lines corresponds to an electron dropping from a certain energy level (greater than 1) to the first energy level. == See also == * Bohr model * H-alpha * Hydrogen spectral series * K-alpha * Lyman-alpha line * Lyman continuum photon * Moseley's law * Rydberg formula * Balmer series ==References== Category:Emission spectroscopy Category:Hydrogen physics Replacing the energy in the above formula with the expression for the energy in the hydrogen atom where the initial energy corresponds to energy level n and the final energy corresponds to energy level m, : \frac{1}{\lambda} = \frac{E_\text{i} - E_\text{f}}{12398.4\,\text{eV Å}} = R_\text{H} \left(\frac{1}{m^2} - \frac{1}{n^2} \right) Where RH is the same Rydberg constant for hydrogen from Rydberg's long known formula. This equation is valid for all hydrogen-like species, i.e. atoms having only a single electron, and the particular case of hydrogen spectral lines is given by Z=1. ==Series== ===Lyman series ( = 1)=== In the Bohr model, the Lyman series includes the lines emitted by transitions of the electron from an outer orbit of quantum number n > 1 to the 1st orbit of quantum number n' = 1. The Lyman-alpha line, typically denoted by Ly-α, is a spectral line of hydrogen (or, more generally, of any one-electron atom) in the Lyman series. In hydrogen, its wavelength of 1215.67 angstroms ( or ), corresponding to a frequency of about , places Lyman-alpha in the ultraviolet (UV) part of the electromagnetic spectrum. For the connection between Bohr, Rydberg, and Lyman, one must replace m with 1 to obtain : \frac{1}{\lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) which is Rydberg's formula for the Lyman series. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. The Lyman limit is the short-wavelength end of the hydrogen Lyman series, at . The greater the difference in the principal quantum numbers, the higher the energy of the electromagnetic emission. ==History== thumb|upright=1.3|The Lyman series The first line in the spectrum of the Lyman series was discovered in 1906 by physicist Theodore Lyman, who was studying the ultraviolet spectrum of electrically excited hydrogen gas. This energy is equivalent to the Rydberg constant. == See also == * Balmer Limit * Lyman-alpha emitter * Lyman-alpha forest * Lyman-break galaxy * Lyman series * Rydberg formula ==References== Category:Atomic physics Bohr found that the electron bound to the hydrogen atom must have quantized energy levels described by the following formula, : E_n = - \frac{m_e e^4}{2(4\pi\varepsilon_0\hbar)^2}\,\frac{1}{n^2} = - \frac{13.6\,\text{eV}}{n^2}. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. For example, the line is called "Lyman-alpha" (Ly-α), while the line is called "Paschen-delta" (Pa-δ). thumb|Energy level diagram of electrons in hydrogen atom There are emission lines from hydrogen that fall outside of these series, such as the 21 cm line. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. DOI: https://doi.org/10.18434/T4W30F 3 102.57220 4 97.253650 5 94.974287 6 93.780331 7 93.0748142 8 92.6225605 9 92.3150275 10 92.0963006 11 91.9351334 ∞, the Lyman limit 91.1753 ==Explanation and derivation== In 1914, when Niels Bohr produced his Bohr model theory, the reason why hydrogen spectral lines fit Rydberg's formula was explained. Lyman-alpha radiation had previously been detected from other galaxies, but due to interference from the Sun, the radiation from the Milky Way was not detectable. ==The Lyman series== The version of the Rydberg formula that generated the Lyman series was: {1 \over \lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) \qquad \left( R_\text{H} \approx 1.0968{\times}10^7\,\text{m}^{-1} \approx \frac{13.6\,\text{eV}}{hc} \right) where n is a natural number greater than or equal to 2 (i.e., ). All the wavelengths in the Lyman series are in the ultraviolet band.. These observed spectral lines are due to the electron making transitions between two energy levels in an atom. It is the first spectral line in the Balmer series and is emitted when an electron falls from a hydrogen atom's third- to second-lowest energy level. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). For the Lyman series the naming convention is: *n = 2 to n = 1 is called Lyman- alpha, *n = 3 to n = 1 is called Lyman-beta, etc. H-alpha has a wavelength of 656.281 nm, is visible in the red part of the electromagnetic spectrum, and is the easiest way for astronomers to trace the ionized hydrogen content of gas clouds.

+116.0

26.9

0.166666666

4.16

3

E

A helium-neon laser (used in supermarket scanners) emits light at $632.8 \mathrm{~nm}$. Calculate the frequency of this light.

thumb|Helium–neon laser at the University of Chemnitz, Germany A helium–neon laser or He-Ne laser, is a type of gas laser whose high energetic medium gain medium consists of a mixture of ratio(between 5:1 and 20:1) of helium and neon at a total pressure of about 1 torr inside of a small electrical discharge. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. ==History of He-Ne laser development== The first He-Ne lasers emitted infrared at 1150 nm, and were the first gas lasers and the first lasers with continuous wave output. The excited helium atoms collide with neon atoms, exciting some of them to the state that radiates 632.8 nm. The precise wavelength of red He-Ne lasers is 632.991 nm in a vacuum, which is refracted to about 632.816 nm in air. Absolute stabilization of the laser's frequency (or wavelength) as fine as 2.5 parts in 1011 can be obtained through use of an iodine absorption cell. thumb|Energy levels in a He-Ne Laser|512x330px thumb|Ring He-Ne Laser The mechanism producing population inversion and light amplification in a He-Ne laser plasma originates with inelastic collision of energetic electrons with ground-state helium atoms in the gas mixture. However, a laser that operated at visible wavelengths was much more in demand, and a number of other neon transitions were investigated to identify ones in which a population inversion can be achieved. Without helium, the neon atoms would be excited mostly to lower excited states, responsible for non-laser lines. thumb|right|300 px|A 337nm wavelength and 170 μJ pulse energy 20 Hz cartridge nitrogen laser A nitrogen laser is a gas laser operating in the ultraviolet rangeC. The optical cavity of the laser usually consists of two concave mirrors or one plane and one concave mirror: one having very high (typically 99.9%) reflectance, and the output coupler mirror allowing approximately 1% transmission. frame|Schematic diagram of a helium–neon laser Commercial He-Ne lasers are relatively small devices compared to other gas lasers, having cavity lengths usually ranging from 15 to 50 cm (but sometimes up to about 1 meter to achieve the highest powers), and optical output power levels ranging from 0.5 to 50 mW. It was developed at Bell Telephone Laboratories in 1962, 18 months after the pioneering demonstration at the same laboratory of the first continuous infrared He-Ne gas laser in December 1960. ==Construction and operation== The gain medium of the laser, as suggested by its name, is a mixture of helium and neon gases, in approximately a 10:1 ratio, contained at low pressure in a glass envelope. The 633 nm line was found to have the highest gain in the visible spectrum, making this the wavelength of choice for most He-Ne lasers. A blue laser emits electromagnetic radiation with a wavelength between 400 and 500 nanometers, which the human eye sees in the visible spectrum as blue or violet. The Nike laser at the United States Naval Research Laboratory in Washington, DC is a 56-beam, 4–5 kJ per pulse electron beam pumped krypton fluoride excimer laser which operates in the ultraviolet at 248 nm with pulsewidths of a few nanoseconds. A neon laser with no helium can be constructed, but it is much more difficult without this means of energy coupling. Violet light's 405nm short wavelength, on the visible spectrum, causes fluorescence in some chemicals, like radiation in the ultraviolet ("black light") spectrum (wavelengths less than 400 nm). == History == thumb|445nm - 450nm Blue Laser (middle) Prior to the 1960s and until the late 1990s, gas and argon-ion lasers were common; suffering from poor efficiencies(0.01%) and large sizes. Laser gyroscopes have employed He-Ne lasers operating at 633 nm in a ring laser configuration. * Frequency-resolved electro-absorption gating (FREAG) ==References== * *R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and D. J. Kane, "Measuring Ultrashort Laser Pulses in the Time-Frequency Domain Using Frequency-Resolved Optical Gating," Review of Scientific Instruments 68, 3277-3295 (1997). ==External links== *FROG Page by Rick Trebino (co-inventor of FROG) Category:Nonlinear optics Category:Lasers Category:Optical metrology They most commonly emit light at 473 nm, which is produced by frequency doubling of 946 nm laser radiation from a diode-pumped Nd:YAG or Nd:YVO4 crystal. Conversion efficiency for producing 473 nm laser radiation is inefficient with some of the best lab produced results coming in at 10-15% efficient at converting 946 nm laser radiation to 473 nm laser radiation. 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. Stimulated emissions are known from over 100 μm in the far infrared to 540 nm in the visible.

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4500

D

What is the uncertainty of the momentum of an electron if we know its position is somewhere in a $10 \mathrm{pm}$ interval?

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. 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. Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). 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. 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. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. * Problem 1 – If the photon has a short wavelength, and therefore, a large momentum, the position can be measured accurately. Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. 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 uncertainty principle also says that eliminating uncertainty about position maximizes uncertainty about momentum, and eliminating uncertainty about momentum maximizes uncertainty about position. On the other hand, the standard deviation of the position is \sigma_x = \frac{x_0}{\sqrt{2}} \sqrt{1+\omega_0^2 t^2} such that the uncertainty product can only increase with time as \sigma_x(t) \sigma_p(t) = \frac{\hbar}{2} \sqrt{1+\omega_0^2 t^2} ==Additional uncertainty relations== ===Systematic and statistical errors=== The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation \sigma. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. In doing so, we find the momentum of the particle to > arbitrary accuracy by conservation of momentum. Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. He imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it. This precision may be quantified by the standard deviations, \sigma_x=\sqrt{\langle \hat{x}^2 \rangle-\langle \hat{x}\rangle^2} \sigma_p=\sqrt{\langle \hat{p}^2 \rangle-\langle \hat{p}\rangle^2}. Heisenberg's uncertainty relation is one of the fundamental results in quantum mechanics. If the photon has a long wavelength and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely. 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. A more robust representation of measurement uncertainty in such cases can be fashioned from intervals.Manski, C.F. (2003); Partial Identification of Probability Distributions, Springer Series in Statistics, Springer, New YorkFerson, S., V. Kreinovich, J. Hajagos, W. Oberkampf, and L. Ginzburg (2007); Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty, Sandia National Laboratories SAND 2007-0939 An interval [a, b] is different from a rectangular or uniform probability distribution over the same range in that the latter suggests that the true value lies inside the right half of the range [(a + b)/2, b] with probability one half, and within any subinterval of [a, b] with probability equal to the width of the subinterval divided by b − a.

0.33333333

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0.3359

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Using the Bohr theory, calculate the ionization energy (in electron volts and in $\mathrm{kJ} \cdot \mathrm{mol}^{-1}$ ) of singly ionized helium.

To convert from "value of ionization energy" to the corresponding "value of molar ionization energy", the conversion is: *:1 eV = 96.48534 kJ/mol *:1 kJ/mol = 0.0103642688 eV == References == === WEL (Webelements) === As quoted at http://www.webelements.com/ from these sources: * J.E. Huheey, E.A. Keiter, and R.L. Keiter in Inorganic Chemistry : Principles of Structure and Reactivity, 4th edition, HarperCollins, New York, USA, 1993. These tables list values of molar ionization energies, measured in kJ⋅mol−1. 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 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. For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). In physics, ionization energy is usually expressed in electronvolts (eV) or joules (J). == Numerical values == For each atom, the column marked 1 is the first ionization energy to ionize the neutral atom, the column marked 2 is the second ionization energy to remove a second electron from the +1 ion, the column marked 3 is the third ionization energy to remove a third electron from the +2 ion, and so on. "use" and "WEL" give ionization energy in the unit kJ/mol; "CRC" gives atomic ionization energy in the unit eV. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 H hydrogen use 1312.0 WEL 1312.0 CRC 13.59844 2 He helium use 2372.3 5250.5 WEL 2372.3 5250.5 CRC 24.58738 54.41776 3 Li lithium use 520.2 7298.1 11815.0 WEL 520.2 7298.1 11815.0 CRC 5.39171 75.64009 122.45435 4 Be beryllium use 899.5 1757.1 14848.7 21006.6 WEL 899.5 1757.1 14848.7 21006.6 CRC 9.32269 18.21115 153.89620 217.71858 5 B boron use 800.6 2427.1 3659.7 25025.8 32826.7 WEL 800.6 2427.1 3659.7 25025.8 32826.7 CRC 8.29803 25.15484 37.93064 259.37521 340.22580 6 C carbon use 1086.5 2352.6 4620.5 6222.7 37831 47277.0 WEL 1086.5 2352.6 4620.5 6222.7 37831 47277.0 CRC 11.26030 24.38332 47.8878 64.4939 392.087 489.99334 7 N nitrogen use 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 WEL 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 CRC 14.53414 29.6013 47.44924 77.4735 97.8902 552.0718 667.046 8 O oxygen use 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 WEL 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 CRC 13.61806 35.11730 54.9355 77.41353 113.8990 138.1197 739.29 871.4101 9 F fluorine use 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 WEL 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 CRC 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 953.9112 1103.1176 10 Ne neon use 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 WEL 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 CRC 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 1195.8286 1362.1995 11 Na sodium use 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 159076 WEL 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 CRC 5.13908 47.2864 71.6200 98.91 138.40 172.18 208.50 264.25 299.864 1465.121 1648.702 12 Mg magnesium use 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 169988 189368 WEL 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 CRC 7.64624 15.03528 80.1437 109.2655 141.27 186.76 225.02 265.96 328.06 367.50 1761.805 1962.6650 13 Al aluminium use 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 42647 201266 222316 WEL 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 CRC 5.98577 18.82856 28.44765 119.992 153.825 190.49 241.76 284.66 330.13 398.75 442.00 2085.98 2304.1410 14 Si silicon use 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 45962 50502 235196 257923 WEL 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 CRC 8.15169 16.34585 33.49302 45.14181 166.767 205.27 246.5 303.54 351.12 401.37 476.36 523.42 2437.63 2673.182 15 P phosphorus use 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 46261 54110 59024 271791 296195 WEL 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 CRC 10.48669 19.7694 30.2027 51.4439 65.0251 220.421 263.57 309.60 372.13 424.4 479.46 560.8 611.74 2816.91 3069.842 16 S sulfur use 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 48710 54460 62930 68216 311048 337138 WEL 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 CRC 10.36001 23.3379 34.79 47.222 72.5945 88.0530 280.948 328.75 379.55 447.5 504.8 564.44 652.2 707.01 3223.78 3494.1892 17 Cl chlorine use 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 51068 57119 63363 72341 78095 352994 380760 WEL 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 CRC 12.96764 23.814 39.61 53.4652 67.8 97.03 114.1958 348.28 400.06 455.63 529.28 591.99 656.71 749.76 809.40 3658.521 3946.2960 18 Ar argon use 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 52002 59653 66199 72918 82473 88576 397605 427066 WEL 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 CRC 15.75962 27.62967 40.74 59.81 75.02 91.009 124.323 143.460 422.45 478.69 538.96 618.26 686.10 755.74 854.77 918.03 4120.8857 4426.2296 19 K potassium use 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 54490 60730 68950 75900 83080 93400 99710 444880 476063 WEL 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 CRC 4.34066 31.63 45.806 60.91 82.66 99.4 117.56 154.88 175.8174 503.8 564.7 629.4 714.6 786.6 861.1 968 1033.4 4610.8 4934.046 20 Ca calcium use 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 63410 70110 78890 86310 94000 104900 111711 494850 527762 WEL 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 CRC 6.11316 11.87172 50.9131 67.27 84.50 108.78 127.2 147.24 188.54 211.275 591.9 657.2 726.6 817.6 894.5 974 1087 1157.8 5128.8 5469.864 21 Sc scandium use 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 WEL 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 CRC 6.5615 12.79967 24.75666 73.4894 91.65 110.68 138.0 158.1 180.03 225.18 249.798 687.36 756.7 830.8 927.5 1009 1094 1213 1287.97 5674.8 6033.712 22 Ti titanium use 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 639294 WEL 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 CRC 6.8281 13.5755 27.4917 43.2672 99.30 119.53 140.8 170.4 192.1 215.92 265.07 291.500 787.84 863.1 941.9 1044 1131 1221 1346 1425.4 6249.0 6625.82 23 V vanadium use 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 661050 699144 WEL 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 CRC 6.7462 14.66 29.311 46.709 65.2817 128.13 150.6 173.4 205.8 230.5 255.7 308.1 336.277 896.0 976 1060 1168 1260 1355 1486 1569.6 6851.3 7246.12 24 Cr chromium use 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 166090 721870 761733 WEL 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 CRC 6.7665 16.4857 30.96 49.16 69.46 90.6349 160.18 184.7 209.3 244.4 270.8 298.0 354.8 384.168 1010.6 1097 1185 1299 1396 1496 1634 1721.4 7481.7 7894.81 25 Mn manganese use 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 172500 181380 785450 827067 WEL 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 CRC 7.43402 15.63999 33.668 51.2 72.4 95.6 119.203 194.5 221.8 248.3 286.0 314.4 343.6 403.0 435.163 1134.7 1224 1317 1437 1539 1644 1788 1879.9 8140.6 8571.94 26 Fe iron use 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 173600 188100 195200 851800 895161 WEL 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 CRC 7.9024 16.1878 30.652 54.8 75.0 99.1 124.98 151.06 233.6 262.1 290.2 330.8 361.0 392.2 457 489.256 1266 1358 1456 1582 1689 1799 1950 2023 8828 9277.69 27 Co cobalt use 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 178100 189300 204500 214100 920870 966023 WEL 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 CRC 7.8810 17.083 33.50 51.3 79.5 102.0 128.9 157.8 186.13 275.4 305 336 379 411 444 511.96 546.58 1397.2 1504.6 1603 1735 1846 1962 2119 2219.0 9544.1 10012.12 28 Ni nickel use 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 182700 194000 205600 221400 231490 992718 1039668 WEL 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 CRC 7.6398 18.16884 35.19 54.9 76.06 108 133 162 193 224.6 321.0 352 384 430 464 499 571.08 607.06 1541 1648 1756 1894 2011 2131 2295 2399.2 10288.8 10775.40 29 Cu copper use 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 184900 198800 210500 222700 239100 249660 1067358 1116105 WEL 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 CRC 7.72638 20.29240 36.841 57.38 79.8 103 139 166 199 232 265.3 369 401 435 484 520 557 633 670.588 1697 1804 1916 2060 2182 2308 2478 2587.5 11062.38 11567.617 30 Zn zinc use 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 WEL 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 CRC 9.3942 17.96440 39.723 59.4 82.6 108 134 174 203 238 274 310.8 419.7 454 490 542 579 619 698 738 1856 31 Ga gallium use 578.8 1979.3 2963 6180 WEL 578.8 1979.3 2963 6180 CRC 5.99930 20.5142 30.71 64 32 Ge germanium use 762 1537.5 3302.1 4411 9020 WEL 762 1537.5 3302.1 4411 9020 CRC 7.8994 15.93462 34.2241 45.7131 93.5 33 As arsenic use 947.0 1798 2735 4837 6043 12310 WEL 947.0 1798 2735 4837 6043 12310 CRC 9.7886 18.633 28.351 50.13 62.63 127.6 34 Se selenium use 941.0 2045 2973.7 4144 6590 7880 14990 WEL 941.0 2045 2973.7 4144 6590 7880 14990 CRC 9.75238 21.19 30.8204 42.9450 68.3 81.7 155.4 35 Br bromine use 1139.9 2103 3470 4560 5760 8550 9940 18600 WEL 1139.9 2103 3470 4560 5760 8550 9940 18600 CRC 11.81381 21.8 36 47.3 59.7 88.6 103.0 192.8 36 Kr krypton use 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 90400 96300 101400 111100 116290 282500 296200 311400 326200 WEL 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 CRC 13.99961 24.35985 36.950 52.5 64.7 78.5 111.0 125.802 230.85 268.2 308 350 391 447 492 541 592 641 786 833 884 937 998 1051 1151 1205.3 2928 3070 3227 3381 37 Rb rubidium use 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 WEL 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 CRC 4.17713 27.285 40 52.6 71.0 84.4 99.2 136 150 277.1 38 Sr strontium use 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 31270 WEL 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 CRC 5.6949 11.03013 42.89 57 71.6 90.8 106 122.3 162 177 324.1 39 Y yttrium use 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 19900 36090 WEL 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 CRC 6.2171 12.24 20.52 60.597 77.0 93.0 116 129 146.2 191 206 374.0 40 Zr zirconium use 640.1 1270 2218 3313 7752 9500 WEL 640.1 1270 2218 3313 7752 9500 CRC 6.63390 13.13 22.99 34.34 80.348 41 Nb niobium use 652.1 1380 2416 3700 4877 9847 12100 WEL 652.1 1380 2416 3700 4877 9847 12100 CRC 6.75885 14.32 25.04 38.3 50.55 102.057 125 42 Mo molybdenum use 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 93400 98420 104400 121900 127700 133800 139800 148100 154500 WEL 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 CRC 7.09243 16.16 27.13 46.4 54.49 68.8276 125.664 143.6 164.12 186.4 209.3 230.28 279.1 302.60 544.0 570 636 702 767 833 902 968 1020 1082 1263 1323 1387 1449 1535 1601 43 Tc technetium use 702 1470 2850 WEL 702 1470 2850 CRC 7.28 15.26 29.54 44 Ru ruthenium use 710.2 1620 2747 WEL 710.2 1620 2747 CRC 7.36050 16.76 28.47 45 Rh rhodium use 719.7 1740 2997 WEL 719.7 1740 2997 CRC 7.45890 18.08 31.06 46 Pd palladium use 804.4 1870 3177 WEL 804.4 1870 3177 CRC 8.3369 19.43 32.93 47 Ag silver use 731.0 2070 3361 WEL 731.0 2070 3361 CRC 7.5762 21.49 34.83 48 Cd cadmium use 867.8 1631.4 3616 WEL 867.8 1631.4 3616 CRC 8.9938 16.90832 37.48 49 In indium use 558.3 1820.7 2704 5210 WEL 558.3 1820.7 2704 5210 CRC 5.78636 18.8698 28.03 54 50 Sn tin use 708.6 1411.8 2943.0 3930.3 7456 WEL 708.6 1411.8 2943.0 3930.3 7456 CRC 7.3439 14.63225 30.50260 40.73502 72.28 51 Sb antimony use 834 1594.9 2440 4260 5400 10400 WEL 834 1594.9 2440 4260 5400 10400 CRC 8.6084 16.53051 25.3 44.2 56 108 52 Te tellurium use 869.3 1790 2698 3610 5668 6820 13200 WEL 869.3 1790 2698 3610 5668 6820 13200 CRC 9.0096 18.6 27.96 37.41 58.75 70.7 137 53 I iodine use 1008.4 1845.9 3180 WEL 1008.4 1845.9 3180 CRC 10.45126 19.1313 33 54 Xe xenon use 1170.4 2046.4 3099.4 WEL 1170.4 2046.4 3099.4 CRC 12.1298 21.20979 32.1230 55 Cs caesium use 375.7 2234.3 3400 WEL 375.7 2234.3 3400 CRC 3.89390 23.15745 56 Ba barium use 502.9 965.2 3600 WEL 502.9 965.2 3600 CRC 5.21170 10.00390 57 La lanthanum use 538.1 1067 1850.3 4819 5940 WEL 538.1 1067 1850.3 4819 5940 CRC 5.5769 11.060 19.1773 49.95 61.6 58 Ce cerium use 534.4 1050 1949 3547 6325 7490 WEL 534.4 1050 1949 3547 6325 7490 CRC 5.5387 10.85 20.198 36.758 65.55 77.6 59 Pr praseodymium use 527 1020 2086 3761 5551 WEL 527 1020 2086 3761 5551 CRC 5.473 10.55 21.624 38.98 57.53 60 Nd neodymium use 533.1 1040 2130 3900 WEL 533.1 1040 2130 3900 CRC 5.5250 10.73 22.1 40.41 61 Pm promethium use 540 1050 2150 3970 WEL 540 1050 2150 3970 CRC 5.582 10.90 22.3 41.1 62 Sm samarium use 544.5 1070 2260 3990 WEL 544.5 1070 2260 3990 CRC 5.6436 11.07 23.4 41.4 63 Eu europium use 547.1 1085 2404 4120 WEL 547.1 1085 2404 4120 CRC 5.6704 11.241 24.92 42.7 64 Gd gadolinium use 593.4 1170 1990 4250 WEL 593.4 1170 1990 4250 CRC 6.1501 12.09 20.63 44.0 65 Tb terbium use 565.8 1110 2114 3839 WEL 565.8 1110 2114 3839 CRC 5.8638 11.52 21.91 39.79 66 Dy dysprosium use 573.0 1130 2200 3990 WEL 573.0 1130 2200 3990 CRC 5.9389 11.67 22.8 41.47 67 Ho holmium use 581.0 1140 2204 4100 WEL 581.0 1140 2204 4100 CRC 6.0215 11.80 22.84 42.5 68 Er erbium use 589.3 1150 2194 4120 WEL 589.3 1150 2194 4120 CRC 6.1077 11.93 22.74 42.7 69 Tm thulium use 596.7 1160 2285 4120 WEL 596.7 1160 2285 4120 CRC 6.18431 12.05 23.68 42.7 70 Yb ytterbium use 603.4 1174.8 2417 4203 WEL 603.4 1174.8 2417 4203 CRC 6.25416 12.1761 25.05 43.56 71 Lu lutetium use 523.5 1340 2022.3 4370 6445 WEL 523.5 1340 2022.3 4370 6445 CRC 5.4259 13.9 20.9594 45.25 66.8 72 Hf hafnium use 658.5 1440 2250 3216 WEL 658.5 1440 2250 3216 CRC 6.82507 14.9 23.3 33.33 73 Ta tantalum use 761 1500 WEL 761 1500 CRC 7.5496 74 W tungsten use 770 1700 WEL 770 1700 CRC 7.8640 75 Re rhenium use 760 1260 2510 3640 WEL 760 1260 2510 3640 CRC 7.8335 76 Os osmium use 840 1600 WEL 840 1600 CRC 8.4382 77 Ir iridium use 880 1600 WEL 880 1600 CRC 8.9670 78 Pt platinum use 870 1791 WEL 870 1791 CRC 8.9587 18.563 28https://dept.astro.lsa.umich.edu/~cowley/ionen.htm archived 79 Au gold use 890.1 1980 WEL 890.1 1980 CRC 9.2255 20.5 30 80 Hg mercury use 1007.1 1810 3300 WEL 1007.1 1810 3300 CRC 10.43750 18.756 34.2 81 Tl thallium use 589.4 1971 2878 WEL 589.4 1971 2878 CRC 6.1082 20.428 29.83 82 Pb lead use 715.6 1450.5 3081.5 4083 6640 WEL 715.6 1450.5 3081.5 4083 6640 CRC 7.41666 15.0322 31.9373 42.32 68.8 83 Bi bismuth use 703 1610 2466 4370 5400 8520 WEL 703 1610 2466 4370 5400 8520 CRC 7.2856 16.69 25.56 45.3 56.0 88.3 84 Po polonium use 812.1 WEL 812.1 CRC 8.417 85 At astatine use 899.003 WEL 920 CRC talk Originally quoted as 9.31751(8) eV. 86 Rn radon use 1037 WEL 1037 CRC 10.74850 87 Fr francium use 380 WEL 380 CRC 4.0727 talk give 4.0712±0.00004 eV (392.811(4) kJ/mol) 88 Ra radium use 509.3 979.0 WEL 509.3 979.0 CRC 5.2784 10.14716 89 Ac actinium use 499 1170 WEL 499 1170 CRC 5.17 12.1 90 Th thorium use 587 1110 1930 2780 WEL 587 1110 1930 2780 CRC 6.3067 11.5 20.0 28.8 91 Pa protactinium use 568 WEL 568 CRC 5.89 92 U uranium use 597.6 1420 WEL 597.6 1420 CRC 6.19405 14.72 93 Np neptunium use 604.5 WEL 604.5 CRC 6.2657 94 Pu plutonium use 584.7 WEL 584.7 CRC 6.0262 95 Am americium use 578 WEL 578 CRC 5.9738 96 Cm curium use 581 WEL 581 CRC 5.9915 97 Bk berkelium use 601 WEL 601 CRC 6.1979 98 Cf californium use 608 WEL 608 CRC 6.2817 99 Es einsteinium use 619 WEL 619 CRC 6.42 100 Fm fermium use 627 WEL 627 CRC 6.50 101 Md mendelevium use 635 WEL 635 CRC 6.58 102 No nobelium use 642 WEL CRC 6.65 103 Lr lawrencium use 470 WEL CRC 4.9 104 Rf rutherfordium use 580 WEL CRC 6.0 == Notes == * Values from CRC are ionization energies given in the unit eV; other values are molar ionization energies given in the unit kJ/mol. This in turn makes its ionization energies increase by 18 kJ/mol−1. There are two main ways in which ionization energy is calculated. * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. == External links == * NIST Atomic Spectra Database Ionization Energies == See also == *Molar ionization energies of the elements Category:Properties of chemical elements Category:Chemical element data pages 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. However, due to experimental limitations, the adiabatic ionization energy is often difficult to determine, whereas the vertical detachment energy is easily identifiable and measurable. ==Analogs of ionization energy to other systems== While the term ionization energy is largely used only for gas-phase atomic, cationic, or molecular species, there are a number of analogous quantities that consider the amount of energy required to remove an electron from other physical systems. ===Electron binding energy=== thumb|500px|Binding energies of specific atomic orbitals as a function of the atomic number. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the "reduced" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). 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. The ionization energy is the lowest binding energy for a particular atom (although these are not all shown in the graph). ===Solid surfaces: work function=== Work function is the minimum amount of energy required to remove an electron from a solid surface, where the work function for a given surface is defined by the difference :W = -e\phi - E_{\rm F}, where is the charge of an electron, is the electrostatic potential in the vacuum nearby the surface, and is the Fermi level (electrochemical potential of electrons) inside the material. ==Note== ==See also== * Rydberg equation, a calculation that could determine the ionization energies of hydrogen and hydrogen-like elements. In general, the computation for the Nth ionization energy requires calculating the energies of Z-N+1 and Z-N electron systems. The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. Calculating these energies exactly is not possible except for the simplest systems (i.e. hydrogen and hydrogen-like elements), primarily because of difficulties in integrating the electron correlation terms. * Electron pairing energies: Half-filled subshells usually result in higher ionization energies. The adiabatic ionization is the diagonal transition to the vibrational ground state of the ion. In chemistry, it is expressed as the energy to ionize a mole of atoms or molecules, usually as kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol). Some values for elements of the third period are given in the following table: Successive ionization energy values / kJ mol−1 (96.485 kJ mol−1 ≡ 1 eV) Element First Second Third Fourth Fifth Sixth Seventh Na 496 4,560 Mg 738 1,450 7,730 Al 577 1,816 2,881 11,600 Si 786 1,577 3,228 4,354 16,100 P 1,060 1,890 2,905 4,950 6,270 21,200 S 1,000 2,295 3,375 4,565 6,950 8,490 27,107 Cl 1,256 2,260 3,850 5,160 6,560 9,360 11,000 Ar 1,520 2,665 3,945 5,770 7,230 8,780 12,000 Large jumps in the successive molar ionization energies occur when passing noble gas configurations.

0.54

292

0.2553

89,034.79

54.394

E

When an excited nucleus decays, it emits a $\gamma$ ray. The lifetime of an excited state of a nucleus is of the order of $10^{-12} \mathrm{~s}$. What is the uncertainty in the energy of the $\gamma$ ray produced?

The emission of a gamma ray from an excited nucleus typically requires only 10−12 seconds. Gamma rays from radioactive decay are in the energy range from a few kiloelectronvolts (keV) to approximately 8 megaelectronvolts (MeV), corresponding to the typical energy levels in nuclei with reasonably long lifetimes. In this type of decay, an excited nucleus emits a gamma ray almost immediately upon formation.It is now understood that a nuclear isomeric transition, however, can produce inhibited gamma decay with a measurable and much longer half-life. As in optical spectroscopy (see Franck–Condon effect) the absorption of gamma rays by a nucleus is especially likely (i.e., peaks in a "resonance") when the energy of the gamma ray is the same as that of an energy transition in the nucleus. Those excited states that lie below the separation energy for protons (Sp) decay by γ emission towards the ground state of daughter B. Then the excited decays to the ground state (see nuclear shell model) by emitting gamma rays in succession of 1.17 MeV followed by . Gamma decay is also a mode of relaxation of many excited states of atomic nuclei following other types of radioactive decay, such as beta decay, so long as these states possess the necessary component of nuclear spin. Because subatomic particles mostly have far shorter wavelengths than atomic nuclei, particle physics gamma rays are generally several orders of magnitude more energetic than nuclear decay gamma rays. Such nuclei have half-lifes that are more easily measurable, and rare nuclear isomers are able to stay in their excited state for minutes, hours, days, or occasionally far longer, before emitting a gamma ray. The decay energy is the energy change of a nucleus having undergone a radioactive decay. In some cases, the gamma emission spectrum of the daughter nucleus is quite simple, (e.g. /) while in other cases, such as with (/ and /), the gamma emission spectrum is complex, revealing that a series of nuclear energy levels exist. ===Particle physics=== Gamma rays are produced in many processes of particle physics. The rate of gamma decay is also slowed when the energy of excitation of the nucleus is small. thumb|350px|The decay of a proton rich nucleus A populates excited states of a daughter nucleus B by β+ emission or electron capture (EC). It is thought to be the principal absorption mechanism for gamma rays in the intermediate energy range 100 keV to 10 MeV. Gamma rays are produced by a number of astronomical processes in which very high-energy electrons are produced. Any gamma energy in excess of the equivalent rest mass of the two particles (totaling at least 1.02 MeV) appears as the kinetic energy of the pair and in the recoil of the emitting nucleus. The energy spectrum of gamma rays can be used to identify the decaying radionuclides using gamma spectroscopy. This is part and parcel of the general realization that many gamma rays produced in astronomical processes result not from radioactive decay or particle annihilation, but rather in non-radioactive processes similar to X-rays. However, when emitted gamma rays carry essentially all of the energy of the atomic nuclear de-excitation that produces them, this energy is also sufficient to excite the same energy state in a second immobilized nucleus of the same type. ==Applications== Gamma rays provide information about some of the most energetic phenomena in the universe; however, they are largely absorbed by the Earth's atmosphere. If the annihilating electron and positron are at rest, each of the resulting gamma rays has an energy of ~ 511 keV and frequency of ~ . Gamma rays are approximately 50% of the total energy output. There is no lower limit to the energy of photons produced by nuclear reactions, and thus ultraviolet or lower energy photons produced by these processes would also be defined as "gamma rays".

71

2.567

1.154700538

460.5

7

E

Calculate the wavelength and the energy of a photon associated with the series limit of the Lyman series.

The Lyman limit is the short-wavelength end of the hydrogen Lyman series, at . This energy is equivalent to the Rydberg constant. == See also == * Balmer Limit * Lyman-alpha emitter * Lyman-alpha forest * Lyman-break galaxy * Lyman series * Rydberg formula ==References== Category:Atomic physics All the wavelengths in the Lyman series are in the ultraviolet band.. The wavelengths in the Lyman series are all ultraviolet: n Wavelength (nm) 2 121.56701Kramida, A., Ralchenko, Yu., Reader, J., and NIST ASD Team (2019). In physics and chemistry, the Lyman series is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1 (where n is the principal quantum number), the lowest energy level of the electron. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. For the connection between Bohr, Rydberg, and Lyman, one must replace m with 1 to obtain : \frac{1}{\lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) which is Rydberg's formula for the Lyman series. This also means that the inverse of the Rydberg constant is equal to the Lyman limit. Therefore, each wavelength of the emission lines corresponds to an electron dropping from a certain energy level (greater than 1) to the first energy level. == See also == * Bohr model * H-alpha * Hydrogen spectral series * K-alpha * Lyman-alpha line * Lyman continuum photon * Moseley's law * Rydberg formula * Balmer series ==References== Category:Emission spectroscopy Category:Hydrogen physics Lyman-alpha radiation had previously been detected from other galaxies, but due to interference from the Sun, the radiation from the Milky Way was not detectable. ==The Lyman series== The version of the Rydberg formula that generated the Lyman series was: {1 \over \lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) \qquad \left( R_\text{H} \approx 1.0968{\times}10^7\,\text{m}^{-1} \approx \frac{13.6\,\text{eV}}{hc} \right) where n is a natural number greater than or equal to 2 (i.e., ). In hydrogen, its wavelength of 1215.67 angstroms ( or ), corresponding to a frequency of about , places Lyman-alpha in the ultraviolet (UV) part of the electromagnetic spectrum. The greater the difference in the principal quantum numbers, the higher the energy of the electromagnetic emission. ==History== thumb|upright=1.3|The Lyman series The first line in the spectrum of the Lyman series was discovered in 1906 by physicist Theodore Lyman, who was studying the ultraviolet spectrum of electrically excited hydrogen gas. DOI: https://doi.org/10.18434/T4W30F 3 102.57220 4 97.253650 5 94.974287 6 93.780331 7 93.0748142 8 92.6225605 9 92.3150275 10 92.0963006 11 91.9351334 ∞, the Lyman limit 91.1753 ==Explanation and derivation== In 1914, when Niels Bohr produced his Bohr model theory, the reason why hydrogen spectral lines fit Rydberg's formula was explained. Replacing the energy in the above formula with the expression for the energy in the hydrogen atom where the initial energy corresponds to energy level n and the final energy corresponds to energy level m, : \frac{1}{\lambda} = \frac{E_\text{i} - E_\text{f}}{12398.4\,\text{eV Å}} = R_\text{H} \left(\frac{1}{m^2} - \frac{1}{n^2} \right) Where RH is the same Rydberg constant for hydrogen from Rydberg's long known formula. There is also a more comfortable notation when dealing with energy in units of electronvolts and wavelengths in units of angstroms, : \lambda = \frac{12398.4\,\text{eV}}{E_\text{i} - E_\text{f}} Å. The Lyman-alpha line, typically denoted by Ly-α, is a spectral line of hydrogen (or, more generally, of any one-electron atom) in the Lyman series. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. Because of the spin–orbit interaction, the Lyman-alpha line splits into a fine-structure doublet with the wavelengths of 1215.668 and 1215.674 angstroms. In physical cosmology, the photon epoch was the period in the evolution of the early universe in which photons dominated the energy of the universe. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). This equation is valid for all hydrogen-like species, i.e. atoms having only a single electron, and the particular case of hydrogen spectral lines is given by Z=1. ==Series== ===Lyman series ( = 1)=== In the Bohr model, the Lyman series includes the lines emitted by transitions of the electron from an outer orbit of quantum number n > 1 to the 1st orbit of quantum number n' = 1. The energy of an emitted photon corresponds to the energy difference between the two states.

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Another application of the relationship given in Problem $1-48$ has to do with the excitedstate energies and lifetimes of atoms and molecules. If we know that the lifetime of an excited state is $10^{-9} \mathrm{~s}$, then what is the uncertainty in the energy of this state?

After about 85 years of existence of the uncertainty relation this problem was solved recently by Lorenzo Maccone and Arun K. Pati. 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). The lifetime of a system in an excited state is usually short: spontaneous or induced emission of a quantum of energy (such as a photon or a phonon) usually occurs shortly after the system is promoted to the excited state, returning the system to a state with lower energy (a less excited state or the ground state). Heisenberg's uncertainty relation is one of the fundamental results in quantum mechanics. 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 Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. The Heisenberg–Robertson–Schrödinger uncertainty relation was proved at the dawn of quantum formalism and is ever-present in the teaching and research on quantum mechanics. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. A system of highly excited atoms can form a long-lived condensed excited state e.g. a condensed phase made completely of excited atoms: Rydberg matter. == Perturbed gas excitation == A collection of molecules forming a gas can be considered in an excited state if one or more molecules are elevated to kinetic energy levels such that the resulting velocity distribution departs from the equilibrium Boltzmann distribution. 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). In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. Everett's Dissertation proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki-Birula and Mycielski. In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. "One-parameter class of uncertainty relations based on entropy power". By giving the atom additional energy (for example, by absorption of a photon of an appropriate energy), the electron moves into an excited state (one with one or more quantum numbers greater than the minimum possible). An atom in a high excited state is termed a Rydberg atom. One can prove an improved version of the Heisenberg–Robertson uncertainty relation which reads as : \Delta A \Delta B \ge \frac{ \pm \frac{i}{2} \langle \Psi|[A, B]|\Psi \rangle }{1- \frac{1}{2} | \langle \Psi|( \frac{A}{\Delta A} \pm i \frac{B}{\Delta B} )| {\bar \Psi} \rangle|^2 }. The other non-trivial stronger uncertainty relation is given by : \Delta A^2 + \Delta B^2 \ge \frac{1}{2}| \langle {\bar \Psi}_{A+B} |(A + B)| \Psi \rangle|^2, where | {\bar \Psi}_{A+B} \rangle is a unit vector orthogonal to |\Psi \rangle . The Heisenberg–Robertson uncertainty relation follows from the above uncertainty relation. ==Remarks== In quantum theory, one should distinguish between the uncertainty relation and the uncertainty principle. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations. And since the whole oscillatory process (from r_{min} to infinity and back) is periodic, it is logical that this quantum mechanical problem has a stationary solution. === Second case === thumb|(a) Potential and wave function (on an arbitrary scale along the vertical axis) corresponding to zero energy, for the second case of the Wigner von Neumann SSC, A=1.5 (b) A=15. The potential would then be equal (with the corrected arithmetical error in the original article):Stillinger, F. H. & Herrick, D. R. Bound states in the continuum.

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One of the most powerful modern techniques for studying structure is neutron diffraction. This technique involves generating a collimated beam of neutrons at a particular temperature from a high-energy neutron source and is accomplished at several accelerator facilities around the world. If the speed of a neutron is given by $v_{\mathrm{n}}=\left(3 k_{\mathrm{B}} T / m\right)^{1 / 2}$, where $m$ is the mass of a neutron, then what temperature is needed so that the neutrons have a de Broglie wavelength of $50 \mathrm{pm}$ ?

The following is a detailed classification: === Thermal === A thermal neutron is a free neutron with a kinetic energy of about 0.025 eV (about 4.0×10−21 J or 2.4 MJ/kg, hence a speed of 2.19 km/s), which is the energy corresponding to the most probable speed at a temperature of 290 K (17 °C or 62 °F), the mode of the Maxwell–Boltzmann distribution for this temperature, Epeak = 1/2 k T. The momentum and wavelength of the neutron are related through the de Broglie relation. The large wavelength of slow neutrons allows for the large cross section. == Neutron energy distribution ranges == Neutron energy range names Neutron energy Energy range 0.0 – 0.025 eV Cold (slow) neutrons 0.025 eV Thermal neutrons (at 20°C) 0.025–0.4 eV Epithermal neutrons 0.4–0.5 eV Cadmium neutrons 0.5–10 eV Epicadmium neutrons 10–300 eV Resonance neutrons 300 eV–1 MeV Intermediate neutrons 1–20 MeV Fast neutrons > 20 MeV Ultrafast neutrons But different ranges with different names are observed in other sources. Efficient neutron optical components are being developed and optimized to remedy this lack. ===Resonance=== :*Refers to neutrons which are strongly susceptible to non-fission capture by U-238. :*1 eV to 300 eV ===Intermediate=== :*Neutrons that are between slow and fast :*Few hundred eV to 0.5 MeV. ===Fast=== : A fast neutron is a free neutron with a kinetic energy level close to 1 MeV (100 TJ/kg), hence a speed of 14,000 km/s or higher. Engineering diffraction refers to a sub-field of neutron scattering which investigates microstructural features that influence the mechanical properties of materials. Hydrogen Deuterium Beryllium Carbon Oxygen Uranium Mass of kernels u 1 2 9 12 16 238 Energy decrement \xi 1 0.7261 0.2078 0.1589 0.1209 0.0084 Number of Collisions 18 25 86 114 150 2172 ===Distribution of neutron velocities once moderated=== After sufficient impacts, the speed of the neutron will be comparable to the speed of the nuclei given by thermal motion; this neutron is then called a thermal neutron, and the process may also be termed thermalization. The wavelength of the neutrons used for reflectivity are typically on the order of 0.2 to 1 nm (2 to 10 Å). The characteristic neutron temperature of several-MeV neutrons is several tens of billions kelvin. Qualitatively, the higher the temperature, the higher the kinetic energy of the free neutrons. After a number of collisions with nuclei (scattering) in a medium (neutron moderator) at this temperature, those neutrons which are not absorbed reach about this energy level. According to the equipartition theorem, the average kinetic energy, \bar{E}, can be related to temperature, T, via: :\bar{E}=\frac{1}{2}m_n \langle v^2 \rangle=\frac{3}{2}k_B T, where m_n is the neutron mass, \langle v^2 \rangle is the average squared neutron speed, and k_B is the Boltzmann constant. However the range of neutrons from fission follows a Maxwell–Boltzmann distribution from 0 to about 14 MeV in the center of momentum frame of the disintegration, and the mode of the energy is only 0.75 MeV, meaning that fewer than half of fission neutrons qualify as "fast" even by the 1 MeV criterion.Byrne, J. Neutrons, Nuclei, and Matter, Dover Publications, Mineola, New York, 2011, (pbk.) The neutron detection temperature, also called the neutron energy, indicates a free neutron's kinetic energy, usually given in electron volts. Neutron reflectometry is a neutron diffraction technique for measuring the structure of thin films, similar to the often complementary techniques of X-ray reflectivity and ellipsometry. The term temperature is used, since hot, thermal and cold neutrons are moderated in a medium with a certain temperature. The main limitations of the use of slow neutrons is the low flux and the lack of efficient optical devices (in the case of CN and VCN). Neutron is a medium-lift two-stage launch vehicle under development by Rocket Lab. Announced on 1 March 2021, the vehicle is being designed to be capable of delivering a payload of to low Earth orbit in a partially reusable configuration, and will focus on the growing megaconstellation satellite delivery market. This is only slightly modified in a real moderator due to the speed (energy) dependence of the absorption cross-section of most materials, so that low-speed neutrons are preferentially absorbed,Neutron scattering lengths and cross sections V.F. Sears, Neutron News 3, No. 3, 26-37 (1992) so that the true neutron velocity distribution in the core would be slightly hotter than predicted. ==Reactor moderators== In a thermal-neutron reactor, the nucleus of a heavy fuel element such as uranium absorbs a slow-moving free neutron, becomes unstable, and then splits ("fissions") into two smaller atoms ("fission products"). The neutron energy distribution is then adapted to the Maxwell distribution known for thermal motion. :Cold (slow) neutrons are subclassified into cold (CN), very cold (VCN), and ultra-cold (UCN) neutrons, each having particular characteristics in terms of their optical interactions with matter. This is done through numerous collisions with (in general) slower-moving and thus lower- temperature particles like atomic nuclei and other neutrons. A neutron research facility is most commonly a big laboratory operating a large-scale neutron source that provides thermal neutrons to a suite of research instruments.

0.1800

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The temperature of the fireball in a thermonuclear explosion can reach temperatures of approximately $10^7 \mathrm{~K}$. What value of $\lambda_{\max }$ does this correspond to?

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 specific heat capacity has a sharp peak as the temperature approaches the lambda point. The entropy of vaporization of at its boiling point has the extraordinarily high value of 136.9 J/(K·mol). Hence, \lambda_{\rm th} = \frac{h}{\sqrt{2\pi m k_{\mathrm B} T}} , where h is the Planck constant, is the mass of a gas particle, k_{\mathrm B} is the Boltzmann constant, and is the temperature of the gas. The lambda point is the temperature at which normal fluid helium (helium I) makes the transition to superfluid helium II (approximately 2.17 K at 1 atmosphere). The thermometer indicates the current temperature, and the highest and lowest temperatures since the last reset. ==Description== Six's Maximum and Minimum thermometer consists of a U-shaped glass tube with two separate temperature scales set along each arm of the U. The current temperature is 23 degrees Celsius, the maximum recorded is 25, and the minimum is 15; both read from the base of the small markers in each arm of the U tube. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. The plasma parameter is a dimensionless number, denoted by capital Lambda, Λ. In thermodynamics, Trouton's rule states that the entropy of vaporization is almost the same value, about 85–88 J/(K·mol), for various kinds of liquids at their boiling points.Compare 85 J/(K·mol) in and 88 J/(K·mol) in The entropy of vaporization is defined as the ratio between the enthalpy of vaporization and the boiling temperature. Accessed April 2011 thumb|200px|right|Detail of the thermometer bulbs of the maximum-minimum thermometer shown above. The kelvin, symbol K, is a unit of measurement for temperature. The point's name derives from the graph (pictured) that results from plotting the specific heat capacity as a function of temperature (for a given pressure in the above range, in the example shown, at 1 atmosphere), which resembles the Greek letter lambda \lambda. The numerical value of an absolute temperature, , on the 1848 scale is related to the absolute temperature of the melting point of water, , and the absolute temperature of the boiling point of water, , by * (1848 scale) = 100 () / () On this scale, an increase of 222 degrees always means an approximate doubling of absolute temperature regardless of the starting temperature. In this case, the plasma parameter is given by:Chen, F.F., Introduction to Plasma Physics and Controlled Fusion, (Springer, New York, 2006) \Lambda = 4\pi n_\text{e}\lambda_\text{D}^3 where * ne is the number density of electrons, * λD is the Debye length. If is the number of dimensions, and the relationship between energy () and momentum () is given by E=ap^s (with and being constants), then the thermal wavelength is defined as \lambda_{\mathrm{th}}=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{\mathrm B}T}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} , where is the Gamma function. Electronic thermometers often include a maximum-minimum registering feature. ==See also== * Maximum minimum temperature system ==References== * Two hundred years of the Six's Self Registering Thermometer Austin and McConnell, Notes and Records of the Royal Society of London * Volume. 35, No. 1, July., 1980 at JSTOR * A History of the Thermometer and Its Uses in Meteorology by Amit Batra, Johns Hopkins University Press, 1966; * The Construction of a Thermometer by James Six, Nimbus Publishing Ltd,1980; ==External links== * Article on Six's thermometer at the Museum of the History of Science at Florence, Italy * Explanation of the working of Six's thermometer * Category:Thermometers If the temperature rises, the maximum scale marker will be pushed. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. A similar way of stating this (Trouton's ratio) is that the latent heat is connected to boiling point roughly as : \frac{L_\text{vap}}{T_\text{boiling}} \approx 85{-}88\ \frac{\text{J}}{\text{K} \cdot \text{mol}}. Six's maximum and minimum thermometer is a registering thermometer that can record the maximum and minimum temperatures reached over a period of time, for example 24 hours. Hence the heat capacity was measured within 2 nK below the transition in an experiment included in a Space Shuttle payload in 1992.

4943

4.86

3.0

0.65625

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C

Show that l'Hôpital's rule amounts to forming a Taylor expansion of both the numerator and the denominator. Evaluate the limit $$ \lim _{x \rightarrow 0} \frac{\ln (1+x)-x}{x^2} $$ both ways.

Applying L'Hôpital's rule and finding the derivatives with respect to of the numerator and the denominator yields as expected. Applying L'Hôpital's rule a single time still results in an indeterminate form. Limitations of the Taylor rule include. The \ln(1 + x) = x approximation is used here. Applying L'Hopital's rule shows that f'(a) := \lim_{x\to a}\frac{f(x)-f(a)}{x-a} = \lim_{x\to a}\frac{h(x)}{g(x)} = \lim_{x\to a}f'(x). == See also == * L'Hôpital controversy == Notes == == References == === Sources === * * * * * Category:Articles containing proofs Category:Theorems in calculus Category:Theorems in real analysis Category:Limits (mathematics) For example, to evaluate a limit involving , convert the difference of two functions to a quotient: : \begin{align} \lim_{x\to 1}\left(\frac{x}{x-1}-\frac1{\ln x}\right) & = \lim_{x\to 1}\frac{x\cdot\ln x -x+1}{(x-1)\cdot\ln x} & \quad (1) \\\\[6pt] & = \lim_{x\to 1}\frac{\ln x}{\frac{x-1}{x}+\ln x} & \quad (2) \\\\[6pt] & = \lim_{x\to 1}\frac{x\cdot\ln x}{x-1+x\cdot\ln x} & \quad (3) \\\\[6pt] & = \lim_{x\to 1}\frac{1+\ln x}{1+1+\ln x} & \quad (4) \\\\[6pt] & = \lim_{x\to 1}\frac{1+\ln x}{2+\ln x} \\\\[6pt] & = \frac{1}{2}, \end{align} where L'Hôpital's rule is applied when going from (1) to (2) and again when going from (3) to (4). L'Hôpital's rule can be used on indeterminate forms involving exponents by using logarithms to "move the exponent down". The limit \lim_{x\to 0^+}x\cdot\ln x is of the indeterminate form , but as shown in an example above, l'Hôpital's rule may be used to determine that :\lim_{x\to 0^+}x\cdot\ln x = 0. * Here is an example involving the indeterminate form (see below), which is rewritten as the form : \lim_{x\to 0^+}x \ln x =\lim_{x\to 0^+} \frac{\ln x}{\frac{1}{x}} = \lim_{x\to 0^+} \frac{\frac{1}{x}}{-\frac{1}{x^2}} = \lim_{x\to 0^+} -x = 0. If is twice-differentiable in a neighborhood of and that its second derivative is continuous on this neighbourhood, then \begin{align} \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2} &= \lim_{h\to 0}\frac{f'(x+h)-f'(x-h)}{2h} \\\\[4pt] &= \lim_{h\to 0}\frac{f(x+h) + f(x-h)}{2} \\\\[4pt] &= f(x). \end{align} * Sometimes L'Hôpital's rule is invoked in a tricky way: suppose converges as and that e^x\cdot f(x) converges to positive or negative infinity. L'Hôpital's rule states that for functions and which are differentiable on an open interval except possibly at a point contained in , if \lim_{x\to c}f(x)=\lim_{x\to c}g(x)=0 \text{ or } \pm\infty, and g'(x) e 0 for all in with , and \lim_{x\to c}\frac{f'(x)}{g'(x)} exists, then :\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}. Again, an alternative approach is to multiply numerator and denominator by x^{1/2} before applying L'Hôpital's rule: \lim_{x\to\infty} \frac{x^\frac{1}{2}+x^{-\frac{1}{2}}}{x^\frac{1}{2}-x^{-\frac{1}{2}}} = \lim_{x\to\infty} \frac{x+1}{x-1} = \lim_{x\to\infty} \frac{1}{1} = 1. L'Hôpital's rule (, ), also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives. Extract of page 321 == General form == The general form of L'Hôpital's rule covers many cases. L'Hôpital's rule then states that the slope of the curve when is the limit of the slope of the tangent to the curve as the curve approaches the origin, provided that this is defined. == Proof of L'Hôpital's rule == ===Special case=== The proof of L'Hôpital's rule is simple in the case where and are continuously differentiable at the point and where a finite limit is found after the first round of differentiation. A common pitfall is using L'Hôpital's rule with some circular reasoning to compute a derivative via a difference quotient. It is not a proof of the general L'Hôpital's rule because it is stricter in its definition, requiring both differentiability and that c be a real number. Thus, since \lim_{x\to c} \frac{f(x)}{g(x)} = \frac{0}{0} and \lim_{x\to c} \frac{f'(x)}{g'(x)} exists, L'Hôpital's rule still holds. === Derivative of denominator is zero === The necessity of the condition that g'(x) e 0 near c can be seen by the following counterexample due to Otto Stolz. The limit in the conclusion is not indeterminate because g'(c) e 0. It follows that :\int_0^x \frac{dt}{1+t}=\int_0^x \left(1-t+t^2-\cdots+(-t)^{n-1}+\frac{(-t)^n}{1+t}\right)\ dt and by termwise integration, :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots+(-1)^{n-1}\frac{x^n}{n}+(-1)^n \int_0^x \frac{t^n}{1+t}\ dt. Repeatedly apply L'Hôpital's rule until the exponent is zero (if is an integer) or negative (if is fractional) to conclude that the limit is zero. In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots In summation notation, :\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n.

-0.5

0.178

4.85

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A

Evaluate the series $$ S=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2^n} $$

The Mercator series provides an analytic expression of the natural logarithm: \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n \;=\; \ln (1+x). It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. The constant can also be explicitly defined by the following infinite sums: : 0.187859\ldots = \sum_{k=1}^{\infty} (-1)^k (k^{1/k} - 1) = \sum_{k=1}^{\infty} \left((2k)^{1/(2k)} - (2k-1)^{1/(2k-1)}\right). As another example, by Mercator series \ln(2) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots. This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. In contrast, as r approaches −1 the sum of the first several terms of the geometric series starts to converge to 1/2 but slightly flips up or down depending on whether the most recently added term has a power of r that is even or odd. The constant relates to the divergent series: :\sum_{k=1}^{\infty} (-1)^k k^{1/k}. In mathematics, an alternating series is an infinite series of the form \sum_{n=0}^\infty (-1)^n a_n or \sum_{n=0}^\infty (-1)^{n+1} a_n with for all . In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots In summation notation, :\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n. The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. The series converges to the natural logarithm (shifted by 1) whenever -1 . ==History== The series was discovered independently by Johannes Hudde and Isaac Newton. But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for \tfrac 1 2 \ln(2): \begin{align} & {} \quad \left(1-\frac{1}{2}\right)-\frac{1}{4} +\left(\frac{1}{3}-\frac{1}{6}\right) -\frac{1}{8}+\left(\frac{1}{5} -\frac{1}{10}\right)-\frac{1}{12}+\cdots \\\\[8pt] & = \frac{1}{2}-\frac{1}{4}+\frac{1}{6} -\frac{1}{8}+\frac{1}{10}-\frac{1}{12} +\cdots \\\\[8pt] & = \frac{1}{2}\left(1-\frac{1}{2} + \frac{1}{3} -\frac{1}{4}+\frac{1}{5}- \frac{1}{6}+ \cdots\right)= \frac{1}{2} \ln(2). \end{align} == Series acceleration == In practice, the numerical summation of an alternating series may be sped up using any one of a variety of series acceleration techniques. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2. :\sum_{k=0}^{n} (-2)^k As a series of real numbers it diverges, so in the usual sense it has no sum. Therefore, an alternating series is also a unit series when -1 < r < 0 and a + r = 1 (for example, coefficient a = 1.7 and common ratio r = -0.7). In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs. The closed form geometric series 1 / (1 - r) is the black dashed line. See for example Grandi's series: 1 − 1 + 1 − 1 + ···. Using the geometric series closed form as before :0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac{a}{1-r} \;=\; \frac{49/64}{1-1/64} \;=\; \frac{49/64}{63/64} \;=\; \frac{49}{63} \;=\; \frac{7}{9}. Alternatively, one can start with the finite geometric series (t e -1) :1-t+t^2-\cdots+(-t)^{n-1}=\frac{1-(-t)^n}{1+t} which gives :\frac1{1+t}=1-t+t^2-\cdots+(-t)^{n-1}+\frac{(-t)^n}{1+t}. That flipping behavior near r = −1 is illustrated in the adjacent image showing the first 11 terms of the geometric series with a = 1 and |r| < 1\.

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1.22

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Calculate the percentage difference between $\ln (1+x)$ and $x$ for $x=0.0050$

thumb|400px|The logarithmic decrement can be obtained e.g. as ln(x1/x3). A percentage point or percent point is the unit for the arithmetic difference between two percentages. Mistakenly using percentage as the unit for the standard deviation is confusing, since percentage is also used as a unit for the relative standard deviation, i.e. standard deviation divided by average value (coefficient of variation). == Related units == * Percentage (%) 1 part in 100 * Per mille (‰) 1 part in 1,000 *Basis point (bp) difference of 1 part in 10,000 *Permyriad (‱) 1 part in 10,000 * Per cent mille (pcm) 1 part in 100,000 * Baker percentage == See also == * Parts-per notation * Per-unit system * Percent point function * Relative change and difference == References == Category:Mathematical terminology Category:Probability assessment Category:Units of measurement ru:Процент#Процентный пункт In this case, the reciprocal transform of the percentage-point difference would be 1/(20pp) = 1/0.20 = 5. In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f, :(\ln f)' = \frac{f'}{f} \quad \implies \quad f' = f \cdot (\ln f)'. thumbnail|upright=1.3|Plot of logit(x) in the domain of 0 to 1, where the base of the logarithm is e. In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion : -\ln(1-p) = p + \frac{p^2}{2} + \frac{p^3}{3} + \cdots. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions. Mathematically, the logit is the inverse of the standard logistic function \sigma(x) = 1/(1+e^{-x}), so the logit is defined as :\operatorname{logit} p = \sigma^{-1}(p) = \ln \frac{p}{1-p} \quad \text{for} \quad p \in (0,1). Percentage-point differences are one way to express a risk or probability. After the first occurrence, some writers abbreviate by using just "point" or "points". ==Differences between percentages and percentage points== Consider the following hypothetical example: In 1980, 50 percent of the population smoked, and in 1990 only 40 percent of the population smoked. Differentiating by applying the chain and the sum rules yields :\frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}, and, after rearranging, yields :f'(x) = f(x)\times \Bigg\\{\frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}\Bigg\\}= g(x)h(x)\times \Bigg\\{\frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}\Bigg\\}=g'(x)h(x)+g(x)h'(x), which is the product rule for derivatives. ===Quotients=== A natural logarithm is applied to a quotient of two functions :f(x)=\frac{g(x)}{h(x)}\,\\! to transform the division into a subtraction :\ln(f(x))=\ln\Bigg(\frac{g(x)}{h(x)}\Bigg)=\ln(g(x))-\ln(h(x))\,\\! For measurements involving percentages as a unit, such as, growth, yield, or ejection fraction, statistical deviations and related descriptive statistics, including the standard deviation and root-mean-square error, the result should be expressed in units of percentage points instead of percentage. Percentage solution may refer to: * Mass fraction (or "% w/w" or "wt.%"), for percent mass * Volume fraction (or "% v/v" or "vol.%"), volume concentration, for percent volume * "Mass/volume percentage" (or "% m/v") in biology, for mass per unit volume; incorrectly used to denote mass concentration (chemistry). Fisher described the logarithmic distribution in a paper that used it to model relative species abundance. ==See also== * Poisson distribution (also derived from a Maclaurin series) ==References== ==Further reading== * * Category:Discrete distributions Category:Logarithms Thus, the logit is a type of function that maps probability values from (0, 1) to real numbers in (-\infty, +\infty), akin to the probit function. ==Definition== If is a probability, then is the corresponding odds; the of the probability is the logarithm of the odds, i.e.: :\operatorname{logit}(p)=\ln\left( \frac{p}{1-p} \right) =\ln(p)-\ln(1-p)=-\ln\left( \frac{1}{p}-1\right)=2\operatorname{atanh}(2p-1) The base of the logarithm function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm with base is the one most often used. The cumulative distribution function is : F(k) = 1 + \frac{\Beta(p; k+1,0)}{\ln(1-p)} where B is the incomplete beta function. The application of natural logarithms results in (with capital sigma notation) :\ln (f(x))=\sum_i\alpha_i(x)\cdot \ln(f_i(x)), and after differentiation, :\frac{f'(x)}{f(x)}=\sum_i\left[\alpha_i'(x)\cdot \ln(f_i(x))+\alpha_i(x)\cdot \frac{f_i'(x)}{f_i(x)}\right]. Because of this, the logit is also called the log-odds since it is equal to the logarithm of the odds \frac{p}{1-p} where is a probability. In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. In fact, the is the quantile function of the logistic distribution, while the is the quantile function of the normal distribution.

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Calculate the reduced mass of a nitrogen molecule in which both nitrogen atoms have an atomic mass of 14.00.

The molecular formula C14H14 (molar mass: 182.26 g/mol, exact mass: 182.1096 u) may refer to: * Bibenzyl * Cyclotetradecaheptaene, or [14]annulene Category:Molecular formulas The molecular formula C11H14N2 (molar mass : 174.24 g/mol, exact mass : 174.115698) may refer to : * 6-(2-Aminopropyl)indole * Gramine * 5-IT * α-Methyltryptamine * N-Methyltryptamine The molecular formula C24H14 (molar mass: 302.37 g/mol, exact mass: 302.1096 u) may refer to: * Dibenzopyrenes * Zethrene, or dibenzo[de,mn]naphthacene The molecular formula CH4N2S (molar mass: 76.12 g/mol, exact mass: 76.0095 u) may refer to: * Ammonium thiocyanate * Thiourea The reduction of nitro compounds are chemical reactions of wide interest in organic chemistry. The reaction can also be effected through radical reaction with tributyltin hydride and a radical initiator, AIBN as an example.T. V. (Babu) RajanBabu, Philip C. Bulman Page, Benjamin R. Buckley, "Tri-n-butylstannane" Encyclopedia of Reagents for Organic Synthesis 2004, John Wiley & Sons. doi:10.1002/047084289X.rt181.pub2 ===Reduction to amines=== 180px|Generalization of the reduction of a nitroalkane to an amine Aliphatic nitro compounds can be reduced to aliphatic amines by several reagents: * Catalytic hydrogenation using platinum(IV) oxide (PtO2) or Raney nickel * Iron metal in refluxing acetic acid * Samarium diiodide *Raney nickel, platinum on carbon, or zinc dust and formic acid or ammonium formate α,β-Unsaturated nitro compounds can be reduced to saturated amines by: * Catalytic hydrogenation over palladium-on-carbon * Iron metal * Lithium aluminium hydride (Note: Hydroxylamines and oximes are typical impurities.) *Lithium borohydride or sodium borohydride and trimethylsilyl chloride *Red-Al ===Reduction to hydroxylamines=== Aliphatic nitro compounds can be reduced to aliphatic hydroxylamines using diborane. :180px|Generalization of the reduction of a nitroalkane to a hydroxylamine The reaction can also be carried out with zinc dust and ammonium chloride: : R-NO2 \+ 4 NH4Cl + 2 Zn → R-NH-OH + 2 ZnCl2 \+ 4 NH3 \+ H2O ===Reduction to oximes=== 180px|Generalization of the reduction of a nitroalkane to an oxime Nitro compounds are typically reduced to oximes using metal salts, such as tin(II) chloride or chromium(II) chloride. The nitro group was one of the first functional groups to be reduced. Illustrated by the selective reduction of dinitrophenol to the nitroaminophenol. (Excess zinc will reduce the azo group to a hydrazino compound.) ==Aliphatic nitro compounds== ===Reduction to hydrocarbons=== 180px|Generalization of the reduction of a nitroalkane to an alkane Hydrodenitration (replacement of a nitro group with hydrogen) is difficult to achieve but can be effected by catalytic hydrogenation over platinum on silica gel at high temperatures. Most useful is the reduction of aryl nitro compounds. ==Aromatic nitro compounds== ===Reduction to anilines=== :200px|Generalization of the reduction of a nitroarene to aniline The reduction of nitroaromatics is conducted on an industrial scale. Alkyl and aryl nitro compounds behave differently. (See below) ===Reduction to hydroxylamines=== Several methods have been described for the production of aryl hydroxylamines from aryl nitro compounds: * Raney nickel and hydrazine at 0-10 °C * Electrolytic reduction * Zinc metal in aqueous ammonium chloride * Catalytic Rhodium on carbon with excess hydrazine monohydrate at room temperature ===Reduction to hydrazine compounds=== Treatment of nitroarenes with excess zinc metal results in the formation of N,N'-diarylhydrazine. ===Reduction to azo compounds=== 240px|Generalization of the reduction of a nitroarene to an azo compound Treatment of aromatic nitro compounds with metal hydrides gives good yields of azo compounds. * Tin(II) chloride * Titanium(III) chloride * Samarium *Hydroiodic acid Metal hydrides are typically not used to reduce aryl nitro compounds to anilines because they tend to produce azo compounds. Many methods exist, such as: * Catalytic hydrogenation using: Raney nickel or palladium-on- carbon, platinum(IV) oxide, or Urushibara nickel. The conversion can be effected by many reagents. For example, one could use: * Lithium aluminium hydride * Zinc metal with sodium hydroxide. * Sodium hydrosulfite * Sodium sulfide (or hydrogen sulfide and base). Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions * Iron in acidic media. Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions

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Two narrow slits are illuminated with red light of wavelength $694.3 \mathrm{~nm}$ from a laser, producing a set of evenly placed bright bands on a screen located $3.00 \mathrm{~m}$ beyond the slits. If the distance between the bands is $1.50 \mathrm{~cm}$, then what is the distance between the slits?

In the Fraunhofer approximation, with the observer far away from the slits, the difference in path length to the two slits can be seen from the image to be \Delta S={a} \sin \theta Maxima in the intensity occur if this path length difference is an integer number of wavelengths. a \sin \theta = n \lambda where * n is an integer that labels the order of each maximum, * \lambda is the wavelength, * a is the distance between the slits, and * \theta is the angle at which constructive interference occurs. For an array of slits, positions of the minima and maxima are not changed, the fringes visible on a screen however do become sharper, as can be seen in the image. right|frame|2-slit and 5-slit diffraction of red laser light ===Mathematical description=== To calculate this intensity pattern, one needs to introduce some more sophisticated methods. As the distance between the measured point of diffraction and the obstruction point increases, the diffraction patterns or results predicted converge towards those of Fraunhofer diffraction, which is more often observed in nature due to the extremely small wavelength of visible light. ==Multiple narrow slits== ===A simple quantitative description=== right|thumb|Diagram of a two slit diffraction problem, showing the angle to the first minimum, where a path length difference of a half wavelength causes destructive interference. The wave at a screen some distance away from the plane of the slits is given by the sum of the waves emanating from each of the slits. thumb|right|Interference pattern of double slits, where the slit width is one third the wavelength. The simplest case is that of two narrow slits, spaced a distance \ a apart. The result is the Fraunhofer approximation, which is only valid very far away from the object S \approx L + \frac{x^2}{2L}+\frac{x a}{2L} Depending on the size of the diffraction object, the distance to the object and the wavelength of the wave, the Fresnel approximation, the Fraunhofer approximation or neither approximation may be valid. Such treatments are applied to a wave passing through one or more slits whose width is specified as a proportion of the wavelength. If the distance to each source is an integer plus one half of a wavelength, there will be complete destructive interference. If is the location at which the intensity of the diffraction pattern is being computed, the slit extends from x' = -a/2 to +a/2\,, and from y'=-\infty to \infty. Now, substituting in \frac{2\pi}{\lambda} = k, the intensity (squared amplitude) I of the diffracted waves at an angle θ is given by: I(\theta) = I_0 {\left[ \operatorname{sinc} \left( \frac{\pi a}{\lambda} \sin \theta \right) \right] }^2 ==Multiple slits== right|frame|Double-slit diffraction of red laser light right|frame|2-slit and 5-slit diffraction Let us again start with the mathematical representation of Huygens' principle. To determine the maxima and minima in the amplitude we must determine the path difference to the first slit and to the second one. In other words, the distance to the target is much larger than the diffraction width on the target. The distance r from the slot is: r = \sqrt{\left(x - x^\prime\right)^2 + y^{\prime2} + z^2} r = z \left(1 + \frac{\left(x - x^\prime\right)^2 + y^{\prime2}}{z^2}\right)^\frac{1}{2} Assuming Fraunhofer diffraction will result in the conclusion z \gg \big|\left(x - x^\prime\right)\big|. The C-band is located between the short wavelengths (S) band (1460–1530 nm) and the long wavelengths (L) band (1565–1625 nm). The W band of the microwave part of the electromagnetic spectrum ranges from 75 to 110 GHz, wavelength ≈2.7–4 mm. Consider a monochromatic complex plane wave \Psi^\prime of wavelength λ incident on a slit of width a. Optical Fiber Communications article in rp- photonics' Encyclopedia of Laser Physics and Technology (accessed Nov. 11 2010) The C-band is located around the absorption minimum in optical fiber, where the loss reaches values as good as 0.2 dB/km, as well as an atmospheric transmission window (see figures). Since the pulse duration from these oscillators is about 3 ns, the laser linewidth performance is near the limit allowed by the Heisenberg uncertainty principle. ==See also== *Laser *Fabry–Perot interferometer *Beam divergence *Multiple-prism dispersion theory *Multiple-prism grating laser oscillator *N-slit interferometric equation *Oscillator linewidth *Solid state dye lasers ==References== Linewidth thumb|Absorption in fiber in the range 900–1700 nm with a minimum at the C-band thumb|Transmittance of the atmosphere around the C-band In infrared optical communications, C-band (C for "conventional") refers to the wavelength range 1530–1565 nm, which corresponds to the amplification range of erbium doped fiber amplifiers (EDFAs). For definiteness let us say we are diffracting light and we are interested in what the intensity looks like on a screen a distance L away from the object. Laser linewidth is the spectral linewidth of a laser beam.

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Calculate the energy and wavelength associated with an $\alpha$ particle that has fallen through a potential difference of $4.0 \mathrm{~V}$. Take the mass of an $\alpha$ particle to be $6.64 \times 10^{-27} \mathrm{~kg}$.

Alpha particles have a typical kinetic energy of 5 MeV (or ≈ 0.13% of their total energy, 110 TJ/kg) and have a speed of about 15,000,000 m/s, or 5% of the speed of light. When an atom emits an alpha particle in alpha decay, the atom's mass number decreases by four due to the loss of the four nucleons in the alpha particle. Due to the mechanism of their production in standard alpha radioactive decay, alpha particles generally have a kinetic energy of about 5 MeV, and a velocity in the vicinity of 4% of the speed of light. With a typical kinetic energy of 5 MeV; the speed of emitted alpha particles is 15,000 km/s, which is 5% of the speed of light. However, decay alpha particles only have energies of around 4 to 9 MeV above the potential at infinity, far less than the energy needed to overcome the barrier and escape. The energy of alpha particles emitted varies, with higher energy alpha particles being emitted from larger nuclei, but most alpha particles have energies of between 3 and 7 MeV (mega- electron-volts), corresponding to extremely long and extremely short half- lives of alpha-emitting nuclides, respectively. For example, performing the calculation for uranium-232 shows that alpha particle emission releases 5.4 MeV of energy, while a single proton emission would require 6.1 MeV. This energy is roughly the weight of the alpha (4 u) divided by the weight of the parent (typically about 200 u) times the total energy of the alpha. Computing the total disintegration energy given by the equation E_{di} = (m_\text{i} - m_\text{f} - m_\text{p})c^2 where is the initial mass of the nucleus, is the mass of the nucleus after particle emission, and is the mass of the emitted (alpha-)particle, one finds that in certain cases it is positive and so alpha particle emission is possible, whereas other decay modes would require energy to be added. This energy is a substantial amount of energy for a single particle, but their high mass means alpha particles have a lower speed than any other common type of radiation, e.g. β particles, neutrons.N.B. The energy needed to bring an alpha particle from infinity to a point near the nucleus just outside the range of the nuclear force's influence is generally in the range of about 25 MeV. Most of the disintegration energy becomes the kinetic energy of the alpha particle, although to fulfill conservation of momentum, part of the energy goes to the recoil of the nucleus itself (see atomic recoil). However, since the mass numbers of most alpha-emitting radioisotopes exceed 210, far greater than the mass number of the alpha particle (4), the fraction of the energy going to the recoil of the nucleus is generally quite small, less than 2%. An alpha particle is identical to the nucleus of a helium-4 atom, which consists of two protons and two neutrons. The radiated energy is approximately 2.8MeV. Such alpha particles are termed "long range alphas" since at their typical energy of 16 MeV, they are at far higher energy than is ever produced by alpha decay. An alpha particle with a speed of 1.5×107 m/s within a nuclear diameter of approximately 10−14 m will collide with the barrier more than 1021 times per second. These disintegration energies, however, are substantially smaller than the repulsive potential barrier created by the interplay between the strong nuclear and the electromagnetic force, which prevents the alpha particle from escaping. For example, one of the heaviest naturally occurring isotopes, ^238U -> ^234Th + ^4He (ignoring charges): : Qα = -931.5 (234.043 601 + 4.002 603 254 13 - 238.050 788 2) = 4.2699 MeV Note that the decay energy will be divided between the alpha-particle and the heavy recoiling daughter so that the kinetic energy of the alpha particle (Tα) will be slightly less: Tα = (234.043 601 / 238.050 788 2) 4.2699 = 4.198 MeV, (note this is for the 238gU to 238gTh reaction, which in this case has the branching ratio of 79%). To the adjacent pictures: According to the energy-loss curve by Bragg, it is recognizable that the alpha particle indeed loses more energy on the end of the trace.Magazine "nuclear energy" (III/18 (203) special edition, Volume 10, Issue 2 /1967. ==Anti-alpha particle== In 2011, members of the international STAR collaboration using the Relativistic Heavy Ion Collider at the U.S. Department of Energy's Brookhaven National Laboratory detected the antimatter partner of the helium nucleus, also known as the anti-alpha. . The symbol for the alpha particle is α or α2+. Alpha radiation has a high linear energy transfer (LET) coefficient, which is about one ionization of a molecule/atom for every angstrom of travel by the alpha particle.

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Calculate the number of photons in a $2.00 \mathrm{~mJ}$ light pulse at (a) $1.06 \mu \mathrm{m}$

Retrieved 9 December 2015 Using the intensity distribution together with Mandel's formula which describes the probability of the number of photon counts registered by a photodetector, the statistical distribution of photons in thermal light can be obtained. The formula describes the probability of observing n photon counts and is given by : P(n) = \int_{0}^{\infty} \frac{{\left ( \epsilon I \right )}^{n}}{n!} e^{-\epsilon I} P(I) dI The factor \epsilon = \frac{\eta}{h u} where \eta is the quantum efficiency describes the efficiency of the photon counter. In these experiments, light incident on the photodetector generates photoelectrons and a counter registers electrical pulses generating a statistical distribution of photon counts. SI units for these quantities are summarized in the table below. ==See also== *Single-photon source *Visible Light Photon Counter *Transition edge sensor *Superconducting nanowire single-photon detector *Time-correlated single photon counting *Oversampled binary image sensor ==References== Category:Optical metrology Category:Photonics Category:Particle detectors Photon counting is a technique in which individual photons are counted using a single-photon detector (SPD). A single-photon detector emits a pulse of signal for each detected photon. thumb|470px|Diagramm of operation of a photonic radar. Photon statistics is the theoretical and experimental study of the statistical distributions produced in photon counting experiments, which use photodetectors to analyze the intrinsic statistical nature of photons in a light source. Evidence of the sub-Poissonian nature of light is shown by obtaining a negative intensity correlation as was shown in. ==References== Category:Optical metrology Category:Photonics thumb|right|220 px|Schematic of an ion-to-photon detector with a conversion dynode. A study that compared photon-counting mammography to the state-wide average of the North Rhine-Westphalian mammography screening program in Germany reported a slightly improved diagnostic performance at a dose that was 40% of conventional technologies. However, if it is known that a single photon was detected, the center of the impulse response can be evaluated to precisely determine its arrival time. These photons are then detected by the photomultiplier tube. Photon-counting mammography was introduced commercially in 2003 and was the first widely available application of photon-counting detector technology in medical x-ray imaging. Photon- counting mammography was introduced commercially in 2003. In photon-counting mammography, contrast-enhanced imaging has been focused on iodine imaging. === Tomosynthesis === Photon-counting breast tomosynthesis has been developed to a prototype state. thumb|230px|The three leading Slepian sequences for T=1000 and 2WT=6. Thus, the excess noise factor of a photon-counting detector is unity, and the achievable signal-to-noise ratio for a fixed number of photons is generally higher than the same detector without photon counting. Photon-counting detectors, on the other hand, are fast enough to register single photon events. The flux per unit solid angle is the photon intensity. The advent of ultrafast photodetectors has made it possible to measure the sub-Poissonian nature of light. Photons that arrive during this interval may not be detected.

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The force constant of ${ }^{35} \mathrm{Cl}^{35} \mathrm{Cl}$ is $319 \mathrm{~N} \cdot \mathrm{m}^{-1}$. Calculate the fundamental vibrational frequency

The fundamental is the frequency at which the entire wave vibrates. The Rayleigh's quotient represents a quick method to estimate the natural frequency of a multi-degree-of-freedom vibration system, in which the mass and the stiffness matrices are known. The fundamental frequency is defined as its reciprocal: When the units of time are seconds, the frequency is in s^{-1}, also known as Hertz. ===Fundamental frequency of a pipe=== For a pipe of length L with one end closed and the other end open the wavelength of the fundamental harmonic is 4L, as indicated by the first two animations. 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 natural frequency, or fundamental frequency, 0, can be found using the following equation: \, }} where: * = stiffness of the spring * = mass * 0 = natural frequency in radians per second. Each step represents a frequency ratio of , or 70.6 cents. 17-ET is the tuning of the regular diatonic tuning in which the tempered perfect fifth is equal to 705.88 cents, as shown in Figure 1 (look for the label "17-TET"). ==History and use== Alexander J. Ellis refers to a tuning of seventeen tones based on perfect fourths and fifths as the Arabic scale.Ellis, Alexander J. (1863). Hence, Therefore, using the relation where v is the speed of the wave, the fundamental frequency can be found in terms of the speed of the wave and the length of the pipe: If the ends of the same pipe are now both closed or both opened as in the last two animations, the wavelength of the fundamental harmonic becomes 2L . Conversely 34-ET is a subdivision of 17-ET. ==References== Sources * == External links == * "The 17-tone Puzzle — And the Neo-medieval Key that Unlocks It" by George Secor * Libro y Programa Tonalismo, heptadecatonic system applications (in Spanish) * Georg Hajdu's 1992 ICMC paper on the 17-tone piano project * , by Wongi Hwang Category:Equal temperaments Category:Microtonality Chladni's law, named after Ernst Chladni, relates the frequency of modes of vibration for flat circular surfaces with fixed center as a function of the numbers m of diametric (linear) nodes and n of radial (circular) nodes. The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In other words, 2^{24/41} \approx 1.50042 is a better approximation to the ratio 3/2 = 1.5 than either 2^{17/29} \approx 1.50129 or 2^{7/12} \approx 1.49831. ==History and use== Although 41-ET has not seen as wide use as other temperaments such as 19-ET or 31-ET , pianist and engineer Paul von Janko built a piano using this tuning, which is on display at the Gemeentemuseum in The Hague. It is stated as the equation : f = C (m + 2n)^p where C and p are coefficients which depend on the properties of the plate.. thumb|Chladni figures, used for studying vibrationsFor flat circular plates, p is roughly 2, but Chladni's law can also be used to describe the vibrations of cymbals, handbells, and church bells in which case p can vary from 1.4 to 2.4.. The latter can be elucidated by the following 3-DOF example. == Example – 3DOF == As an example, we can consider a 3-degree-of-freedom system in which the mass and the stiffness matrices of them are known as follows: M = \begin{bmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 3 \end{bmatrix} \; , \quad K = \begin{bmatrix} 3 & -1 & 0 \\\ -1 & 3 & -2 \\\ 0 & -2 & 2 \end{bmatrix} To get an estimation of the lowest natural frequency we choose a trial vector of static displacement obtained by loading the system with a force proportional to the masses: \textbf{F} = k\begin{bmatrix} m_1 \\\ m_2 \\\ m_3 \end{bmatrix} = 1 \begin{bmatrix} 1 \\\ 1 \\\ 3 \end{bmatrix} Thus, the trial vector will become \textbf{u} = K^{-1}\textbf{F} = \begin{bmatrix} 2.5 \\\ 6.5 \\\ 8 \end{bmatrix} that allow us to calculate the Rayleigh's quotient: R = \frac{\textbf{u}^T\,K\,\textbf{u}}{\textbf{u}^T\,M\,\textbf{u}} = \cdots = 0.137214 Thus, the lowest natural frequency, calculated by means of Rayleigh's quotient is: w_\text{Ray} = 0.370424 Using a calculation tool is pretty fast to verify how much it differs from the "real" one. This is also expressed as: \,}} where: * 0 = natural frequency (SI unit: hertz) * = length of the string (SI unit: metre) * = mass per unit length of the string (SI unit: kg/m) * = tension on the string (SI unit: newton) ==See also== *Greatest common divisor *Hertz *Missing fundamental *Natural frequency *Oscillation *Harmonic series (music)#Terminology *Pitch detection algorithm *Scale of harmonics ==References== Category:Musical tuning Category:Acoustics Category:Fourier analysis While doing a modal analysis, the frequency of the 1st mode is the fundamental frequency. Moreover, the equation allow us to calculate the natural frequency only if the eigenvector (as well as any other displacement vector) \textbf{u}_{m} is known. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. The fundamental may be created by vibration over the full length of a string or air column, or a higher harmonic chosen by the player. To determine the natural frequency in Hz, the omega value is divided by 2. By the same method as above, the fundamental frequency is found to be ==In music== In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. The fundamental frequency is considered the first harmonic and the first partial. In music, 41 equal temperament, abbreviated 41-TET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally sized steps (equal frequency ratios).

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$$ \text {Calculate the energy of a photon for a wavelength of } 100 \mathrm{pm} \text { (about one atomic diameter). } $$

To find the photon energy in electronvolts using the wavelength in micrometres, the equation is approximately :E\text{ (eV)} = \frac{1.2398}{\lambda\text{ (μm)}} since hc/e=1.239 \; 841 \; 984... \times 10^{-6} eVm where h is Planck's constant, c is the speed of light in m/sec, and e is the electron charge. The photon energy of near infrared radiation at 1 μm wavelength is approximately 1.2398 eV. ==Examples== An FM radio station transmitting at 100 MHz emits photons with an energy of about 4.1357 × 10−7 eV. Additionally, E = \frac{hc}{\lambda} where *E is photon energy *λ is the photon's wavelength *c is the speed of light in vacuum *h is the Planck constant The photon energy at 1 Hz is equal to 6.62607015 × 10−34 J That is equal to 4.135667697 × 10−15 eV === Electronvolt === Energy is often measured in electronvolts. The amount of energy is directly proportional to the photon's electromagnetic frequency and thus, equivalently, is inversely proportional to the wavelength. Photon energy can be expressed using any unit of energy. Very-high-energy gamma rays have photon energies of 100 GeV to over 1 PeV (1011 to 1015 electronvolts) or 16 nanojoules to 160 microjoules. Photon energy is the energy carried by a single photon. These wavelengths correspond to photon energies of down to . Equivalently, the longer the photon's wavelength, the lower its energy. The higher the photon's frequency, the higher its energy. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. This minuscule amount of energy is approximately 8 × 10−13 times the electron's mass (via mass-energy equivalence). Light's wavenumber is proportional to frequency \textstyle \frac{1}{\lambda}=\frac{f}{c}, and therefore also proportional to light's quantum energy E. As one joule equals 6.24 × 1018 eV, the larger units may be more useful in denoting the energy of photons with higher frequency and higher energy, such as gamma rays, as opposed to lower energy photons as in the optical and radio frequency regions of the electromagnetic spectrum. ==Formulas== === Physics === Photon energy is directly proportional to frequency. The centre wavelength is the power-weighted mean wavelength: : \lambda_c = \frac {1}{P_{total}} \int p(\lambda) \lambda\, d\lambda And the total power is: :P_{total}=\int p(\lambda) d\lambda where p(\lambda) is the power spectral density, for example in W/nm. The wavelengths in the solar spectrum range from approximately 0.3-2.0 μm. During photosynthesis, specific chlorophyll molecules absorb red-light photons at a wavelength of 700 nm in the photosystem I, corresponding to an energy of each photon of ≈ 2 eV ≈ 3 × 10−19 J ≈ 75 kBT, where kBT denotes the thermal energy. 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. Among the units commonly used to denote photon energy are the electronvolt (eV) and the joule (as well as its multiples, such as the microjoule). Johannes (Janne) Robert Rydberg (; 8 November 1854 - 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom. == Biography == Rydberg was born 8 November 1854 in Halmstad in southern Sweden, the only child of Sven Rydberg and Maria Anderson Rydberg. Spectral irradiance of wavelengths in the solar spectrum. A minimum of 48 photons is needed for the synthesis of a single glucose molecule from CO2 and water (chemical potential difference 5 × 10−18 J) with a maximal energy conversion efficiency of 35%. ==See also== *Photon *Electromagnetic radiation *Electromagnetic spectrum *Planck constant *Planck–Einstein relation *Soft photon ==References== Category:Foundational quantum physics Category:Electromagnetic spectrum Category:Photons

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A proton and a negatively charged $\mu$ meson (called a muon) can form a short-lived species called a mesonic atom. The charge of a muon is the same as that on an electron and the mass of a muon is $207 m_{\mathrm{e}}$. Assume that the Bohr theory can be applied to such a mesonic atom and calculate the ground-state energy, the radius of the first Bohr orbit, and the energy and frequency associated with the $n=1$ to $n=2$ transition in a mesonic atom.

Unlike baryonic molecules, which form the nuclei of all elements in nature save hydrogen-1, a mesonic molecule has yet to be definitively observed. The Bohr model also has difficulty with, or else fails to explain: * Much of the spectra of larger atoms. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the "reduced" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). In kaonic hydrogen this strong contribution was found to be repulsive, shifting the ground state energy by 283 ± 36 (statistical) ± 6 (systematic) eV, thus making the system unstable with a resonance width of 541 ± 89 (stat) ± 22 (syst) eV (decay into Λπ and ΣπYiguang Yan, Kaonic hydrogen atom and kaon-proton scattering length, ). System Radius Hydrogen 1.00054\, a_0 Positronium 2 a_0 Muonium 1.0048\, a_0 He+ a_0/2 Li2+ a_0/3 ==See also== * Bohr magneton * Rydberg energy ==References== == External links == * Length Scales in Physics: the Bohr Radius Category:Atomic physics Category:Physical constants Category:Units of length Category:Niels Bohr Category:Atomic radius In atomic physics, the Bohr model or Rutherford–Bohr model of the atom, presented by Niels Bohr and Ernest Rutherford in 1913, consists of a small, dense nucleus surrounded by orbiting electrons. Any one of these constants can be written in terms of any of the others using the fine-structure constant \alpha : :r_{\mathrm{e}} = \alpha \frac{\lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0. ==Hydrogen atom and similar systems== The Bohr radius including the effect of reduced mass in the hydrogen atom is given by : \ a_0^* \ = \frac{m_\text{e}}{\mu}a_0 , where \mu = m_\text{e} m_\text{p} / (m_\text{e} + m_\text{p}) is the reduced mass of the electron–proton system (with m_\text{p} being the mass of proton). The CODATA value of the Bohr radius (in SI units) is ==History== In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction. 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. A heavy Rydberg system consists of a weakly bound positive and negative ion orbiting their common centre of mass. Kaonic hydrogen is an exotic atom consisting of a negatively charged kaon orbiting a proton. The Bohr model is a relatively primitive model of the hydrogen atom, compared to the valence shell model. This outer electron should be at nearly one Bohr radius from the nucleus. A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. A mesonic molecule is a set of two or more mesons bound together by the strong force. Schrödinger employed de Broglie's matter waves, but sought wave solutions of a three-dimensional wave equation describing electrons that were constrained to move about the nucleus of a hydrogen-like atom, by being trapped by the potential of the positive nuclear charge. == Electron energy levels == thumb|Models depicting electron energy levels in hydrogen, helium, lithium, and neon The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. The Bohr formula properly uses the reduced mass of electron and proton in all situations, instead of the mass of the electron, :m_\text{red} = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = m_\mathrm{e} \frac{1}{1 + m_\mathrm{e}/m_\mathrm{p}}. :The smallest possible value of r in the hydrogen atom () is called the Bohr radius and is equal to: :: r_1 = \frac{\hbar^2}{k_\mathrm{e} e^2 m_\mathrm{e}} \approx 5.29 \times 10^{-11}~\mathrm{m}. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. In 1921, following the work of chemists and others involved in work on the periodic table, Bohr extended the model of hydrogen to give an approximate model for heavier atoms. This result can be generalized to other systems, such as positronium (an electron orbiting a positron) and muonium (an electron orbiting an anti-muon) by using the reduced mass of the system and considering the possible change in charge. Starting from the angular momentum quantum rule as Bohr admits is previously given by Nicholson in his 1912 paper, Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions.

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$$ \beta=2 \pi c \tilde{\omega}_{\mathrm{obs}}\left(\frac{\mu}{2 D}\right)^{1 / 2} $$ Given that $\tilde{\omega}_{\mathrm{obs}}=2886 \mathrm{~cm}^{-1}$ and $D=440.2 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}$ for $\mathrm{H}^{35} \mathrm{Cl}$, calculate $\beta$.

But these variables can be calculated with GSW. thumb|440x440px|This is the 2020 average for the haline contraction coefficient β. The GSW beta(SA,CT,p) function can calculate β when the absolute salinity (SA), conserved temperature (CT) and the pressure are known. \\\ & = \left ( \frac{(1 - 0.0436) (3.827 \times 10^{26}\ \mbox{W})} {0.9 (5.670 \times 10^{-8}\ \mbox{W/m}^2\mbox{K}^4) 16 \cdot 3.142 (3.959 \times 10^{11}\ \mbox{m})^2} \right )^{\frac{1}{4}} \\\ & = 173.7\ \mbox{K} \end{align} See: K | mean_motion= / day | observation_arc=130.38 yr (47622 d) | uncertainty=0 | moid= | jupiter_moid= | tisserand=3.331 }} Mathilde (minor planet designation: 253 Mathilde) is an asteroid in the intermediate asteroid belt, approximately 50 kilometers in diameter, that was discovered by Austrian astronomer Johann Palisa at Vienna Observatory on 12 November 1885. The molecular formula C25H27N (molar mass: 341.49 g/mol, exact mass: 341.2143 u) may refer to: * JWH-184 * JWH-196 The haline contraction coefficient is constant when a water parcel moves adiabatically along the isobars. == Application == The amount that density is influenced by a change in salinity or temperature can be computed from the density formula that is derived from the thermal wind balance. \rho = \rho_0 \left( \alpha \Theta + \beta S_A \right) The Brunt–Väisälä frequency can also be defined when β is known, in combination with α, Θ and S_A. A high β means that the increase in density is more than when β is low.thumb|435x435px|This graph shows the 2020 average salinity in an intersection in the Atlantic ocean at 30W. HD 142415 b is an exoplanet with the semi-amplitude of 51.3 ± 2.3 m/s. With these two coefficients, the density ratio can be calculated. The molecular formula C25H38O2 (molar mass: 370.57 g/mol, exact mass: 370.2872 u) may refer to: * CBD-DMH, or DMH-CBD * Dimethylheptylpyran * JWH-051 * Penmesterol, or penmestrol * Variecolol 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 Haline contraction coefficient, abbreviated as β, is a coefficient that describes the change in ocean density due to a salinity change, while the potential temperature and the pressure are kept constant. It is a parameter in the Equation Of State (EOS) of the ocean. β is also described as the saline contraction coefficient and is measured in [kg]/[g] in the EOS that describes the ocean. The subscripts Θ and p indicate that β is defined at constant potential temperature Θ and constant pressure p. This determines the contribution of the temperature and salinity to the density of a water parcel. β is called a contraction coefficient, because when salinity increases, water becomes denser, and if the temperature increases, water becomes less dense. == Definition == Τhe haline contraction coefficient is defined as: \beta = \frac{1}{\rho} \frac{\partial \rho}{\partial S_A}\Bigg |_{\Theta,p} where ρ is the density of a water parcel in the ocean and S_A is the absolute salinity. This is the thermodynamic equation of state. β is the salinity variant of the thermal expansion coefficient α, where the density changes due to a change in temperature instead of salinity. The direction of the mixing and whether the mixing is temperature- or salinity-driven can be determined from the density difference and the Brunt-Väisälä frequency. == Computation == β can be computed when the conserved temperature, the absolute salinity and the pressure are known from a water parcel. This equation relates the thermodynamic properties of the ocean (density, temperature, salinity and pressure). This means that changing the salinity will have a large effect on the density when the haline contraction coefficient is high. === Physical examples === β is not a constant, it mostly changes with latitude and depth. This frequency is a measure of the stratification of a fluid column and is defined over depth as: N^2 = g \left( \alpha \frac{\partial \Theta}{\partial z} - \beta \frac{\partial S_A}{\partial z} \right). These equations are based on empirical thermodynamic properties. This water is dense, because it is cold. β around Antarctica is relatively high. At locations where salinity is high, as in the tropics, β is low and where salinity is low, β is high.

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Two narrow slits separated by $0.10 \mathrm{~mm}$ are illuminated by light of wavelength $600 \mathrm{~nm}$. What is the angular position of the first maximum in the interference pattern? If a detector is located $2.00 \mathrm{~m}$ beyond the slits, what is the distance between the central maximum and the first maximum?

thumb|right|Interference pattern of double slits, where the slit width is one third the wavelength. Angular distance or angular separation, also known as apparent distance or apparent separation, denoted \theta, is the angle between the two sightlines, or between two point objects as viewed from an observer. Given that \delta_A-\delta_B\ll 1 and \alpha_A-\alpha_B\ll 1, at a second-order development it turns that \cos\delta_A\cos\delta_B \frac{(\alpha_A - \alpha_B)^2}{2} \approx \cos^2\delta_A \frac{(\alpha_A - \alpha_B)^2}{2}, so that :\theta \approx \sqrt{\left[(\alpha_A - \alpha_B)\cos\delta_A\right]^2 + (\delta_A-\delta_B)^2} === Small angular distance: planar approximation === thumb|Planar approximation of angular distance on sky If we consider a detector imaging a small sky field (dimension much less than one radian) with the y-axis pointing up, parallel to the meridian of right ascension \alpha, and the x-axis along the parallel of declination \delta, the angular separation can be written as: : \theta \approx \sqrt{\delta x^2 + \delta y^2} where \delta x = (\alpha_A - \alpha_B)\cos\delta_A and \delta y=\delta_A-\delta_B. Note that the y-axis is equal to the declination, whereas the x-axis is the right ascension modulated by \cos\delta_A because the section of a sphere of radius R at declination (latitude) \delta is R' = R \cos\delta_A (see Figure). ==See also== * Milliradian * Gradian * Hour angle * Central angle * Angle of rotation * Angular diameter * Angular displacement * Great-circle distance * ==References== * CASTOR, author(s) unknown. thumb|250px|An illustration of how position angle is estimated through a telescope eyepiece; the primary star is at center. 300px|right|thumb|Jack Huntington .500 Maximum The .500 Maximum, also known as .500 Linebaugh Maximum and .500 Linebaugh Long, is a revolver cartridge developed by John Linebaugh. thumb|Local and global maxima and minima for cos(3πx)/x, 0.1≤ x ≤1.1 In mathematical analysis, the maximum and minimum of a function are, respectively, the largest and smallest value taken by the function. The minimum distance is the distance along a great circle that runs through and . thumb|upright=1.35|The five red points are the maxima of the set of all the red and yellow points. (See figure at right) x−x Unique global maximum over the positive real numbers at x = 1/e. x3/3 − x First derivative x2 − 1 and second derivative 2x. If it is, output the point as one of the maximal points, and remember its -coordinate as the greatest seen so far. If it is, output the point as one of the maximal points, and remember its -coordinate as the greatest seen so far. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the largest (or smallest) one. Although the first derivative (3x2) is 0 at x = 0, this is an inflection point. (2nd derivative is 0 at that point.) \sqrt[x]{x} Unique global maximum at x = e. It is calculated in a plane that contains the sphere center and the great circle, :: d_{s,t}=R\theta_{s,t} where is the angular distance of two points viewed from the center of the sphere, measured in radians. A position along the great circle is ::\mathbf{s}(\theta) = \cos\theta \mathbf{s}+\sin\theta \mathbf{s}_\perp,\quad 0\le\theta\le 2\pi. The maxima of a point set are all the maximal points of . Therefore, \mathbf{n_A} \cdot \mathbf{n_B} = \cos\delta_A \cos\alpha_A \cos\delta_B \cos\alpha_B + \cos\delta_A \sin\alpha_A \cos\delta_B \sin\alpha_B + \sin\delta_A \sin\delta_B \equiv \cos\theta then: :\theta = \cos^{-1}\left[\sin\delta_A \sin\delta_B + \cos\delta_A \cos\delta_B \cos(\alpha_A - \alpha_B)\right] === Small angular distance approximation === The above expression is valid for any position of A and B on the sphere. Therefore, the greatest area attainable with a rectangle of 200 feet of fencing is ==Functions of more than one variable== thumb|right|The global maximum is the point at the top thumb|right|Counterexample: The red dot shows a local minimum that is not a global minimum For functions of more than one variable, similar conditions apply. In general, if an ordered set S has a greatest element m, then m is a maximal element of the set, also denoted as \max(S). The problem of finding all maximal points, sometimes called the problem of the maxima or maxima set problem, has been studied as a variant of the convex hull and orthogonal convex hull problems. Let O indicate the observer on Earth, assumed to be located at the center of the celestial sphere. The value of the function at a maximum point is called the of the function, denoted \max(f(x)), and the value of the function at a minimum point is called the of the function.

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$$ \text { If we locate an electron to within } 20 \mathrm{pm} \text {, then what is the uncertainty in its speed? } $$

* Grabe, M ., Measurement Uncertainties in Science and Technology, Springer 2005. Uncertainty of measurement results. Evaluating the Uncertainty of 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 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 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. 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. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. Uncertainty principle of Heisenberg, 1927. Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology. One way to quantify the precision of the position and momentum is the standard deviation σ. The measurement uncertainty is often taken as the standard deviation of a state-of-knowledge probability distribution over the possible values that could be attributed to a measured quantity. * Possolo A and Iyer H K 2017 Concepts and tools for the evaluation of measurement uncertainty Rev. Sci. Instrum.,88 011301 (2017). * Introduction to evaluating uncertainty of measurement * JCGM 200:2008. "Quantifying uncertainty in analytical measurement". Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The book draws its name from the uncertainty principle, which states that, in quantum mechanics, there exists pairs of quantities, such as position and velocity, in which you cannot know the precise value of both at the same time. In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a measured quantity. Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. The uncertainty has two components, namely, bias (related to accuracy) and the unavoidable random variation that occurs when making repeated measurements (related to precision). In this manner, said Einstein, one could measure the energy emitted and the time it was released with any desired precision, in contradiction to the uncertainty principle." In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet.

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The mean temperature of the earth's surface is $288 \mathrm{~K}$. What is the maximum wavelength of the earth's blackbody radiation?

This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrumsPrinciples of Planetary Climate by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. Notice that a gray (flat spectrum) ball where ({1-\alpha}) ={\overline{\varepsilon}} comes to the same temperature as a black body no matter how dark or light gray. ====Effective temperature of Earth==== Substituting the measured values for the Sun and Earth yields: :T_{\rm S} = 5778 \ \mathrm{K},NASA Sun Fact Sheet :R_{\rm S} = 6.96 \times 10^8 \ \mathrm{m}, :D = 1.496 \times 10^{11} \ \mathrm{m}, :\alpha = 0.306 \ With the average emissivity \overline{\varepsilon} set to unity, the effective temperature of the Earth is: :T_{\rm E} = 254.356\ \mathrm{K} or −18.8 °C. Solving for the wavelength \lambda in millimetres, and using kelvins for the temperature yields: : ===Parameterization by frequency=== Another common parameterization is by frequency. Only 25 percent of the energy in the black-body spectrum is associated with wavelengths shorter than the value given by the peak-wavelength version of Wien's law. thumb|upright=1.45|Planck blackbody spectrum parameterized by wavelength, fractional bandwidth (log wavelength or log frequency), and frequency, for a temperature of 6000 K. Notice that for a given temperature, different parameterizations imply different maximal wavelengths. The Sun, with an effective temperature of approximately 5800 K, is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well. The formula is given, where E is the radiant heat emitted from a unit of area per unit time, T is the absolute temperature, and is the Stefan–Boltzmann constant. ==Equations== ===Planck's law of black-body radiation=== Planck's law states that :B_ u(T) = \frac{2h u^3}{c^2}\frac{1}{e^{h u/kT} - 1}, where :B_{ u}(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency u radiation per unit frequency at thermal equilibrium at temperature T. Units: power / [area × solid angle × frequency]. :h is the Planck constant; :c is the speed of light in vacuum; :k is the Boltzmann constant; : u is the frequency of the electromagnetic radiation; :T is the absolute temperature of the body. An Introductory Survey, second edition, Elsevier, Amsterdam, , exercise 4.6, pages 119–120. ===Cosmology=== The cosmic microwave background radiation observed today is the most perfect black-body radiation ever observed in nature, with a temperature of about 2.7 K. Application of Wien's law to human-body emission results in a peak wavelength of :\lambda_\text{peak} = \frac{2.898 \times 10^{-3}~\text{K} \cdot \text{m}}{305~\text{K}} = 9.50~\mu\text{m}. The relevant math is detailed in the next section. ==Derivation from Planck's law== ===Parameterization by wavelength=== Planck's law for the spectrum of black body radiation predicts the Wien displacement law and may be used to numerically evaluate the constant relating temperature and the peak parameter value for any particular parameterization. The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. The theory even predicted that all bodies would emit most of their energy in the ultraviolet range, clearly contradicted by the experimental data which showed a different peak wavelength at different temperatures (see also Wien's law). thumb|303px|As the temperature increases, the peak of the emitted black-body radiation curve moves to higher intensities and shorter wavelengths. An equilibrium temperature of 255 K on Earth yields a skin temperature of 214 K, which compares with a tropopause temperature of 209 K. == References == Category:Temperature Category:Atmospheric radiation At a typical room temperature of 293 K (20 °C), the maximum intensity is for . ===Stefan–Boltzmann law=== By integrating B_ u(T)\cos(\theta) over the frequency the radiance L (units: power / [area * solid angle] ) is : L=\frac{2\pi^5}{15} \frac{k^4 T^4}{c^2h^3} \frac{1}{\pi}= \sigma T^4 \frac{\cos(\theta)}{\pi} by using \int_0^\infty dx\, \frac{x^3}{e^x - 1}=\frac{\pi^4}{15} with x \equiv \frac{h u}{k T} and with \sigma \equiv \frac{2\pi^5}{15} \frac{k^4}{c^2h^3}=5.670373 \times 10^{-8} \frac{W}{m^2 K^4} being the Stefan–Boltzmann constant. The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. This is an inverse relationship between wavelength and temperature. *The effective temperature of the Sun is 5778 Kelvin. The intensity maximum for this is given by : u_\text{peak} = T \times 5.879 \times 10^{10} \ \mathrm{Hz}/\mathrm{K}. For the same temperature, but parameterizing by frequency, the frequency for maximal spectral radiance is with corresponding wavelength . For wavelength λ, it is: B_{\lambda} (T) = \frac{2 ck_{\mathrm{B}} T}{\lambda^4}, where B_{\lambda} is the spectral radiance, the power emitted per unit emitting area, per steradian, per unit wavelength; c is the speed of light; k_{\mathrm{B}} is the Boltzmann constant; and T is the temperature in kelvin. At a typical room temperature of 293 K (20 °C), the maximum intensity is at . The two factors combined give the characteristic maximum wavelength. Commonly a wavelength parameterization is used and in that case the black body spectral radiance (power per emitting area per solid angle) is: :u_{\lambda}(\lambda,T) = {2 h c^2\over \lambda^5}{1\over e^{h c/\lambda kT}-1}.

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The power output of a laser is measured in units of watts (W), where one watt is equal to one joule per second. $\left(1 \mathrm{~W}=1 \mathrm{~J} \cdot \mathrm{s}^{-1}\right.$.) What is the number of photons emitted per second by a $1.00 \mathrm{~mW}$ nitrogen laser? The wavelength emitted by a nitrogen laser is $337 \mathrm{~nm}$.

thumb|right|300 px|A 337nm wavelength and 170 μJ pulse energy 20 Hz cartridge nitrogen laser A nitrogen laser is a gas laser operating in the ultraviolet rangeC. The wall-plug efficiency of the nitrogen laser is low, typically 0.1% or less, though nitrogen lasers with efficiency of up to 3% have been reported in the literature. Duarte and L. W. Hillman, Dye Laser Principles (Academic, New York, 1990) Chapter 6. * measurement of air pollution (Lidar) * Matrix-assisted laser desorption/ionization * List of laser articles ==External links== *Professor Mark Csele's Homebuilt Lasers Page *Example of TEA Laser prototype *Sam's lasers FAQ/Home Built nitrogen (N2) laser * an update of the Amateur Scientist column, on page 122 of the June, 1974 issue of Scientific American *Compact High-Power N2 Laser: Circuit Theory and Design Adolph Schwab & Fritz Hollinger IEEE Journal of Quantum Electronics, QE-12, No. 3, March 1966, p.183 ==References== Category:Gas lasers Air, which is 78% nitrogen, can be used, but more than 0.5% oxygen poisons the laser. == Optics == Nitrogen lasers can operate within a resonator cavity, but due to the typical gain of 2 every 20 mm they more often operate on superluminescence alone; though it is common to put a mirror at one end such that the output is emitted from the opposite end. Pulse durations vary from a few hundred picoseconds (at 1 atmosphere partial pressure of nitrogen) to about 30 nanoseconds at reduced pressure (typically some dozens of Torr), though FWHM pulsewidths of 6 to 8 ns are typical. === Amateur construction === The transverse discharge nitrogen laser has long been a popular choice for amateur home construction, owing to its simple construction and simple gas handling. With about 11 ns the UV generation, ionisation, and electron capture are in a similar speed regime as the nitrogen laser pulse duration and thus as fast electric must be applied. === Excitation by electron impact === The upper laser level is excited efficiently by electrons with more than 11 eV, best energy is 15 eV. The wall-plug efficiency is the product of the following three efficiencies: * electrical: TEA laser * gain medium: This is the same for all nitrogen lasers and thus has to be at least 3% ** inversion by electron impact is 10 to 1 due to Franck–Condon principle ** energy lost in the lower laser level: 40% * optical: More stimulated emission than spontaneous emission ==Gain medium== The gain medium is nitrogen molecules in the gas phase. This nicely matches the typical rise times of 1×10−8 s and typical currents of 1×103 A occurring in nitrogen lasers. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. ==History of He-Ne laser development== The first He-Ne lasers emitted infrared at 1150 nm, and were the first gas lasers and the first lasers with continuous wave output. Measured nitrogen laser pulses are so long that the second step is unimportant. The precise wavelength of red He-Ne lasers is 632.991 nm in a vacuum, which is refracted to about 632.816 nm in air. The nitrogen laser is a three-level laser. 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. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. Absolute stabilization of the laser's frequency (or wavelength) as fine as 2.5 parts in 1011 can be obtained through use of an iodine absorption cell. thumb|Energy levels in a He-Ne Laser|512x330px thumb|Ring He-Ne Laser The mechanism producing population inversion and light amplification in a He-Ne laser plasma originates with inelastic collision of energetic electrons with ground-state helium atoms in the gas mixture. In contrast to more typical four-level lasers, the upper laser level of nitrogen is directly pumped, imposing no speed limits on the pump. Higher voltages mean shorter pulses. ==Typical devices== The gas pressure in a nitrogen laser ranges from a few mbar to as much as several bar. The laser diode rate equations model the electrical and optical performance of a laser diode. thumb|250px|Laser modules (bottom to top: 405, 445, 520, 532, 635, and 660 nm) Laser science or laser physics is a branch of optics that describes the theory and practice of lasers. thumb|Helium–neon laser at the University of Chemnitz, Germany A helium–neon laser or He-Ne laser, is a type of gas laser whose high energetic medium gain medium consists of a mixture of ratio(between 5:1 and 20:1) of helium and neon at a total pressure of about 1 torr inside of a small electrical discharge. However, in high-power He-Ne lasers having a particularly long cavity, superluminescence at 3.39 μm can become a nuisance, robbing power from the stimulated emission medium, often requiring additional suppression. Willett, Introduction to Gas Lasers: Population Inversion Mechanisms (Pergamon, New York,1974). (typically 337.1 nm) using molecular nitrogen as its gain medium, pumped by an electrical discharge.

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Sirius, one of the hottest known stars, has approximately a blackbody spectrum with $\lambda_{\max }=260 \mathrm{~nm}$. Estimate the surface temperature of Sirius.

The brighter component, termed Sirius A, is a main-sequence star of spectral type early A, with an estimated surface temperature of 9,940 K. The Sun, with an effective temperature of approximately 5800 K, is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well. Because there is no internal heat source, Sirius B will steadily cool as the remaining heat is radiated into space over the next two billion years or so. Sirius is the brightest star in the night sky. To the naked eye, it often appears to be flashing with red, white, and blue hues when near the horizon. == Observation == With an apparent magnitude of −1.46, Sirius is the brightest star in the night sky, almost twice as bright as the second-brightest star, Canopus. This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrumsPrinciples of Planetary Climate by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. When the star's or planet's net emissivity in the relevant wavelength band is less than unity (less than that of a black body), the actual temperature of the body will be higher than the effective temperature. An Introductory Survey, second edition, Elsevier, Amsterdam, , exercise 4.6, pages 119–120. ===Cosmology=== The cosmic microwave background radiation observed today is the most perfect black-body radiation ever observed in nature, with a temperature of about 2.7 K. This is a list of hottest stars so far discovered (excluding degenerate stars), arranged by decreasing temperature. Because of its declination of roughly −17°, Sirius is a circumpolar star from latitudes south of 73° S. The effective temperature or effective radiative emission temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation.Stull, R. (2000). Sirius is gradually moving closer to the Solar System; it is expected to increase in brightness slightly over the next 60,000 years to reach a peak magnitude of −1.68. Notice that a gray (flat spectrum) ball where ({1-\alpha}) ={\overline{\varepsilon}} comes to the same temperature as a black body no matter how dark or light gray. ====Effective temperature of Earth==== Substituting the measured values for the Sun and Earth yields: :T_{\rm S} = 5778 \ \mathrm{K},NASA Sun Fact Sheet :R_{\rm S} = 6.96 \times 10^8 \ \mathrm{m}, :D = 1.496 \times 10^{11} \ \mathrm{m}, :\alpha = 0.306 \ With the average emissivity \overline{\varepsilon} set to unity, the effective temperature of the Earth is: :T_{\rm E} = 254.356\ \mathrm{K} or −18.8 °C. In that year, Sirius will come to within 1.6 degrees of the south celestial pole. The outer atmosphere of Sirius B is now almost pure hydrogen—the element with the lowest mass—and no other elements are seen in its spectrum. === Apparent third star === Since 1894, irregularities have been tentatively observed in the orbits of Sirius A and B with an apparent periodicity of 6–6.4 years. With a visual apparent magnitude of −1.46, Sirius is almost twice as bright as Canopus, the next brightest star. Sirius appears bright because of its intrinsic luminosity and its proximity to the Solar System. A star near the middle of the spectrum, such as the modest Sun or the giant Capella radiates more energy per unit of surface area than the feeble red dwarf stars or the bloated supergiants, but much less than such a white or blue star as Vega or Rigel. == Planet == ===Blackbody temperature=== To find the effective (blackbody) temperature of a planet, it can be calculated by equating the power received by the planet to the known power emitted by a blackbody of temperature . Sirius A is about twice as massive as the Sun () and has an absolute visual magnitude of +1.43. The visible star is now sometimes known as Sirius A. When compared to the Sun, the proportion of iron in the atmosphere of Sirius A relative to hydrogen is given by \textstyle\ \left[\frac{\ce{Fe}}{\ce{H}}\right] = 0.5\ , meaning iron is 316% as abundant as in the Sun's atmosphere. The stars with temperatures higher than 60,000 K are included. ==List== Star name Effective Temperature (K) Mass () Luminosity () Spectral type Distance Ref. WR 102 210,000 16.1 380,000 WO2 8,610 WR 142 200,000 28.6 912,000 WO2 5,400 LMC195-1 200,000 WO2 160,000 BAT99-123 170,000 158,000 WO3 ~160,000 WR 93b 160,000 8.1 110,000 WO3 7,470 [HC2007] 31 160,000?

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A ground-state hydrogen atom absorbs a photon of light that has a wavelength of $97.2 \mathrm{~nm}$. It then gives off a photon that has a wavelength of $486 \mathrm{~nm}$. What is the final state of the hydrogen atom?

In optical excitation, the incident photon is absorbed by the target atom, resulting in a precise final state energy. thumb|A hydrogen atom with proton and electron spins aligned (top) undergoes a flip of the electron spin, resulting in emission of a photon with a 21 cm wavelength (bottom) The hydrogen line, 21 centimeter line, or H I line is a spectral line that is created by a change in the energy state of solitary, electrically neutral hydrogen atoms. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. A photon in this energy range, with a frequency that coincides with that of one of the lines in the Lyman or Werner bands, can be absorbed by H2, placing the molecule in an excited electronic state. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. Although the initial step excites to a broad range of intermediate states, the precision inherent in the optical excitation process means that the laser light only interacts with a specific subset of atoms in a particular state, exciting to the chosen final state. == Hydrogenic potential == thumb|right|Figure 3. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). The record wavelength for hydrogen is λ = 73 cm for H253α, implying atomic diameters of a few microns, and for carbon, λ = 18 metres, from C732α, from atoms with a diameter of 57 micron. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. The transition from the state 1^2S_{1/2} with mj=-1/2 to the state 2^2P_{3/2} with mj=-1/2 is only allowed for light with polarization along the z axis (quantization axis) of the atom. This is a quantum state change between the two hyperfine levels of the hydrogen 1 s ground state. A hydrogen atom consists of an electron orbiting its nucleus. In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. This two-step photodissociation process, known as the Solomon process, is one of the main mechanisms by which molecular hydrogen is destroyed in the interstellar medium. thumb|Electronic and vibrational levels of the hydrogen molecule In reference to the figure shown, Lyman-Werner photons are emitted as described below: *A hydrogen molecule can absorb a far- ultraviolet photon (11.2 eV < energy of the photon < 13.6 eV) and make a transition from the ground electronic state X to excited state B (Lyman) or C (Werner). A state that cannot absorb an incident photon is called a dark state. The new state with energy E_3 of the atom no longer interacts with the laser simply because no photons of the right frequency are present to induce a transition to a different level. A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, n. An atom in a Rydberg state has a valence electron in a large orbit far from the ion core; in such an orbit, the outermost electron feels an almost hydrogenic, Coulomb potential, UC from a compact ion core consisting of a nucleus with Z protons and the lower electron shells filled with Z-1 electrons. In hydrogen the binding energy is given by: : E_\text{B} = -\frac{\rm Ry}{n^2}, where Ry = 13.6 eV is the Rydberg constant. Electromagnetically induced transparency was used in combination with strong interactions between two atoms excited in Rydberg state to provide medium that exhibits strongly nonlinear behaviour at the level of individual optical photons. However, the atom can still decay spontaneously into a third state by emitting a photon of a different frequency.

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It turns out that the solution of the Schrödinger equation for the Morse potential can be expressed as $$ G(v)=\tilde{\omega}_{\mathrm{e}}\left(v+\frac{1}{2}\right)-\tilde{\omega}_{\mathrm{e}} \tilde{x}_{\mathrm{e}}\left(v+\frac{1}{2}\right)^2 $$ Chapter 5 / The Harmonic Oscillator and Vibrational Spectroscopy where $$ \tilde{x}_{\mathrm{e}}=\frac{h c \tilde{\omega}_{\mathrm{e}}}{4 D} $$ Given that $\tilde{\omega}_{\mathrm{e}}=2886 \mathrm{~cm}^{-1}$ and $D=440.2 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}$ for $\mathrm{H}^{35} \mathrm{Cl}$, calculate $\tilde{x}_{\mathrm{e}}$ and $\tilde{\omega}_{\mathrm{e}} \tilde{x}_{\mathrm{e}}$.

With that, the one-dimensional Schrödinger equation that describes on S^3 the quantum motion of an electric charge dipole perturbed by the trigonometric Rosen–Morse potential, produced by another electric charge dipole, takes the form of {{NumBlk|:| \left(-\frac{\hbar^2 c^2}{R^2}\frac{{\mathrm d}^2}{{\mathrm d}\chi^2}+V^{(\ell+1,\alpha Z/2,1)}_{\mbox{tRM}}(\chi)\right)U_{K\ell}^{(\alpha Z)}(\chi)=\frac{\hbar^2c^2}{R^2}\left({\epsilon}_{\ell n}^{(\alpha Z)}\right)^2 U_{K\ell }^{(\alpha Z)}(\chi), |}} (\chi)=\frac{\hbar^2 c^2}{R^2}\frac{\ell(\ell+1)}{\sin^2\chi}-2\frac{\hbar^2 c^2}{R^2}\alpha Z\cot\chi, |}} Because of the relationship, K-\ell=n, with n being the node number of the wave function, one could change labeling of the wave functions, U^{(b)}_{K\ell}(\chi), to the more familiar in the literature, U^{(b)}_{\ell n}(\chi). For this reason, the wave equation which transforms upon the variable change, \Psi_{K\ell m}(\chi,\theta,\varphi)=\frac{U^{(b)}_{K\ell}(\chi)}{\sin\chi}Y_{\ell}^m(\theta,\varphi), into the familiar one-dimensional Schrödinger equation with the V_{tRM}^{(\ell+1,b,1)}(\chi) trigonometric Rosen–Morse potential, {{NumBlk|:| -\frac{\hbar^2c^2}{R^2}\left[\frac{{\mathrm d}^2}{{\mathrm d}\chi^2}+\frac{\ell(\ell+1)}{\sin^2\chi}\right]U^{(b)}_{K\ell}(\chi)-2b\cot\chi U^{(b)}_{K\ell}(\chi)=\frac{\hbar^2c^2}{R^2}\left[(K+1)^2-\frac{b^2}{(K+1)^2}\right]U^{(b)}_{K\ell}(\chi), |}} in reality describes quantum motion of a charge dipole perturbed by the field due to another charge dipole, and not the motion of a single charge within the field produced by another charge. Changing in () variables as one observes that the \psi_{K\ell}(\chi) function satisfies the one-dimensional Schrödinger equation with the \csc^2\chi potential according to {{NumBlk|:| -\frac{\hbar^2c^2}{R^2}\left[\frac{{\mathrm d}^2}{{\mathrm d}\chi^2}+\frac{\ell(\ell+1)}{\sin^2\chi}\right]\psi_{K\ell}(\chi)=\frac{\hbar^2c^2}{R^2}(K+1)^2\psi_{K\ell}(\chi). |}} The one-dimensional potential in the latter equation, in coinciding with the Rosen–Morse potential in () for a=\ell+1 and b=0, clearly reveals that for integer a values, the first term of this potential takes its origin from the centrifugal barrier on S^3. This potential is approximately a Morse potential with 16\pi^{2} e^{8|x|} The asymptotic of the energies depend on the quantum number as E_n = \frac{4\pi^2 n^2}{W^2(ne^{-1})} , where is the Lambert W function. == References == * * G. Sierra, A physics pathway to the Riemann hypothesis, arXiv:math-ph/1012.4264, 2010. 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. For the case of the Riemann zeros Wu and Sprung and others have shown that the potential can be written implicitly in terms of the Gamma function and zeroth- order Bessel function. f^{-1} (x)=\frac{2}{\sqrt{4x+1} } +\frac{1}{4\pi } \int_{-\sqrt{x} }^{\sqrt{x}}\frac{dr}{\sqrt{x-r^2} } \left( \frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +\frac{ir}{2} \right) -\ln \pi \right) -\sum\limits_{n=1}^\infty \frac{\Lambda (n)}{2\sqrt{n} } J_0 \left( \sqrt{x} \ln n\right) and that the density of states of this Hamiltonian is just the Delsarte's formula for the Riemann zeta function and defined semiclassically as \frac{1}{\sqrt \pi} \frac{d^{1/2}}{dx^{1/2}}f^{-1}(x)= \sum_{n=0}^{\infty}\delta (x-E_{n}) \begin{align} \sum_{n=0}^{\infty }\delta \left( x-\gamma _{n} \right) + \sum_{n=0}^{\infty }\delta \left( x+\gamma _{n} \right) ={}& \frac{1}{2\pi } \frac{\zeta }{\zeta } \left( \frac{1}{2} +ix\right) +\frac{1}{2\pi } \frac{\zeta '}{\zeta } \left( \frac{1}{2} -ix\right) -\frac{\ln \pi }{2\pi } \\\\[10pt] &{} +\frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +i\frac{x}{2} \right) \frac{1}{4\pi } +\frac{\Gamma '}{\Gamma } \left( \frac{1}{4} -i\frac{x}{2} \right) \frac{1}{4\pi } +\frac{1}{\pi } \delta \left( x-\frac{i}{2} \right) + \frac{1}{\pi } \delta \left( x + \frac{i}{2} \right) \end{align} here they have taken the derivative of the Euler product on the critical line \frac{1}{2}+is ; also they use the Dirichlet generating function \frac{\zeta ' (s)}{\zeta(s)}= -\sum_{n=1}^{\infty} \Lambda (n) e^{-slnn} . \\! | 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. Wu and Sprung also showed that the zeta-regularized functional determinant is the Riemann Xi-function \frac{ \xi(s)}{\xi(0)} = \frac{\det(H-s(1-s)+\frac{1}{4})}{\det(H+\frac{1}{4})} The main idea inside this problem is to recover the potential from spectral data as in some inverse spectral problems in this case the spectral data is the Eigenvalue staircase, which is a quantum property of the system, the inverse of the potential then, satisfies an Abel integral equation (fractional calculus) which can be immediately solved to obtain the potential. == Asymptotics== For large x if we take only the smooth part of the eigenvalue staircase N(E) \sim \frac{\sqrt{E} }{2\pi } \log \left( \frac{\sqrt{E} }{2\pi e} \right) , then the potential as |x| \to \infty is positive and it is given by the asymptotic expression f(-x) = f(x) \sim 4\pi^2 e^2 \left( \frac{2\epsilon \sqrt{\pi } x+B}{A(\epsilon )} \right) ^{2 / \epsilon } with A(\epsilon ) = \frac{\Gamma{\left( \frac{3+\epsilon }{2} \right)}}{\Gamma{\left( 1 + \frac{\epsilon }{2} \right)}} and B = A(0) in the limit \epsilon \to 0 . 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. e^{-\frac{(x-\xi)^2}{2\omega^2}} \int_{-\infty}^{\alpha\left(\frac{x-\xi}{\omega}\right)} \frac{1}{\sqrt{2 \pi}} e^{-\frac{t^2}{2}}\ dt| cdf =\Phi\left(\frac{x-\xi}{\omega}\right)-2T\left(\frac{x-\xi}{\omega},\alpha\right) T(h,a) is Owen's T function| mean =\xi + \omega\delta\sqrt{\frac{2}{\pi}} where \delta = \frac{\alpha}{\sqrt{1+\alpha^2}}| median =| mode =\xi + \omega m_o(\alpha) | variance =\omega^2\left(1 - \frac{2\delta^2}{\pi}\right)| skewness =\gamma_1 = \frac{4-\pi}{2} \frac{\left(\delta\sqrt{2/\pi}\right)^3}{ \left(1-2\delta^2/\pi\right)^{3/2}}| kurtosis =2(\pi - 3)\frac{\left(\delta\sqrt{2/\pi}\right)^4}{\left(1-2\delta^2/\pi\right)^2}| entropy =| mgf =M_X\left(t\right)=2\exp\left(\xi t+\frac{\omega^2t^2}{2}\right)\Phi\left(\omega\delta t\right)| cf =M_X\left(i\delta\omega t\right)| char =e^{i t \xi -t^2\omega^2/2}\left(1+i\, \textrm{Erfi}\left(\frac{\delta\omega t}{\sqrt{2}}\right)\right)| }} In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness. ==Definition== Let \phi(x) denote the standard normal probability density function :\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} with the cumulative distribution function given by :\Phi(x) = \int_{-\infty}^{x} \phi(t)\ dt = \frac{1}{2} \left[ 1 + \operatorname{erf} \left(\frac{x}{\sqrt{2}}\right)\right], where "erf" is the error function. {\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. In mathematical physics, the Wu–Sprung potential, named after Hua Wu and Donald Sprung, is a potential function in one dimension inside a Hamiltonian H = p^2 + f(x) with the potential defined by solving a non-linear integral equation defined by the Bohr–Sommerfeld quantization conditions involving the spectral staircase, the energies E_n and the potential f(x) . \oint p \, dq =2 \pi n(E)= 4\int_0^a dx \sqrt{E_n - f(x)} here is a classical turning point so E = f(a) = f(-a) , the quantum energies of the model are the roots of the Riemann Xi function \xi{\left( \frac{1}{2} + i \sqrt{E_n}\right)} = 0 and f(x)=f(-x) . The trigonometric Rosen–Morse potential, named after the physicists Nathan Rosen and Philip M. Morse, is among the exactly solvable quantum mechanical potentials. ==Definition== In dimensionless units and modulo additive constants, it is defined as where r is a relative distance, \lambda is an angle rescaling parameter, and R is so far a matching length parameter. Similarly, denoting the sum of the ω functions by W : W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}), the potential energy V can be written as : V = \frac{W}{Y}. ===Lagrange equation=== The Lagrange equation for the rth variable \varphi_{r} is : \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) = \frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}} -\frac{\partial V}{\partial \varphi_{r}}. In eqs. ()-() one recognizes the one-dimensional wave equation with the trigonometric Rosen–Morse potential in () for a=\ell+1 and 2b=\alpha Z. > In this way, the cotangent term of the trigonometric Rosen–Morse potential > could be derived from the Gauss law on S^3 in combination with the > superposition principle, and could be interpreted as a dipole potential > generated by a system consisting of two opposite fundamental charges. In this manner, the complete trigonometric > Rosen–Morse potential could be derived from first principles. 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}}. 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. Introducing elliptic coordinates, : x = a \cosh \xi \cos \eta, : y = a \sinh \xi \sin \eta, the potential energy can be written as : V(\xi, \eta) = \frac{-\mu_{1}}{a\left( \cosh \xi - \cos \eta \right)} - \frac{\mu_{2}}{a\left( \cosh \xi + \cos \eta \right)} = \frac{-\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)}{a\left( \cosh^{2} \xi - \cos^{2} \eta \right)}, and the kinetic energy as : T = \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right) \left( \dot{\xi}^{2} + \dot{\eta}^{2} \right). In general, although Wu and Sprung considered only the smooth part, the potential is defined implicitly by f^{-1}(x)= \sqrt \pi \frac{d^{1/2}}{dx^{1/2}}N(x) ; with being the eigenvalue staircase N(x) = \sum_{n=0}^\infty H(x - E_{n}) and is the Heaviside step function. Inverting, taking the square root and separating the variables yields 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}}}. ==References== ==Further reading== * Category:Classical mechanics When transcribed to the current notations and units, the partition function in presents itself as, , \quad p=\beta \frac{\hbar c}{R}\alpha^2Z^2. \end{align} |}} The infinite integral has first been treated by means of partial integration giving, & = \int_0^\infty x^2e^{-ax^2}e^{\frac{p}{x^2}}{\mathrm d}x \\\ & = -\frac{1}{2a}\int_0^\infty x e^{\frac{p}{x^2}}{\mathrm d}e^{-ax^2}\\\ & = -\frac{1}{2a}xe^{\frac{p}{x^2}}e^{-ax^2}|_0^\infty + \frac{1}{2a}\int_0^\infty e^{-ax^2}{\mathrm d}\, \left( x e^{\frac{p}{x^2}}\right)\\\ & = \frac{1}{2a} \int_0^\infty e^{-ax^2 +\frac{p}{x^2}}{\mathrm d}x + \frac{2p}{2a}\int_0^\infty e^{-ax^2 +\frac{p}{x^2}}{\mathrm d}\left(\frac{1}{x}\right). \end{align} |}} Then the argument of the exponential under the sign of the integral has been cast as, {x}, \end{align} |}} thus reaching the following intermediate result, \int_0^\infty e^{-z^2}{\mathrm d} \left( x + \frac{2p}{x}\right) . \end{align} |}} As a next step the differential has been represented as -2i\sqrt{p} \right) z + \frac{1}{2} \left(\frac{1}{\sqrt{a}}+2i\sqrt{p} \right) z^\ast, \end{align} |}} an algebraic manipulation which allows to express the partition function in () in terms of the \mbox{erf}(u) function of complex argument according to, -2i\sqrt{p}\right) e^{2i\sqrt{ap}}\int_{\Gamma} e^{-z^2}{\mathrm d}z + \left( \frac{1}{\sqrt{a}} +2i\sqrt{p} \right) e^{-2i\sqrt{ap}}\int_{\Gamma} e^{- \left( z^\ast\right)^2} {\mathrm d}z^\ast \right], \\\ \end{align} |}} where \Gamma is an arbitrary path on the complex plane starting in zero and ending in u\to \infty.

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In the infrared spectrum of $\mathrm{H}^{127} \mathrm{I}$, there is an intense line at $2309 \mathrm{~cm}^{-1}$. Calculate the force constant of $\mathrm{H}^{127} \mathrm{I}$.

In a laboratory setting, the hydrogen line parameters have been more precisely measured as: : λ = 21.106114054160(30) cm : ν = 1420405751.768(2) Hz in a vacuum. Such a signal would not be overwhelmed by the H I line itself, or by any of its harmonics. ==See also== * Balmer series * Chronology of the universe * Dark Ages Radio Explorer * Hydrogen spectral series * H-alpha, the visible red spectral line with wavelength of 656.28 nanometers * Rydberg formula * Timeline of the Big Bang ==Footnotes== ==References== ==Further reading== ===Cosmology=== * * * * * * * * ==External links== * * — PAST experiment description * * * Category:Hydrogen physics Category:Emission spectroscopy Category:Radio astronomy Category:Physical cosmology Category:Astrochemistry Category:Hydrogen Available: http://physics.nist.gov/constants. The value of this constant is given here as 1/137.035999206 (note the difference in the last three digits). The constant is expressed for either hydrogen as R_\text{H}, or at the limit of infinite nuclear mass as R_\infty. Approximate value of Value of In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter alpha), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between elementary charged particles. Since the measurement of an absolute intensity in an experiment can be challenging, the ratio of different spectral line intensities can be used to achieve information about the plasma, as well. == Theory == The emission intensity density of an atomic transition from the upper state to the lower state is: P_{u \rightarrow l} = N_u \ \hbar \omega_{u \rightarrow l} \ A_{u \rightarrow l} , where: * N_u is the density of ions in the upper state, * \hbar \omega_{u \rightarrow l} is the energy of the emitted photon, which is the product of the Planck constant and the transition frequency, * A_{u \rightarrow l} is the Einstein coefficient for the specific transition. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series, but Niels Bohr later showed that its value could be calculated from more fundamental constants according to his model of the atom. In hydrogen, its wavelength of 1215.67 angstroms ( or ), corresponding to a frequency of about , places Lyman-alpha in the ultraviolet (UV) part of the electromagnetic spectrum. At higher energies, such as the scale of the Z boson, about 90 GeV, one instead measures an effective ≈ 1/127. In either case, the constant is used to express the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from a hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing a hydrogen atom from its ground state. Its numerical value is approximately , with a relative uncertainty of The constant was named by Arnold Sommerfeld, who introduced it in 1916 Equation 12a, "rund 7·" (about ...) when extending the Bohr model of the atom. quantified the gap in the fine structure of the spectral lines of the hydrogen atom, which had been measured precisely by Michelson and Morley in 1887. In spectroscopy, the Rydberg constant, symbol R_\infty for heavy atoms or R_\text{H} for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. When applied to the Planck relation, this gives: :\lambda = \frac {1}{ u} \cdot c = \frac {h}{E} \cdot c \approx \frac{\; 4.1357 \cdot 10^{-15} \ \mathrm{eV}\cdot\text{s} \;}{5.874\,33 \cdot 10^{-6}\ \mathrm{eV}}\, \cdot\, 2.9979 \cdot 10^8 \ \mathrm{m} \cdot \mathrm{s}^{-1} \approx 0.211\,06\ \mathrm{m} = 21.106\ \mathrm{cm}\; where is the wavelength of an emitted photon, is its frequency, is the photon energy, is the Planck constant, and is the speed of light. The Lyman-alpha line, typically denoted by Ly-α, is a spectral line of hydrogen (or, more generally, of any one-electron atom) in the Lyman series. The electromagnetic radiation producing this line has a frequency of (1.42 GHz), which is equivalent to a wavelength of in a vacuum. Why the constant should have this value is not understood, but there are a number of ways to measure its value. ==Definition== In terms of other fundamental physical constants, may be defined as: \alpha = \frac{e^2}{2 \varepsilon_0 h c} = \frac{e^2}{4 \pi \varepsilon_0 \hbar c} , where * is the elementary charge (); * is the Planck constant (); * is the reduced Planck constant, (6.62607015×10−34 J⋅Hz−1/2π) * is the speed of light (); * is the electric constant (). Specifically, they found that :\frac{\ \Delta \alpha\ }{\alpha} ~~ \overset{\underset{\mathsf{~def~}}{}}{=} ~~ \frac{\ \alpha _\mathrm{prev}-\alpha _\mathrm{now}\ }{\alpha_\mathrm{now}} ~~=~~ \left(-5.7\pm 1.0 \right) \times 10^{-6} ~. The first physical interpretation of the fine-structure constant was as the ratio of the velocity of the electron in the first circular orbit of the relativistic Bohr atom to the speed of light in the vacuum. The hydrogen spectral series can be expressed simply in terms of the Rydberg constant for hydrogen R_\text{H} and the Rydberg formula. The Rydberg constant for hydrogen may be calculated from the reduced mass of the electron: : R_\text{H} = R_\infty \frac{ m_\text{e} m_\text{p} }{ m_\text{e}+m_\text{p} } \approx 1.09678 \times 10^7 \text{ m}^{-1} , where * m_\text{e} is the mass of the electron, * m_\text{p} is the mass of the nucleus (a proton). === Rydberg unit of energy === The Rydberg unit of energy is equivalent to joules and electronvolts in the following manner: :1 \ \text{Ry} \equiv h c R_\infty = \frac{m_\text{e} e^4}{8 \varepsilon_{0}^{2} h^2} = \frac{e^2}{8 \pi \varepsilon_{0} a_0} = 2.179\;872\;361\;1035(42) \times 10^{-18}\ \text{J} \ = 13.605\;693\;122\;994(26)\ \text{eV}. === Rydberg frequency === :c R_\infty = 3.289\;841\;960\;2508(64) \times 10^{15}\ \text{Hz} . === Rydberg wavelength === :\frac 1 {R_\infty} = 9.112\;670\;505\;824(17) \times 10^{-8}\ \text{m}. The strength of the electromagnetic interaction varies with the strength of the energy field. | In the fields of electrical engineering and solid-state physics, the fine- structure constant is one fourth the product of the characteristic impedance of free space, ~ Z_0 = \mu_0 c = \sqrt{\frac{\mu_0}{\varepsilon_0}} , and the conductance quantum, G_0 = \frac{2 e^2}{ h }: \alpha = \tfrac{1}{4} Z_0 G_0\ .

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Calculate the percentage difference between $e^x$ and $1+x$ for $x=0.0050$

It follows that is transcendental over . ==Computation== When computing (an approximation of) the exponential function near the argument , the result will be close to 1, and computing the value of the difference e^x-1 with floating-point arithmetic may lead to the loss of (possibly all) significant figures, producing a large calculation error, possibly even a meaningless result. That is, \frac{d}{dx}e^x = e^x \quad\text{and}\quad e^0=1. Mistakenly using percentage as the unit for the standard deviation is confusing, since percentage is also used as a unit for the relative standard deviation, i.e. standard deviation divided by average value (coefficient of variation). == Related units == * Percentage (%) 1 part in 100 * Per mille (‰) 1 part in 1,000 *Basis point (bp) difference of 1 part in 10,000 *Permyriad (‱) 1 part in 10,000 * Per cent mille (pcm) 1 part in 100,000 * Baker percentage == See also == * Parts-per notation * Per-unit system * Percent point function * Relative change and difference == References == Category:Mathematical terminology Category:Probability assessment Category:Units of measurement ru:Процент#Процентный пункт thumb|Graph of f(x) = e^x (blue) with its linear approximation P_1(x) = 1 + x (red) at a = 0. In this case, the reciprocal transform of the percentage-point difference would be 1/(20pp) = 1/0.20 = 5. For example, if the exponential is computed by using its Taylor series e^x = 1 + x + \frac {x^2}2 + \frac{x^3}6 + \cdots + \frac{x^n}{n!} + \cdots, one may use the Taylor series of e^x-1: e^x-1=x+\frac {x^2}2 + \frac{x^3}6+\cdots +\frac{x^n}{n!}+\cdots. The equation \tfrac{d}{dx}e^x = e^x means that the slope of the tangent to the graph at each point is equal to its -coordinate at that point. ==Relation to more general exponential functions== The exponential function f(x) = e^x is sometimes called the natural exponential function for distinguishing it from the other exponential functions. A percentage point or percent point is the unit for the arithmetic difference between two percentages. In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable. If v e 0, the relative error is : \eta = \frac{\epsilon}{|v|} = \left| \frac{v-v_\text{approx}}{v} \right| = \left| 1 - \frac{v_\text{approx}}{v} \right|, and the percent error (an expression of the relative error) is :\delta = 100\%\times\eta = 100\%\times\left| \frac{v-v_\text{approx}}{v} \right|. 420px|thumb| As x approaches zero from the right, the magnitude of the rate of change of 1/x increases. The derivative (rate of change) of the exponential function is the exponential function itself. Percentage-point differences are one way to express a risk or probability. For measurements involving percentages as a unit, such as, growth, yield, or ejection fraction, statistical deviations and related descriptive statistics, including the standard deviation and root-mean-square error, the result should be expressed in units of percentage points instead of percentage. thumb|200px|right|Exponential functions with bases 2 and 1/2 The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). The base of the exponential function, its value at 1, e = \exp(1), is a ubiquitous mathematical constant called Euler's number. After the first occurrence, some writers abbreviate by using just "point" or "points". ==Differences between percentages and percentage points== Consider the following hypothetical example: In 1980, 50 percent of the population smoked, and in 1990 only 40 percent of the population smoked. Euler's number is the unique base for which the constant of proportionality is 1, since \ln(e) = 1, so that the function is its own derivative: \frac{d}{dx} e^x = e^x \ln (e) = e^x. Exponential constant may refer to: * e (mathematical constant) * The growth or decay constant in exponential growth or exponential decay, respectively. For real numbers and , a function of the form f(x) = a b^{cx + d} is also an exponential function, since it can be rewritten as a b^{c x + d} = \left(a b^d\right) \left(b^c\right)^x. ==Formal definition== right|thumb|The exponential function (in blue), and the sum of the first terms of its power series (in red). The natural exponential is hence denoted by x\mapsto e^x or x\mapsto \exp x. Percentage solution may refer to: * Mass fraction (or "% w/w" or "wt.%"), for percent mass * Volume fraction (or "% v/v" or "vol.%"), volume concentration, for percent volume * "Mass/volume percentage" (or "% m/v") in biology, for mass per unit volume; incorrectly used to denote mass concentration (chemistry).

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Calculate (a) the wavelength and kinetic energy of an electron in a beam of electrons accelerated by a voltage increment of $100 \mathrm{~V}$

# Quantitatively, where the intensities of diffracted beams are recorded as a function of incident electron beam energy to generate the so-called I–V curves. Following Kunio Fujiwara and Archibald Howie, the relationship between the total energy of the electrons and the wavevector is written as: :E=\frac{h^2k^2}{2m^*} with m^*=m_0 + \frac{E}{2c^2} where h is Planck's constant, m^* is a relativistic effective mass used to cancel out the relativistic terms for electrons of energy E with c the speed of light and m_0 the rest mass of the electron. Typically the energy of the electrons is written in electronvolts (eV), the voltage used to accelerate the electrons; the actual energy of each electron is this voltage times the electron charge. framed|Geometry of electron beam in precession electron diffraction. Relativistic electron beams are streams of electrons moving at relativistic speeds. The Schrödinger equation combines the kinetic energy of waves and the potential energy due to, for electrons, the Coulomb potential. 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. First, as previously noted, the electron must absorb an amount of energy equivalent to the energy difference between the electron's current energy level and an unoccupied, higher energy level in order to be promoted to that energy level. The energy released is equal to the difference in energy levels between the electron energy states. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. This can be done by photoexcitation (PE), where the electron absorbs a photon and gains all its energy or by collisional excitation (CE), where the electron receives energy from a collision with another, energetic electron. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. Combining this with the aforementioned Lorentz correction yields: I_\mathbf{g}^{kinematical} \propto I_\mathbf{g}^{experimental} \cdot g\sqrt{1-\frac{g}{2R_o}} \cdot \int\limits_0^{A_\mathbf{g}}J_0(2x)\, dx where A_\mathbf{g} = \frac{2 \pi t F_\mathbf{g}}{k} , t is the sample thickness, and k is the wave-vector of the electron beam. Johannes (Janne) Robert Rydberg (; 8 November 1854 - 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom. == Biography == Rydberg was born 8 November 1854 in Halmstad in southern Sweden, the only child of Sven Rydberg and Maria Anderson Rydberg. 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. For context, the typical energy of a chemical bond is a few eV; electron diffraction involves electrons up to 5,000,000 eV. Electron diffraction refers to changes in the direction of electron beams due to interactions with atoms. The acceleration voltage is an important quantity for the design of microwave cavities for particle accelerators. The following considerations are generalized for time-dependent fields. == Longitudinal voltage == The longitudinal effective acceleration voltage is given by the kinetic energy gain experienced by a particle with velocity \beta c along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge, V_\parallel(\beta) = \frac 1 q \vec e_s \cdot \int \vec F_L(s,t) \,\mathrm{d}s = \frac 1 q \vec e_s \cdot \int \vec F_L(s, t = \frac{s}{\beta c}) \,\mathrm d s . These include: # The simplest approximation using the de Broglie wavelength for electrons, where only the geometry is considered and often Bragg's law is invoked, a far- field or Fraunhofer approach. One of the methods is to use the concept of dressed particle. == See also == * Energy level * Mode (electromagnetism) == References == Category:Electron *Developments in the convergent-beam electron diffraction approach.

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Suppose that $10.0 \mathrm{~mol} \mathrm{C}_2 \mathrm{H}_6(\mathrm{~g})$ is confined to $4.860 \mathrm{dm}^3$ at $27^{\circ} \mathrm{C}$. Predict the pressure exerted by the ethane from the perfect gas.

Natural gas from different gas fields varies in ethane content from less than 1% to more than 6% by volume. The bond parameters of ethane have been measured to high precision by microwave spectroscopy and electron diffraction: rC−C = 1.528(3) Å, rC−H = 1.088(5) Å, and ∠CCH = 111.6(5)° by microwave and rC−C = 1.524(3) Å, rC−H = 1.089(5) Å, and ∠CCH = 111.9(5)° by electron diffraction (the numbers in parentheses represents the uncertainties in the final digits). ===Atmospheric and extraterrestrial=== Ethane occurs as a trace gas in the Earth's atmosphere, currently having a concentration at sea level of 0.5 ppb,Trace gases (archived). At standard temperature and pressure, ethane is a colorless, odorless gas. Today, ethane is an important petrochemical feedstock and is separated from the other components of natural gas in most well-developed gas fields. Global ethane quantities have varied over time, likely due to flaring at natural gas fields. Atmosphere.mpg.de. Retrieved on 2011-12-08. though its preindustrial concentration is likely to have been only around 0.25 part per billion since a significant proportion of the ethane in today's atmosphere may have originated as fossil fuels. Ethane was discovered dissolved in Pennsylvanian light crude oil by Edmund Ronalds in 1864. ==Properties== At standard temperature and pressure, ethane is a colorless, odorless gas. As far back as 1890–1891, chemists suggested that ethane molecules preferred the staggered conformation with the two ends of the molecule askew from each other. ==Production== After methane, ethane is the second-largest component of natural gas. : C2H5O• → CH3• + CH2O Some minor products in the incomplete combustion of ethane include acetaldehyde, methane, methanol, and ethanol. Ethane ( , ) is an organic chemical compound with chemical formula . Computer simulations of the chemical kinetics of ethane combustion have included hundreds of reactions. In 2006, Dale Cruikshank of NASA/Ames Research Center (a New Horizons co-investigator) and his colleagues announced the spectroscopic discovery of ethane on Pluto's surface. ==Chemistry== Ethane can be viewed as two methyl groups joined, that is, a dimer of methyl groups. This error was corrected in 1864 by Carl Schorlemmer, who showed that the product of all these reactions was in fact ethane. Solid ethane exists in several modifications. Ethane is most efficiently separated from methane by liquefying it at cryogenic temperatures. This page provides supplementary chemical data on ethane. == Material Safety Data Sheet == The handling of this chemical may incur notable safety precautions. : C2H5• + O2 → C2H5OO• : C2H5OO• + HR → C2H5OOH + •R : C2H5OOH → C2H5O• + •OH The principal carbon-containing products of incomplete ethane combustion are single-carbon compounds such as carbon monoxide and formaldehyde. Ethane can also be separated from petroleum gas, a mixture of gaseous hydrocarbons produced as a byproduct of petroleum refining. Like many hydrocarbons, ethane is isolated on an industrial scale from natural gas and as a petrochemical by-product of petroleum refining. The chemistry of ethane involves chiefly free radical reactions. They mistook the product of these reactions for the methyl radical (), of which ethane () is a dimer. In fact, ethane's global warming potential largely results from its conversion in the atmosphere to methane.Hodnebrog, Øivind; Dalsøren, Stig B. and Myrhe, Gunnar; ‘Lifetimes, direct and indirect radiative forcing, and globalwarming potentials of ethane (C2H6), propane (C3H8),and butane (C4H10)’; Atmospheric Science Letters; 2018;19:e804 It has been detected as a trace component in the atmospheres of all four giant planets, and in the atmosphere of Saturn's moon Titan.

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E

Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the standard enthalpy of solution of $\mathrm{AgCl}(\mathrm{s})$ in water from the enthalpies of formation of the solid and the aqueous ions.

The solubility product, Ksp, for AgCl in water is at room temperature, which indicates that only 1.9 mg (that is, \sqrt{1.77\times 10^{-10}} \ \mathrm{mol}) of AgCl will dissolve per liter of water. The chloride content of an aqueous solution can be determined quantitatively by weighing the precipitated AgCl, which conveniently is non-hygroscopic since AgCl is one of the few transition metal chlorides that are unreactive toward water. The standard enthalpy of formation is then determined using Hess's law. The Saha equation describes the degree of ionization for any gas in thermal equilibrium as a function of the temperature, density, and ionization energies of the atoms. Silver chloride is a chemical compound with the chemical formula AgCl. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). :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. In physics, the Saha ionization equation is an expression that relates the ionization state of a gas in thermal equilibrium to the temperature and pressure. Examples are given in the following sections. == Ionic compounds: Born–Haber cycle == For ionic compounds, the standard enthalpy of formation is equivalent to the sum of several terms included in the Born–Haber cycle. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. Above 7.5 GPa, silver chloride transitions into a monoclinic KOH phase. K (? °C), ? K (? °C), ? This is true for all enthalpies of formation. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. For a gas composed of a single atomic species, the Saha equation is written: :\frac{n_{i+1}n_e}{n_i} = \frac{2}{\lambda^{3}}\frac{g_{i+1}}{g_i}\exp\left[-\frac{(\epsilon_{i+1}-\epsilon_i)}{k_B T}\right] where: * n_i is the density of atoms in the i-th state of ionization, that is with i electrons removed. * g_i is the degeneracy of states for the i-ions * \epsilon_i is the energy required to remove i electrons from a neutral atom, creating an i-level ion. * n_e is the electron density * \lambda is the thermal de Broglie wavelength of an electron ::\lambda \ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{h^2}{2\pi m_e k_B T}} * m_e is the mass of an electron * T is the temperature of the gas * h is Planck's constant The expression (\epsilon_{i+1}-\epsilon_i) is the energy required to remove the (i+1)^{th} electron. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. '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. 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. "Standard potential of the silver-silver chloride electrode".

+65.49

36

130.41

76

0.166666666

A

Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the change in chemical potential of a perfect gas when its pressure is increased isothermally from $1.8 \mathrm{~atm}$ to $29.5 \mathrm{~atm}$ at $40^{\circ} \mathrm{C}$.

For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. 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. 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. * A partial pressure of 101.325 kPa (absolute) (1 atm, 1.01325 bar) for each gaseous reagent. 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. 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). A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. But even if the heat capacity is strictly a function of temperature for a given gas, it might be assumed constant for purposes of calculation if the temperature and heat capacity variations are not too large, which would lead to the assumption of a calorically perfect gas (see below). In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. 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. (i) Indicates values calculated from ideal gas thermodynamic functions. Table 363, Evaporation of Metals :The equations are described as reproducing the observed pressures to a satisfactory degree of approximation. == 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. It can be proved that an ideal gas (i.e. satisfying the ideal gas equation of state, PV = nRT ) is either calorically perfect or thermally perfect. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general "perfect gas" definition. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. :The Nernst equation will then give potentials at concentrations, pressures, and temperatures other than standard. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. When pressure and temperature are variable, only I-1 of I components have independent values for chemical potential and Gibbs' phase rule follows. 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 .

0.086

7166.67

7.3

-131.1

14.5115

C

Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. 3.1(a) Calculate the change in entropy when $25 \mathrm{~kJ}$ of energy is transferred reversibly and isothermally as heat to a large block of iron at $100^{\circ} \mathrm{C}$.

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). 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. * The heat capacity of the gas from the boiling point to room temperature. 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). Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. 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. 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). Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. 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. 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. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. In these equations is the base of natural logarithms, is the Planck constant, is the Boltzmann constant and the absolute temperature. ' is the ideal gas constant in units of (bar·L)/(mol·K). The amount of energy added equals , with representing the change in temperature. Therefore, the heat capacity ratio in this example is 1.4. 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. 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. * Individual Gas Constants and the Universal Gas Constant – Engineering Toolbox Category:Ideal gas Category:Physical constants Category:Amount of substance Category:Statistical mechanics Category:Thermodynamics 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 (). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . Equilibrium occurs when the temperature is equal to the melting point T = T_f so that :\Delta G_{\text{fus}} = \Delta H_{\text{fus}} - T_f \times \Delta S_{\text{fus}} = 0, and the entropy of fusion is the heat of fusion divided by the melting point: : \Delta S_{\text{fus}} = \frac {\Delta H_{\text{fus}}} {T_f} ==Helium== Helium-3 has a negative entropy of fusion at temperatures below 0.3 K. Helium-4 also has a very slightly negative entropy of fusion below 0.8 K. 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.

240

14.34457

36.0

67

-2

D

Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. A sample consisting of $3.00 \mathrm{~mol}$ of diatomic perfect gas molecules at $200 \mathrm{~K}$ is compressed reversibly and adiabatically until its temperature reaches $250 \mathrm{~K}$. Given that $C_{V, \mathrm{~m}}=27.5 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, calculate $q$.

Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. 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). 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. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. *A conversion factor is included into the original first coefficients of the equations to provide the pressure in pascals (CR2: 5.006, SMI: -0.875). 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). From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. In addition, other factors come into play and dominate during a compression cycle if they have a greater impact on the final calculated result than whether or not C_P was held constant. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. 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 (). When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general "perfect gas" definition.

27

59.4

0.0029

+7.3

0

E

Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. For the reaction $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(\mathrm{l})+3 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_2(\mathrm{~g})+3 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$, $\Delta_{\mathrm{r}} U^\ominus=-1373 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298 \mathrm{~K}$, calculate $\Delta_{\mathrm{r}} H^{\ominus}$.

==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. 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 . :*(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). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . ==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. {{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 value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . The USSA1976 acknowledges that this value is not consistent with the cited values for the Avogadro constant and the Boltzmann constant. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. * Individual Gas Constants and the Universal Gas Constant – Engineering Toolbox Category:Ideal gas Category:Physical constants Category:Amount of substance Category:Statistical mechanics Category:Thermodynamics To convert from L^2bar/mol^2 to m^6 Pa/mol^2, divide by 10. a (L2bar/mol2) b (L/mol) Acetic acid 17.7098 0.1065 Acetic anhydride 20.158 0.1263 Acetone 16.02 0.1124 Acetonitrile 17.81 0.1168 Acetylene 4.516 0.0522 Ammonia 4.225 0.0371 Aniline 29.14 0.1486 Argon 1.355 0.03201 Benzene 18.24 0.1193 Bromobenzene 28.94 0.1539 Butane 14.66 0.1226 1-Butanol 20.94 0.1326 2-Butanone 19.97 0.1326 Carbon dioxide 3.640 0.04267 Carbon disulfide 11.77 0.07685 Carbon monoxide 1.505 0.0398500 Carbon tetrachloride 19.7483 0.1281 Chlorine 6.579 0.05622 Chlorobenzene 25.77 0.1453 Chloroethane 11.05 0.08651 Chloromethane 7.570 0.06483 Cyanogen 7.769 0.06901 Cyclohexane 23.11 0.1424 Cyclopropane 8.34 0.0747 Decane 52.74 0.3043 1-Decanol 59.51 0.3086 Diethyl ether 17.61 0.1344 Diethyl sulfide 19.00 0.1214 Dimethyl ether 8.180 0.07246 Dimethyl sulfide 13.04 0.09213 Dodecane 69.38 0.3758 1-Dodecanol 75.70 0.3750 Ethane 5.562 0.0638 Ethanethiol 11.39 0.08098 Ethanol 12.18 0.08407 Ethyl acetate 20.72 0.1412 Ethylamine 10.74 0.08409 Ethylene 4.612 0.0582 Fluorine 1.171 0.0290 Fluorobenzene 20.19 0.1286 Fluoromethane 4.692 0.05264 Freon 10.78 0.0998 Furan 12.74 0.0926 Germanium tetrachloride 22.90 0.1485 Helium 0.0346 0.0238 Heptane 31.06 0.2049 1-Heptanol 38.17 0.2150 Hexane 24.71 0.1735 1-Hexanol 31.79 0.1856 Hydrazine 8.46 0.0462 Hydrogen 0.2476 0.02661 Hydrogen bromide 4.510 0.04431 Hydrogen chloride 3.716 0.04081 Hydrogen cyanide 11.29 0.0881 Hydrogen fluoride 9.565 0.0739 Hydrogen iodide 6.309 0.0530 Hydrogen selenide 5.338 0.04637 Hydrogen sulfide 4.490 0.04287 Isobutane 13.32 0.1164 Iodobenzene 33.52 0.1656 Krypton 2.349 0.03978 Mercury 8.200 0.01696 Methane 2.283 0.04278 Methanol 9.649 0.06702 Methylamine 7.106 0.0588 Neon 0.2135 0.01709 Neopentane 17.17 0.1411 Nitric oxide 1.358 0.02789 Nitrogen 1.370 0.0387 Nitrogen dioxide 5.354 0.04424 Nitrogen trifluoride 3.58 0.0545 Nitrous oxide 3.832 0.04415 Octane 37.88 0.2374 1-Octanol 44.71 0.2442 Oxygen 1.382 0.03186 Ozone 3.570 0.0487 Pentane 19.26 0.146 1-Pentanol 25.88 0.1568 Phenol 22.93 0.1177 Phosphine 4.692 0.05156 Propane 8.779 0.08445 1-Propanol 16.26 0.1079 2-Propanol 15.82 0.1109 Propene 8.442 0.0824 Pyridine 19.77 0.1137 Pyrrole 18.82 0.1049 Radon 6.601 0.06239 Silane 4.377 0.05786 Silicon tetrafluoride 4.251 0.05571 Sulfur dioxide 6.803 0.05636 Sulfur hexafluoride 7.857 0.0879 Tetrachloromethane 20.01 0.1281 Tetrachlorosilane 20.96 0.1470 Tetrafluoroethylene 6.954 0.0809 Tetrafluoromethane 4.040 0.0633 Tetrafluorosilane 5.259 0.0724 Tetrahydrofuran 16.39 0.1082 Tin tetrachloride 27.27 0.1642 Thiophene 17.21 0.1058 Toluene 24.38 0.1463 1-1-1-Trichloroethane 20.15 0.1317 Trichloromethane 15.34 0.1019 Trifluoromethane 5.378 0.0640 Trimethylamine 13.37 0.1101 Water 5.536 0.03049 Xenon 4.250 0.05105 ==Units== 1 J·m3/mol2 = 1 m6·Pa/mol2 = 10 L2·bar/mol2 1 L2atm/mol2 = 0.101325 J·m3/mol2 = 0.101325 Pa·m6/mol2 1 dm3/mol = 1 L/mol = 1 m3/kmol (where kmol is kilomoles = 1000 moles) ==References== Category:Gas laws Constants (Data Page) :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). 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. 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. Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. This disparity is not a significant departure from accuracy, and USSA1976 uses this value of R∗ for all the calculations of the standard atmosphere. 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. : R_{\rm specific} = \frac{R}{M} Just as the molar gas constant can be related to the Boltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by the molecular mass of the gas. As pressure is defined as force per area of measurement, the gas equation can also be written as: :R = \frac{ \dfrac{\mathrm{force}}{\mathrm{area}} \times \mathrm{volume} } { \mathrm{amount} \times \mathrm{temperature} } Area and volume are (length)2 and (length)3 respectively.

0.11

209.1

'-1368.0'

1.51

1.88

C

Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2$ occupies a fixed volume of $15.0 \mathrm{dm}^3$ at $300 \mathrm{~K}$. When it is supplied with $2.35 \mathrm{~kJ}$ of energy as heat its temperature increases to $341 \mathrm{~K}$. Assume that $\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $w$.

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 (). 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. 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. 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== * * . 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.. 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. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. 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). The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. 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. 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. 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. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . == 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. 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. 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. Therefore, the heat capacity ratio in this example is 1.4. 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 value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. Van der Waals' doctoral research culminated in an 1873 dissertation that provided a semi- quantitative theory describing the gas-liquid change of state and the origin of a critical temperature, Over de Continuïteit van den Gas- en Vloeistoftoestand (Dutch; in English, On the Continuity of the Gas and Liquid State).

0

11000

15.0

8.87

0.1792

A

The density of a gaseous compound was found to be $1.23 \mathrm{kg} \mathrm{m}^{-3}$ at $330 \mathrm{K}$ and $20 \mathrm{kPa}$. What is the molar mass of the compound?

The molar mass of molecules of these elements is the molar mass of the atoms multiplied by the number of atoms in each molecule: :\begin{array}{lll} M(\ce{H2}) &= 2\times 1.00797(7) \times M_\mathrm{u} &= 2.01588(14) \text{ g/mol} \\\ M(\ce{S8}) &= 8\times 32.065(5) \times M_\mathrm{u} &= 256.52(4) \text{ g/mol} \\\ M(\ce{Cl2}) &= 2\times 35.453(2) \times M_\mathrm{u} &= 70.906(4) \text{ g/mol} \end{array} == Molar masses of compounds == The molar mass of a compound is given by the sum of the relative atomic mass of the atoms which form the compound multiplied by the molar mass constant : :M = M_{\rm u} M_{\rm r} = M_{\rm u} \sum_i {A_{\rm r}}_i. Examples are: \begin{array}{ll} M(\ce{NaCl}) &= \bigl[22.98976928(2) + 35.453(2)\bigr] \times 1 \text{ g/mol} \\\ &= 58.443(2) \text{ g/mol} \\\\[4pt] M(\ce{C12H22O11}) &= \bigl[12 \times 12.0107(8) + 22 \times 1.00794(7) + 11 \times 15.9994(3)\bigr] \times 1 \text{ g/mol} \\\ &= 342.297(14) \text{ g/mol} \end{array} An average molar mass may be defined for mixtures of compounds. In chemistry, the molar mass () of a chemical compound is defined as the ratio between the mass and the amount of substance (measured in moles) of any sample of said compound. The molar mass of a compound in g/mol thus is equal to the mass of this number of molecules of the compound in grams. == Molar masses of elements == The molar mass of atoms of an element is given by the relative atomic mass of the element multiplied by the molar mass constant, For normal samples from earth with typical isotope composition, the atomic weight can be approximated by the standard atomic weight or the conventional atomic weight. :\begin{array}{lll} M(\ce{H}) &= 1.00797(7) \times M_\mathrm{u} &= 1.00797(7) \text{ g/mol} \\\ M(\ce{S}) &= 32.065(5) \times M_\mathrm{u} &= 32.065(5) \text{ g/mol} \\\ M(\ce{Cl}) &= 35.453(2) \times M_\mathrm{u} &= 35.453(2) \text{ g/mol} \\\ M(\ce{Fe}) &= 55.845(2) \times M_\mathrm{u} &= 55.845(2) \text{ g/mol} \end{array} Multiplying by the molar mass constant ensures that the calculation is dimensionally correct: standard relative atomic masses are dimensionless quantities (i.e., pure numbers) whereas molar masses have units (in this case, grams per mole). The molar mass is an average of many instances of the compound, which often vary in mass due to the presence of isotopes. As an example, the average molar mass of dry air is 28.97 g/mol.The Engineering ToolBox Molecular Mass of Air == Related quantities == Molar mass is closely related to the relative molar mass () of a compound, to the older term formula weight (F.W.), and to the standard atomic masses of its constituent elements. Thus, for example, the average mass of a molecule of water is about 18.0153 daltons, and the molar mass of water is about 18.0153 g/mol. Molar masses typically vary between: :1–238 g/mol for atoms of naturally occurring elements; : for simple chemical compounds; : for polymers, proteins, DNA fragments, etc. * Molar mass: chemistry second-level course. If represents the mass fraction of the solute in solution, and assuming no dissociation of the solute, the molar mass is given by :M = {{wK_\text{b}}\over{\Delta T}}.\ == See also == * Mole map (chemistry) == References == ==External links== * HTML5 Molar Mass Calculator web and mobile application. The mass average molar mass is calculated by \bar{M}_w=\frac{\sum_i N_iM_i^2}{\sum_i N_iM_i} where is the number of molecules of molecular mass . The molar mass is appropriate for converting between the mass of a substance and the amount of a substance for bulk quantities. * Online Molar Mass Calculator with the uncertainty of M and all the calculations shown * Molar Mass Calculator Online Molar Mass and Elemental Composition Calculator * Stoichiometry Add-In for Microsoft Excel for calculation of molecular weights, reaction coefficients and stoichiometry. The molecular formula C23H21NO (molar mass: 327.42 g/mol, exact mass: 327.1623 u) may refer to: * JWH-015 * JWH-073 * JWH-120 Category:Molecular formulas In the International System of Units (SI), the coherent unit of molar mass is kg/mol. The molecular formula C13H28O (molar mass: 200.36 g/mol, exact mass: 200.2140 u) may refer to: * 2,2,4,4-Tetramethyl-3-t-butyl-pentane-3-ol, or tri-tert- butylcarbinol * 1-Tridecanol In polymer chemistry, the molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species () and the molar mass () of that species.I. Katime "Química Física Macromolecular". The molar mass of any element or compound is its relative atomic mass (atomic weight) multiplied by the molar mass constant. Here, is the relative molar mass, also called formula weight. Gram atomic mass is another term for the mass, in grams, of one mole of atoms of that element. The molar mass constant was thus given by :M_{\text{u}} = {\text{molar mass }[M( ^{12}\mathrm{C} )]\over \text{relative atomic weight }[A_{\text{r}}( ^{12}\mathrm{C} )]} = {{12\ {\rm g/mol}}\over 12}=1\ \rm g/mol The molar mass constant is related to the mass of a carbon-12 atom in grams: :m({}^{12}{\text{C}}) = \frac{12 \times M_{\text{u}}}{N_{\text{A}}} The Avogadro constant being a fixed value, the mass of a carbon-12 atom depends on the accuracy and precision of the molar mass constant. The molecular formula C12H15N5O3 (molar mass: 277.28 g/mol, exact mass: 277.1175 u) may refer to: * Entecavir (ETV) * Queuine (Q)

169

4.56

258.14

1.8

57.2

A

Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A sample of $4.50 \mathrm{~g}$ of methane occupies $12.7 \mathrm{dm}^3$ at $310 \mathrm{~K}$. Calculate the work that would be done if the same expansion occurred reversibly.

In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. We assume the expansion occurs without exchange of heat (adiabatic expansion). The reaction depends on a delicate balance between methane pressure and catalyst concentration, and consequently more work is being done to further improve yields. ==References== Category:Organometallic chemistry Category:Organic chemistry Category:Chemistry Category:Methane The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. Despite this, if the density is fairly low and intermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, ), which reflects the relatively constant difference in work done during expansion for constant pressure vs. constant volume conditions. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. 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. A methane reformer is a device based on steam reforming, autothermal reforming or partial oxidation and is a type of chemical synthesis which can produce pure hydrogen gas from methane using a catalyst. But even if the heat capacity is strictly a function of temperature for a given gas, it might be assumed constant for purposes of calculation if the temperature and heat capacity variations are not too large, which would lead to the assumption of a calorically perfect gas (see below). For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. An experimental value should be used rather than one based on this approximation, where possible. Methane functionalization is the process of converting methane in its gaseous state to another molecule with a functional group, typically methanol or acetic acid, through the use of transition metal catalysts. 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). Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. For example, a comparison of calculations for one compression stage of an axial compressor (one with variable C_P and one with constant C_P) may produce a deviation small enough to support this approach. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at .

-167

1.8

6.64

0.01961

0.396

A

Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A strip of magnesium of mass $15 \mathrm{~g}$ is placed in a beaker of dilute hydrochloric acid. Calculate the work done by the system as a result of the reaction. The atmospheric pressure is 1.0 atm and the temperature $25^{\circ} \mathrm{C}$.

The quantity of thermodynamic work is defined as work done by the system on its surroundings. Thermodynamic work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. Given only the initial state and the final state of the system, one can only say what the total change in internal energy was, not how much of the energy went out as heat, and how much as work. Such work done by compression is thermodynamic work as here defined. An electric discharge through hydrogen gas at low pressure (20 pascals) containing pieces of magnesium can produce MgH. The reaction that produces it is either or Mg + H → MgH. The work done by the system on its surroundings is calculated from the quantities that constitute the whole cycle.Lavenda, B.H. (2010). Work done on a thermodynamic system, by devices or systems in the surroundings, is performed by actions such as compression, and includes shaft work, stirring, and rubbing. The reaction of Mg atoms with (dihydrogen gas) is actually endothermic and proceeds when magnesium atoms are excited electronically. Otherwise in these stars, below any magnesium silicate clouds where the temperature is hotter, the concentration of MgH is proportional to the square root of the pressure, and concentration of magnesium, and 10−4236/T. MgH is the second most abundant magnesium containing gas (after atomic magnesium) in the deeper hotter parts of planets and brown dwarfs. Thermodynamic work is defined for the purposes of thermodynamic calculations about bodies of material, known as thermodynamic systems. The work is due to change of system volume by expansion or contraction of the system. First, one assumes that the given reaction at constant temperature and pressure is the only one that is occurring. Bulk properties of the MgH gas include enthalpy of formation of 229.79 kJ mol−1, entropy 193.20 J K−1 mol−1 and heat capacity of 29.59 J K−1 mol−1. Magnesium monohydride is a molecular gas with formula MgH that exists at high temperatures, such as the atmospheres of the Sun and stars. A complete reaction takes 20 to 24 hours at 1,200 °C." Thermodynamic work done by a thermodynamic system on its surroundings is defined so as to comply with this principle. Several kinds of thermodynamic work are especially important. Atmospheric Thermodynamics. When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. Both the temperature change ∆T of the water and the height of the fall ∆h of the weight mg were recorded. As a result, the work done by the system also depends on the initial and final states.

17.4

-1.5

1.5

14.34457

48

B

Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. For a van der Waals gas, $\pi_T=a / V_{\mathrm{m}}^2$. Calculate $\Delta U_{\mathrm{m}}$ for the isothermal expansion of nitrogen gas from an initial volume of $1.00 \mathrm{dm}^3$ to $24.8 \mathrm{dm}^3$ at $298 \mathrm{~K}$.

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 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. 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. 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. 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. 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.. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. 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 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. 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. It is of some historical interest to point out that Van der Waals, in his Nobel prize lecture, gave credit to Laplace for the argument that pressure is reduced proportional to the square of the density. ===Statistical thermodynamics derivation=== The canonical partition function Z of an ideal gas consisting of N = nNA identical (non-interacting) particles, is: : Z = \frac{z^N}{N!}\quad \hbox{with}\quad z = \frac{V}{\Lambda^3} where \Lambda is the thermal de Broglie wavelength, : \Lambda = \sqrt{\frac{h^2}{2\pi m k T}} with the usual definitions: h is the Planck constant, m the mass of a particle, k the Boltzmann constant and T the absolute temperature. 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 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. 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. 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. This apparent discrepancy is resolved in the context of vapour–liquid equilibrium: at a particular temperature, there exist two points on the Van der Waals isotherm that have the same chemical potential, and thus a system in thermodynamic equilibrium will appear to traverse a straight line on the p–V diagram as the ratio of vapour to liquid changes. 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. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. Van der Waals' doctoral research culminated in an 1873 dissertation that provided a semi- quantitative theory describing the gas-liquid change of state and the origin of a critical temperature, Over de Continuïteit van den Gas- en Vloeistoftoestand (Dutch; in English, On the Continuity of the Gas and Liquid State). 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). 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 ().

313

122

0.6

3.333333333

131

E

Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. Take nitrogen to be a van der Waals gas with $a=1.352 \mathrm{dm}^6 \mathrm{~atm} \mathrm{\textrm {mol } ^ { - 2 }}$ and $b=0.0387 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, and calculate $\Delta H_{\mathrm{m}}$ when the pressure on the gas is decreased from $500 \mathrm{~atm}$ to $1.00 \mathrm{~atm}$ at $300 \mathrm{~K}$. For a van der Waals gas, $\mu=\{(2 a / R T)-b\} / C_{p, \mathrm{~m}}$. Assume $C_{p, \mathrm{~m}}=\frac{7}{2} R$.

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. 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. The table below for a selection of gases uses the following conventions: * T_c: critical temperature [K] * P_c: critical pressure [Pa] * v_c: critical specific volume [m3⋅kg−1] * R: gas constant (8.314 J⋅K−1⋅mol−1) * \mu: Molar mass [kg⋅mol−1] Substance P_c [Pa] T_c [K] v_c [m3/kg] Z_c H2O 647.3 0.23 4He 5.2 0.31 He 5.2 0.30 H2 33.2 0.30 Ne 44.5 0.29 N2 126.2 0.29 Ar 150.7 0.29 Xe 289.7 0.29 O2 154.8 0.291 CO2 304.2 0.275 SO2 430.0 0.275 CH4 190.7 0.285 C3H8 370.0 0.267 ==See also== *Van der Waals equation *Equation of state *Compressibility factors *Johannes Diderik van der Waals equation *Noro- Frenkel law of corresponding states ==References== ==External links== * Properties of Natural Gases. 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. 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. 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). 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. 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. 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. 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. 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. 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. 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 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. 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. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. 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 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 = . "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. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases.

30

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Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the molar entropy of a constant-volume sample of neon at $500 \mathrm{~K}$ given that it is $146.22 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ at $298 \mathrm{~K}$.

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. 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 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. 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). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . 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. 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. According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Substituting this into the above equation along with V=[\mathrm{g}]/\rho\, and \gamma = 5/3\, for an ideal monatomic gas one finds : K = \frac{k_{B}T}{(\rho/\mu m_{H})^{2/3}}, where \mu\, is the mean molecular weight of the gas or plasma; and m_{H}\, is the mass of the Hydrogen atom, which is extremely close to the mass of the proton, m_{p}\,, the quantity more often used in astrophysical theory of galaxy clusters. It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per amount of substance, i.e. the pressure–volume product, rather than energy per temperature increment per particle. As a consequence, the SI value of the molar gas constant is exactly . This quantity relates to the thermodynamic entropy as : \Delta S = 3/2 \ln K . : R_{\rm specific} = \frac{R}{M} Just as the molar gas constant can be related to the Boltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by the molecular mass of the gas. 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. 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. Instead of a mole the constant can be expressed by considering the normal cubic meter. * The heat capacity of the gas from the boiling point to room temperature. 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. Otherwise, we can also say that: :\mathrm{force} = \frac{ \mathrm{mass} \times \mathrm{length} } { (\mathrm{time})^2 } Therefore, we can write R as: :R = \frac{ \mathrm{mass} \times \mathrm{length}^2 } { \mathrm{amount} \times \mathrm{temperature} \times (\mathrm{time})^2 } And so, in terms of SI base units: :R = . ==Relationship with the Boltzmann constant== The Boltzmann constant kB (alternatively k) may be used in place of the molar gas constant by working in pure particle count, N, rather than amount of substance, n, since :R = N_{\rm A} k_{\rm B},\, where NA is the Avogadro constant. 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).

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C

Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the final temperature of a sample of argon of mass $12.0 \mathrm{~g}$ that is expanded reversibly and adiabatically from $1.0 \mathrm{dm}^3$ at $273.15 \mathrm{~K}$ to $3.0 \mathrm{dm}^3$.

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). 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. * J.V. Iribarne and W.L. Godson, Atmospheric Thermodynamics, published by D. Reidel Publishing Company, Dordrecht, Holland, 1973, 222 pages Category:Atmospheric thermodynamics Category:Atmospheric temperature Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. 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. 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). To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. The constant-pressure adiabatic flame temperature of such substances in air is in a relatively narrow range around 1950 °C. 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). The molecular-scale temperature is the defining property of the U.S. Standard Atmosphere, 1962. We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. 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 will depend on the latent heat release as: T_e \approx T + \frac{L_v}{c_{pd}} r where: * L_v : latent heat of evaporation (2400 kJ/kg at 25°C to 2600 kJ/kg at −40°C) * c_{pd} : specific heat at constant pressure for air (≈ 1004 J/(kg·K)) Tables exist for exact values of the last two coefficients. == See also == * Wet-bulb temperature * Potential temperature * Atmospheric thermodynamics * Equivalent potential temperature ==Bibliography== * M Robitzsch, Aequivalenttemperatur und Aequivalentthemometer, Meteorologische Zeitschrift, 1928, pp. 313-315. There are two types adiabatic flame temperature: constant volume and constant pressure, depending on how the process is completed. 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. 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 (). The following figure illustrates that the effects of dissociation tend to lower the adiabatic flame temperature. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. thumb|upright|Ethanol burning with its spectrum depicted In the study of combustion, the adiabatic flame temperature is the temperature reached by a flame under ideal conditions. Temperatures in the atmosphere decrease with height at an average rate of 6.5°C (11.7°F) per kilometer. Adiabatic flame temperature (constant pressure) of common fuels Fuel Oxidizer 1 bar 20 °C T_\text{ad} T_\text{ad} Fuel Oxidizer 1 bar 20 °C (°C) (°F) Acetylene () Air 2500 4532 Acetylene () Oxygen 3480 6296 Butane () Air 2231 4074 Cyanogen () Oxygen 4525 8177 Dicyanoacetylene () Oxygen 4990 9010 Ethane () Air 1955 3551 Ethanol () Air 2082 3779Flame Temperature Analysis and NOx Emissions for Different Fuels Gasoline Air 2138 3880 Hydrogen () Air 2254 4089 Magnesium (Mg) Air 1982 3600 Methane () Air 1963 3565CRC Handbook of Chemistry and Physics, 96th Edition, p. 15-51 Methanol () Air 1949 3540 Naphtha Air 2533 4591 Natural gas Air 1960 3562 Pentane () Air 1977 3591 Propane () Air 1980 3596 Methylacetylene () Air 2010 3650 Methylacetylene () Oxygen 2927 5301 Toluene () Air 2071 3760 Wood Air 1980 3596 Kerosene Air 2093Power Point Presentation: Flame Temperature, Hsin Chu, Department of Environmental Engineering, National Cheng Kung University, Taiwan 3801 Light fuel oil Air 2104 3820 Medium fuel oil Air 2101 3815 Heavy fuel oil Air 2102 3817 Bituminous Coal Air 2172 3943 Anthracite Air 2180 3957 Anthracite Oxygen ≈3500Analysis of oxy-fuel combustion power cycle utilizing a pressurized coal combustor by Jongsup Hong et al., MIT, which cites .

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A

Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the change in entropy when $25 \mathrm{~kJ}$ of energy is transferred reversibly and isothermally as heat to a large block of iron at $0^{\circ} \mathrm{C}$.

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. 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 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 vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. * The heat capacity of the gas from the boiling point to room temperature. 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. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. 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. According to Trouton's rule, the entropy of vaporization (at standard pressure) of most liquids has similar values. Equilibrium occurs when the temperature is equal to the melting point T = T_f so that :\Delta G_{\text{fus}} = \Delta H_{\text{fus}} - T_f \times \Delta S_{\text{fus}} = 0, and the entropy of fusion is the heat of fusion divided by the melting point: : \Delta S_{\text{fus}} = \frac {\Delta H_{\text{fus}}} {T_f} ==Helium== Helium-3 has a negative entropy of fusion at temperatures below 0.3 K. Helium-4 also has a very slightly negative entropy of fusion below 0.8 K. In astrophysics, what is referred to as "entropy" is actually the adiabatic constant derived as follows. This quantity relates to the thermodynamic entropy as : \Delta S = 3/2 \ln K . According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. The concept of entropy developed in response to the observation that a certain amount of functional energy released from combustion reactions is always lost to dissipation or friction and is thus not transformed into useful work. The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. In thermodynamics, the entropy of fusion is the increase in entropy when melting a solid substance. At standard pressure , the value is denoted as and normally expressed in joules per mole-kelvin, J/(mol·K). Thus, using the above description, we can calculate the entropy change ΔS for the passage of the quantity of heat Q from the temperature T1, through the "working body" of fluid, which was typically a body of steam, to the temperature T2 as shown below: If we make the assignment: : S= \frac {Q}{T} Then, the entropy change or "equivalence-value" for this transformation is: : \Delta S = S_{\rm final} - S_{\rm initial} \, which equals: : \Delta S = \left(\frac {Q}{T_2} - \frac {Q}{T_1}\right) and by factoring out Q, we have the following form, as was derived by Clausius: : \Delta S = Q\left(\frac {1}{T_2} - \frac {1}{T_1}\right) ==1856 definition== In 1856, Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form: :\int \frac{\delta Q}{T} = -N where N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Changes in entropy are associated with phase transitions and chemical reactions. In these equations is the base of natural logarithms, is the Planck constant, is the Boltzmann constant and the absolute temperature. ' is the ideal gas constant in units of (bar·L)/(mol·K). 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 Jaynes (1957) Information theory and statistical mechanics II, Physical Review 108:171 the statistical thermodynamic entropy can be seen as just a particular application of Shannon's information entropy to the probabilities of particular microstates of a system occurring in order to produce a particular macrostate. ==Popular use== The term entropy is often used in popular language to denote a variety of unrelated phenomena.

0

-3.0

5.85

92

10

D

Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2$ occupies a fixed volume of $15.0 \mathrm{dm}^3$ at $300 \mathrm{~K}$. When it is supplied with $2.35 \mathrm{~kJ}$ of energy as heat its temperature increases to $341 \mathrm{~K}$. Assume that $\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $\Delta H$.

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 (). 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. 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== * * . 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. 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.. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. 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. "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. 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 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. 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. 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. == 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. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. 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. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. 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 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). Therefore, the heat capacity ratio in this example is 1.4. 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. 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.

30

5

205.0

0.66666666666

+3.03

E

Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. Concerns over the harmful effects of chlorofluorocarbons on stratospheric ozone have motivated a search for new refrigerants. One such alternative is 2,2-dichloro-1,1,1-trifluoroethane (refrigerant 123). Younglove and McLinden published a compendium of thermophysical properties of this substance (J. Phys. Chem. Ref. Data 23, 7 (1994)), from which properties such as the Joule-Thomson coefficient $\mu$ can be computed. Compute $\mu$ at 1.00 bar and $50^{\circ} \mathrm{C}$ given that $(\partial H / \partial p)_T=-3.29 \times 10^3 \mathrm{~J} \mathrm{MPa}^{-1} \mathrm{~mol}^{-1}$ and $C_{p, \mathrm{~m}}=110.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$.

Property Value Ozone depletion potential (ODP) 0.44 (CCl3F = 1) Global warming potential (GWP: 100-year) 5,860 \- 7,670 (CO2 = 1) Atmospheric lifetime 1,020 \- 1,700 years == See also == * IPCC list of greenhouse gases * List of refrigerants == References == Category:Ozone depletion Category:Greenhouse gases Category:Refrigerants Category:Chlorofluorocarbons ==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 1,1,1,2-Tetrafluoroethane (also known as norflurane (INN), R-134a, Klea®134a, Freon 134a, Forane 134a, Genetron 134a, Green Gas, Florasol 134a, Suva 134a, or HFC-134a) is a hydrofluorocarbon (HFC) and haloalkane refrigerant with thermodynamic properties similar to R-12 (dichlorodifluoromethane) but with insignificant ozone depletion potential and a lower 100-year global warming potential (1,430, compared to R-12's GWP of 10,900). Even though 1,1,1,2-Tetrafluoroethane has insignificant ozone depletion potential (ozone layer) and negligible acidification potential (acid rain), it has a 100-year global warming potential (GWP) of 1430 and an approximate atmospheric lifetime of 14 years. Retrieved 21 August 2011. == History and environmental impacts == 1,1,1,2-Tetrafluoroethane was introduced in the early 1990s as a replacement for dichlorodifluoromethane (R-12), which has massive ozone depleting properties. Some of the properties of cyclic ozone have been predicted theoretically. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. This colorless gas is of interest as a more environmentally friendly (lower GWP; global warming potential) refrigerant in air conditioners. It has the formula CFCHF and a boiling point of −26.3 °C (−15.34 °F) at atmospheric pressure. The Society of Automotive Engineers (SAE) has proposed that it be best replaced by a new fluorochemical refrigerant HFO-1234yf (CFCF=CH) in automobile air-conditioning systems.HFO-1234yf A Low GWP Refrigerant For MAC . Chloropentafluoroethane is a chlorofluorocarbon (CFC) once used as a refrigerant and also known as R-115 and CFC-115. A phaseout and transition to HFO-1234yf and other refrigerants, with GWPs similar to CO2, began in 2012 within the automotive market. == Uses == 1,1,1,2-Tetrafluoroethane is a non-flammable gas used primarily as a "high-temperature" refrigerant for domestic refrigeration and automobile air conditioners. It has also been studied as a potential inhalational anesthetic, but it is nonanaesthetic at doses used in inhalers. == See also == * List of refrigerants * Tetrabromoethane * Tetrachloroethane == References == == External links == * * European Fluorocarbons Technical Committee (EFCTC) * MSDS at Oxford University * Concise International Chemical Assessment Document 11, at inchem.org * Pressure temperature calculator * * R134a 2 phase computer cooling Category:Fluoroalkanes Category:Refrigerants Category:Automotive chemicals Category:Propellants Category:Airsoft Category:Excipients Category:Greenhouse gases Category:GABAA receptor positive allosteric modulators Category:General anesthetics It should have more energy than ordinary ozone. It has been speculated that, if cyclic ozone could be made in bulk, and if it proved to have good stability properties, it could be added to liquid oxygen to improve the specific impulse of rocket fuel. 1-Chloro-3,3,3-trifluoropropene (HFO-1233zd) is the unsaturated chlorofluorocarbon with the formula HClC=C(H)CF3. Currently, the possibility of cyclic ozone is confirmed within diverse theoretical approaches. ==References== ==External links== * Category:Allotropes of oxygen Category:Hypothetical chemical compounds Category:Three-membered rings Category:Homonuclear triatomic molecules It would differ from ordinary ozone in how those three oxygen atoms are arranged. Its production and consumption has been banned since 1 January 1996 under the Montreal Protocol because of its high ozone depletion potential and very long lifetime when released into the environment.Ozone Depleting Substances List (Montreal Protocol) CFC-115 is also a potent greenhouse gas. ==Atmospheric properties== The atmospheric abundance of CFC-115 rose from 8.4 parts per trillion (ppt) in year 2010 to 8.7 ppt in 2020 based on analysis of air samples gathered from sites around the world. In ordinary ozone, the atoms are arranged in a bent line; in cyclic ozone, they would form an equilateral triangle. Thus it was included in the IPCC list of greenhouse gases. thumb|left|200px|HFC-134a atmospheric concentration since year 1995. Cyclic ozone has not been made in bulk, although at least one researcher has attempted to do so using lasers.

200

29.9

252.8

6.6

0.84

B

A gas at $250 \mathrm{~K}$ and $15 \mathrm{~atm}$ has a molar volume 12 per cent smaller than that calculated from the perfect gas law. Calculate the molar volume of the gas.

The molar volume of an ideal gas at 100 kPa (1 bar) is : at 0 °C, : at 25 °C. The molar volume of an ideal gas at 1 atmosphere of pressure is : at 0 °C, : at 25 °C. == Crystalline solids == For crystalline solids, the molar volume can be measured by X-ray crystallography. This gives rise to the molar volume of a gas, which at STP (273.15 K, 1 atm) is about 22.4 L. This follows from above where the specific volume is the reciprocal of the density of a substance: V_{\rm m,i} = {M_i \over \rho_i^0} = M_i v_i == Ideal gases == For ideal gases, the molar volume is given by the ideal gas equation; this is a good approximation for many common gases at standard temperature and pressure. It shows the relationship between the pressure, volume, and temperature for a fixed mass (quantity) of gas: :PV = k_5 T This can also be written as: : \frac {P_1V_1}{T_1}= \frac {P_2V_2}{T_2} With the addition of Avogadro's law, the combined gas law develops into the ideal gas law: :PV = nRT :where :*P is pressure :*V is volume :*n is the number of moles :*R is the universal gas constant :*T is temperature (K) :The proportionality constant, now named R, is the universal gas constant with a value of 8.3144598 (kPa∙L)/(mol∙K). The ideal gas equation can be rearranged to give an expression for the molar volume of an ideal gas: V_{\rm m} = \frac{V}{n} = \frac{RT}{P} Hence, for a given temperature and pressure, the molar volume is the same for all ideal gases and is based on the gas constant: R = , or about . The relation is given by :V \propto n\,, or :\frac{V_1}{n_1}=\frac{V_2}{n_2} \, :where n is equal to the number of molecules of gas (or the number of moles of gas). ==Combined and ideal gas laws== The Combined gas law or General Gas Equation is obtained by combining Boyle's Law, Charles's law, and Gay-Lussac's Law. An equivalent formulation of this law is: :PV = Nk_\text{B}T :where :*P is the pressure :*V is the volume :*N is the number of gas molecules :*kB is the Boltzmann constant (1.381×10−23J·K−1 in SI units) :*T is the temperature (K) These equations are exact only for an ideal gas, which neglects various intermolecular effects (see real gas). It is equal to the molar mass (M) divided by the mass density (ρ): V_{\text{m}} = \frac{M}{\rho} The molar volume has the SI unit of cubic metres per mole (m3/mol), although it is more typical to use the units cubic decimetres per mole (dm3/mol) for gases, and cubic centimetres per mole (cm3/mol) for liquids and solids. ==Definition== thumb|Change in volume with increasing ethanol fraction. 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. However, the ideal gas law is a good approximation for most gases under moderate pressure and temperature. # If the number of gas molecules and the temperature remain constant, then the pressure is inversely proportional to the volume. 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. The molar volume of a substance i is defined as its molar mass divided by its density ρi0: V_{\rm m,i} = {M_i\over\rho_i^0} For an ideal mixture containing N components, the molar volume of the mixture is the weighted sum of the molar volumes of its individual components. The basic gas laws had been discovered by the end of the 18th century, when scientists began to realize that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases. In chemistry and related fields, the molar volume, symbol Vm, or \tilde V of a substance is the ratio of the volume occupied by a substance to the amount of substance, usually given at a given temperature and pressure. # If the temperature and volume remain constant, then the pressure of the gas changes is directly proportional to the number of molecules of gas present. He observed that volume of a given mass of a gas is inversely proportional to its pressure at a constant temperature. This law has the following important consequences: # If temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of gas. # If the temperature changes and the number of gas molecules are kept constant, then either pressure or volume (or both) will change in direct proportion to the temperature. ==Other gas laws== ;Graham's law: states that the rate at which gas molecules diffuse is inversely proportional to the square root of the gas density at constant temperature. The statement of Charles's law is as follows: the volume (V) of a given mass of a gas, at constant pressure (P), is directly proportional to its temperature (T). In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles.

1.2

2

0.686

0

22.2036033112

A

Calculate the mass of water vapour present in a room of volume $400 \mathrm{m}^3$ that contains air at $27^{\circ} \mathrm{C}$ on a day when the relative humidity is 60 per cent.

The density of humid air is found by:Shelquist, R (2009) Equations - Air Density and Density Altitude \rho_\text{humid air} = \frac{p_\text{d}}{R_\text{d} T} + \frac{p_\text{v}}{R_\text{v} T} = \frac{p_\text{d}M_\text{d} + p_\text{v}M_\text{v}}{R T} where: *\rho_\text{humid air}, density of the humid air (kg/m3) *p_\text{d}, partial pressure of dry air (Pa) *R_\text{d}, specific gas constant for dry air, 287.058J/(kg·K) *T, temperature (K) *p_\text{v}, pressure of water vapor (Pa) *R_\text{v}, specific gas constant for water vapor, 461.495J/(kg·K) *M_\text{d}, molar mass of dry air, 0.0289652kg/mol *M_\text{v}, molar mass of water vapor, 0.018016kg/mol *R, universal gas constant, 8.31446J/(K·mol) The vapor pressure of water may be calculated from the saturation vapor pressure and relative humidity. The theoretical value for water vapor is 19.6, but due to vapor condensation the water vapor density dependence is highly variable and is not well approximated by this formula. This can be seen by using the ideal gas law as an approximation. ==Dry air== The density of dry air can be calculated using the ideal gas law, expressed as a function of temperature and pressure: \begin{align} \rho &= \frac{p}{R_\text{specific} T}\\\ R_\text{specific} &= \frac{R}{M} = \frac{k_{\rm B}}{m}\\\ \rho &= \frac{pM}{RT} = \frac{pm}{k_{\rm B}T}\\\ \end{align} where: *\rho, air density (kg/m3)In the SI unit system. The density of humid air may be calculated by treating it as a mixture of ideal gases. This occurs because the molar mass of water vapor (18g/mol) is less than the molar mass of dry airas dry air is a mixture of gases, its molar mass is the weighted average of the molar masses of its components (around 29g/mol). Air is given a vapour density of one. For this use, air has a molecular weight of 28.97 atomic mass units, and all other gas and vapour molecular weights are divided by this number to derive their vapour density.HazMat Math: Calculating Vapor Density . * At 20°C and 101.325kPa, dry air has a density of 1.2041 kg/m3. * At 70°F and 14.696psi, dry air has a density of 0.074887lb/ft3. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. It is found by: p_\text{v} = \phi p_\text{sat} where: *p_\text{v}, vapor pressure of water *\phi, relative humidity (0.0–1.0) *p_\text{sat}, saturation vapor pressure The saturation vapor pressure of water at any given temperature is the vapor pressure when relative humidity is 100%. The following table illustrates the air density–temperature relationship at 1 atm or 101.325 kPa: ==Humid air== thumb|right|400px|Effect of temperature and relative humidity on air density The addition of water vapor to air (making the air humid) reduces the density of the air, which may at first appear counter-intuitive. *R_\text{specific}, the specific gas constant for dry air, which using the values presented above would be approximately in J⋅kg−1⋅K−1. Depending on the measuring instruments used, different sets of equations for the calculation of the density of air can be applied. 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. Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. However, other units can be used. *p, absolute pressure (Pa) *T, absolute temperature (K) *R is the gas constant, in J⋅K−1⋅mol−1 *M is the molar mass of dry air, approximately in kg⋅mol−1. *k_{\rm B} is the Boltzmann constant, in J⋅K−1 *m is the molecular mass of dry air, approximately in kg. One formula is Tetens' equation fromShelquist, R (2009) Algorithms - Schlatter and Baker used to find the saturation vapor pressure is: p_\text{sat} = 6.1078 \times 10^{\frac{7.5 T}{T + 237.3}} where: *p_\text{sat}, saturation vapor pressure (hPa) *T, temperature (°C) See vapor pressure of water for other equations. Therefore: * At IUPAC standard temperature and pressure (0°C and 100kPa), dry air has a density of approximately 1.2754kg/m3. 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). So when water molecules (water vapor) are added to a given volume of air, the dry air molecules must decrease by the same number, to keep the pressure or temperature from increasing. At 101.325kPa (abs) and , air has a density of approximately , which is about that of water, according to the International Standard Atmosphere (ISA).

22

+116.0

3930.0

6.2

2

D

Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. The constant-volume heat capacity of a gas can be measured by observing the decrease in temperature when it expands adiabatically and reversibly. If the decrease in pressure is also measured, we can use it to infer the value of $\gamma=C_p / C_V$ and hence, by combining the two values, deduce the constant-pressure heat capacity. A fluorocarbon gas was allowed to expand reversibly and adiabatically to twice its volume; as a result, the temperature fell from $298.15 \mathrm{~K}$ to $248.44 \mathrm{~K}$ and its pressure fell from $202.94 \mathrm{kPa}$ to $81.840 \mathrm{kPa}$. Evaluate $C_p$.

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 (). Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. 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. . 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. There are two types adiabatic flame temperature: constant volume and constant pressure, depending on how the process is completed. 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). 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. 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. 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). 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). This relation may be used to show the heat capacities may be expressed in terms of the heat capacity ratio () and the gas constant (): : C_P = \frac{\gamma n R}{\gamma - 1} \quad \text{and} \quad C_V = \frac{n R}{\gamma - 1}, === Relation with degrees of freedom === The classical equipartition theorem predicts that the heat capacity ratio () for an ideal gas can be related to the thermally accessible degrees of freedom () of a molecule by : \gamma = 1 + \frac{2}{f},\quad \text{or} \quad f = \frac{2}{\gamma - 1}. Therefore, the heat capacity ratio in this example is 1.4. 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 . 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). This result can be explained through Le Chatelier's principle. ==See also== *Flame speed ==References== == External links == === General information === * * Computation of adiabatic flame temperature * Adiabatic flame temperature === Tables === * adiabatic flame temperature of hydrogen, methane, propane and octane with oxygen or air as oxidizers * * Temperature of a blue flame and common materials === Calculators === * Online adiabatic flame temperature calculator using Cantera * Adiabatic flame temperature program * Gaseq, program for performing chemical equilibrium calculations. The constant-pressure adiabatic flame temperature of such substances in air is in a relatively narrow range around 1950 °C. 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. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. However, when the gas density is sufficiently high and intermolecular forces are important, thermodynamic expressions may sometimes be used to accurately describe the relationship between the two heat capacities, as explained below. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. 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.

41.40

0.36

0.66666666666

0.011

1.6

A