Datasets:

natnitaract

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SciBench-TruthfulQA-RAG

like 1

A coin is flipped 10 times and the sequence of heads and tails is observed. What is the number of possible 10-tuplets that result in four heads and six tails?

thumb|upright=1.35|Coin of Tennes. The Philippine ten-centavo coin (10¢) coin is a denomination of the Philippine peso. In the 1954/55 National Hunt season Four Ten won three of his first four races including two wide-margin victories at Warwick Racecourse in December and January. Four Ten won a total of nine races between his second Gold Cup attempt and his retirement. Commenting on the horse's tendency to jump to the left, Kirkpatrick explained that Four Ten would be better suited by a left-handed track and would probably contest both the Gold Cup and the Grand National. Four Ten (1946 - 1971) was a British Thoroughbred racehorse who won the 1954 Cheltenham Gold Cup. The Hundred-dollar, Hundred-digit Challenge problems are 10 problems in numerical mathematics published in 2002 by . The horse ran six times in point-to-points winning four times consecutively before falling in a hunter chase. A $100 prize was offered to whoever produced the most accurate solutions, measured up to 10 significant digits. He imagined that B and C toss their dice in groups of six, and said that A was most favorable because it required a 6 in only one toss, while B and C required a 6 in each of their tosses. In the early part of the following season, Four Ten further established himself as a high-class steeplechaser with an "impressive" win over three miles at Cheltenham in November, beating E.S.B. and Mariner's Log. The Newton–Pepys problem is a probability problem concerning the probability of throwing sixes from a certain number of dice. A 6×4 or six-by-four is a vehicle with three axles, with a drivetrain delivering power to two wheel ends on two of them.International ProStar ES Class 8 truck: Axle configurations It is a form of four-wheel driveNACFE (North American Council for Freight Efficiency) Executive Report – 6x2 (Dead Axle) Tractors "A typical three axle Class 8 tractor today is equipped with two rear drive axles (“live” tandem) and is commonly referred to as a 6 X 4 configuration meaning that it has four-wheel drive capability." The name of the republic, the date and denomination are all on the obverse. ==== New Generation Currency Series ==== The BSP announced in 2017 that the ten-centavo coin would not be included in this series, and that it was dropping the coin from circulation. Mariner's Log took over the lead, but Four Ten, relishing the heavy ground, moved up to challenge at the last and drew away up the run-in to win by four lengths. In general, if P(n) is the probability of throwing at least n sixes with 6n dice, then: :P(n)=1-\sum_{x=0}^{n-1}\binom{6n}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{6n-x}\, . If r is the total number of dice selecting the 6 face, then P(r \ge k ; n, p) is the probability of having at least k correct selections when throwing exactly n dice. Strange originally used Four Ten as a hunter before training the horse himself for races on the amateur point-to-point circuit. These results may be obtained by applying the binomial distribution (although Newton obtained them from first principles). The BSP Series coin will still be used until that series is demonetized. ===Version history=== English Series (1958–1967) Pilipino Series (1969–1974) Ang Bagong Lipunan Series (1975–1983) Flora and Fauna Series (1983–1994) BSP Coin Series (1995–2017) Obverse centre|frameless|101x101px centre|frameless|101x101px centre|frameless|101x101px centre|frameless|102x102px centre|frameless|101x101px Reverse centre|frameless|100x100px centre|frameless|102x102px centre|frameless|102x102px centre|frameless|104x104px centre|frameless|101x101px ==References== Category:Currencies of the Philippines Category:Ten-cent coins The issues from 1979 to 1982 featured a mintmark underneath the 10 centavo. ==== Flora and Fauna Series ==== From 1983 to 1994, a new coin was issued with Baltazar again faced to the left in profile, and the denomination was moved to the reverse with the date on the front. Around it was the inscription 'Rey de Espana' (King of Spain) and the denomination as 10 Cs. de Po. (10 centimos of peso).http://worldcoingallery.com/countries/display.php?image=nmc2/142-148&desc;=Philippines km148 10 Centimos (1880-1885)&query;=Philippines === United States administration=== thumb|left|10 centavos issued 1907-1945 In 1903, the 10-centavo coin equivalent to was minted for the Philippines, weighing of 0.9 fine silver.

210

0.020

7.0

72

0.405

A

Among nine orchids for a line of orchids along one wall, three are white, four lavender, and two yellow. How many color displays are there?

Bulbophyllum concolor is a species of orchid in the genus Bulbophyllum. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia concolor Bulbophyllum bicolor is a species of orchid in the genus Bulbophyllum. Bulbophyllum tricolor is a species of orchid in the genus Bulbophyllum. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia tricolor Bulbophyllum bicoloratum is a species of orchid in the genus Bulbophyllum. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia *Photos of Bulbophyllum bicoloratum == External links == * * bicoloratum Category:Articles containing video clips Category:Plants described in 1924 Pabstiella tricolor is a species of orchid plant. == References == tricolor It is found only in Hong Kong and isolated parts of southeast China and northern Vietnam. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia bicolor Category:Plants described in 1830 7 Colors (a.k.a. Filler) is a puzzle game, designed by Dmitry Pashkov. The game was published by Infogrames for MS-DOS, Amiga, and NEC PC-9801. ==Reception== ==References== ==External links== * Category:1991 video games Category:Amiga games Category:DOS games Category:Hot B games Category:Infogrames games Category:Multiplayer and single-player video games Category:NEC PC-9801 games Category:Puzzle video games Category:Video games developed in Russia It was developed by the Russian company in 1991. Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor

1.7

3.42

1260.0

0.84

0.333333333333333

C

A survey was taken of a group's viewing habits of sporting events on TV during I.I-5 the last year. Let $A=\{$ watched football $\}, B=\{$ watched basketball $\}, C=\{$ watched baseball $\}$. The results indicate that if a person is selected at random from the surveyed group, then $P(A)=0.43, P(B)=0.40, P(C)=0.32, P(A \cap B)=0.29$, $P(A \cap C)=0.22, P(B \cap C)=0.20$, and $P(A \cap B \cap C)=0.15$. Find $P(A \cup B \cup C)$.

The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. One may resolve this overlap by the principle of inclusion- exclusion, or, in this case, by simply finding the probability of the complementary event and subtracting it from 1, thus: : Pr(at least one "1") = 1 − Pr(no "1"s) := 1 − Pr([no "1" on 1st trial] and [no "1" on 2nd trial] and ... and [no "1" on 8th trial]) := 1 − Pr(no "1" on 1st trial) × Pr(no "1" on 2nd trial) × ... × Pr(no "1" on 8th trial) := 1 −(5/6) × (5/6) × ... × (5/6) := 1 − (5/6)8 := 0.7674... ==See also== *Logical complement *Exclusive disjunction *Binomial probability ==References== ==External links== *Complementary events - (free) page from probability book of McGraw-Hill Category:Experiment (probability theory) That is, for an event A, :P(A^c) = 1 - P(A). In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. Therefore, the probability of an event's complement must be unity minus the probability of the event. It may be tempting to say that : Pr(["1" on 1st trial] or ["1" on second trial] or ... or ["1" on 8th trial]) := Pr("1" on 1st trial) + Pr("1" on second trial) + ... + P("1" on 8th trial) := 1/6 + 1/6 + ... + 1/6 := 8/6 := 1.3333... For two events to be complements, they must be collectively exhaustive, together filling the entire sample space. Equivalently, the probabilities of an event and its complement must always total to 1. In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. Cengage Learning 2007, , p. 229 () The event A and its complement [not A] are mutually exclusive and exhaustive. Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. By contrast, in the example above the law of total probability applies, since the event X = 0.5 is included into a family of events X = x where x runs over (−1,1), and these events are a partition of the probability space. On the other hand, conditioning on an event B is well-defined, provided that \mathbb{P}(B) eq 0, irrespective of any partition that may contain B as one of several parts. ===Conditional distribution=== Given X = x, the conditional distribution of Y is : \mathbb{P} ( Y=y | X=x ) = \frac{ \binom 3 y \binom 7 {x-y} }{ \binom{10}x } = \frac{ \binom x y \binom{10-x}{3-y} }{ \binom{10}3 } for 0 ≤ y ≤ min ( 3, x ). The probability (with respect to some probability measure) that an event S occurs is the probability that S contains the outcome x of an experiment (that is, it is the probability that x \in S). The 2010 Women's Junior World Handball Championship (17th tournament) took place in South Korea from July 17 to July 31. ==Preliminary round== ===Group A=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ===Group B=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ===Group C=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ===Group D=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ==Main round== ===Group I=== \---- \---- \---- \---- \---- \---- \---- \---- ===Group II=== \---- \---- \---- \---- \---- \---- \---- \---- ==President's Cup== ===21st–24th=== \---- ====23rd/24th==== ====21st/22nd==== ===17th–20th=== \---- ====19th/20th==== ====17th/18th==== ===13th–16th=== \---- ====15th/16th==== ====13th/14th==== ==Placement matches== ===11th/12th=== ===9th/10th=== ===7th/8th=== ===5th/6th=== ==Final round== ===Semifinals=== \---- ===Bronze medal match=== ===Gold medal match=== ==Ranking and statistics== ===Final ranking=== Rank Team Image:Gold medal icon.svg Image:Silver medal icon.svg Image:Bronze medal icon.svg 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2010 Junior Women's World Champions 150px|border|Norway Norway First title ;Team roster Silje Solberg, Christine Homme, Veronica Kristiansen, Hanna Yttereng, Mai Marcussen, Stine Bredal Oftedal, Mari Molid, Maja Jakobsen, Sanna Solberg, Nora Mørk, Guro Rundbråten, Silje Katrine Svendsen, Ellen Marie Folkvord, Hilde Kamperud, Kristin Nørstebø, Susann Iren Hall. ===All Star Team=== *Goalkeeper: *Left wing: *Left back: *Pivot: *Centre back: *Right back: *Right wing: Chosen by team officials and IHF experts: IHF.info ===Other awards=== *Most Valuable Player: *Top Goalscorer: 75 goals ===Top goalkeepers=== Rank Name Team Saves Shots % 1 Jessica Oliveira 127 325 39.1% 2 Marta Žderić 122 314 38.9% 3 Nele Kurzke 93 285 32.6% 4 Marina Vukčević 85 220 38.6% 5 Guro Rundbråten 82 198 41.4% 6 Shuk Yee Wong 80 301 26.6% 7 Marija Colic 75 213 35.2% 8 Preeyanut Bureeruk 73 266 27.4% 9 Elena Fomina 71 194 36.6% 9 Sori Park 71 260 27.3% Source: ihf.info ===Top goalscorers=== Rank Name Team Goals Shots % 1 Nathalie Hagman 75 96 78.1% 2 Laura van den Heijden 72 109 66.1% 3 Ryu Eun-hee 63 109 57.8% 4 Milena Knežević 62 124 50.0% 5 Jelena Živković 59 103 57.3% 6 Lee Eun-bi 58 87 66.7% 7 Tatiana Khmyrova 55 80 68.8% 7 Luciana Mendoza 55 87 63.2% 9 Marta Lopez 52 91 57.1% 10 Ana Martinez 49 92 53.3% Source: ihf.info ==References== Tournament Summary ==External links== *XVII Women's Junior World Championship at IHF.info Category:International handball competitions hosted by South Korea Women's Junior World Handball Championship, 2010 2010 Category:Women's handball in South Korea Junior World Handball Championship Category:July 2010 sports events in South Korea Other events are proper subsets of the sample space that contain multiple elements. In the latter two examples the law of total probability is irrelevant, since only a single event (the condition) is given. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event defines a complementary event, namely the complementary set (the event occurring), and together these define a Bernoulli trial: did the event occur or not? The expectation of this random variable is equal to the (unconditional) probability, E ( P ( Y ≤ 1/3 | X ) ) = P ( Y ≤ 1/3 ), namely, : 1 \cdot \mathbb{P} (X<0.5) + 0 \cdot \mathbb{P} (X=0.5) + \frac13 \cdot \mathbb{P} (X>0.5) = 1 \cdot \frac16 + 0 \cdot \frac13 + \frac13 \cdot \left( \frac16 + \frac13 \right) = \frac13, which is an instance of the law of total probability E ( P ( A | X ) ) = P ( A ). American football win probability estimates often include whether a team is home or away, the down and distance, score difference, time remaining, and field position. Win probability added is the change in win probability, often how a play or team member affected the probable outcome of the game. ==Current research== Current research work involves measuring the accuracy of win probability estimates, as well as quantifying the uncertainty in individual estimates.

0.65625

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D

A grade school boy has five blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left to his right pocket, what is the probability of his then drawing a blue marble from his right pocket?

thumb|right|Four bags with three marbles per bag gives twelve marbles (4 × 3 = 12). thumb|Animation for the multiplication 2 × 3 = 6. thumb|right|4 × 5 = 20. At the end of each puzzle, the marbles that have been guided into their proper bins are returned to the player. Thus, likelihood of landing on a particular space in Blue Marble Game can't be exactly calculated like in Monopoly, as players can choose to go to different spaces on the board based on current game conditions. The aim is to ensure that each marble arrives in the bin of the same color as the marble. For example, four bags with three marbles each can be thought of as: :[4 bags] × [3 marbles per bag] = 12 marbles. Marble Drop is a puzzle video game published by Maxis on February 28, 1997. == Gameplay == Players are given an initial set of marbles that are divided evenly into six colors: red, orange, yellow, green, blue, and purple, with two more colors available to purchase: black and silver (steel). Steel (silver) balls are 20 percent of the price of colored marbles and can be used as test marbles or to help release a catch instead of using a valuable colored marble; additionally, there are steel-coloured exit bins in the final puzzle. Players must determine how the marble will travel through the puzzle, and how its journey will change the puzzle for the next marble. It can be played by 2 to 4 players.Blue Marble Game instruction booklet ==Gameplay== Players move around the board in order to buy property, build buildings on the properties, pay rent to other players, and earn a salary. Blue Marble Game (부루마불게임) is a Korean board game similar to Monopoly manufactured by Si-Yat-Sa. Black marbles are very expensive, but change to the correct color when they arrive in a bin. ==Reception== Marble Drop received lukewarm reception upon release. While Monopoly is traditionally played across locations in a single city, the Blue Marble Game features cities from across the world; its title is a reference to The Blue Marble photograph taken by the crew of Apollo 17, and its description of the Earth as seen from space. Lost marbles must be purchased when they are needed to complete a puzzle. There is no mortgage system in the Blue Marble Game. ===Statistics=== In Blue Marble Game, there is no space that sends the player to the deserted island other than the deserted island space itself. More marble has been extracted from the over 650 quarry sites near Carrara than from any other place. Marble died of cancer on September 11, 2015 at the age of 48.Roy Marble, Iowa's scoring leader, dies at 48 Marble's son, Devyn, followed in his father's footsteps to Iowa and the NBA. Devyn and his father were the first father-son duo in Big Ten history to each score 1,000 points.Iowa's Devyn Marble joins father Roy in scoring 1,000th point for Hawkeye Marble came into the news again in 2021 when his family expressed displeasure at the retirement of Luka Garza's jersey number (announced after the last game of the season on March 7), noting that they felt hurt and disrespected by the move upon the fact that Marble's number was not retired; Marble, alongside Murray Wier and Chuck Darling, are considered the best players to not have their jersey number retired by Iowa. The layers of marble are interbedded with schists and quartzites. These marbles are picked up and dropped by the players into funnels leading to a series of rails, switches, traps and other devices which grow more complex as the game progresses. Some of them, such as the Maffioli, who rented some quarries north of Carrara, in the Torano area, or, around 1490, Giovanni Pietro Buffa, who bought marble on credit from local quarrymen and then resold it on the Venetian market, were able to create a dense commercial network, exporting the marble even to distant locations. thumb|264x264px|The distinct green colour of the middle slab is a result of an abundance of serpentine minerals|alt=An image of some slabs of connemara marble against a wall. thumb|Sample sheets, 2016 Carrara marble, Luna marble to the Romans, is a type of white or blue-grey marble popular for use in sculpture and building decor.

+10

35.91

0.444444444444444

-1270

2.3

C

A faculty leader was meeting two students in Paris, one arriving by train from Amsterdam and the other arriving by train from Brussels at approximately the same time. Let $A$ and $B$ be the events that the respective trains are on time. Suppose we know from past experience that $P(A)=0.93, P(B)=0.89$, and $P(A \cap B)=0.87$. Find $P(A \cup B)$.

The probability of one or both events occurring is denoted P(A ∪ B) and in general, it equals P(A) + P(B) – P(A ∩ B). Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. When A and B are mutually exclusive, .Stats: Probability Rules. Cengage Learning 2007, , p. 229 () The event A and its complement [not A] are mutually exclusive and exhaustive. For two events to be complements, they must be collectively exhaustive, together filling the entire sample space. In consequence, mutually exclusive events have the property: P(A ∩ B) = 0.intmath.com; Mutually Exclusive Events. On 21 April 2012 at 18:30 local time (16:30 UTC), two trains were involved in a head-on collision at Westerpark, near Sloterdijk, in the west of Amsterdam, Netherlands. An event defines a complementary event, namely the complementary set (the event occurring), and together these define a Bernoulli trial: did the event occur or not? Therefore, the probability of an event's complement must be unity minus the probability of the event. The probability that at least one of the events will occur is equal to one.Scott Bierman. In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. Formally said, the intersection of each two of them is empty (the null event): A ∩ B = ∅. In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. It is estimated that at the moment of the collision the intercity was travelling at and the local train at about . The probabilities of the individual events (red, and club) are multiplied rather than added. The local train was travelling between Amsterdam and whilst the Intercity train was travelling between and . In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. That is, for an event A, :P(A^c) = 1 - P(A). Therefore, two mutually exclusive events cannot both occur. Given an event, the event and its complementary event define a Bernoulli trial: did the event occur or not? In physics, and in particular relativity, an event is the instantaneous physical situation or occurrence associated with a point in spacetime (that is, a specific place and time).

37.9

0.95

'-3.141592'

6.3

-167

B

What is the number of possible four-letter code words, selecting from the 26 letters in the alphabet?

The Code 39 specification defines 43 characters, consisting of uppercase letters (A through Z), numeric digits (0 through 9) and a number of special characters (-, ., $, /, +, %, and space). The alphabet for Modern English is a Latin-script alphabet consisting of 26 letters, each having an upper- and lower-case form. The English alphabet has 5 vowels, 19 consonants, and 2 letters (Y and W) that can function as consonants or vowels. 26 (twenty-six) is the natural number following 25 and preceding 27. == In mathematics == thumb|Poster designed to depict the speciality of the number 26 *26 is the only integer that is one greater than a square (5 + 1) and one less than a cube (3 − 1). *26 is a telephone number, specifically, the number of ways of connecting 5 points with pairwise connections. A FourCC ("four-character code") is a sequence of four bytes (typically ASCII) used to uniquely identify data formats. This code is traditionally mapped to the * character in barcode fonts and will often appear with the human-readable representation alongside the barcode. thumb|Code 39 Characters As a generality, the location of the two wide bars can be considered to encode a number between 1 and 10, and the location of the wide space (which has four possible positions) can be considered to classify the character into one of four groups (from left to right): Letters(+30) (U–Z), Digits(+0) (1–9,0), Letters(+10) (A–J), and Letters(+20) (K–T). Lower case letters, additional punctuation characters and control characters are represented by sequences of two characters of Code 39. Microsoft and Windows developers refer to their four-byte identifiers as FourCCs or Four-Character Codes. 262px|right|thumb|*WIKIPEDIA* encoded in Code 39 Code 39 (also known as Alpha39, Code 3 of 9, Code 3/9, Type 39, USS Code 39, or USD-3) is a variable length, discrete barcode symbology defined in ISO/IEC 16388:2007. Code Details Nr Character Encoding Nr Character Encoding Nr Character Encoding Nr Character Encoding 0 NUL %U 32 [space] [space] 64 @ %V 96 ` %W 1 SOH $A 33 ! * GOD=26=G7+O15+D4 in Simple6,74 English7,74 Gematria8,74 ('The Key': A=1, B2, C3, ..., Z26). Because there are only six letters in the Letters(+30) group (letters 30–35, or U–Z), the other four positions in this group (36–39) are used to represent three symbols (dash, period, space) as well as the start/stop character. However, Code 39 is still used by some postal services (although the Universal Postal Union recommends using Code 128 in all casesAs one example of an international standard, see ), and can be decoded with virtually any barcode reader. Depending on the way one counts, the West Frisian alphabet contains between 25 and 32 characters. ==Letters== West Frisian alphabet Upper case vowels and vowels with diacritics A Â E Ê É I/Y O Ô U Û Ú Lower case vowels and vowels with diacritics a â e ê é i/y o ô u û ú Vowel Pronunciation / Upper case letters B C D F G H J K L M N P Q R S T V W X Z Lower case letters b c d f g h j k l m n p q r s t v w x z Letter Pronunciation ==Alphabetical order== In alphabetical listings both I and Y are usually found between H and J. *The number of the last letter of the English alphabet, Z. In base ten, 26 is the smallest positive integer that is not a palindrome to have a square (262 = 676) that is a palindrome. == In science == *The atomic number of iron. This table outlines the Code 39 specification. Setting aside one of these characters as a start and stop pattern left 39 characters, which was the origin of the name Code 39. The two wide bars, out of five possible positions, encode a number between 1 and 10 using a two-out-of-five code with the following numeric equivalence: 1, 2, 4, 7, 0. RealMedia files also use four-character codes, however, the actual codes used differ from those found in AVI or QuickTime files. All these fields are four-character codes known as OSType. *In a normal deck of cards, there are 26 red cards and 26 black cards.

358800

3.2

1068.0

0.0384

0.3333333

A

The change in molar internal energy when $\mathrm{CaCO}_3(\mathrm{~s})$ as calcite converts to another form, aragonite, is $+0.21 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate the difference between the molar enthalpy and internal energy changes when the pressure is 1.0 bar given that the densities of the polymorphs are $2.71 \mathrm{~g} \mathrm{~cm}^{-3}$ and $2.93 \mathrm{~g} \mathrm{~cm}^{-3}$, respectively.

However, the sea level, temperature, and calcium carbonate saturation state of the surrounding system also determine which polymorph of calcium carbonate (aragonite, low-magnesium calcite, high-magnesium calcite) will form. Aragonite will change to calcite over timescales of days or less at temperatures exceeding 300 °C, and vaterite is even less stable. ==Etymology== Calcite is derived from the German , a term from the 19th century that came from the Latin word for lime, (genitive ) with the suffix -ite used to name minerals. thumb|235px|Crystal structure of calcite Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate (CaCO3). Calcite, obtained from an 80 kg sample of Carrara marble, is used as the IAEA-603 isotopic standard in mass spectrometry for the calibration of δ18O and δ13C. Calcite defines hardness 3 on the Mohs scale of mineral hardness, based on scratch hardness comparison. The molecular formula SrCO3 (molar mass: 147.63 g/mol, exact mass: 147.8904 u) may refer to: * Strontianite * Strontium carbonate Other polymorphs of calcium carbonate are the minerals aragonite and vaterite. Twinning, cleavage and crystal forms are often given in morphological units. ==Properties == The diagnostic properties of calcite include a defining Mohs hardness of 3, a specific gravity of 2.71 and, in crystalline varieties, a vitreous luster. An aragonite sea contains aragonite and high-magnesium calcite as the primary inorganic calcium carbonate precipitates. The molecular formula C3H8O10P2 (molar mass: 266.035 g/mol) may refer to: * 1,3-Bisphosphoglyceric acid (1,3-BPG) * 2,3-Bisphosphoglyceric acid (2,3-BPG) Category:Molecular formulas The molecular formula C3H7O7P (molar mass: 186.06 g/mol, exact mass: 185.9929 u) may refer to: * 2-Phosphoglyceric acid, or 2-phosphoglycerate * 3-Phosphoglyceric acid The molecular formula C12H15N2O3PS (molar mass: 298.30 g/mol, exact mass: 298.0541 u) may refer to: * Phoxim * Quinalphos José María Patoni, San Juan del Río, Durango (Mexico) ==See also== *Carbonate rock *Ikaite, CaCO3·6H2O *List of minerals *Lysocline *Manganoan calcite, (Ca,Mn)CO3 *Monohydrocalcite, CaCO3·H2O *Nitratine *Ocean acidification *Ulexite ==References== ==Further reading== * Category:Calcium minerals Category:Carbonate minerals Category:Limestone Category:Optical materials Category:Transparent materials Category:Calcite group Category:Cave minerals Category:Trigonal minerals Category:Minerals in space group 167 Category:Evaporite Category:Luminescent minerals Category:Polymorphism (materials science) Category:Bastet Calcite is also more soluble at higher pressures. However, crystallization of calcite has been observed to be dependent on the starting pH and concentration of magnesium in solution. Calcite in limestone is divided into low-magnesium and high-magnesium calcite, with the dividing line placed at a composition of 4% magnesium. Due to its acidity, carbon dioxide has a slight solubilizing effect on calcite. These processes can be traced by the specific carbon isotope composition of the calcites, which are extremely depleted in the 13C isotope, by as much as −125 per mil PDB (δ13C). ==In Earth history== Calcite seas existed in Earth's history when the primary inorganic precipitate of calcium carbonate in marine waters was low-magnesium calcite (lmc), as opposed to the aragonite and high- magnesium calcite (hmc) precipitated today. The chemical conditions of the seawater must be notably high in magnesium content relative to calcium (high Mg/Ca ratio) for an aragonite sea to form. As ocean acidification causes pH to drop, carbonate ion concentrations will decline, potentially reducing natural calcite production. ==Gallery== File:Calcite-Mottramite-cktsu-45b.jpg|Calcite with mottramite File:Erbenochile eye.JPG|Trilobite eyes employed calcite File:CalciteEchinosphaerites.jpg|Calcite crystals inside a test of the cystoid Echinosphaerites aurantium (Middle Ordovician, northeastern Estonia) File:Calcite-Dolomite-Gypsum-159389.jpg|Rhombohedrons of calcite that appear almost as books of petals, piled up 3-dimensionally on the matrix File:Calcite-Hematite-Chalcopyrite-176263.jpg|Calcite crystal canted at an angle, with little balls of hematite and crystals of chalcopyrite both on its surface and included just inside the surface of the crystal File:GeopetalCarboniferousNV.jpg|Thin section of calcite crystals inside a recrystallized bivalve shell in a biopelsparite File:OoidSurface01.jpg|Grainstone with calcite ooids and sparry calcite cement; Carmel Formation, Middle Jurassic, of southern Utah, USA. File:Calcite-Aragonite-Sulphur-69380.jpg|Several well formed milky white casts, made up of many small sharp calcite crystals, from the sulfur mines at Agrigento, Sicily File:Calcite-tch21c.jpg|Reddish rhombohedral calcite crystals from China. Likewise, the occurrence of calcite seas is controlled by the same suite of factors controlling aragonite seas, with the most obvious being a low seawater Mg/Ca ratio (Mg/Ca < 2), which occurs during intervals of rapid seafloor spreading.

-0.28

+2.35

1.16

83.81

28

A

Estimate the molar volume of $\mathrm{CO}_2$ at $500 \mathrm{~K}$ and 100 atm by treating it as a van der Waals gas.

Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . The 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. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. 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). Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. The molar van der Waals volume should not be confused with the molar volume of the substance. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The 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 density of solid helium at 1.1 K and 66 atm is , corresponding to a molar volume V = . However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. The 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. 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. 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. 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.

432.07

7.82

0.66666666666

0.366

-0.347

D

Suppose the concentration of a solute decays exponentially along the length of a container. Calculate the thermodynamic force on the solute at $25^{\circ} \mathrm{C}$ given that the concentration falls to half its value in $10 \mathrm{~cm}$.

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. In terms of separate decay constants, the total half-life T _{1/2} can be shown to be :T_{1/2} = \frac{\ln 2}{\lambda _c} = \frac{\ln 2}{\lambda _1 + \lambda _2}. The exponential time-constant for the process is \tau = R \, C , so the half-life is R \, C \, \ln(2) . This plot shows decay for decay constant () of 25, 5, 1, 1/5, and 1/25 for from 0 to 5. The enthalpy change in this process, normalized by the mole number of solute, is evaluated as the molar integral heat of dissolution. The integral heat of dissolution is defined for a process of obtaining a certain amount of solution with a final concentration. For example, polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days. == Solution of the differential equation == The equation that describes exponential decay is :\frac{dN}{dt} = -\lambda N or, by rearranging (applying the technique called separation of variables), :\frac{dN}{N} = -\lambda dt. The solution to this equation (see derivation below) is: :N(t) = N_0 e^{-\lambda t}, where is the quantity at time , is the initial quantity, that is, the quantity at time . == Measuring rates of decay== === Mean lifetime === If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. The solution to this equation is given in the previous section, where the sum of \lambda _1 + \lambda _2\, is treated as a new total decay constant \lambda _c. :N(t) = N_0 e^{-(\lambda _1 + \lambda _2) t} = N_0 e^{-(\lambda _c) t}. Concentration of X in solvent A/concentration of X in solvent B=Kď If C1 denotes the concentration of solute X in solvent A & C2 denotes the concentration of solute X in solvent B; Nernst's distribution law can be expressed as C1/C2 = Kd. The molar differential enthalpy change of dissolution is: :\Delta_\text{diss}^{d} H= \left(\frac{\partial \Delta_\text{diss} H}{\partial \Delta n_i}\right)_{T,p,n_B} where is the infinitesimal variation or differential of mole number of the solute during dissolution. For a non-ideal solution it is an excess molar quantity. ==Energetics== Dissolution by most gases is exothermic. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac{dN}{dt} = -\lambda N. In thermochemistry, the enthalpy of solution (heat of solution or enthalpy of solvation) is the enthalpy change associated with the dissolution of a substance in a solvent at constant pressure resulting in infinite dilution. The value of the enthalpy of solvation is the sum of these individual steps. The half-life can be written in terms of the decay constant, or the mean lifetime, as: :t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln (2). The temperature of the solution eventually decreases to match that of the surroundings. The dilution between two concentrations of the solute is associated to an intermediary heat of dilution by mole of solute. ==Dilution and Dissolution== The process of dissolution and the process of dilution are closely related to each other. In thermochemistry, the heat of dilution, or enthalpy of dilution, refers to the enthalpy change associated with the dilution process of a component in a solution at a constant pressure. The enthalpy change in this process, normalized by the mole number of solute, is evaluated as the molar integral heat of dilution. Thus, the amount of material left is 2−1 = 1/2 raised to the (whole or fractional) number of half-lives that have passed. Reactions whose rate depends only on the concentration of one reactant (known as first-order reactions) consequently follow exponential decay.

200

17

0.15

0.18162

0.14

B

A container is divided into two equal compartments (Fig. 5.8). One contains $3.0 \mathrm{~mol} \mathrm{H}_2(\mathrm{~g})$ at $25^{\circ} \mathrm{C}$; the other contains $1.0 \mathrm{~mol} \mathrm{~N}_2(\mathrm{~g})$ at $25^{\circ} \mathrm{C}$. Calculate the Gibbs energy of mixing when the partition is removed. Assume perfect behaviour.

Upon removal of the dividing partition, they expand into a final common volume (the sum of the two initial volumes), and the entropy of mixing \Delta S_{mix} is given by :\Delta S_{mix} = -nR(x_1\ln x_1 + x_2\ln x_2)\,. where R is the gas constant, n the total number of moles and x_i the mole fraction of component i\,, which initially occupies volume V_i = x_iV\,. In thermodynamics, the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in a thermodynamic state of internal equilibrium, are mixed without chemical reaction by the thermodynamic operation of removal of impermeable partition(s) between them, followed by a time for establishment of a new thermodynamic state of internal equilibrium in the new unpartitioned closed system. Again, the same equations for the entropy of mixing apply, but only for homogeneous, uniform phases. ==Mixing under other constraints== ===Mixing with and without change of available volume=== In the established customary usage, expressed in the lead section of this article, the entropy of mixing comes from two mechanisms, the intermingling and possible interactions of the distinct molecular species, and the change in the volume available for each molecular species, or the change in concentration of each molecular species. This is not true for the corresponding Gibbs free energies however. == Ideal and regular mixtures == An ideal mixture is any in which the arithmetic mean (with respect to mole fraction) of the two pure substances is the same as that of the final mixture. We again obtain the entropy of mixing on multiplying by the Boltzmann constant k_\text{B}. This explains why enthalpy of mixing is typically experimentally determined. ==Relation to the Gibbs free energy of mixing== The excess Gibbs free energy of mixing can be related to the enthalpy of mixing by the ușe of the Gibbs-Helmholtz equation: :\left( \frac{\partial ( \Delta G^E/T ) } {\partial T} \right)_p = - \frac {\Delta H^E} {T^2} = - \frac {\Delta H_{mix}} {T^2} or equivalently :\left( \frac{\partial ( \Delta G^E/T ) } {\partial (1/T)} \right)_p = \Delta H^E = \Delta H_{mix} In these equations, the excess and total enthalpies of mixing are equal because the ideal enthalpy of mixing is zero. We can regard the mixing process as allowing the contents of the two originally separate contents to expand into the combined volume of the two conjoined containers. For an ideal gas mixture or an ideal solution, there is no enthalpy of mixing (\Delta H_\text{mix} \,), so that the Gibbs free energy of mixing is given by the entropy term only: :\Delta G_\text{mix} = - T\Delta S_\text{mix} For an ideal solution, the Gibbs free energy of mixing is always negative, meaning that mixing of ideal solutions is always spontaneous. The entropy of mixing is entirely accounted for by the diffusive expansion of each material into a final volume not initially accessible to it. Introduction to Modern Thermodynamics, Wiley, Chichester, , pages 197-199. ==See also== * CALPHAD * Enthalpy of mixing * Gibbs energy ==Notes== ==References== == External links == * Online lecture Category:Statistical mechanics Category:Thermodynamic entropy See also: Gibbs paradox, in which it would seem that "mixing" two samples of the same gas would produce entropy. ===Application to solutions=== If the solute is a crystalline solid, the argument is much the same. A version of Statistical Mechanics for Students of Physics and Chemistry, Cambridge University Press, Cambridge UK, pages 163-164 the conflating of the just-mentioned two mechanisms for the entropy of mixing is well established in customary terminology, but can be confusing unless it is borne in mind that the independent variables are the common initial and final temperature and total pressure; if the respective partial pressures or the total volume are chosen as independent variables instead of the total pressure, the description is different. ====Mixing with each gas kept at constant partial volume, with changing total volume==== In contrast to the established customary usage, "mixing" might be conducted reversibly at constant volume for each of two fixed masses of gases of equal volume, being mixed by gradually merging their initially separate volumes by use of two ideal semipermeable membranes, each permeable only to one of the respective gases, so that the respective volumes available to each gas remain constant during the merge. This quantity combines two physical effects—the enthalpy of mixing, which is a measure of the energy change, and the entropy of mixing considered here. Therefore, \Delta S_\text{mix} = -\left(\frac{\partial \Delta G_\text{mix}}{\partial T}\right)_P is negative for mixing of these two equilibrium phases. In the general case of mixing non-ideal materials, however, the total final common volume may be different from the sum of the separate initial volumes, and there may occur transfer of work or heat, to or from the surroundings; also there may be a departure of the entropy of mixing from that of the corresponding ideal case. In ideal mixtures, the enthalpy of mixing is null. Enthalpy of mixing can often be ignored in calculations for mixtures where other heat terms exist, or in cases where the mixture is ideal. Among other important thermodynamic simplifications, this means that enthalpy of mixing is zero: \Delta H_{mix,ideal}=0. Principles and Applications.(MacMillan 1969) p.263 For binary mixtures the entropy of random mixing can be considered as a function of the mole fraction of one component. The lowest value is when the mole fraction is 0.5 for a mixture of two components, or 1/n for a mixture of n components. thumb|The entropy of mixing for an ideal solution of two species is maximized when the mole fraction of each species is 0.5. ==Solutions and temperature dependence of miscibility== ===Ideal and regular solutions=== The above equation for the entropy of mixing of ideal gases is valid also for certain liquid (or solid) solutions—those formed by completely random mixing so that the components move independently in the total volume. In non-ideal mixtures, the thermodynamic activity of each component is different from its concentration by multiplying with the activity coefficient. In this case, the increase in entropy is entirely due to the irreversible processes of expansion of the two gases, and involves no heat or work flow between the system and its surroundings. ===Gibbs free energy of mixing=== The Gibbs free energy change \Delta G_\text{mix} = \Delta H_\text{mix} - T\Delta S_\text{mix} determines whether mixing at constant (absolute) temperature T and pressure p is a spontaneous process.

362880

-6.9

0.7854

-1368

-1.78

B

What is the mean speed, $\bar{c}$, of $\mathrm{N}_2$ molecules in air at $25^{\circ} \mathrm{C}$ ?

CO-0.40-0.22 is a high velocity compact gas cloud near the centre of the Milky Way. The gas is moving away from Earth at speeds ranging from 20 to 120 km/s. The name followed the precedent set by CO-0.02-0.02, which is another high velocity compact cloud in the central molecular zone. The molecular formula C20H24N2O (molar mass: 308.42 g/mol, exact mass: 308.1889 u) may refer to: * Affinisine * Indecainide Category:Molecular formulas The differences in the velocity, termed velocity dispersion, of the gas is unusually high at 100 km/s. Rotational correlation time (\tau_c) is the average time it takes for a molecule to rotate one radian. The Honda CM250 is a parallel twin cylinder air-cooled OHC four-stroke cruiser motorcycle produced by the Honda corporation from 1981–1983 with a top speed of 85 mph and delivering 70mpg.Secondhand Buying Guide: Honda CM125C/200T/250T (Motor Cycle News 9/01/91) The 234cc North American market variant was coded as the CM250C and was the precursor to the current Honda CMX250C, also known as the Honda Rebel 250. It is 200 light years away from the centre in the central molecular zone. The gas cloud includes carbon monoxide and hydrogen cyanide molecules. The cloud is 0.2° away from Sgr C to the galactic southeast. The spectral lines of carbon monoxide reveal that the gas is dense, and warm and fairly opaque. {m} | meanlimits = \bar c \pm 3\sqrt{\bar c} | meanstatistic = \bar c_i = \sum_{j=1}^n \mbox{no. of defects for } x_{ij} }} In statistical quality control, the c-chart is a type of control chart used to monitor "count"-type data, typically total number of nonconformities per unit. Air-Speed, Inc. was a commuter airline in the United States from the 1970s based at Hanscom Field in Bedford, Massachusetts. Subsequent theoretical studies of the gas cloud and nearby IMBH candidates have re-opened the possibility, though no observational evidence for existence of an IMBH has been reported. The molecular cloud has a mass of 4,000 solar masses. For example, the \tau_c = 1.7 ps for water, and 100 ps for a pyrroline nitroxyl radical in a DMSO-water mixture. In solution, rotational correlation times are in the order of picoseconds. Another example of this naming convention is CO–0.30–0.07. ==References== Category:Molecular clouds Category:Sagittarius (constellation) Category:Intermediate-mass black holes The European market variant was identified as the CM250TB. ==Description== The CM250TB is based on the Honda Superdream CB250N engine but with a five-speed and not six-speed gearbox.Secondhand Buying Guide: Honda CM125C/200T/250T (Motor Cycle News 9/01/91) The model is instead characterised by its North American cruiser styling with stepped seat, high handlebars, 'megaphone' exhaust silencers, teardrop-shaped tank and many chromium-plated and polished alloy parts. Rotational correlation times of probe molecules in media have been measured by fluorescence lifetime or for radicals, from the linewidths of electron spin resonances. == References == Category:Molecular dynamics Category:Nuclear magnetic resonance Rotational correlation times are employed in the measurement of microviscosity (viscosity at the molecular level) and in protein characterization. Rotational correlation times may be measured by rotational (microwave), dielectric, and nuclear magnetic resonance (NMR) spectroscopy.

164

475

0.9522

0.14

-11.2

B

Caesium (m.p. $29^{\circ} \mathrm{C}$, b.p. $686^{\circ} \mathrm{C}$ ) was introduced into a container and heated to $500^{\circ} \mathrm{C}$. When a hole of diameter $0.50 \mathrm{~mm}$ was opened in the container for $100 \mathrm{~s}$, a mass loss of $385 \mathrm{mg}$ was measured. Calculate the vapour pressure of liquid caesium at $500 \mathrm{~K}$.

The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. 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. According to Monteith and Unsworth, "Values of saturation vapour pressure from Tetens' formula are within 1 Pa of exact values up to 35 °C." 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). On the computation of saturation vapour pressure. The (unattributed) constants are given as {| class="wikitable" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). == 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. Boca Raton, Florida, 2003; Section 4, Properties of the Elements and Inorganic Compounds; Vapor Pressure of the Metallic Elements :The equations reproduce the observed pressures to an accuracy of ±5% or better. Applied Meteorology 6: 203-204. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2 ::P = 0.61078 \exp\left(\frac{21.875 T}{T + 265.5}\right). ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation ==References== Category:Meteorological concepts Category:Thermodynamic equations As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. At boiling temperatures if Raoult's law applies, the total pressure becomes: :Ptot = x1 P o1T + x2 P o2T + ... etc. Table 363, Evaporation of Metals :The equations are described as reproducing the observed pressures to a satisfactory degree of approximation. Recall from the first section that vapor pressures of liquids are very dependent on temperature. 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. *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. Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. The concentration of a vapor in contact with its liquid, especially at equilibrium, is often expressed in terms of vapor pressure, which will be a partial pressure (a part of the total gas pressure) if any other gas(es) are present with the vapor. The chemical equation for this reaction is :CaCO3 \+ heat → CaO + CO2 This reaction can take place at anywhere above 840 °C (1544 °F), but is generally considered to occur at 900 °C(1655 °F) (at which temperature the partial pressure of CO2 is 1 atmosphere), but a temperature around 1000 °C (1832 °F) (at which temperature the partial pressure of CO2 is 3.8 atmospheresCRC Handbook of Chemistry and Physics, 54th Ed, p F-76) is usually used to make the reaction proceed quickly.Parkes, G.D. and Mellor, J.W. (1939). The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. An understanding of vapour pressure is also relevant in explaining high altitude breathing and cavitation. ==Approximation formulas == There are many published approximations for calculating saturated vapour pressure over water and over ice. AP, Amsterdam. http://store.elsevier.com/Principles-of-Environmental-Physics/John- Monteith/isbn-9780080924793/ ::P = 0.61078 \exp\left(\frac{17.27 T}{T + 237.3}\right), where temperature is in degrees Celsius (°C) and saturation vapor pressure is in kilopascals (kPa). With exhaust gas temperatures as low as 120 °C and lime temperature at kiln outlet in 80 °C range the heat loss of the regenerative kiln is minimal, fuel consumption is as low as 3.6 MJ/kg.

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To get an idea of the distance dependence of the tunnelling current in STM, suppose that the wavefunction of the electron in the gap between sample and needle is given by $\psi=B \mathrm{e}^{-\kappa x}$, where $\kappa=\left\{2 m_{\mathrm{e}}(V-E) / \hbar^2\right\}^{1 / 2}$; take $V-E=2.0 \mathrm{eV}$. By what factor would the current drop if the needle is moved from $L_1=0.50 \mathrm{~nm}$ to $L_2=0.60 \mathrm{~nm}$ from the surface?

Tunneling current is exponentially dependent on the separation of the sample and the tip, typically reducing by an order of magnitude when the separation is increased by 1 Å (0.1 nm). How the tunneling current depends on distance between the two electrodes is contained in the tunneling matrix element : M_{\mu u} = \int_{z > z_0} \psi^S_\mu \,U_T\, {\psi^T_ u}^* \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z. The current due to an applied voltage V (assume tunneling occurs from the sample to the tip) depends on two factors: 1) the number of electrons between the Fermi level EF and EF − eV in the sample, and 2) the number among them which have corresponding free states to tunnel into on the other side of the barrier at the tip. The exponential dependence of the tunneling current on the separation of the electrodes comes from the very wave functions that leak through the potential step at the surface and exhibit exponential decay into the classically forbidden region outside of the material. Then the tunneling current is simply the convolution of the densities of states of the sample surface and the tip: : I_t \propto \int_0^{eV} \rho_S(E_F - eV + \varepsilon) \, \rho_T(E_F + \varepsilon) \, d \varepsilon. Because of this, even when tunneling occurs from a non-ideally sharp tip, the dominant contribution to the current is from its most protruding atom or orbital. === Tunneling between two conductors === thumb|300x300px|Negative sample bias V raises its electronic levels by e⋅V. Using the modified Bardeen transfer Hamiltonian method, which treats tunneling as a perturbation, the tunneling current (I) is found to be where f\left(E\right) is the Fermi distribution function, \rho_s and \rho_T are the density of states (DOS) in the sample and tip, respectively, and M_{\mu u} is the tunneling matrix element between the modified wavefunctions of the tip and the sample surface. The tunneling electric current will be a small fraction of the impinging current. The main result was that the tunneling current is proportional to the local density of states of the sample at the Fermi level taken at the position of the center of curvature of a spherically symmetric tip (s-wave tip model). Using the WKB approximation, equations (5) and (7), we obtain:R. J. Hamers, “STM on Semiconductors,” from Scanning Tunneling Microscopy I, Springer Series in Surface Sciences 20, Ed. by H. -J. Güntherodt and R. Wiesendanger, Berlin: Springer-Verlag, 1992. The tunneling current can be related to the density of available or filled states in the sample. The tunneling current from a single level is therefore : j_t = \left[\frac{4k\kappa}{k^2 + \kappa^2}\right]^2 \, \frac{\hbar k}{m_e}\,e^{-2\kappa w}, where both wave vectors depend on the level's energy E, k = \tfrac{1}{\hbar} \sqrt{2m_eE}, and \kappa = \tfrac{1}{\hbar}\sqrt{2m_e(U - E)}. Since both the tunneling current, equation (5), and the conductance, equation (7), depend on the tip DOS and the tunneling transition probability, T, quantitative information about the sample DOS is very difficult to obtain. Since the tip-sample bias range in tunneling experiments is limited to \pm\phi/e, where \phi is the apparent barrier height, STM and STS only sample valence electron states. If at some point the tunneling current is below the set level, the tip is moved towards the sample, and conversely. For these ideal assumptions, the tunneling conductance is directly proportional to the sample DOS. With the height of the tip fixed, the electron tunneling current is then measured as a function of electron energy by varying the voltage between the tip and the sample (the tip to sample voltage sets the electron energy). The resulting tunneling current is a function of the tip position, applied voltage, and the local density of states (LDOS) of the sample. The current of tunneling electrons at each instance is therefore proportional to |c_ u(t + \mathrm{d}t)|^2 - |c_ u(t)|^2 divided by \mathrm{d}t, which is the time derivative of |c_ u(t)|^2, : \Gamma_{\mu \to u}\ \overset{\text{def}}{=}\ \frac{\mathrm{d}}{\mathrm{d}t} |c_ u(t)|^2 = \frac{2\pi}{\hbar} |M_{\mu u}|^2\frac{\sin\big[(E^S_\mu - E^T_ u) \tfrac{t}{\hbar}\big]}{\pi(E^S_\mu - E^T_ u)}. The tunneling current is therefore the sum of little contributions over all these energies of the product of three factors: 2e \cdot \rho_S(E_F - eV + \varepsilon)\,\mathrm{d}\varepsilon representing available electrons, f(E_F - eV + \varepsilon) - f(E_F + \varepsilon) for those that are allowed to tunnel, and the probability factor \Gamma for those that will actually tunnel: : I_t = \frac{4 \pi e}{\hbar} \int_{-\infty}^{+\infty} [f(E_F - eV + \varepsilon) - f(E_F + \varepsilon)] \, \rho_S(E_F - eV + \varepsilon) \, \rho_T(E_F + \varepsilon) \, |M|^2 \, d \varepsilon. There is however also an important correction to the elastic component of the tunneling current at the onset. As the tip is moved across the surface in a discrete x–y matrix, the changes in surface height and population of the electronic states cause changes in the tunneling current.

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Calculate the typical wavelength of neutrons that have reached thermal equilibrium with their surroundings at $373 \mathrm{~K}$.

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. 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. 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. 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. 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 term temperature is used, since hot, thermal and cold neutrons are moderated in a medium with a certain temperature. 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. 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. Introduction of the Theory of Thermal Neutron Scattering. https://books.google.com/books?id=KUVD8KJt7_0C&dq;=thermal- neutron+reactor&pg;=PR9 thus scattering neutrons by nuclear forces, some nuclides are scattered large. The neutron energy distribution is then adapted to the Maxwell distribution known for thermal motion. The critical temperature is the transition point between these two regimes, and at this critical temperature, the thermal wavelength will be approximately equal to the interparticle distance. The neutron fluence is defined as the neutron flux integrated over a certain time period, so its usual unit is cm−2 (neutrons per centimeter squared). Qualitatively, the higher the temperature, the higher the kinetic energy of the free neutrons. 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. A thermal-neutron reactor is a nuclear reactor that uses slow or thermal neutrons. The measured quantity is the difference in the number of gamma rays emitted within a solid angle between the two neutron spin states. ("Thermal" does not mean hot in an absolute sense, but means in thermal equilibrium with the medium it is interacting with, the reactor's fuel, moderator and structure, which is much lower energy than the fast neutrons initially produced by fission.) The momentum and wavelength of the neutron are related through the de Broglie relation. Thermal neutrons have a different and sometimes much larger effective neutron absorption cross-section for a given nuclide than fast neutrons, and can therefore often be absorbed more easily by an atomic nucleus, creating a heavier, often unstable isotope of the chemical element as a result. The neutron detection temperature, also called the neutron energy, indicates a free neutron's kinetic energy, usually given in electron volts. Earth atmospheric neutron flux, apparently from thunderstorms, can reach levels of 3·10−2 to 9·10+1 cm−2 s−1. The usual unit is cm−2s−1 (neutrons per centimeter squared per second).

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Calculate the separation of the $\{123\}$ planes of an orthorhombic unit cell with $a=0.82 \mathrm{~nm}, b=0.94 \mathrm{~nm}$, and $c=0.75 \mathrm{~nm}$.

Bravais lattice Primitive orthorhombic Base-centered orthorhombic Body-centered orthorhombic Face-centered orthorhombic Pearson symbol oP oS oI oF Unit cell Orthohombic, simple Orthohombic, base-centered Orthohombic, body-centered Orthohombic, face- centered For the base-centered orthorhombic lattice, the primitive cell has the shape of a right rhombic prism;See , row oC, column Primitive, where the cell parameters are given as a1 = a2, α = β = 90° it can be constructed because the two-dimensional centered rectangular base layer can also be described with primitive rhombic axes. Assuming that the particle is spherical, a simple analysis to calculate critical separation particle sizes can be established. Carl Friedrich Gauss, in his treatise Allgemeine Theorie des Erdmagnetismus, presented a method, the Gauss separation algorithm, of partitioning the magnetic field vector, B(r, \theta, \phi), measured over the surface of a sphere into two components, internal and external, arising from electric currents (per the Biot–Savart law) flowing in the volumes interior and exterior to the spherical surface, respectively. thumb|Projections of K-cells onto the plane (from k=1 to 6). Only the edges of the higher-dimensional cells are shown. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. Note that the length a of the primitive cell below equals \frac{1}{2} \sqrt{a^2+b^2} of the conventional cell above. ==Crystal classes== The orthorhombic crystal system class names, examples, Schönflies notation, Hermann-Mauguin notation, point groups, International Tables for Crystallography space group number, orbifold notation, type, and space groups are listed in the table below. thumb|350px|right|The three possible plane-line relationships in three dimensions. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal. ==Bravais lattices== There are four orthorhombic Bravais lattices: primitive orthorhombic, base-centered orthorhombic, body-centered orthorhombic, and face-centered orthorhombic. In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Bravais lattice Rectangular Centered rectangular Pearson symbol op oc Unit cell 100px 100px ==See also== *Crystal structure *Crystal system *Overview of all space groups ==References== ==Further reading== * * Category:Crystal systems thumb|A partially demolished factory with dominating cyclonic separators Cyclonic separation is a method of removing particulates from an air, gas or liquid stream, without the use of filters, through vortex separation. The sides and edges of a k-cell need not be equal in (Euclidean) length; although the unit cube (which has boundaries of equal Euclidean length) is a 3-cell, the set of all 3-cells with equal-length edges is a strict subset of the set of all 3-cells. == Notes == == References == * * Category:Basic concepts in set theory Category:Compactness (mathematics) Intl Orb. Cox. Type Example Primitive Base- centered Face-centered Body-centered 16–24 Rhombic disphenoidal D2 (V) 222 222 [2,2]+ Enantiomorphic Epsomite P222, P2221, P21212, P212121 C2221, C222 F222 I222, I212121 25–46 Rhombic pyramidal C2v mm2 *22 [2] Polar Hemimorphite, bertrandite Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2 Cmm2, Cmc21, Ccc2 Amm2, Aem2, Ama2, Aea2 Fmm2, Fdd2 Imm2, Iba2, Ima2 47–74 Rhombic dipyramidal D2h (Vh) mmm *222 [2,2] Centrosymmetric Olivine, aragonite, marcasite Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma Cmcm, Cmca, Cmmm, Cccm, Cmme, Ccce Fmmm, Fddd Immm, Ibam, Ibca, Imma == In two dimensions == In two dimensions there are two orthorhombic Bravais lattices: primitive rectangular and centered rectangular. One such method for separating cuticles from a rock matrix is acid maceration, which involves soaking the sample in agents such as dilute hydrogen peroxide or hydrochloric and hydrofluoric acid (known as the HCI/HF protocol) to break down the matrix. A 2-cell is the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid. (Shown in each case is only a portion of the plane, which extends infinitely far.) It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Substituting the equation for the line into the equation for the plane gives :((\mathbf{l_0} + \mathbf{l}\ d) - \mathbf{p_0})\cdot\mathbf{n} = 0. A k-cell is a higher-dimensional version of a rectangle or rectangular solid. In vision-based 3D reconstruction, a subfield of computer vision, depth values are commonly measured by so-called triangulation method, which finds the intersection between light plane and ray reflected toward camera. The method employs spherical harmonics.

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Calculate the moment of inertia of an $\mathrm{H}_2 \mathrm{O}$ molecule around the axis defined by the bisector of the $\mathrm{HOH}$ angle (3). The $\mathrm{HOH}$ bond angle is $104.5^{\circ}$ and the bond length is $95.7 \mathrm{pm}$.

When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. Description Figure Moment(s) of inertia Point mass M at a distance r from the axis of rotation. The moment of inertia I_\mathbf{C} about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the polar moment of inertia. Height (h) and breadth (b) are the linear measures, except for circles, which are effectively half-breadth derived, r === Sectional areas moment calculated thus === # Square: I_{xx}=I_{yy}=\frac{b^4}{12} # Rectangular: I_{xx}=\frac{bh^3}{12} and; I_{yy}=\frac{hb^3}{12} # Triangular: I_{xx}=\frac{bh^3}{36} # Circular: I_{xx}=I_{yy}=\frac{1}{4} {\pi} r^4=\frac{1}{64} {\pi} d^4 == Motion in a fixed plane == === Point mass === The moment of inertia about an axis of a body is calculated by summing mr^2 for every particle in the body, where r is the perpendicular distance to the specified axis. File:moment of inertia thick cylinder h.svg I_z = \frac{1}{2} m \left(r_2^2 + r_1^2\right) = m r_2^2 \left(1-t+\frac{t^2}{2}\right) Classical Mechanics - Moment of inertia of a uniform hollow cylinder . A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved. frameless|upright I = M r^2 Two point masses, m1 and m2, with reduced mass μ and separated by a distance x, about an axis passing through the center of mass of the system and perpendicular to the line joining the two particles. frameless|upright I = \frac{ m_1 m_2 }{ m_1 \\! + \\! m_2 } x^2 = \mu x^2 Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about its center. The moment of inertia relative to the axis , which is at a distance from the center of mass along the x-axis, is :I = \int \left[(x - D)^2 + y^2\right] \, dm. Expanding the brackets yields :I = \int (x^2 + y^2) \, dm + D^2 \int dm - 2D\int x\, dm. The moment of inertia relative to the z-axis is then :I_\mathrm{cm} = \int (x^2 + y^2) \, dm. This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses each multiplied by the square of its perpendicular distance to an axis . This shows that the inertia matrix can be used to calculate the moment of inertia of a body around any specified rotation axis in the body. == Inertia tensor == For the same object, different axes of rotation will have different moments of inertia about those axes. The moment of inertia of a body with the shape of the cross- section is the second moment of this area about the z-axis perpendicular to the cross-section, weighted by its density. File:moment of inertia solid rectangular prism.png I_h = \frac{1}{12} m \left(w^2+d^2\right) I_w = \frac{1}{12} m \left(d^2+h^2\right) I_d = \frac{1}{12} m \left(w^2+h^2\right) Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal. thumb|Lewis Structure of H2O indicating bond angle and bond length Water () is a simple triatomic bent molecule with C2v molecular symmetry and bond angle of 104.5° between the central oxygen atom and the hydrogen atoms. * The moment of inertia of a thin disc of constant thickness s, radius R, and density \rho about an axis through its center and perpendicular to its face (parallel to its axis of rotational symmetry) is determined by integration. Moment of inertia of potentially tilted cuboids. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. The parallel axis theorem, also known as Huygens-Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes. ==Mass moment of inertia== thumb|right|The mass moment of inertia of a body around an axis can be determined from the mass moment of inertia around a parallel axis through the center of mass. A product of inertia term such as I_{12} is obtained by the computation I_{12} = \mathbf{e}_1\cdot\mathbf{I}\cdot\mathbf{e}_2, and can be interpreted as the moment of inertia around the x-axis when the object rotates around the y-axis. This shows that the moment of inertia of the body is the sum of each of the mr^2 terms, that is I_P = \sum_{i=1}^N m_i r_i^2. In order to obtain the moment of inertia IS in terms of the moment of inertia IR, introduce the vector d from S to the center of mass R, : \begin{align} I_S & = \int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R}+\mathbf{d})\cdot (\mathbf{r}-\mathbf{R}+\mathbf{d}) \, dV \\\ & = \int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R})\cdot (\mathbf{r}-\mathbf{R})dV + 2\mathbf{d}\cdot\left(\int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R}) \, dV\right) + \left(\int_V \rho(\mathbf{r}) \, dV\right)\mathbf{d}\cdot\mathbf{d}. \end{align} The first term is the moment of inertia IR, the second term is zero by definition of the center of mass, and the last term is the total mass of the body times the square magnitude of the vector d. * The moment of inertia of a thin rod with constant cross-section s and density \rho and with length \ell about a perpendicular axis through its center of mass is determined by integration. File:moment of inertia thin cylinder.png I \approx m r^2 \,\\!

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The data below show the temperature variation of the equilibrium constant of the reaction $\mathrm{Ag}_2 \mathrm{CO}_3(\mathrm{~s}) \rightleftharpoons \mathrm{Ag}_2 \mathrm{O}(\mathrm{s})+\mathrm{CO}_2(\mathrm{~g})$. Calculate the standard reaction enthalpy of the decomposition. $\begin{array}{lllll}T / \mathrm{K} & 350 & 400 & 450 & 500 \\ K & 3.98 \times 10^{-4} & 1.41 \times 10^{-2} & 1.86 \times 10^{-1} & 1.48\end{array}$

The standard enthalpy change ΔH⚬ is essentially the enthalpy change when the stoichiometric coefficients in the reaction are considered as the amounts of reactants and products (in mole); usually, the initial and final temperature is assumed to be 25 °C. Petrucci, Harwood and Herring, pages 241–243 Finally the reaction enthalpy may be estimated using bond energies for the bonds which are broken and formed in the reaction of interest. The standard enthalpy of formation is then determined using Hess's law. *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. If the enthalpies for each step can be measured, then their sum gives the enthalpy of the overall single reaction. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). Accordingly, the calculation of standard enthalpy of reaction is the most established way of quantifying the conversion of chemical potential energy into thermal energy. ==Enthalpy of reaction for standard conditions defined and measured== The standard enthalpy of a reaction is defined so as to depend simply upon the standard conditions that are specified for it, not simply on the conditions under which the reactions actually occur. 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. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. 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 combustion of ethane gas refers to the reaction C2H6 (g) + (7/2) O2 (g) → 2 CO2 (g) + 3 H2O (l). For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. The enthalpy of reaction can then be analyzed by applying Hess's Law, which states that the sum of the enthalpy changes for a number of individual reaction steps equals the enthalpy change of the overall reaction. 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. The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. However O2 is an element in its standard state, so that \Delta_{\text{f}} H^{\ominus }(\text{O}_2) = 0, and the heat of reaction is simplified to :\Delta_{\text{r}} H^{\ominus } = [\Delta_{\text{f}} H^{\ominus }(\text{CO}_2) + 2\Delta_{\text{f}} H^{\ominus } (\text{H}_2{}\text{O})] - \Delta_{\text{f}} H^{\ominus }(\text{CH}_4), which is the equation in the previous section for the enthalpy of combustion \Delta_{\text{comb}}H^{\ominus }. == Key concepts for enthalpy calculations == *When a reaction is reversed, the magnitude of ΔH stays the same, but the sign changes. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. * Standard enthalpy of combustion is the enthalpy change when one mole of an organic compound reacts with molecular oxygen (O2) to form carbon dioxide and liquid water. 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. For this reason it is important to note which standard concentration value is being used when consulting tables of enthalpies of formation. ==Introduction== Two initial thermodynamic systems, each isolated in their separate states of internal thermodynamic equilibrium, can, by a thermodynamic operation, be coalesced into a single new final isolated thermodynamic system. It is also possible to evaluate the enthalpy of one reaction from the enthalpies of a number of other reactions whose sum is the reaction of interest, and these not need be formation reactions.

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The osmotic pressures of solutions of poly(vinyl chloride), PVC, in cyclohexanone at $298 \mathrm{~K}$ are given below. The pressures are expressed in terms of the heights of solution (of mass density $\rho=0.980 \mathrm{~g} \mathrm{~cm}^{-3}$ ) in balance with the osmotic pressure. Determine the molar mass of the polymer. $\begin{array}{llllll}c /\left(\mathrm{g} \mathrm{dm}^{-3}\right) & 1.00 & 2.00 & 4.00 & 7.00 & 9.00 \\ h / \mathrm{cm} & 0.28 & 0.71 & 2.01 & 5.10 & 8.00\end{array}$

The mass- average molecular mass, , is also related to the fractional monomer conversion, , in step-growth polymerization (for the simplest case of linear polymers formed from two monomers in equimolar quantities) as per Carothers' equation: :\bar{X}_w=\frac{1+p}{1-p} \quad \bar{M}_w=\frac{M_o\left(1+p\right)}{1-p}, where is the molecular mass of the repeating unit. ===Z-average molar mass=== The z-average molar mass is the third moment or third power average molar mass, which is calculated by \bar{M}_z=\frac{\sum M_i^3 N_i} {\sum M_i^2 N_i}\quad The z-average molar mass can be determined with ultracentrifugation. If the relationship between molar mass and the hydrodynamic volume changes (i.e., the polymer is not exactly the same shape as the standard) then the calibration for mass is in error. Combining these two equations gives an expression for the molar mass in terms of the vapour density for conditions of known pressure and temperature: :M = {{RT\rho}\over{p}} . === Freezing-point depression === The freezing point of a solution is lower than that of the pure solvent, and the freezing-point depression () is directly proportional to the amount concentration for dilute solutions. 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. It is determined by measuring the molecular mass of polymer molecules, summing the masses, and dividing by . \bar{M}_n=\frac{\sum_i N_iM_i}{\sum_i N_i} The number average molecular mass of a polymer can be determined by gel permeation chromatography, viscometry via the (Mark–Houwink equation), colligative methods such as vapor pressure osmometry, end-group determination or proton NMR.Polymer Molecular Weight Analysis by 1H NMR Spectroscopy Josephat U. Izunobi and Clement L. Higginbotham J. Chem. Educ., 2011, 88 (8), pp 1098–1104 High number-average molecular mass polymers may be obtained only with a high fractional monomer conversion in the case of step-growth polymerization, as per the Carothers' equation. ===Mass average molar mass=== The mass average molar mass (often loosely termed weight average molar mass) is another way of describing the molar mass of a polymer. The molar mass distribution of a polymer may be modified by polymer fractionation. == Definitions of molar mass average == Different average values can be defined, depending on the statistical method applied. The molar mass distribution of a polymer sample depends on factors such as chemical kinetics and work-up procedure. 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". This is particularly important in polymer science, where different polymer molecules may contain different numbers of monomer units (non-uniform polymers). == Average molar mass of mixtures == The average molar mass of mixtures \overline{M} can be calculated from the mole fractions of the components and their molar masses : :\overline{M} = \sum_i x_i M_i. * Molar mass: chemistry second-level course. The molecular formula C3H7Cl (molar mass: 78.54 g/mol, exact mass: 78.0236 u) may refer to: * Isopropyl chloride * n-Propyl chloride, also known as 1-propyl chloride or 1-chloropropane 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. The quantity a in the expression for the viscosity average molar mass varies from 0.5 to 0.8 and depends on the interaction between solvent and polymer in a dilute solution. 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 molecular formula C2H3ClO (molar mass: 78.50 g/mol, exact mass: 77.9872 u) may refer to: * Acetyl chloride * Chloroacetaldehyde * Chloroethylene oxide 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. About 57% of the mass of PVC is chlorine. 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. By dissolving a polymer an insoluble high molar mass fraction may be filtered off resulting in a large reduction in and a small reduction in , thus reducing dispersity. ===Number average molar mass=== The number average molar mass is a way of determining the molecular mass of a polymer. The molecular formula C3H4Cl2O2 (molar mass: 142.97 g/mol, exact mass: 141.9588 u) may refer to: * Chloroethyl chloroformate * Dalapon 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 . * 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.

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Estimate the wavelength of electrons that have been accelerated from rest through a potential difference of $40 \mathrm{kV}$.

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. The wavelength corresponding to the mean photon energy is given by :\lambda_{\langle E \rangle} = ( cm K)/T\;. ==Recommendation to de-emphasize== Marr and Wilkin (2012) contend that the widespread teaching of Wien's displacement law in introductory courses is undesirable, and it would be better replaced by alternate material. Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. Solving for the wavelength \lambda in millimetres, and using kelvins for the temperature yields: : ===Parameterization by frequency=== Another common parameterization is by frequency. Using Wien's law, one finds a peak emission per nanometer (of wavelength) at a wavelength of about 500 nm, in the green portion of the spectrum near the peak sensitivity of the human eye.Walker, J. Fundamentals of Physics, 8th ed., John Wiley and Sons, 2008, p. 891. . 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. 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). This is a consequence of the strong statement of Wien's law. ==Frequency-dependent formulation== For spectral flux considered per unit frequency d u (in hertz), Wien's displacement law describes a peak emission at the optical frequency u_\text{peak} given by: : u_\text{peak} = { x \over h} k\,T \approx (5.879 \times 10^{10} \ \mathrm{Hz/K}) \cdot T or equivalently :h u_\text{peak} = x\, k\, T \approx (2.431 \times 10^{-4} \ \mathrm{eV/K}) \cdot T where is a constant resulting from the maximization equation, is the Boltzmann constant, is the Planck constant, and is the absolute temperature. From the Planck constant h and the Boltzmann constant k, Wien's constant b can be obtained. ==Peak differs according to parameterization== Constants for different parameterizations of Wien's law Parameterized by x_\mathrm{peak} b (μm⋅K) Wavelength, \lambda 2898 \log\lambda or \log u 3670 Frequency, u 5099 Other characterizations of spectrum Parameterized by x b (μm⋅K) Mean photon energy 5327 10% percentile 2195 25% percentile 2898 50% percentile 4107 70% percentile 5590 90% percentile 9376 The results in the tables above summarize results from other sections of this article. 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. 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. A formula may be derived mathematically for the rate of scattering when a beam of electrons passes through a material. ==The interaction picture== Define the unperturbed Hamiltonian by H_0, the time dependent perturbing Hamiltonian by H_1 and total Hamiltonian by H. Walter Kaufmann's measurement of the electron charge-to- mass ratio for different velocities of the electron. 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. In case of constant perturbation,c_{k'}^{(1)} is calculated by :c_{k'}^{(1)}=\frac{\lang\ k'|H_1|k\rang }{E_{k'}-E_k}(1-e^{i(E_{k'} - E_k)t/\hbar}) :|c_{k'}(t)|^2= |\lang\ k'|H_1|k\rang |^2\frac {sin ^2(\frac {E_{k'}-E_k} {2 \hbar}t)} { ( \frac {E_{k'} -E_k} {2 \hbar} ) ^2 }\frac {1}{\hbar^2} Using the equation which is :\lim_{\alpha \rightarrow \infty} \frac{1}{\pi} \frac{sin^2(\alpha x)}{\alpha x^2}= \delta(x) The transition rate of an electron from the initial state k to final state k' is given by :P(k,k')=\frac {2 \pi} {\hbar} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) where E_k and E_{k'} are the energies of the initial and final states including the perturbation state and ensures the \delta-function indicate energy conservation. ==The scattering rate== The scattering rate w(k) is determined by summing all the possible finite states k' of electron scattering from an initial state k to a final state k', and is defined by :w(k)=\sum_{k'}P(k,k')=\frac {2 \pi} {\hbar} \sum_{k'} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) The integral form is :w(k)=\frac {2 \pi} {\hbar} \frac {L^3} {(2 \pi)^3} \int d^3k' |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) ==References== * * Category:Semiconductor technology This is perhaps a more intuitive way of presenting "wavelength of peak emission". Thus, \textstyle \frac{1}{\lambda}=\frac{E}{hc} (in this formula the h represents Planck's constant). This is an inverse relationship between wavelength and temperature. Contrary to the then known cathode rays which reached speeds only up to 0.3c, c being the speed of light, Becquerel rays reached velocities up to 0.9c. With the emission now considered per unit frequency, this peak now corresponds to a wavelength about 76% longer than the peak considered per unit wavelength. Formally, the wavelength version of Wien's displacement law states that the spectral radiance of black-body radiation per unit wavelength, peaks at the wavelength \lambda_\text{peak} given by: :\lambda_\text{peak} = \frac{b}{T} where is the absolute temperature and is a constant of proportionality called Wien's displacement constant, equal to or .

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The standard enthalpy of formation of $\mathrm{H}_2 \mathrm{O}(\mathrm{g})$ at $298 \mathrm{~K}$ is $-241.82 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Estimate its value at $100^{\circ} \mathrm{C}$ given the following values of the molar heat capacities at constant pressure: $\mathrm{H}_2 \mathrm{O}(\mathrm{g}): 33.58 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} ; \mathrm{H}_2(\mathrm{~g}): 28.82 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} ; \mathrm{O}_2(\mathrm{~g})$ : $29.36 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$. Assume that the heat capacities are independent of temperature.

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. 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. . An ideal gas has the equation of state: P V = n R T\, where :P = pressure :V = volume :n = number of moles :R = universal gas constant(Gas constant) :T = temperature The ideal gas equation of state can be arranged to give: : V = n R T / P\, or \, n R = P V / T The following partial derivatives are obtained from the above equation of state: :\left(\frac{\partial V}{\partial T}\right)_{P}\ = \frac {n R}{P}\ = \left(\frac{V P}{T}\right)\left(\frac{1}{P}\right) = \frac{V}{T} :\left(\frac{\partial V}{\partial P}\right)_{T}\ = - \frac {n R T}{P^2}\ = - \frac {P V}{P^2}\ = - \frac{V}{P} The following simple expressions are obtained for thermal expansion coefficient \alpha : :\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\ = \frac{1}{V}\left(\frac{V}{T}\right) :\alpha= 1 / T \, and for isothermal compressibility \beta_{T}: :\beta_{T}= - \frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\ = - \frac{1}{V}\left( - \frac{V}{P}\right) :\beta_{T}= 1 / P \, One can now calculate C_{P} - C_{V}\, for ideal gases from the previously-obtained general formula: :C_{P} - C_{V}= V T\frac{\alpha^{2}}{\beta_{T}}\ = V T\frac{(1 / T)^2}{1 / P} = \frac{V P}{T} Substituting from the ideal gas equation gives finally: :C_{P} - C_{V} = n R\, where n = number of moles of gas in the thermodynamic system under consideration and R = universal gas constant. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? "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. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? In an isenthalpic process, the enthalpy is constant. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? The corresponding expression for the ratio of specific heat capacities remains the same since the thermodynamic system size- dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ?

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The standard potential of the cell $\mathrm{Pt}(\mathrm{s})\left|\mathrm{H}_2(\mathrm{~g})\right| \mathrm{HBr}(\mathrm{aq})|\operatorname{AgBr}(\mathrm{s})| \mathrm{Ag}(\mathrm{s})$ was measured over a range of temperatures, and the data were found to fit the following polynomial: $$ E_{\text {cell }}^{\bullet} / \mathrm{V}=0.07131-4.99 \times 10^{-4}(T / \mathrm{K}-298)-3.45 \times 10^{-6}(\mathrm{~T} / \mathrm{K}-298)^2 $$ The cell reaction is $\operatorname{AgBr}(\mathrm{s})+\frac{1}{2} \mathrm{H}_2(\mathrm{~g}) \rightarrow \mathrm{Ag}(\mathrm{s})+\mathrm{HBr}(\mathrm{aq})$. Evaluate the standard reaction Gibbs energy, enthalpy, and entropy at $298 \mathrm{~K}$.

==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. -\beta\left[\log{\frac{x}{\sigma}}\right]^2} \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right) | cdf =1-e^{-\alpha\log{\frac{x}{\sigma}}-\beta[\log{\frac{x}{\sigma}}]^2}| mean =\sigma+\tfrac{\sigma}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-1+\alpha}{\sqrt{2\beta}}\right) where H_n(x) is the "probabilists' Hermite polynomials"| median =\sigma \left(e^{\frac{-\alpha+\sqrt{\alpha^2+\beta\log{16}}}{2\beta}}\right)| mode =| variance = \left(\sigma^2+\tfrac{2\sigma^2}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-2+\alpha}{\sqrt{2\beta}}\right)\right)-\mu^2 | skewness =| kurtosis =| entropy =| mgf =| char =| }} In probability, statistics, economics, and actuarial science, the Benini distribution is a continuous probability distribution that is a statistical size distribution often applied to model incomes, severity of claims or losses in actuarial applications, and other economic data.A. Sen and J. Silber (2001). {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup | cdf = F(x)=\frac{1}{\Gamma(k)} \gamma\left(k, \frac{x}{\theta}\right) | mean = k \theta | median = No simple closed form | mode = (k - 1)\theta \text{ for } k \geq 1, 0 \text{ for } k < 1 | variance = k \theta^2 | skewness = \frac{2}{\sqrt{k}} | kurtosis = \frac{6}{k} | entropy = \begin{align} k &\+ \ln\theta + \ln\Gamma(k)\\\ &\+ (1 - k)\psi(k) \end{align} | mgf = (1 - \theta t)^{-k} \text{ for } t < \frac{1}{\theta} | char = (1 - \theta it)^{-k} | parameters2 = | support2 = x \in (0, \infty) | pdf2 = f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x } | cdf2 = F(x)=\frac{1}{\Gamma(\alpha)} \gamma(\alpha, \beta x) | mean2 = \frac{\alpha}{\beta} | median2 = No simple closed form | mode2 = \frac{\alpha - 1}{\beta} \text{ for } \alpha \geq 1\text{, }0 \text{ for } \alpha < 1 | variance2 = \frac{\alpha}{\beta^2} | skewness2 = \frac{2}{\sqrt{\alpha}} | kurtosis2 = \frac{6}{\alpha} | entropy2 = \begin{align} \alpha &\- \ln \beta + \ln\Gamma(\alpha)\\\ &\+ (1 - \alpha)\psi(\alpha) \end{align} | mgf2 = \left(1 - \frac{t}{\beta}\right)^{-\alpha} \text{ for } t < \beta | char2 = \left(1 - \frac{it}{\beta}\right)^{-\alpha} | moments = k = \frac{E[X]^2}{V[X]} \quad \quad \theta = \frac{V[X]}{E[X]} \quad \quad | moments2 = \alpha = \frac{E[X]^2}{V[X]} \beta = \frac{E[X]}{V[X]} | fisher = I(k, \theta) = \begin{pmatrix}\psi^{(1)}(k) & \theta^{-1} \\\ \theta^{-1} & k \theta^{-2}\end{pmatrix} | fisher2 = I(\alpha, \beta) = \begin{pmatrix}\psi^{(1)}(\alpha) & -\beta^{-1} \\\ -\beta^{-1} & \alpha \beta^{-2}\end{pmatrix} }} In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). \\! | 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. This page provides supplementary chemical data on Ytterbium(III) chloride == Structure and properties data == Structure and properties Standard reduction potential, E° 1.620 eV Crystallographic constants a = 6.73 b = 11.65 c = 6.38 β = 110.4° Effective nuclear charge 3.290 Bond strength 1194±7 kJ/mol Bond length 2.434 (Yb-Cl) Bond angle 111.5° (Cl-Yb-Cl) Magnetic susceptibility 4.4 μB == Thermodynamic properties == Phase behavior Std enthalpy change of fusionΔfusH ~~o~~ 58.1±11.6 kJ/mol Std entropy change of fusionΔfusS ~~o~~ 50.3±10.1 J/(mol•K) Std enthalpy change of atomizationΔatH ~~o~~ 1166.5±4.3 kJ/mol Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −959.5±3.0 kJ/mol Standard molar entropy S ~~o~~ solid 163.5 J/(mol•K) Heat capacity cp 101.4 J/(mol•K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid −212.8 kJ/mol Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas −651.1±5.0 kJ/mol == Spectral data == UV-Vis λmax 268 nm Extinction coefficient 303 M−1cm−1 IR Major absorption bands ν1 = 368.0 cm−1 ν2 = 178.4 cm−1 ν3 = 330.7 cm−1 ν4 = 117.8 cm−1 MS Ionization potentials from electron impact YbCl3+ = 10.9±0.1 eV YbCl2+ = 11.6±0.1 eV YbCl+ = 14.3±0.1 eV ==References== Category:Chemical data pages Category:Chemical data pages cleanup :*(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). :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Interpolated approximations and bounds are all of the form : u(k) \approx \tilde{g}(k) u_{L\infty}(k) + (1 - \tilde{g}(k)) u_U(k) where \tilde{g} is an interpolating function running monotonically from 0 at low k to 1 at high k, approximating an ideal, or exact, interpolator g(k): :g(k) = \frac{ u_U(k) - u(k)}{ u_U(k) - u_{L\infty}(k)} For the simplest interpolating function considered, a first-order rational function :\tilde{g}_1(k) = \frac{k}{b_0 + k} the tightest lower bound has :b_0 = \frac{\frac{8}{405} + e^{-\gamma} \log 2 - \frac{\log^2 2}{2}}{e^{-\gamma} - \log 2 + \frac{1}{3}} - \log 2 \approx 0.143472 and the tightest upper bound has :b_0 = \frac{e^{-\gamma} - \log 2 + \frac{1}{3}}{1 - \frac{e^{-\gamma} \pi^2}{12}} \approx 0.374654 The interpolated bounds are plotted (mostly inside the yellow region) in the log–log plot shown. Above 750 K Tc values may be in error by 10 K or more. Nowadays, statistical software, such as the R programming language, and functions available in many spreadsheet programs compute values of the t-distribution and its inverse without tables. ==See also== * F-distribution * Folded-t and half-t distributions * Hotelling's T-squared distribution * Multivariate Student distribution * Standard normal table (Z-distribution table) * t-statistic * Tau distribution, for internally studentized residuals * Wilks' lambda distribution * Wishart distribution * Modified half-normal distribution with the pdf on (0, \infty) is given as f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}}, where \Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\\\(1,0)\end{matrix};z \right) denotes the Fox–Wright Psi function. ==Notes== ==References== * * * * ==External links== * *Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term "Student's distribution") * First Students on page 112. (DOE contract 95‑831). : p(k,\theta \mid p, q, r, s) = \frac{1}{Z} \frac{p^{k-1} e^{-\theta^{-1} q}}{\Gamma(k)^r \theta^{k s}}, where Z is the normalizing constant with no closed-form solution. Berg and Pedersen found more terms: : u(k) = k - \frac{1}{3} + \frac{8}{405 k} + \frac{184}{25515 k^2} + \frac{2248}{3444525 k^3} - \frac{19006408}{15345358875 k^4} - O\left(\frac{1}{k^5}\right) + \cdots 320px|thumb| Two gamma distribution median asymptotes which are conjectured to be bounds (upper solid red and lower dashed red), of the from u(k) \approx 2^{-1/k}(A + k), and an interpolation between them that makes an approximation (dotted red) that is exact at k = 1 and has maximum relative error of about 0.6%. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. Partial sums of these series are good approximations for high enough k; they are not plotted in the figure, which is focused on the low-k region that is less well approximated. If α is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the following series expansion: :F(x;\alpha,\beta) = 1-\sum_{i=0}^{\alpha-1} \frac{(\beta x)^i}{i!} e^{-\beta x} = e^{-\beta x} \sum_{i=\alpha}^{\infty} \frac{(\beta x)^i}{i!}. === Characterization using shape k and scale θ === A random variable X that is gamma-distributed with shape k and scale θ is denoted by :X \sim \Gamma(k, \theta) \equiv \operatorname{Gamma}(k, \theta) [[Image:Gamma- PDF-3D.png|thumb|right|320px|Illustration of the gamma PDF for parameter values over k and x with θ set to 1, 2, 3, 4, 5 and 6\. A bias-corrected variant of the estimator for the scale θ is : \tilde{\theta} = \frac{N}{N - 1} \hat{\theta} A bias correction for the shape parameter k is given as : \tilde{k} = \hat{k} - \frac{1}{N} \left(3 \hat{k} - \frac{2}{3} \left(\frac{\hat{k}}{1 + \hat{k}}\right) - \frac{4}{5} \frac{\hat{k}}{(1 + \hat{k})^2} \right) ====Bayesian minimum mean squared error==== With known k and unknown θ, the posterior density function for theta (using the standard scale-invariant prior for θ) is : P(\theta \mid k, x_1, \dots, x_N) \propto \frac 1 \theta \prod_{i=1}^N f(x_i; k, \theta) Denoting : y \equiv \sum_{i=1}^Nx_i , \qquad P(\theta \mid k, x_1, \dots, x_N) = C(x_i) \theta^{-N k-1} e^{-y/\theta} Integration with respect to θ can be carried out using a change of variables, revealing that 1/θ is gamma-distributed with parameters α = Nk, β = y. :\int_0^\infty \theta^{-Nk - 1 + m} e^{-y/\theta}\, d\theta = \int_0^\infty x^{Nk - 1 - m} e^{-xy} \, dx = y^{-(Nk - m)} \Gamma(Nk - m) \\! The Kullback–Leibler divergence (KL-divergence), of Gamma(αp, βp) ("true" distribution) from Gamma(αq, βq) ("approximating" distribution) is given byW.D. Penny, [www.fil.ion.ucl.ac.uk/~wpenny/publications/densities.ps KL-Divergences of Normal, Gamma, Dirichlet, and Wishart densities] : \begin{align} D_{\mathrm{KL}}(\alpha_p,\beta_p; \alpha_q, \beta_q) = {} & (\alpha_p-\alpha_q) \psi(\alpha_p) - \log\Gamma(\alpha_p) + \log\Gamma(\alpha_q) \\\ & {} + \alpha_q(\log \beta_p - \log \beta_q) + \alpha_p\frac{\beta_q-\beta_p}{\beta_p}. \end{align} Written using the k, θ parameterization, the KL-divergence of Gamma(kp, θp) from Gamma(kq, θq) is given by : \begin{align} D_{\mathrm{KL}}(k_p,\theta_p; k_q, \theta_q) = {} & (k_p-k_q)\psi(k_p) - \log\Gamma(k_p) + \log\Gamma(k_q) \\\ & {} + k_q(\log \theta_q - \log \theta_p) + k_p \frac{\theta_p - \theta_q}{\theta_q}. \end{align} ===Laplace transform=== The Laplace transform of the gamma PDF is :F(s) = (1 + \theta s)^{-k} = \frac{\beta^\alpha}{(s + \beta)^\alpha} . ==Related distributions== ===General=== * Let X_1, X_2, \ldots, X_n be n independent and identically distributed random variables following an exponential distribution with rate parameter λ, then \sum_i X_i ~ Gamma(n, 1/λ) where n is the shape parameter and λ is the rate, and \bar{X} = \frac{1}{n} \sum_i X_i \sim \operatorname{Gamma}(n, n\lambda) where the rate changes nλ. For the two parameter model, the quantile function (inverse cdf) is : F^{-1}(u) = \sigma \exp \sqrt{-\frac{1}{\beta} \log(1-u)}, \quad 0 < u < 1\. ==Related distributions== * If X \sim \mathrm{Benini}(\alpha,0,\sigma)\,, then X has a Pareto distribution with x_\mathrm{m}=\sigma * If X \sim \mathrm{Benini}(0,\tfrac{1}{2\sigma^2},1), then X \sim e^U where U \sim \mathrm{Rayleigh}(\sigma) ==Software== The (two parameter) Benini distribution density, probability distribution, quantile function and random number generator is implemented in the VGAM package for R, which also provides maximum likelihood estimation of the shape parameter. In particular for integer valued degrees of freedom u we have: For u >1 even, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 5 \cdot 3} { 2 \sqrt{ u}( u -2)( u -4)\cdots 4 \cdot 2\,}\cdot For u >1 odd, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 4 \cdot 2} {\pi \sqrt{ u}( u -2)( u -4)\cdots 5 \cdot 3\,}\cdot\\! This can be derived using the exponential family formula for the moment generating function of the sufficient statistic, because one of the sufficient statistics of the gamma distribution is ln(x). ===Information entropy=== The information entropy is : \begin{align} \operatorname{H}(X) & = \operatorname{E}[-\ln(p(X))] \\\\[4pt] & = \operatorname{E}[-\alpha \ln(\beta) + \ln(\Gamma(\alpha)) - (\alpha-1)\ln(X) + \beta X] \\\\[4pt] & = \alpha - \ln(\beta) + \ln(\Gamma(\alpha)) + (1-\alpha)\psi(\alpha). \end{align} In the k, θ parameterization, the information entropy is given by : \operatorname{H}(X) =k + \ln(\theta) + \ln(\Gamma(k)) + (1-k)\psi(k). ===Kullback–Leibler divergence=== thumb|right|320px|Illustration of the Kullback–Leibler (KL) divergence for two gamma PDFs.

1.3

-21.2

260.0

0.829

0.0547

B

In an industrial process, nitrogen is heated to $500 \mathrm{~K}$ in a vessel of constant volume. If it enters the vessel at $100 \mathrm{~atm}$ and $300 \mathrm{~K}$, what pressure would it exert at the working temperature if it behaved as a perfect gas?

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. thumb|Nitrogen is a liquid under -195.8 degrees Celsius (77K). thumb|A medium- sized dewar is being filled with liquid nitrogen by a larger cryogenic storage tank. thumb|right|A gas regulator attached to a nitrogen cylinder Industrial gases are the gaseous materials that are manufactured for use in industry. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. The pressure melting point is nearly a constant 0 °C at pressures above the triple point at 611.7 Pa, where water can exist in only the solid or liquid phases, through atmospheric pressure (100 kPa) until about 10 MPa. All of these substances are also provided as a gas (not a vapor) at the 200 bar pressure in a gas cylinder because that pressure is above their critical pressure. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. Springer Science+Business Media LLC (2007) This is a logical dividing line, since the normal boiling points of the so-called permanent gases (such as helium, hydrogen, neon, nitrogen, oxygen, and normal air) lie below 120K while the Freon refrigerants, hydrocarbons, and other common refrigerants have boiling points above 120K. The pressure melting point of ice is the temperature at which ice melts at a given pressure. 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. 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. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. A pressure reactor, sometimes referred to as a pressure tube, or a sealed tube, is a chemical reaction vessel which can conduct a reaction under pressure. Pressure vessels for gas storage may also be classified by volume. A pressure reactor is a special application of a pressure vessel. However, the idea of a perfect gas model is often invoked as a combination of the ideal gas equation of state with specific additional assumptions regarding the variation (or nonvariation) of the heat capacity with temperature. == Perfect gas nomenclature == The terms perfect gas and ideal gas are sometimes used interchangeably, depending on the particular field of physics and engineering. However, pressure can speed up the desired reaction and only impacts decomposition when it involves the release of a gas or a reaction with a gas in the vessel. With increasing pressure above 10 MPa, the pressure melting point decreases to a minimum of −21.9 °C at 209.9 MPa. The term “industrial gases” is sometimes narrowly defined as just the major gases sold, which are: nitrogen, oxygen, carbon dioxide, argon, hydrogen, acetylene and helium. This is because the internal energy of an ideal gas is at most a function of temperature, as shown by the thermodynamic equation \left({{\partial U} \over {\partial V}}\right)_T = T\left({{\partial S} \over {\partial V}}\right)_T - P = T\left({{\partial P} \over {\partial T}}\right)_V - P, which is exactly zero when P = nRT / V . A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric 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.

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For instance, suppose that one opens an individual retirement account (IRA) at age 25 and makes annual investments of $\$ 2000$ thereafter in a continuous manner. Assuming a rate of return of $8 \%$, what will be the balance in the IRA at age 65 ?

For those 65 and over, 11.6% of retirement accounts have balances of at least $1 million, more than twice that of the $407,581 average (shown). For those 65 and over, 11.6% of retirement accounts have balances of at least $1 million, more than twice that of the $407,581 average (shown). The return over the five-year period for such an investor would be ($19.90 + $5.78) / $14.21 − 1 = 80.72%, and the arithmetic average rate of return would be 80.72%/5 = 16.14% per year. ==See also== * Annual percentage yield * Average for a discussion of annualization of returns * Capital budgeting * Compound annual growth rate * Compound interest * Dollar cost averaging * Economic value added * Effective annual rate * Effective interest rate * Expected return * Holding period return * Internal rate of return * Modified Dietz method * Net present value * Rate of profit * Return of capital * Return on assets * Return on capital * Returns (economics) * Simple Dietz method * Time value of money * Time-weighted return * Value investing * Yield ==Notes== ==References== ==Further reading== * A. A. Groppelli and Ehsan Nikbakht. As another example, a two-year return of 10% converts to an annualized rate of return of 4.88% = ((1+0.1)(12/24) − 1), assuming reinvestment at the end of the first year. thumb|Average balances of retirement accounts, for households having such accounts, exceed median net worth across all age groups. thumb|Average balances of retirement accounts, for households having such accounts, exceed median net worth across all age groups. Some highlights from the 2008 data follow: * The average and median IRA account balance was $54,863 and $15,756, respectively, while the average and median IRA individual balance (all accounts from the same person combined) was $69,498 and $20,046. The purpose of this report was to study individual retirement accounts (IRAs) and the account valuations. Also that there are an estimate of 791 taxpayers with IRA account balances between $10,000,000 and $25,000,000. Furthermore, the typical working household has virtually no retirement savings - the median retirement account balance is $2,500 for all working-age households and $14,500 for near-retirement households. * In 2014, the federal government will forgo an estimated $17.45 billion in tax revenue from IRAs, which Congress created to ensure equitable tax treatment for those not covered by employer-sponsored retirement plans. * 98.5% of taxpayers have IRA account balances at $1,000,000 or less. * 1.2% of tax payers have IRA account balances at $1,000,000 to $2,000,000. * 0.2% (or 83,529) taxpayers have IRA account balances of $2,000,000 to $3,000,000 * 0.1% (or 36,171) taxpayers have IRA account balances of $3,000,000 to $5,000,000 * <0.1% (or 7,952) taxpayers have IRA account balances of $5,000,000 to $10,000,000 * <0.1% (or 791) taxpayers have IRA account balances of $10,000,000 to $25,000,000 * <0.1% (or 314) taxpayers have IRA account balances of $25,000,000 or more ===Retirement savings=== While the average (mean) and median IRA individual balance in 2008 were approximately $70,000 and $20,000 respectively, higher balances are not rare. 6.3% of individuals had total balances of $250,000 or more (about 12.5 times the median), and in rare cases, individuals own IRAs with very substantial balances, in some cases $100 million or above (about 5,000 times the median individual balance). An individual retirement account is a type of individual retirement arrangementSee 26 C.F.R. sec. 1.408-4. as described in IRS Publication 590, Individual Retirement Arrangements (IRAs). Actuarial assumptions like 5% interest, 3% salary increases and the UP84 Life Table for mortality are used to calculate a level contribution rate that would create the needed lump sum at retirement age. For example, a person aged 45, who put $4,000 into a traditional IRA this year so far, can either put $2,000 more into this traditional IRA, or $2,000 in a Roth IRA, or some combination of those. The cash balance plan typically offers a lump sum at and often before normal retirement age. * Contributions are concentrated at the maximum amount – of those contributing to an IRA, approximately 40% contributed the maximum (whether contributing to traditional or Roth), and 46.7% contributed close to the maximum (in the $5,000–$6,000 range). * Excluding SEPs and SIMPLEs (i.e., concerning traditional, rollover, and Roth IRAs), 15.1% of individuals holding an IRA contributed to one. The Retirement Age as referred to in paragraph (2) is further increased by 1 (one) year for every subsequent 3 (three) years until it reaches the Retirement Age of 65 (sixty five) years. For example, if the logarithmic return of a security per trading day is 0.14%, assuming 250 trading days in a year, then the annualized logarithmic rate of return is 0.14%/(1/250) = 0.14% x 250 = 35% ===Returns over multiple periods=== When the return is calculated over a series of sub- periods of time, the return in each sub-period is based on the investment value at the beginning of the sub-period. Cash flow example on $1,000 investment Year 1 Year 2 Year 3 Year 4 Dollar return $100 $55 $60 $50 ROI 10% 5.5% 6% 5% ==Uses== * Rates of return are useful for making investment decisions. For example, a return over one month of 1% converts to an annualized rate of return of 12.7% = ((1+0.01)12 − 1). * The continuously compounded rate of return in this example is: :\ln\left(\frac{103.02}{100}\right) = 2.98\%.

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Suppose that a mass weighing $10 \mathrm{lb}$ stretches a spring $2 \mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \mathrm{ft} / \mathrm{s}$, by determining the position of the mass at any later time, calculate the amplitude of the motion.

Since the inertia of the beam can be found from its mass, the spring constant can be calculated. For waves on a string, or in a medium such as water, the amplitude is a displacement. Particle displacement or displacement amplitude is a measurement of distance of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The weight rotates about the axis of the spring, twisting it, instead of swinging like an ordinary pendulum. The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). Thus, for g ≈ 9.8 ms−2, :L \approx \frac{894}{n^2} \; \text{m} | n 71 0.846 0.177 70 0.857 0.182 69 0.870 0.188 68 0.882 0.193 67 0.896 0.199 66 0.909 0.205 65 0.923 0.212 64 0.938 0.218 63 0.952 0.225 62 0.968 0.232 61 0.984 0.240 60 1.000 0.248 Parameters of the pendulum wave in the animation above ---|--- == See also == * Newton's cradle - a set of pendulums constrained to swing along the axis of the apparatus and collide with one another ==References== Category:Pendulums Category:Kinetic art RMS Amplitude . When the oscillatory motion of the balance dies out, the deflection will be proportional to the force: :\theta = FL/\kappa\, To determine F\, it is necessary to find the torsion spring constant \kappa\,. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum. ===Lagrangian=== The spring has the rest length l_0 and can be stretched by a length x. thumb|Video of a model torsion pendulum oscillating A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. The amplitude of sound waves and audio signals (which relates to the volume) conventionally refers to the amplitude of the air pressure in the wave, but sometimes the amplitude of the displacement (movements of the air or the diaphragm of a speaker) is described. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. ==Analysis and interpretation== thumb|300px|2 DOF elastic pendulum with polar coordinate plots. Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by :\delta = \frac{v}{\omega}, :\delta = \frac{p}{\omega z(\mathbf{r},\, s)}. ==See also== *Sound *Sound particle *Particle velocity *Particle acceleration ==References and notes== Related Reading: * * * ==External links== *Acoustic Particle-Image Velocimetry. The value of damping that causes the oscillatory motion to settle quickest is called the critical dampingC_c\,: :C_c = 2 \sqrt{\kappa I}\, ==See also== * Beam (structure) * Slinky, helical toy spring ==References== ==Bibliography== * . Hooke's law is the potential energy of the spring itself: :V_k=\frac{1}{2}kx^2 where k is the spring constant. Strictly speaking, this is no longer amplitude since there is the possibility that a constant (DC component) is included in the measurement. ===Peak-to-peak amplitude=== Peak-to-peak amplitude (abbreviated p–p) is the change between peak (highest amplitude value) and trough (lowest amplitude value, which can be negative). With appropriate circuitry, peak-to- peak amplitudes of electric oscillations can be measured by meters or by viewing the waveform on an oscilloscope. There are various definitions of amplitude (see below), which are all functions of the magnitude of the differences between the variable's extreme values. We will use the word "torsion" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. The general differential equation of motion is: :I\frac{d^2\theta}{dt^2} + C\frac{d\theta}{dt} + \kappa\theta = \tau(t) If the damping is small, C \ll \sqrt{\kappa I}\,, as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system: :f_n = \frac{\omega_n}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}\, Therefore, the period is represented by: :T_n = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{I}{\kappa}}\, The general solution in the case of no drive force (\tau = 0\,), called the transient solution, is: :\theta = Ae^{-\alpha t} \cos{(\omega t + \phi)}\, where: ::\alpha = C/2I\, ::\omega = \sqrt{\omega_n^2 - \alpha^2} = \sqrt{\kappa/I - (C/2I)^2}\, ===Applications=== thumb|Animation of a torsion spring oscillating The balance wheel of a mechanical watch is a harmonic oscillator whose resonant frequency f_n\, sets the rate of the watch. The square of the amplitude is proportional to the intensity of the wave.

9.8

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B

At time $t=0$ a tank contains $Q_0 \mathrm{lb}$ of salt dissolved in 100 gal of water; see Figure 2.3.1. Assume that water containing $\frac{1}{4} \mathrm{lb}$ of salt/gal is entering the tank at a rate of $r \mathrm{gal} / \mathrm{min}$ and that the well-stirred mixture is draining from the tank at the same rate. Set up the initial value problem that describes this flow process. By finding the amount of salt $Q(t)$ in the tank at any time, and the limiting amount $Q_L$ that is present after a very long time, if $r=3$ and $Q_0=2 Q_L$, find the time $T$ after which the salt level is within $2 \%$ of $Q_L$.

Renewal time simply becomes a question how quickly could the inflows of the lake fill the entire volume of the basin (this does still assume the outflows are unchanged). It may therefore be estimated by the steady-state approximation, which specifies that the rate at which it is formed equals the (total) rate at which it is consumed. This case is referred to as diffusion control and, in general, occurs when the formation of product from the activated complex is very rapid and thus the provision of the supply of reactants is rate-determining. == See also == * Product-determining step * Rate-limiting step (biochemistry) ==References== * Category:Chemical kinetics ja:反応速度#律速段階 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. Instead, residence time models developed for gas and fluid dynamics, chemical engineering, and bio-hydrodynamics can be adapted to generate residence times for sub-volumes of lakes. == Renewal time == One useful mathematical model the measurement of how quickly inflows are able to refill a lake. In principle, the time evolution of the reactant and product concentrations can be determined from the set of simultaneous rate equations for the individual steps of the mechanism, one for each step. The observed rate law is :v = k \frac{\ce{[Cl2][H2C2O4]}}{[\ce{H+}]^2[\ce{Cl^-}]}, which implies an activated complex in which the reactants lose 2 + before the rate-determining step. This equation can be derived from the mass conservation equations of two-phase flow, under the assumptions listed below. ==Equation== In a quasi-1D domain, the Buckley–Leverett equation is given by: : \frac{\partial S_w}{\partial t} + \frac{\partial}{\partial x}\left( \frac{Q}{\phi A} f_w(S_w) \right) = 0, where S_w(x,t) is the wetting-phase (water) saturation, Q is the total flow rate, \phi is the rock porosity, A is the area of the cross-section in the sample volume, and f_w(S_w) is the fractional flow function of the wetting phase. That is, the overall rate is determined by the rate of the first step, and (almost) all molecules that react at the first step continue to the fast second step. ===Pre- equilibrium: if the second step were rate-determining=== The other possible case would be that the second step is slow and rate-determining, meaning that it is slower than the first step in the reverse direction: r2 ≪ r−1. In the simplest case the initial step is the slowest, and the overall rate is just the rate of the first step. The rate-determining step is then the step with the largest Gibbs energy difference relative either to the starting material or to any previous intermediate on the diagram. Such a situation in which an intermediate (here ) forms an equilibrium with reactants prior to the rate-determining step is described as a pre-equilibriumPeter Atkins and Julio de Paula, Physical Chemistry (8th ed., W. H. Freeman 2006) p. 814–815. . In this example is formed in one step and reacts in two, so that : \frac{d\ce{[NO3]}}{dt} = r_1 - r_2 - r_{-1} \approx 0. A possible mechanism in two elementary steps that explains the rate equation is: # + → NO + (slow step, rate-determining) # + CO → + (fast step) In this mechanism the reactive intermediate species is formed in the first step with rate r1 and reacts with CO in the second step with rate r2. It roughly expresses the amount of time taken for a substance introduced into a lake to flow out of it again. For a given reaction mechanism, the prediction of the corresponding rate equation (for comparison with the experimental rate law) is often simplified by using this approximation of the rate-determining step. Time of concentration is a concept used in hydrology to measure the response of a watershed to a rain event. The retention time is particularly important where downstream flooding or pollutants are concerned. ==Global retention time== The global retention time for a lake (the overall mean time that water spends in the lake) is calculated by dividing the lake volume by either the mean rate of inflow of all tributaries, or by the mean rate of outflow (ideally including evaporation and seepage). A number of methods can be used to calculate time of concentration, including the Kirpich (1940) and NRCS (1997) methods. This can be important for infrastructure development (design of bridges, culverts, etc.) and management, as well as to assess flood risk such as the ARkStorm-scenario. ==Example== thumb|left|400px|NOAA diagram illustrating the concept underlying time of concentration This image shows the basic principle which leads to determination of the time of concentration. Typically, f_w(S_w) is an 'S'-shaped, nonlinear function of the saturation S_w, which characterizes the relative mobilities of the two phases: : f_w(S_w) = \frac{\lambda_w}{\lambda_w + \lambda_n} = \frac{ \frac{k_{rw}}{\mu_w} }{ \frac{k_{rw}}{\mu_w} + \frac{k_{rn}}{\mu_n} }, where \lambda_w and \lambda_n denote the wetting and non-wetting phase mobilities. k_{rw}(S_w) and k_{rn}(S_w) denote the relative permeability functions of each phase and \mu_w and \mu_n represent the phase viscosities. ==Assumptions== The Buckley–Leverett equation is derived based on the following assumptions: * Flow is linear and horizontal * Both wetting and non-wetting phases are incompressible * Immiscible phases * Negligible capillary pressure effects (this implies that the pressures of the two phases are equal) * Negligible gravitational forces ==General solution== The characteristic velocity of the Buckley-Leverett equation is given by: :U(S_w) = \frac{Q}{\phi A} \frac{\mathrm{d} f_w}{\mathrm{d} S_w}. The Buckley–Leverett equation or the Buckley–Leverett displacement describes an immiscible displacement process, such as the displacement of oil by water, in a one-dimensional or quasi-one-dimensional reservoir.

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Suppose that a mass weighing $10 \mathrm{lb}$ stretches a spring $2 \mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \mathrm{ft} / \mathrm{s}$, by determining the position of the mass at any later time, calculate the phase of the motion.

Since the inertia of the beam can be found from its mass, the spring constant can be calculated. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. When this property is exactly satisfied, the balance spring is said to be isochronous, and the period of oscillation is independent of the amplitude of oscillation. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. thumb|Phase portrait of damped oscillator, with increasing damping strength. When the oscillatory motion of the balance dies out, the deflection will be proportional to the force: :\theta = FL/\kappa\, To determine F\, it is necessary to find the torsion spring constant \kappa\,. In this case, the phase shift is simply the argument shift \tau, expressed as a fraction of the common period T (in terms of the modulo operation) of the two signals and then scaled to a full turn: :\varphi = 2\pi \left[\\!\\!\left[ \frac{\tau}{T} \right]\\!\\!\right]. The stiffness of the spring, its spring coefficient, \kappa\, in N·m/radian^2, along with the balance wheel's moment of inertia, I\, in kg·m2, determines the wheel's oscillation period T\,. The general differential equation of motion is: :I\frac{d^2\theta}{dt^2} + C\frac{d\theta}{dt} + \kappa\theta = \tau(t) If the damping is small, C \ll \sqrt{\kappa I}\,, as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system: :f_n = \frac{\omega_n}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}\, Therefore, the period is represented by: :T_n = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{I}{\kappa}}\, The general solution in the case of no drive force (\tau = 0\,), called the transient solution, is: :\theta = Ae^{-\alpha t} \cos{(\omega t + \phi)}\, where: ::\alpha = C/2I\, ::\omega = \sqrt{\omega_n^2 - \alpha^2} = \sqrt{\kappa/I - (C/2I)^2}\, ===Applications=== thumb|Animation of a torsion spring oscillating The balance wheel of a mechanical watch is a harmonic oscillator whose resonant frequency f_n\, sets the rate of the watch. The term phase can refer to several different things: * It can refer to a specified reference, such as \textstyle \cos(2 \pi f t), in which case we would say the phase of \textstyle x(t) is \textstyle \varphi, and the phase of \textstyle y(t) is \textstyle \varphi - \frac{\pi}{2}. The weight rotates about the axis of the spring, twisting it, instead of swinging like an ordinary pendulum. The value of damping that causes the oscillatory motion to settle quickest is called the critical dampingC_c\,: :C_c = 2 \sqrt{\kappa I}\, ==See also== * Beam (structure) * Slinky, helical toy spring ==References== ==Bibliography== * . The formula above gives the phase as an angle in radians between 0 and 2\pi. A balance spring, or hairspring, is a spring attached to the balance wheel in mechanical timepieces. The term "phase" is also used when comparing a periodic function F with a shifted version G of it. If the shift in t is expressed as a fraction of the period, and then scaled to an angle \varphi spanning a whole turn, one gets the phase shift, phase offset, or phase difference of G relative to F. In physics and mathematics, the phase of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum. ===Lagrangian=== The spring has the rest length l_0 and can be stretched by a length x. Thus, for example, the sum of phase angles is 30° (, minus one full turn), and subtracting 50° from 30° gives a phase of 340° (, plus one full turn). We will use the word "torsion" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. thumb|Video of a model torsion pendulum oscillating A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. The phase \phi(t) is then the angle from the 12:00 position to the current position of the hand, at time t, measured clockwise.

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The logistic model has been applied to the natural growth of the halibut population in certain areas of the Pacific Ocean. ${ }^{12}$ Let $y$, measured in kilograms, be the total mass, or biomass, of the halibut population at time $t$. The parameters in the logistic equation are estimated to have the values $r=0.71 /$ year and $K=80.5 \times 10^6 \mathrm{~kg}$. If the initial biomass is $y_0=0.25 K$, find the biomass 2 years later.

Because of this structure, the model can be considered as the discrete-time analogue of the continuous-time logistic equation for population growth introduced by Verhulst; for comparison, the logistic equation is : \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right), and its solution is : N(t) = \frac{K N(0)}{N(0) + (K - N(0)) e^{-rt}}. ==References== * * * * * Category:Demography Category:Biostatistics Category:Fisheries science Category:Stochastic models As the population approaches its carrying capacity, the rate of growth decreases, and the population trend will become logistic. This S-shaped curve observed in logistic growth is a more accurate model than exponential growth for observing real- life population growth of organisms. ==See also== * Malthusian catastrophe * r/K selection theory ==References== ==Sources== John A. Miller and Stephen B. Harley zoology 4th edition ==External links== * Category:Biology Category:Biology articles needing attention Category:Population ecology Once the carrying capacity, or K, is incorporated to account for the finite resources that a population will be competing for within an environment, the aforementioned equation becomes the following: \frac{dN}{dt}=r_{max}\frac{dN}{dt}=r_{max}N\frac{K-N}{K} A graph of this equation creates an S-shaped curve, which demonstrates how initial population growth is exponential due to the abundance of resources and lack of competition. Here R0 is interpreted as the proliferation rate per generation and K = (R0 − 1) M is the carrying capacity of the environment. If, in a hypothetical population of size N, the birth rates (per capita) are represented as b and death rates (per capita) as d, then the increase or decrease in N during a time period t will be \frac{dN}{dt}=(b-d)N (b-d) is called the 'intrinsic rate of natural increase' and is a very important parameter chosen for assessing the impacts of any biotic or abiotic factor on population growth. The Ricker model, named after Bill Ricker, is a classic discrete population model which gives the expected number N t+1 (or density) of individuals in generation t + 1 as a function of the number of individuals in the previous generation,Ricker (1954) : N_{t+1} = N_t e^{r\left(1-\frac{N_t}{k}\right)}.\, Here r is interpreted as an intrinsic growth rate and k as the carrying capacity of the environment. The Monod equation is a mathematical model for the growth of microorganisms. In the study of age-structured population growth, probably one of the most important equations is the Euler–Lotka equation. The solution is : n_t = \frac{K n_0}{n_0 + (K - n_0) R_0^{-t}}. When the yield coefficient, being the ratio of mass of microorganisms to mass of substrate utilized, becomes very large this signifies that there is deficiency of substrate available for utilization. == Graphical determination of constants == As with the Michaelis–Menten equation graphical methods may be used to fit the coefficients of the Monod equation: * Eadie–Hofstee diagram * Hanes–Woolf plot * Lineweaver–Burk plot == See also == * Activated sludge model (uses the Monod equation to model bacterial growth and substrate utilization) * Bacterial growth * Hill equation (biochemistry) * Hill contribution to Langmuir equation * Langmuir adsorption model (equation with the same mathematical form) * Michaelis–Menten kinetics (equation with the same mathematical form) * Gompertz function * Victor Henry, who first wrote the general equation form in 1901 * Von Bertalanffy function == References == Category:Catalysis Category:Chemical kinetics Category:Environmental engineering Category:Enzyme kinetics Category:Ordinary differential equations Category:Sewerage They will differ between microorganism species and will also depend on the ambient environmental conditions, e.g., on the temperature, on the pH of the solution, and on the composition of the culture medium. == Application notes == The rate of substrate utilization is related to the specific growth rate as follows: : r_s = \mu X/Y where: * X is the total biomass (since the specific growth rate, μ is normalized to the total biomass) * Y is the yield coefficient rs is negative by convention. right|thumb|thumbtime=5|FVCOM simulation of hypersaline sea surface release and propagation under tidal conditions in the northern North Sea The Finite Volume Community Ocean Model (FVCOM; Formerly Finite Volume Coastal Ocean Model) is a prognostic, unstructured-grid, free-surface, 3-D primitive equation coastal ocean circulation model. The equation in discrete time is given by :1 = \sum_{a = 1}^\omega \lambda^{-a}\ell(a)b(a) where \lambda is the discrete growth rate, ℓ(a) is the fraction of individuals surviving to age a and b(a) is the number of offspring born to an individual of age a during the time step. First substitute the definition of the per-capita fertility and divide through by the left hand side: :1 = \frac{s_0b_1}{\lambda} + \frac{s_0s_1b_2}{\lambda^2} + \cdots + \frac{s_0\cdots s_{\omega - 1}b_{\omega}}{\lambda^{\omega}}. The model can be used to predict the number of fish that will be present in a fishery.de Vries et al.Marland Subsequent work has derived the model under other assumptions such as scramble competition,Brännström and Sumpter(2005) within-year resource limited competition or even as the outcome of source-sink Malthusian patches linked by density-dependent dispersal. When c = 1, the Hassell model is simply the Beverton–Holt model. ==See also== * Population dynamics of fisheries ==Notes== ==References== * Brännström A and Sumpter DJ (2005) "The role of competition and clustering in population dynamics" Proc Biol Sci., 272(1576): 2065-72\. Biological exponential growth is the unrestricted growth of a population of organisms, occurring when resources in its habitat are unlimited. The sum is taken over the entire life span of the organism. ==Derivations== ===Lotka's continuous model=== A.J. Lotka in 1911 developed a continuous model of population dynamics as follows. Based on the age demographic of females in the population and female births (since in many cases it is the females that are more limited in the ability to reproduce), this equation allows for an estimation of how a population is growing. Any species growing exponentially under unlimited resource conditions can reach enormous population densities in a short time. The empirical Monod equation is: : \mu = \mu_\max {[S] \over K_s + [S]} where: * μ is the growth rate of a considered microorganism * μmax is the maximum growth rate of this microorganism * [S] is the concentration of the limiting substrate S for growth * Ks is the "half- velocity constant"—the value of [S] when μ/μmax = 0.5 μmax and Ks are empirical (experimental) coefficients to the Monod equation.

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A fluid has density $870 \mathrm{~kg} / \mathrm{m}^3$ and flows with velocity $\mathbf{v}=z \mathbf{i}+y^2 \mathbf{j}+x^2 \mathbf{k}$, where $x, y$, and $z$ are measured in meters and the components of $\mathbf{v}$ in meters per second. Find the rate of flow outward through the cylinder $x^2+y^2=4$, $0 \leqslant z \leqslant 1$.

Thus we find the maximum speed in the flow, , in the low pressure on the sides of the cylinder. The goal is to find the steady velocity vector and pressure in a plane, subject to the condition that far from the cylinder the velocity vector (relative to unit vectors and ) is: :\mathbf{V}=U\mathbf{i}+0\mathbf{j} \,, where is a constant, and at the boundary of the cylinder :\mathbf{V}\cdot\mathbf{\hat n}=0 \,, where is the vector normal to the cylinder surface. Far from the cylinder, the flow is unidirectional and uniform. Then the solution to first-order approximation in terms of the velocity potential is :\phi(r,\theta) = Ur\left(1+ \frac{a^2}{r^2}\right)\cos\theta - \mathrm{M}^2 \frac{Ur}{12} \left[\left( \frac{13 a^2}{r^2} - \frac{6 a^4}{r^4} + \frac{a^6}{r^6}\right) \cos\theta + \left(\frac{a^4}{r^4} - \frac{3a^2}{r^2} \right) \cos 3\theta\right]+ \mathrm{O}\left(\mathrm{M}^4\right) \, where a is the radius of the cylinder. ==Potential flow over a circular cylinder with slight variations== Regular perturbation analysis for a flow around a cylinder with slight perturbation in the configurations can be found in Milton Van Dyke (1975). The low pressure on sides on the cylinder is needed to provide the centripetal acceleration of the flow: :\frac{\partial p}{\partial r}=\frac{\rho V^2}{L} \,, where is the radius of curvature of the flow. The pressure at each point on the wake side of the cylinder will be lower than on the upstream side, resulting in a drag force in the downstream direction. ==Janzen–Rayleigh expansion== The problem of potential compressible flow over circular cylinder was first studied by O. Janzen in 1913O. Now let be the internal pressure coefficient inside the cylinder, then a slight normal velocity due to the slight porousness is given by :\frac{1}{r}\frac{\partial \psi}{\partial \theta} = \varepsilon U \left(C_\mathrm{pi} - C_\mathrm{ps}\right) = \varepsilon U \left(C_\mathrm{pi} +1 - 2\cos 2\theta\right) \quad \text{at } r=a \,, but the zero net flux condition :\int_0^{2\pi} \frac{1}{r}\frac{\partial \psi}{\partial \theta} \,\mathrm{d}\theta = 0 requires that . Velocity vectors. thumb|350px|Close-up view of one quadrant of the flow. Then the solution to first-order approximation is :\psi(r,\theta) = Ur\left(1- \frac{a^2}{r^2}\right)\sin\theta + \varepsilon \frac{Ur}{2} \left( \frac{3a^2}{r^2}\sin \theta - \frac{a^4}{r^4} \sin 3 \theta \right) + \mathrm{O}\left(\varepsilon^2\right) ===Slightly pulsating circle=== Here the radius of the cylinder varies with time slightly so . Then the solution to the first-order approximation is :\psi(r,\theta) = Ur\left(1- \frac{a^2}{r^2}\right)\sin\theta - \varepsilon U \frac{a^3}{r^2} \sin 2\theta+ \mathrm{O}\left(\varepsilon^2\right) \,. ===Corrugated quasi-cylinder=== If the cylinder has variable radius in the axial direction, the -axis, , then the solution to the first-order approximation in terms of the three-dimensional velocity potential is :\phi(r,\theta,z) = Ur\left(1+ \frac{a^2}{r^2}\right)\cos\theta - 2\varepsilon U b \frac{K_1\left(\frac{r}{b}\right)}{K_1'\left(\frac{r}{b}\right)} \cos\theta \sin \frac{z}{b} + \mathrm{O}\left(\varepsilon^2\right) \,, where is the modified Bessel function of the first kind of order one. ==See also== *Joukowsky transform *Kutta condition *Magnus effect ==References== Category:Fluid dynamics A cylinder (or disk) of radius is placed in a two-dimensional, incompressible, inviscid flow. The velocity components (v_r,v_\theta,v_z) of the Rankine vortex, expressed in terms of the cylindrical-coordinate system (r,\theta,z) are given by :v_r=0,\quad v_\theta(r) = \frac{\Gamma}{2\pi}\begin{cases} r/a^2 & r \le a, \\\ 1/ r & r > a \end{cases}, \quad v_z = 0 where \Gamma is the circulation strength of the Rankine vortex. The fluid velocity in the pores \mathbf{v}_a (or short but inaccurately called pore velocity) is related to Darcy velocity by the relation :\mathbf{v}_a = \phi^{-1} \mathbf{q}_a = \phi^{-1} \mathbf{u}_a where a = w, o, g The volumetric flux is an intensive quantity, so it is not good at describing how much fluid is coming per time. The flow is inviscid, incompressible and has constant mass density . In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. This equation is also called the Churchill–Bernstein correlation. ==Heat transfer definition== :\overline{\mathrm{Nu}}_D \ = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\Pr^{1/3}}{\left[1 + (0.4/\Pr)^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad \Pr\mathrm{Re}_D \ge 0.2 where: * \overline{\mathrm{Nu}}_D is the surface averaged Nusselt number with characteristic length of diameter; * \mathrm{Re}_D\,\\! is the Reynolds number with the cylinder diameter as its characteristic length; * \Pr is the Prandtl number. Nomenclature Symbol Description SI Units subscript: phase a, vector component \sigma A{_{\sigma}} vector component \sigma of directional contact surface between two grid cells m2 \mathbf{A} directional contact surface between two (usually neighboring) grid cells m2 {\mathbf{e}_z}{^{3}} unit vector along 3rd axis (z is a reminder here: 3 is z-direction) 1 g acceleration of gravity m/s2 \mathbf{g} acceleration of gravity with direction m/s2 \mathbf{K} absolute permeability as a 3x3 tensor m2 K_{ra} relative permeability of phase a= w, o, g fraction \mathbf{K}_{ra} directional relative permeability (i.e. 3x3 tensor) fraction P_a pressure Pa \mathbf{q}_a volumetric flux (Darcy velocity) through grid cell contact surface m/s {Q_a} volumetric flow rate through grid cell contact surface m3/s \mathbf{v}_a pore (fluid) flow velocity m/s {u_a}^\sigma Darcy (fluid) velocity along axis \sigma m/s \mathbf{u}_a Darcy (fluid) velocity m/s abla gradient operator m−1 \mu_a dynamic viscosity Pa \cdot s \rho_a mass density kg/m3 :\mathbf{u}_a = -\mu_a^{-1} K_{ra} \mathbf{K} \cdot \left( abla P - \rho_a \mathbf{g} \right) where a = w, o, g The present fluid phases are water, oil and gas, and they are represented by the subscript a = w,o,g respectively. With the cylinder blocking some of the flow, must be greater than somewhere in the plane through the center of the cylinder and transverse to the flow. == Comparison with flow of a real fluid past a cylinder == The symmetry of this ideal solution has a stagnation point on the rear side of the cylinder, as well as on the front side. The flow equation in component form (using summation convention) is :u{_a}^\sigma = -\mu_a^{-1} K_{ra}{^\sigma}_{\beta} K{^\beta}_{\gamma} \left( abla^{\gamma} P_a - \rho_a g {e_z}{^{\gamma}} \right) where a = w, o, g where \sigma = 1,2,3 The Darcy velocity \mathbf{u}_a is not the velocity of a fluid particle, but the volumetric flux (frequently represented by the symbol \mathbf{q}_a ) of the fluid stream. In the following, will represent a small positive parameter and is the radius of the cylinder. Then the solution to first-order approximation is :\psi(r,\theta,t) = Ur\left(1- \frac{a^2}{r^2}\right)\sin\theta + \varepsilon Ur\left( \frac{a^2}{Ur} \theta f'(t) - \frac{2 a^2}{r^2} f(t) \sin \theta\right) + \mathrm{O}\left(\varepsilon^2\right) ===Flow with slight vorticity=== In general, the free-stream velocity is uniform, in other words , but here a small vorticity is imposed in the outer flow. ====Linear shear==== Here a linear shear in the velocity is introduced. :\begin{align} \psi &= U \left(y + \frac{1}{2} \varepsilon \frac{y^2}{a}\right)\,, \\\\[3pt] \omega &= - abla^2 \psi = - \varepsilon \frac{U}{a} \quad \text{as } x\rightarrow -\infty\,, \end{align} where is the small parameter. One should not expect much more than 20% accuracy from the above equation due to the wide range of flow conditions that the equation encompasses.

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Suppose that $2 \mathrm{~J}$ of work is needed to stretch a spring from its natural length of $30 \mathrm{~cm}$ to a length of $42 \mathrm{~cm}$. How far beyond its natural length will a force of $30 \mathrm{~N}$ keep the spring stretched?

Let be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). The force an ideal spring would exert is exactly proportional to its extension or compression. An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. When a conventional spring, without stiffness variability features, is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). The change in length may be expressed as \Delta L = \varepsilon L = \frac{F L}{A E}\,. === Spring energy === The potential energy stored in a spring is given by U_\mathrm{el}(x) = \tfrac 1 2 kx^2 which comes from adding up the energy it takes to incrementally compress the spring. Force of fully compressed spring : F_{max} = \frac{E d^4 (L-n d)}{16 (1+ u) (D-d)^3 n} \ where : E – Young's modulus : d – spring wire diameter : L – free length of spring : n – number of active windings : u – Poisson ratio : D – spring outer diameter ==Zero-length springs== thumb|left|120px|Simplified LaCoste suspension using a zero-length spring thumb|upright|Spring length L vs force F graph of ordinary (+), zero-length (0) and negative-length (−) springs with the same minimum length L0 and spring constant "Zero-length spring" is a term for a specially designed coil spring that would exert zero force if it had zero length; if there were no constraint due to the finite wire diameter of such a helical spring, it would have zero length in the unstretched condition. thumb|Hooke's law: the force is proportional to the extension In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its stiffness), and is small compared to the total possible deformation of the spring. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Obviously a coil spring cannot contract to zero length, because at some point the coils touch each other and the spring can't shorten any more. Zero length springs are made by manufacturing a coil spring with built-in tension (A twist is introduced into the wire as it is coiled during manufacture; this works because a coiled spring "unwinds" as it stretches), so if it could contract further, the equilibrium point of the spring, the point at which its restoring force is zero, occurs at a length of zero. Explaining the Power of Springing Bodies, London, 1678. as: ("as the extension, so the force" or "the extension is proportional to the force"). The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. That is, in a line graph of the spring's force versus its length, the line passes through the origin. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. We will use the word "torsion" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. In practice, zero length springs are made by combining a "negative length" spring, made with even more tension so its equilibrium point would be at a "negative" length, with a piece of inelastic material of the proper length so the zero force point would occur at zero length. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils forces the spring to collapse or extend all the coils at the same rate when deformed. ===Simple harmonic motion=== Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like: :F = m a \quad \Rightarrow \quad -k x = m a. : F is the resulting force vector – the magnitude and direction of the restoring force the spring exerts : k is the rate, spring constant or force constant of the spring, a constant that depends on the spring's material and construction. \, The mass of the spring is small in comparison to the mass of the attached mass and is ignored. Big Spring has been explored to a length of 270 m and a depth of 25 m, and Little Spring to a length of 305 m and depth of 45 m. ==References== ==External links== *Retovje Springs on Geopedia * Category:Municipality of Vrhnika Category:Springs of Slovenia Category:Karst springs SRetovje Since the inertia of the beam can be found from its mass, the spring constant can be calculated.

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Find the work done by a force $\mathbf{F}=8 \mathbf{i}-6 \mathbf{j}+9 \mathbf{k}$ that moves an object from the point $(0,10,8)$ to the point $(6,12,20)$ along a straight line. The distance is measured in meters and the force in newtons.

In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. The work done is given by the dot product of the two vectors. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . In physics, work is the energy transferred to or from an object via the application of force along a displacement. The work of the net force is calculated as the product of its magnitude and the particle displacement. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). This is illustrated in the figure to the right: The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. Notice that the work done by gravity depends only on the vertical movement of the object. The dot product of two perpendicular vectors is always zero, so the work , and the magnetic force does not do work. In this case the dot product , where is the angle between the force vector and the direction of movement, that is W = \int_C \mathbf{F} \cdot d\mathbf{s} = Fs\cos\theta. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. If force is changing, or if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point velocity is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). Work is defined by: : \delta W = \mathbf{F}\cdot\mathbf{v}\delta t, Therefore :\frac{\partial W}{\partial t}=\mathbf{F_E} \cdot \,\mathbf{v} ==References== Category:Electromagnetism Therefore, work need only be computed for the gravitational forces acting on the bodies. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. If the force is always directed along this line, and the magnitude of the force is , then this integral simplifies to W = \int_C F\,ds where is displacement along the line. The derivation of the work–energy principle begins with Newton’s second law of motion and the resultant force on a particle. The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of .

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A ball is thrown at an angle of $45^{\circ}$ to the ground. If the ball lands $90 \mathrm{~m}$ away, what was the initial speed of the ball?

thumb|upright=1.5|Spherical pendulum: angles and velocities. 50 meter running target or 50 meter running boar is an ISSF shooting event, shot with a .22-calibre rifle at a target depicting a boar moving sideways across a 10-meter wide opening. If thrown correctly, the changeup will confuse the batter because the human eye cannot discern that the ball is coming significantly slower until it is around 30 feet from the plate. thumb|The target: total Ø = 154.4 mm. 4 ring Ø = 106.4 mm. 9 ring Ø = 26.4 mm. 10 ring Ø = 10.4 mm, height 0.75 m above the floor 50 meter rifle prone (formerly known as one of four free rifle disciplines) is an International Shooting Sport Federation event consisting of 60 shots from the prone position with a .22 Long Rifle (5.6 mm) caliber rifle. A ball moves due to the changes in the pressure of the air surrounding the ball as a result of the kind of pitch thrown. Trajectory of the mass on the sphere can be obtained from the expression for the total energy :E=\underbrace{\left[\frac{1}{2}ml^2\dot\theta^2 + \frac{1}{2}ml^2\sin^2\theta \dot \phi^2\right]}_{T}+\underbrace{ \bigg[-mgl\cos\theta\bigg]}_{V} by noting that the horizontal component of angular momentum L_z = ml^2\sin^2\\!\theta \,\dot\phi is a constant of motion, independent of time. thumb|420px|The typical motion of a pitcher. thumb|Demonstration of pitching techniques In baseball, the pitch is the act of throwing the baseball toward home plate to start a play. Therefore, the ball keeps moving in the path of least resistance, which constantly changes. A changeup is generally thrown 8–15 miles per hour slower than a fastball. That is a consequence of the rotational symmetry of the system around the vertical axis. == Trajectory == thumb|250x250px|Trajectory of a spherical pendulum. It is thrown the same as a fastball, but simply farther back in the hand, which makes it release from the hand slower but still retaining the look of a fastball. While throwing the fastball it is very important to have proper mechanics, because this increases the chance of getting the ball to its highest velocity, making it difficult for the opposing player to hit the pitch. Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of , where is fixed such that r = l. ==Lagrangian mechanics== Routinely, in order to write down the kinetic T=\tfrac{1}{2}mv^2 and potential V parts of the Lagrangian L=T-V in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. thumb|Standard 50m target: total Ø = 154.4 mm. 4 ring Ø = 106.4 mm. 9 ring Ø = 26.4 mm. 10 ring Ø = 10.4 mm International Rifle events that occur in three positions are conducted with an equal number of shots fired from the Kneeling, Prone and Standing positions, although the order has changed over the years. It consists of a mass moving without friction on the surface of a sphere. thumb|The target: total Ø = 154.4 mm. 4 ring Ø = 106.4 mm. 9 ring Ø = 26.4 mm. 10 ring Ø = 10.4 mm, height 0.75 m above the floor 50 meter rifle three positions (formerly known as one of four free rifle disciplines) is an International Shooting Sport Federation event, a miniature version of 300 meter rifle three positions. The two Olympic events are shot with a rimfire rifle at 50m. Typically, pitchers from the set use a high leg kick, but may instead release the ball more quickly by using the slide step. == See also == * First-pitch strike * Bowling – pitching a cricket ball ** Throwing (cricket), a type of bowling more similar to baseball pitching * Pitch (softball) ==References== == External links == * Category:Baseball terminology Category:Articles containing video clips The most common fastball pitches are: * Cutter * Four-seam fastball * Sinker * Split-finger fastball * Two-seam fastball ==Breaking balls== thumb|130px|A common grip of a slider Well-thrown breaking balls have movement, usually sideways or downward. For example, a batter swings at the ball as if it was a 90 mph fastball but it is coming at 75 mph which means he is swinging too early to hit the ball well, making the changeup very effective. In most cases junior shooting is done at either 10m or 50 ft. distances. Most breaking balls are considered off-speed pitches.

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Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A leaky 10-kg bucket is lifted from the ground to a height of $12 \mathrm{~m}$ at a constant speed with a rope that weighs $0.8 \mathrm{~kg} / \mathrm{m}$. Initially the bucket contains $36 \mathrm{~kg}$ of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?

If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The work of the net force is calculated as the product of its magnitude and the particle displacement. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. Therefore, work need only be computed for the gravitational forces acting on the bodies. The small amount of work that occurs over an instant of time is calculated as \delta W = \mathbf{F} \cdot d\mathbf{s} = \mathbf{F} \cdot \mathbf{v}dt where the is the power over the instant . The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. Integral energy is the amount of energy required to remove water from soil with an initial water content \theta_i to water content of \theta_f (where \theta_i > \theta_f). Use this to simplify the formula for work of gravity to, W = -\int^{t_2}_{t_1}\frac{GmM}{r^3}(r\mathbf{e}_r) \cdot \left(\dot{r}\mathbf{e}_r + r\dot{\theta}\mathbf{e}_t\right) dt = -\int^{t_2}_{t_1}\frac{GmM}{r^3}r\dot{r}dt = \frac{GMm}{r(t_2)}-\frac{GMm}{r(t_1)}. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . In addition to completing the dam, work needed was the construction of shipping locks and discharge sluices at the ends of the dam. This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. Substituting the above equations, one obtains: W = Fs = mas = ma\frac{v_2^2-v_1^2}{2a} = \frac{mv_2^2}{2}- \frac{mv_1^2}{2} = \Delta E_\text{k} Other derivation: W = Fs = mas = m\frac{v_2^2 - v_1^2}{2s}s = \frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2 = \Delta E_\text{k} In the general case of rectilinear motion, when the net force is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = \int_{t_1}^{t_2} F \,v \, dt = \int_{t_1}^{t_2} ma \,v \, dt = m \int_{t_1}^{t_2} v \,\frac{dv}{dt}\,dt = m \int_{v_1}^{v_2} v\,dv = \tfrac12 m \left(v_2^2 - v_1^2\right) . ===General derivation of the work–energy principle for a particle=== For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. The work done is given by the dot product of the two vectors. In physics, work is the energy transferred to or from an object via the application of force along a displacement. After the war, work was started on draining the Flevolands, a massive project totalling almost 1000 km2.

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Find the volume of the described solid $S$. The base of $S$ is an elliptical region with boundary curve $9 x^2+4 y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base.

In the end we find the volume is cubic units. ==See also== *Solid of revolution *Disc integration ==References== * *Frank Ayres, Elliott Mendelson. Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. After integrating these two functions with the disk method we would subtract them to yield the desired volume. Wallis's Conical Edge with , In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt{a^2-b^2\cos^2u} where , and are constants. The horizontal plane shows the four quadrants between x- and y-axis. Recycling the subducted slab presents volcanism by flux melting from the mantle wedge. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). Wallis's Conical Edge with right|thumb|600px| Figure 2. thumb|400px|Three axial planes (x=0, y=0, z=0) divide space into eight octants. thumb|right|300px|A volume is approximated by a collection of hollow cylinders. The slab affects the convection and evolution of the Earth's mantle due to the insertion of the hydrous oceanic lithosphere. Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. right|280px|Femisphere The femisphere is a solid that has one single surface, two edges, and four vertices. == Description == The form of the femisphere is reminiscent of that of a sphericon but without straight lines. An octant in solid geometry is one of the eight divisions of a Euclidean three-dimensional coordinate system defined by the signs of the coordinates. thumb|upright=1.75|The figure is a schematic diagram depicting a subduction zone. Dense oceanic lithosphere retreats into the Earth's mantle, while lightweight continental lithospheric material produces active continental margins and volcanic arcs, generating volcanism. For this reason, when rolled over a sphere, it contacts the whole surface area of it in a single revolution.Sphericon Homepage: Femisphere The area of a femisphere of unit radius is S = 4 \pi. This is in contrast to disc integration which integrates along the axis parallel to the axis of revolution. ==Definition== The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the -plane around the -axis. Temperature gradients of subducted slabs depend on the oceanic plate's time and thermal structures. In geology, the slab is a significant constituent of subduction zones . Schaum's Outlines: Calculus. The polar circles of the triangles of a complete quadrilateral form a coaxal system.

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A swimming pool is circular with a $40-\mathrm{ft}$ diameter. The depth is constant along east-west lines and increases linearly from $2 \mathrm{ft}$ at the south end to $7 \mathrm{ft}$ at the north end. Find the volume of water in the pool.

thumb|491px|right|Output from a shallow-water equation model of water in a bathtub. A stream pool, in hydrology, is a stretch of a river or stream in which the water depth is above average and the water velocity is below average.Matthew Chasse, Riffle characteristics in stream investigations == Formation == right|thumb|250px|Stream pool formation. The instantaneous water depth is , with zb(x) the bed level (i.e. elevation of the lowest point in the bed above datum, see the cross-section figure). The shallow- water equations are thus derived. The Swimming Pool (, translit. Therefore, the diver floats at the water's surface. In the case of a horizontal bed, with negligible Coriolis forces, frictional and viscous forces, the shallow-water equations are: \begin{align} \frac{\partial (\rho \eta) }{\partial t} &\+ \frac{\partial (\rho \eta u)}{\partial x} + \frac{\partial (\rho \eta v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta u)}{\partial t} &\+ \frac{\partial}{\partial x}\left( \rho \eta u^2 + \frac{1}{2}\rho g \eta^2 \right) + \frac{\partial (\rho \eta u v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta v)}{\partial t} &\+ \frac{\partial}{\partial y}\left(\rho \eta v^2 + \frac{1}{2}\rho g \eta ^2\right) + \frac{\partial (\rho \eta uv)}{\partial x} = 0. \end{align} Here η is the total fluid column height (instantaneous fluid depth as a function of x, y and t), and the 2D vector (u,v) is the fluid's horizontal flow velocity, averaged across the vertical column. While a vertical velocity term is not present in the shallow-water equations, note that this velocity is not necessarily zero. The trough structure is 7 ft in height, with a width of 7.5 ft. Situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are common, so the shallow-water equations are widely applicable. If the diver rises, by even the most minuscule amount, the pressure on the bubble will decrease, it will expand, it will displace more water, and the diver will become more positively buoyant, rising still more quickly. The trapped air in the straw makes the diver slightly buoyant, and it thus floats. 2\. A stream pool may be bedded with sediment or armoured with gravel, and in some cases the pool formations may have been formed as basins in exposed bedrock formations. This water in turn exerts additional pressure on the air bubble inside the diver; because the air inside the diver is compressible but the water is an incompressible fluid, the air's volume is decreased but the water's volume does not expand, such that the pressure external to the diver a) forces the water already in the diver further inward and b) drives water from outside the diver into the diver. The x-component of the Navier–Stokes equations – when expressed in Cartesian coordinates in the x-direction – can be written as: \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}+ w \frac{\partial u}{\partial z}= -\frac{\partial p}{\partial x} \frac{1}{\rho} + u \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right)+ f_x, where u is the velocity in the x-direction, v is the velocity in the y-direction, w is the velocity in the z-direction, t is time, p is the pressure, ρ is the density of water, ν is the kinematic viscosity, and fx is the body force in the x-direction. For non-moving channel walls the cross-sectional area A in equation () can be written as: A(x,t) = \int_0^{h(x,t)} b(x,h')\, dh', with b(x,h) the effective width of the channel cross section at location x when the fluid depth is h – so for rectangular channels. Pools are often formed on the outside of a bend in a meandering river.http://mostreamteam.org/assets/habitat.pdf == Dynamics == The depth and lack of water velocity often leads to stratification in stream pools, especially in warmer regions. Assuming such a state were to exist at some point, any departure of the diver from its current depth, however small, will alter the pressure exerted on the bubble in the diver due to the change in the weight of the water above it in the vessel. It might be thought that if the weight of displaced water exactly matched the weight of the diver, it would neither rise nor sink, but float in the middle of the container; however, this does not occur in practice. Conversely, should the diver drop by the smallest amount, the pressure will increase, the bubble contract, additional water enter, the diver will become less buoyant, and the rate of the drop will accelerate as the pressure from the water rises still further. In order for shallow-water equations to be valid, the wavelength of the phenomenon they are supposed to model has to be much larger than the depth of the basin where the phenomenon takes place. When the pressure on the container is released, the air expands again, increasing the weight of water displaced and the diver again becomes positively buoyant and floats.

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The orbit of Halley's comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is $36.18 \mathrm{AU}$. [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles.] Find a polar equation for the orbit of Halley's comet. What is the maximum distance from the comet to the sun?

Perihelion distance at recent epochs Epoch Perihelion (AU) 2028 1.310 2022 1.306 2015 1.349 2008 1.355 Comet Borrelly or Borrelly's Comet (official designation: 19P/Borrelly) is a periodic comet, which was visited by the spacecraft Deep Space 1 in 2001. The comet came to perihelion on 18 September 2012, and reached about apparent magnitude 17. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 160P on Seiichi Yoshida's comet list * Elements and Ephemeris for 160P/LINEAR – Minor Planet Center Category:Periodic comets 0160 # Category:Astronomical objects discovered in 2004 170P/Christensen is a periodic comet in the Solar System. For 0 this is an ellipse with = a \cdot \sqrt{1-e^2}|}} For e = 1 this is a parabola with focal length \tfrac{p}{2} For e > 1 this is a hyperbola with = a \cdot \sqrt{e^2-1}|}} The following image illustrates a circle (grey), an ellipse (red), a parabola (green) and a hyperbola (blue) thumb|300px|A diagram of the various forms of the Kepler Orbit and their eccentricities. Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). 1308 Halleria, provisional designation , is a carbonaceous Charis asteroid from the outer regions of the asteroid belt, approximately 43 kilometers in diameter. For the ellipse there is also an apocentre for which the distance to the focus takes the maximal value \tfrac{p}{1-e}. 160P/LINEAR is a periodic comet in the Solar System. It came to perihelion in September 2014 at about apparent magnitude 18. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 170P on Seiichi Yoshida's comet list * Elements and Ephemeris for 170P/Christensen – Minor Planet Center Category:Periodic comets 0170 170P 20050617 Its orbit has an eccentricity of 0.01 and an inclination of 6° with respect to the ecliptic. Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 Mathematically, the distance between a central body and an orbiting body can be expressed as: r(\theta) = \frac{a(1-e^2)}{1+e\cos(\theta)} where: *r is the distance *a is the semi- major axis, which defines the size of the orbit *e is the eccentricity, which defines the shape of the orbit *\theta is the true anomaly, which is the angle between the current position of the orbiting object and the location in the orbit at which it is closest to the central body (called the periapsis). The distance to the focal point is a function of the polar angle relative to the horizontal line as given by the equation () In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. The comet last came to perihelion (closest approach to the Sun) on February 1, 2022 and will next come to perihelion on December 11, 2028. Deep Space 1 returned images of the comet's nucleus from 3400 kilometers away. Red is an elliptical orbit (0 < e < 1). thumb|An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. Alternately, the equation can be expressed as: r(\theta) = \frac{p}{1+e\cos(\theta)} Where p is called the semi-latus rectum of the curve. Lightcurve analysis gave a consolidated rotation period of 6.028 hours with a brightness amplitude between 0.14 and 0.17 magnitude. === Diameter and albedo === According to the surveys carried out by the Infrared Astronomical Satellite IRAS, the Japanese Akari satellite and the NEOWISE mission of NASA's Wide-field Infrared Survey Explorer, Halleria measures between 39.33 and 50.046 kilometers in diameter and its surface has an albedo between 0.0338 and 0.05. Making the substitutions p=\tfrac{|\mathbf{H}|^2}{\alpha} and e=\tfrac{c}{\alpha}, we again arrive at the equation This is the equation in polar coordinates for a conic section with origin in a focal point. We can then define the eccentricity vector associated with the orbit as: \mathbf{e} \triangleq \frac{\mathbf{c}}{\alpha} = \frac{\dot{\mathbf{r}}\times\mathbf{H}}{\alpha} - \mathbf{u} = \frac{\mathbf{v}\times\mathbf{H}}{\alpha} - \frac{\mathbf{r}}{r} = \frac{\mathbf{v}\times(\mathbf{r} \times \mathbf{v})}{\alpha} - \frac{\mathbf{r}}{r} where \mathbf{H} = \mathbf{r} \times \dot{\mathbf{r}} = \mathbf{r} \times \mathbf{v} is the constant angular momentum vector of the orbit, and \mathbf{v} is the velocity vector associated with the position vector \mathbf{r}. For the hyperbola the range for \theta is -\cos^{-1}\left(-\frac{1}{e}\right) < \theta < \cos^{-1}\left(-\frac{1}{e}\right) and for a parabola the range is -\pi < \theta < \pi Using the chain rule for differentiation (), the equation () and the definition of p as \frac {H^2}{\alpha} one gets that the radial velocity component is e \sin \theta|}} and that the tangential component (velocity component perpendicular to V_r) is \cdot (1 + e \cdot \cos \theta)|}} The connection between the polar argument \theta and time t is slightly different for elliptic and hyperbolic orbits.

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If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height in feet $t$ seconds later is given by $y=40 t-16 t^2$. Find the average velocity for the time period beginning when $t=2$ and lasting 0.5 second.

(If the time variable is continuous, the average value during the time period is the integral over the period divided by the length of the duration of the period.) ==See also== Moving average ==References== Category:Means 20 Hrs. 40 Min.: The temporal mean is the arithmetic mean of a series of values over a time period. A simple moving average can be considered to be a sequence of temporal means over periods of equal duration. Timeslip is a horizontally scrolling shooter written by Jon Williams for the Commodore 16 / Commodore Plus/4 computers and published by English Software in 1985. 40+ is a 2019 Maldivian horror comedy film written and directed by Yoosuf Shafeeu and the sequel to his previous comedy film Naughty 40 (2017). 40 Minutes was a BBC TV documentary strand broadcast on BBC Two between 1981 and 1994.BFI | Film & TV Database | 40 MINUTES Some documentaries in the original series were revisited and updated in a 2006 version, Forty Minutes On.BBC Four - Forty Minutes On ==See also== * Sixty Minutes (British TV programme) ==References== Category:BBC television documentaries Category:1981 British television series debuts Category:1994 British television series endings Category:1980s British documentary television series Category:1990s British documentary television series Category:English-language television shows 40 under 40 or Forty under 40 etc. may refer to: * 40 Under 40, annual list published in Fortune magazine * Business Journals Forty Under 40, annual list published by American City Business Journals If a player is hit, they receive a 30 minute penalty. Assuming equidistant measuring or sampling times, it can be computed as the sum of the values over a period divided by the number of values. In addition, if a player is hit five times, a "timeslip" occurs, which is a desynchronisation of all clocks. Your Commodore reviewer summed up Timeslip as the best game he had seen on the C16, and he recommended it without hesitation. The object of the game is to destroy 36 orbs placed within the three sections and synchronize the clocks in all three zones to 00.00 hours. The film ends with everyone laughing. == Cast == * Yoosuf Shafeeu as Ashvani * Ali Seezan as Zahidh * Mohamed Manik as Ajwad * Ahmed Saeed as Ahsan * Sheela Najeeb as Zarifa * Fathimath Azifa as Thaniya * Mohamed Faisal as Akram * Ali Azim as Nadheem * Ahmed Easa as Saiman * Mariyam Shakeela as Gumeyra * Ali Shahid as Zubeiru * Mariyam Shifa as Laila * Irufana Ibrahim as Shamra * Hunaisha Adam Naseer as Mishka * Aminath Ziyadha as Fazna * Aishath Sam'aa as Shaira * Ahmed Bassam as Fairooz * Mariyam Azza in the item number "Lailaa" (Special appearance) ==Development== A sequel to Yoosuf Shafeeu's commercially successful comedy film Naughty 40 (2017) was announced on 31 October 2017. Sections are played one at a time and the player can switch zones at will, leaving the other two frozen in time. ==Reception== Timeslip received mostly positive reviews. I think aviation has a chance to increase intimacy, > understanding, and far-flung friendships thus. 20 Hrs. 40 Min. was the first of two books Earhart would write in her lifetime; the other being 1932's The Fun of It. Filming was commenced on 23 December 2017 in Th. Burunee scheduled to be completed within twenty five days. The game was described by reviewers as "three versions of Scramble rolled into one". ==Gameplay== thumb|left|Atari 8-bit screenshot In Timeslip the player is presented with the screen divided into three sections or time zones. The review in Computer and Video Games magazine was equally positive: "Timeslip's designer and programmer, Jon Williams, has come up with a nifty and exciting little game. C16 owners should raise three cheers for him " ==References== ==External links== *Timeslip at Atari Mania * Category:1985 video games Category:Atari 8-bit family games Category:Commodore 16 and Plus/4 games Category:Horizontally scrolling shooters Category:English Software games Atari 8-bit version followed a year later. Shooting of the film was completed on 21 January 2018. ==Soundtrack== ==Release== The film was initially planned to release on 1 August 2018 though they pushed the release date to the following year citing the political instability in the country in relation to 2018 Maldivian presidential election.

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A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced $1.5 \mathrm{~cm}$ apart. The liver is $15 \mathrm{~cm}$ long and the cross-sectional areas, in square centimeters, are $0,18,58,79,94,106,117,128,63$, 39 , and 0 . Use the Midpoint Rule to estimate the volume of the liver.

thumb|Maximum intensity projection of a PET/CT with choline: Note the physiologic accumulation in the liver, pancreas, kidney, bladder, spleen, bone marrow and salivary glands. The human abdomen is divided into quadrants and regions by anatomists and physicians for the purposes of study, diagnosis, and treatment. The nine regions offer more detailed anatomy and are delineated by two vertical and two horizontal lines. == Quadrants == thumb|right|upright=0.8|Quadrants of the abdomen thumb|right|upright=0.8|Diagram showing which organs (or parts of organs) are in each quadrant of the abdomen The left lower quadrant (LLQ) of the human abdomen is the area left of the midline and below the umbilicus. thumb|Relationship between number of feet, octave and size of an open flue pipe (1′ = 1 foot = about 32 cm) Scaling is the ratio of an organ pipe's diameter to its length. Töpfer reasoned that the cross-sectional area of the pipe was the critical factor, and he chose to vary this by the geometric mean of the ratios 1:2 and 1:4 per octave. The median aperture (also known as the medial aperture, and foramen of Magendie) is an opening of the fourth ventricle at the caudal portion of the roof of the fourth ventricle. Nine regions of the abdomen can be marked using two horizontal and two vertical dividing lines. One of the first authors to publish data on the scaling of organ pipes was Dom Bédos de Celles. The left upper quadrant extends from the umbilical plane to the left ribcage. thumb|upright=1.5|Conical scanning concept. The right upper quadrant extends from umbilical plane to the right ribcage. He established this as a standard scale, or in German, Normalmensur, with the additional stipulation that the internal diameter be at 8′ C (the lowest note of the modern organ compass) and the mouth width one-quarter of the circumference of such a pipe. Nonetheless, the median aperture accounts for most of the outflow of CSF out of the fourt ventricle. This meant that the cross-sectional area varied as 1 : \sqrt{8}. Important organs here are: *Cecum *Appendix *Ascending colon *Right ovary and Fallopian tube *Right ureter ==Regions== thumb|left|upright=0.8|Regions of abdomen thumb|upright=1.8|Regions shown on left in side-by-side comparison with quadrants. The division into four quadrants allows the localisation of pain and tenderness, scars, lumps, and other items of interest, narrowing in on which organs and tissues may be involved. This ratio does not concern the muzzle or face, and thus is distinct from the craniofacial ratio, which compares the size of the cranium to the length of the muzzle. Important organs here are: *the descending colon and sigmoid colon *the left ovary and fallopian tube *the left ureter The left upper quadrant (LUQ) extends from the median plane to the left of the patient, and from the umbilical plane to the left ribcage. The cephalic index of a vertebrate is the ratio between the width (side to side) and length (front to back) of its cranium (skull). Important organs here are: *Stomach *Spleen *Left lobe of liver *Body of pancreas *Left kidney and adrenal gland *Splenic flexure of colon *Parts of transverse and descending colon The right upper quadrant (RUQ) extends from the median plane to the right of the patient, and from the umbilical plane to the right ribcage. The median aperture varies in size. == Anatomy == === Relations === The median foramen on axial images is posterior to the pons and anterior to the caudal cerebellum. The equivalent in other animals is right anterior quadrant.

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A manufacturer of corrugated metal roofing wants to produce panels that are $28 \mathrm{in}$. wide and $2 \mathrm{in}$. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation $y=\sin (\pi x / 7)$ and find the width $w$ of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)

A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the sine trigonometric function, of which it is the graph. It is independent of the circle radius; * a sine function (over a whole number n of half-periods), which can be calculated by computing the sine curve's arclength on those periods, is S = \textstyle \tfrac{1}{n\pi} \int_{0}^{n\pi} \sqrt{1 + (\cos x)^2} dx \approx 1.216... thumb|right|Example with 270° angle With similar opposite arcs joints in the same plane, continuously differentiable: Central angle Sinuosity Degrees Radians Exact Decimal 30° \frac{\pi}{6} \frac{\pi}{3(\sqrt{6}-\sqrt{2})} 1.0115 60° \frac{\pi}{3} \frac{\pi}{3} 1.0472 90° \frac{\pi}{2} \frac{\pi}{2\sqrt{2}} 1.1107 120° \frac{2\cdot\pi}{3} \frac{2\cdot\pi}{3\sqrt{3}} 1.2092 150° \frac{5\cdot\pi}{6} \frac{5\cdot\pi}{3(\sqrt{6}+\sqrt{2})} 1.3552 180° \pi \frac{\pi}{2} 1.5708 210° \frac{7\cdot\pi}{6} \frac{7\cdot\pi}{3(\sqrt{6}+\sqrt{2})} 1.8972 240° \frac{4\cdot\pi}{3} \frac{4\cdot\pi}{3\sqrt{3}} 2.4184 270° \frac{3\cdot\pi}{2} \frac{3\cdot\pi}{2\sqrt{2}} 3.3322 300° \frac{5\cdot\pi}{3} \frac{5\cdot\pi}{3} 5.2360 330° \frac{11\cdot\pi}{6} \frac{11\cdot\pi}{3(\sqrt{6}-\sqrt{2})} 11.1267 ==Rivers== In studies of rivers, the sinuosity index is similar but not identical to the general form given above, being given by: : \text{SI} = \frac{{\text{channel length}}}{{\text{downvalley length}}} The difference from the general form happens because the downvalley path is not perfectly straight. At this position, the top surface of the sine bar is inclined the same amount as the wedge. Some engineering and metalworking reference books contain tables showing the dimension required to obtain an angle from 0-90 degrees, incremented by 1 minute intervals. \sin \left(angle \right) = {perpendicular \over hypotenuse} Angles may be measured or set with this tool. ==Principle== thumb|10-inch and 100-millimetre sine bars. When a sine bar is placed on a level surface the top edge will be parallel to that surface. This is why the frequency of the sine wave increases as one moves to the left in the graph. The sine of the angle of inclination of the wedge is the ratio of the height of the gauge blocks used and the distance between the centers of the cylinders. ==Types== The simplest type consists of a lapped steel bar, at each end of which is attached an accurate cylinder, the axes of the cylinders being mutually parallel and parallel to the upper surface of the bar. These alternative functions are usually known as normalized Fresnel integrals. == Euler spiral == 250px|thumb| Euler spiral . The calculation of the sinuosity is valid in a 3-dimensional space (e.g. for the central axis of the small intestine), although it is often performed in a plane (with then a possible orthogonal projection of the curve in the selected plan; "classic" sinuosity on the horizontal plane, longitudinal profile sinuosity on the vertical plane). thumb|250px|Calculation of sinuosity for an oscillating curve. thumb|Two ski tracks with different degrees of sinuosity on the same slope Sinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve. 300px 300px The Keulegan–Carpenter number is important for the computation of the wave forces on offshore platforms. From the definitions of Fresnel integrals, the infinitesimals and are thus: \begin{align} dx &= C'(t)\,dt = \cos\left(t^2\right)\,dt, \\\ dy &= S'(t)\,dt = \sin\left(t^2\right)\,dt. \end{align} Thus the length of the spiral measured from the origin can be expressed as L = \int_0^{t_0} \sqrt {dx^2 + dy^2} = \int_0^{t_0} dt = t_0. The sine bar is placed over the inclined surface of the wedge. This equation gives a sine wave for a single dimension; thus the generalized equation given above gives the displacement of the wave at a position x at time t along a single line. The resonant frequencies of a string are proportional to: the length between the fixed ends; the tension of the string; and inversely proportional to the mass per unit length of the string. == See also == * Crest (physics) * Damped sine wave * Fourier transform * Harmonic analysis * Harmonic series (mathematics) * Harmonic series (music) * Helmholtz equation * Instantaneous phase * In-phase and quadrature components * Least-squares spectral analysis * Oscilloscope * Phasor * Pure tone * Simple harmonic motion * Sinusoidal model * Wave (physics) * Wave equation * ∿ the sine wave symbol (U+223F) == References == ==Further reading== * Category:Trigonometry Category:Wave mechanics Category:Waves Category:Waveforms Category:Sound Category:Acoustics * The angle is calculated by using the sine rule (a trigonometric function from mathematics). It cannot measure the angle more than 60 degrees. ===Sine table=== A sine table (or sine plate) is a large and wide sine bar, typically equipped with a mechanism for locking it in place after positioning, which is used to hold workpieces during operations. ===Compound sine table=== It is used to measure compound angles of large workpiece. A sine bar consists of a hardened, precision ground body with two precision ground cylinders fixed at the ends. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed. ==Sinusoidal plane wave== == Cosine == The term sinusoid describes any wave with characteristics of a sine wave. The Fresnel integrals and are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (). The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. The simultaneous parametric plot of and is the Euler spiral (also known as the Cornu spiral or clothoid). == Definition == 250px|thumb| Fresnel integrals with arguments instead of converge to instead of .

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The dye dilution method is used to measure cardiac output with $6 \mathrm{mg}$ of dye. The dye concentrations, in $\mathrm{mg} / \mathrm{L}$, are modeled by $c(t)=20 t e^{-0.6 t}, 0 \leqslant t \leqslant 10$, where $t$ is measured in seconds. Find the cardiac output.

In the following test tubes, the blue dye is dissolved in a lower concentration (and at the same time in a smaller amount, since the volume is approximately the same). The concentration of this admixture should be small and the gradient of this concentration should be also small. Dye transfer is a continuous-tone color photographic printing process. Fluorescent dye Color mass (g/mol) Absorb (nm) Emit (nm) ε (M−1cm−1) FluoProbes 390 violet 343 390 479 24 000 FluoProbes 488 green 804 493 519 85 000 FluoProbes 532 yellow 765 532 553 117 000 FluoProbes547H orange 736 557 574 150 000 FluoProbes 594 red 1137 601 627 120 000 FluoProbes647H far-red 761 653 674 250 000 FluoProbes 682 far-red 853 690 709 140 000 FluoProbes 752 near-IR 879 748 772 270 000 FluoProbes 782 near-IR 976 783 800 170 000 Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient The FluoProbes series of fluorescent dyes were developed by Interchim to improve performances of standard fluorophores. Dilution is reduction of concentration, e.g. by adding solvent to a solution. The dye transfer process possesses a larger color gamut and tonal scale than any other process, including inkjet. thumb|Time lapse video of diffusion of a dye dissolved in water into a gel. thumb|Three-dimensional rendering of diffusion of purple dye in water. The dye is absorbed by the paper for one minute after which the matrix is picked up by the farthermost corners and peeled off the paper. Colorimetric analysis is a method of determining the concentration of a chemical element or chemical compound in a solution with the aid of a color reagent. In the test tube with dark blue liquid (in front), the blue dye is dissolved in a high concentration. Another important characteristic of dye transfer is that it allows the practitioner the highest degree of photographic control compared to any other photochemical color print process. * FluoProbes dyes that have comparable excitation and emission spectra to standard fluorophores such as fluoresceins, rhodamines, cyanines Cy2/3/5/5.5/7, are claimed to solve limiting issues observed in some applications such as too high background, insufficient polarity, photobleaching, insufficient brightness, or pH- sensitivity. Recently, colorimetric analyses developed for colorimeters have been adapted for use with plate readers to speed up analysis and reduce the waste stream.Greenan, N. S., R.L. Mulvaney, and G.K. Sims. 1995. The dyes used in the process are very spectrally pure compared to normal coupler-induced photographic dyes, with the exception of the Kodak cyan. The Dyecrete Process is a method of adding dye to permanently color concrete. ==References== *Cited in the Academic Press Dictionary of Science and Technology. Concentrations are often called levels, reflecting the mental schema of levels on the vertical axis of a graph, which can be high or low (for example, "high serum levels of bilirubin" are concentrations of bilirubin in the blood serum that are greater than normal). ==Quantitative notation== There are four quantities that describe concentration: ===Mass concentration=== The mass concentration \rho_i is defined as the mass of a constituent m_i divided by the volume of the mixture V: :\rho_i = \frac {m_i}{V}. This is the diffusion coefficient. thumb|Test tubes with liquid in which a blue dye is dissolved in different concentrations. Note that these should not be called concentrations. ===Normality=== Normality is defined as the molar concentration c_i divided by an equivalence factor f_\mathrm{eq}. Here R is the gas constant, T is the absolute temperature, n is the concentration, the equilibrium concentration is marked by a superscript "eq", q is the charge and φ is the electric potential. The method is widely used in medical laboratories and for industrial purposes, e.g. the analysis of water samples in connection with industrial water treatment. ==Equipment== The equipment required is a colorimeter, some cuvettes and a suitable color reagent. By contrast, to dilute a solution, one must add more solvent, or reduce the amount of solute.

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A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder $100 \mathrm{ft}$ high with a radius of $200 \mathrm{ft}$. The conveyor carries ore at a rate of $60,000 \pi \mathrm{~ft}^3 / \mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time $t$, the pile is $60 \mathrm{ft}$ high, how long will it take for the pile to reach the top of the silo?

For bauxites having more than 10% silica, the Bayer process becomes uneconomic because of the formation of insoluble sodium aluminium silicate, which reduces yield, so another process must be chosen. 1.9-3.6 tons of bauxite is required to produce 1 ton of aluminium oxide. The metal contents in ore shoots are distributed in areas that vary in deposit sizes. == Sizes and structure == The circumference of deposit sizes can range from a few meters, to many kilometres. The extraction process (digestion) converts the aluminium oxide in the ore to soluble sodium aluminate, NaAlO2, according to the chemical equation: :Al2O3 \+ 2 NaOH → 2 NaAlO2 \+ H2O This treatment also dissolves silica, forming sodium silicate : :2 NaOH + SiO2 → Na2SiO3 \+ H2O The other components of Bauxite, however, do not dissolve. thumb|Pouring smelter slag onto the dump, El Teniente thumb|upright|"The ore is mined by a highly developed caving system and carried down to the main transportation level through an elaborate system of ore passes." In the Bayer process, bauxite ore is heated in a pressure vessel along with a sodium hydroxide solution (caustic soda) at a temperature of 150 to 200 °C. A structure may consist of multiple ore shoots with some veins or lodes being as thick as , and extending to thousands of feet horizontally and vertically. == Locations == There are complex stratigraphic historical parameters required in understanding how ore shoots are formed. An ore shoot is a mass of ore deposited in a vein. This is due to a majority of the aluminium in the ore being dissolved in the process. The undissolved waste after the aluminium compounds are extracted, bauxite tailings, contains iron oxides, silica, calcia, titania and some unreacted alumina. The aluminium oxide must be further purified before it can be refined into aluminium metal. == Process == Bauxite ore is a mixture of hydrated aluminium oxides and compounds of other elements such as iron. Over 90% (95-96%) of the aluminium oxide produced is used in the Hall–Héroult process to produce aluminium. == Waste == Red mud is the waste product that is produced in the digestion of bauxite with sodium hydroxide. Year Tonnes Aluminium Price Net Profit Employees 1979 153,537 1575 -1,172,000 1,252 1980 154,740 1770 17,470,000 1,258 1981 153,979 1302 2,941,000 1,359 1982 163,419 1026 -20,698,000 1,452 1983 218,609 1478 -9,665,000 1,651 1984 242,850 1281 1,766,000 1,631 1985 240,835 1072 -24,772,000 1,529 1986 236,332 1160 -18,188,000 1,506 1987 248,365 1496 92,570,000 1,429 1988 257,006 2367 173,040,000 1,770 1989 258,359 1915 118,500,000 1,820 1990 259,408 1635 42,051,000 1,720 1991 258,790 1333 -34,122,000 1,465 1992 241,775 1279 -18,649,000 1,415 1993 267,200 1161 -18,016,000 1,465 The smelter production is in tonnes of saleable metal, the aluminium price is the average London Metal Exchange 3 month in US$/tonne, the Nett Profit/Loss is after tax and NZ$. In 2011 the smelter produced 354,030 saleable tonnes of aluminium, which was its highest ever output at the time. Estimates of the waste stockpiled at the site range up to a quarter of million tonnes. The ore body surrounds the Braden Pipe in a continuous ring with a width of 2000 feet. thumb|Tiwai Point Aluminium Smelter as seen from the top of Bluff Hill The Tiwai Point Aluminium Smelter is an aluminium smelter owned by Rio Tinto Group (79.36%) and the Sumitomo Group (20.64%), via a joint venture called New Zealand Aluminium Smelters (NZAS) Limited. The ore shoot consists of the most valuable part of the ore deposit. Bauxite, the most important ore of aluminium, contains only 30–60% aluminium oxide (Al2O3), the rest being a mixture of silica, various iron oxides, and titanium dioxide. The company investigated sources of large quantities of cheap electricity needed to reduce the alumina recovered from the bauxite into aluminium. The Bayer process is the principal industrial means of refining bauxite to produce alumina (aluminium oxide) and was developed by Carl Josef Bayer. In 2016, an analyst at First New Zealand Capital (FCNZ) utilities said that the smelter was thought to be breaking even, helped by favourable currency rates and low alumina prices. ===Price negotiations, 2019 to 2021=== In October 2019, Rio Tinto announced a strategic review of the Tiwai Point Aluminium Smelter, including a wide range of issues associated with closure. The Deal–Grove model mathematically describes the growth of an oxide layer on the surface of a material.

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A boatman wants to cross a canal that is $3 \mathrm{~km}$ wide and wants to land at a point $2 \mathrm{~km}$ upstream from his starting point. The current in the canal flows at $3.5 \mathrm{~km} / \mathrm{h}$ and the speed of his boat is $13 \mathrm{~km} / \mathrm{h}$. How long will the trip take?

This is often called the hull speed and is a function of the length of the ship V=k\sqrt{L} where constant (k) should be taken as: 2.43 for velocity (V) in kn and length (L) in metres (m) or, 1.34 for velocity (V) in kn and length (L) in feet (ft). This procedure meant a delay of at least 20 minutes for the boat travelling upstream, while the ship heading downstream suffered a delay of about 45 minutes as a result of the manoeuvre. The smaller boat, which is travelling downstream, is moving very fast, driven by the large water sails on either side and is thereby hauling the larger boat upstream against the current. Practical experience showed that free-running, paddle tugboats capable of could achieve a speed of about against a river current of . On the Neckar river with seven chain boats that meant six passing manoeuvres costing at least five hours for those travelling downstream. The boat is pulling itself upstream on a cable laid along the river. So variations in river depth complicated the handling of the vessel considerably.Carl Victor Suppán: Wasserstrassen und Binnenschiffahrt. However, fast currents when rivers were in spate could also be problematic for chain boats. Chain-boat navigationDocument, Volume 1, Issues 1-9, The Commission, US National Waterways Commission, 1909, p. right|thumb|300px|The canal's route is close to the dashed line of the railway across the neck of the peninsula. thumb|293x293px|a ship crossing the Karakum Canal. By the time the attached barges entered the faster-flowing area, the steamer had already sailed past it and was able to generate its full traction again. When travelling downstream the barges were usually just allowed to drift with the current in order to save money, In strong currents, operating a long string of barges was quite dangerous. The electric powered bridge is lifted several times a day to let boats pass through it. == Activities == thumb|left|Sunset on the Shark River Inlet looking west from the drawbridge. The number of trips a boat could make increased, for example, on the Elbe almost three times.Erich Pleißner: "Konzentration der Güterschiffahrt auf der Elbe". These sections of river could be negotiated by anchoring a rope ahead of the boat and then using the crew to haul it upstream.Sigbert Zesewitz, Helmut Düntzsch, Theodor Grötschel: Kettenschiffahrt. On the remaining section with its strong currents the chain boats made a profit in many years of about 30%. The canal tolls reflected these improvements, but if a boat missed the tide it would have to wait in the canal basin for longer than the journey round Hoo would have taken. ==Higham and Strood tunnel== The Higham and Strood tunnel is long, and was the second longest canal tunnel built in the UK (the longest is the Standedge Canal Tunnel). When travelling downstream, boats were either simply propelled along by the current or sails would employ wind power. When heading downstream, however, they were faster and could also haul barges with them. At this point, by the Domfelsen, the river flowed particularly fast. Above 0.3‰ the chain boat has the advantage.

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Find the area bounded by the curves $y=\cos x$ and $y=\cos ^2 x$ between $x=0$ and $x=\pi$.

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. This function has no global maximum or minimum. x| Global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0. cos(x) Infinitely many global maxima at 0, ±2, ±4, ..., and infinitely many global minima at ±, ±3, ±5, .... 2 cos(x) − x Infinitely many local maxima and minima, but no global maximum or minimum. cos(3x)/x with Global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. thumb|right|A graph of the function f(x) = e^{-x^2} and the area between it and the x-axis, (i.e. the entire real line) which is equal to \sqrt{\pi}. In mathematics, the definite integral :\int_a^b f(x)\, dx is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total. The limits of these functions as goes to infinity are known: \int_0^\infty \cos \left(t^2\right)\,dt = \int_0^\infty \sin \left(t^2\right) \, dt = \frac{\sqrt{2\pi}}{4} = \sqrt{\frac{\pi}{8}} \approx 0.6267. right|250px|thumb|The sector contour used to calculate the limits of the Fresnel integrals This can be derived with any one of several methods. thumb|300px|Graph y=ƒ(x) with the x-axis as the horizontal axis and the y-axis as the vertical axis. It is easy to see that the volume of the region under f(x,y) and above z=0 , which is 1 , can be obtained by integrating the area, which is -2\pi\log(z/c^2) , of the circle with radius of value x>0 such that f(x,0)=z between z=0 and z=c^2 . In mathematics, tables of trigonometric functions are useful in a number of areas. As such, these points satisfy x = 0. ==Using equations== If the curve in question is given as y= f(x), the y-coordinate of the y-intercept is found by calculating f(0). 250px|thumb| Plots of and . The y-intercept of ƒ(x) is indicated by the red dot at (x=0, y=1). The simultaneous parametric plot of and is the Euler spiral (also known as the Cornu spiral or clothoid). == Definition == 250px|thumb| Fresnel integrals with arguments instead of converge to instead of . Let \begin{align} y & = xs \\\ dy & = x\,ds. \end{align} Since the limits on as depend on the sign of , it simplifies the calculation to use the fact that is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity. Modern computers and pocket calculators now generate trigonometric function values on demand, using special libraries of mathematical code. Since width is positive, then x>0, and since that implies that Plug in critical point as well as endpoints 0 and into and the results are 2500, 0, and 0 respectively. They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations: S(x) = \int_0^x \sin\left(t^2\right)\,dt, \quad C(x) = \int_0^x \cos\left(t^2\right)\,dt. Category:Trigonometry Category:Numerical analysis 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. The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals. Combining these yields \left ( \int_{-\infty}^\infty e^{-x^2}\,dx \right )^2=\pi, so \int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt{\pi}. ====Complete proof==== To justify the improper double integrals and equating the two expressions, we begin with an approximating function: I(a) = \int_{-a}^a e^{-x^2}dx. As such, these points satisfy y=0. Now retrieve the endpoints by determining the interval to which x is restricted.

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A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of $40^{\circ}$ above the horizontal moves the sled $80 \mathrm{ft}$. Find the work done by the force.

This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. In physics, work is the energy transferred to or from an object via the application of force along a displacement. The work of the net force is calculated as the product of its magnitude and the particle displacement. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Notice that the work done by gravity depends only on the vertical movement of the object. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. The work done is given by the dot product of the two vectors. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. Therefore, work need only be computed for the gravitational forces acting on the bodies. Work is defined by: : \delta W = \mathbf{F}\cdot\mathbf{v}\delta t, Therefore :\frac{\partial W}{\partial t}=\mathbf{F_E} \cdot \,\mathbf{v} ==References== Category:Electromagnetism The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. The work of the torque is calculated as \delta W = \mathbf{T} \cdot \boldsymbol{\omega} \, dt, where the is the power over the instant . Work transfers energy from one place to another or one form to another. ===Derivation for a particle moving along a straight line=== In the case the resultant force is constant in both magnitude and direction, and parallel to the velocity of the particle, the particle is moving with constant acceleration a along a straight line. In this case the dot product , where is the angle between the force vector and the direction of movement, that is W = \int_C \mathbf{F} \cdot d\mathbf{s} = Fs\cos\theta. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance.

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If $R$ is the total resistance of three resistors, connected in parallel, with resistances $R_1, R_2, R_3$, then $$ \frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3} $$ If the resistances are measured in ohms as $R_1=25 \Omega$, $R_2=40 \Omega$, and $R_3=50 \Omega$, with a possible error of $0.5 \%$ in each case, estimate the maximum error in the calculated value of $R$.

The unit was based upon the ohm equal to 109 units of resistance of the C.G.S. system of electromagnetic units. The ohm (symbol: Ω) is the unit of electrical resistance in the International System of Units (SI). Residual-resistivity ratio (also known as Residual-resistance ratio or just RRR) is usually defined as the ratio of the resistivity of a material at room temperature and at 0 K. The formula is a combination of Ohm's law and Joule's law: P=V I =\frac{V^2}{R} = I^2 R, where is the power, is the resistance, is the voltage across the resistor, and is the current through the resistor. The B.A. ohm was intended to be 109 CGS units but owing to an error in calculations the definition was 1.3% too small. A legal ohm, a reproducible standard, was defined by the international conference of electricians at Paris in 1884 as the resistance of a mercury column of specified weight and 106 cm long; this was a compromise value between the B. A. unit (equivalent to 104.7 cm), the Siemens unit (100 cm by definition), and the CGS unit. Uniform field theory is a formula for determining the effective electrical resistance of a parallel wire system. The 3ω-method (3 omega method) or 3ω-technique, is a measurement method for determining the thermal conductivities of bulk material (i.e. solid or liquid) and thin layers. The international ohm is represented by the resistance offered to an unvarying electric current in a mercury column of constant cross-sectional area 106.3 cm long of mass 14.4521 grams and 0 °C. On 21 September 1881 the Congrès internationale des électriciens (international conference of electricians) defined a practical unit of ohm for the resistance, based on CGS units, using a mercury column 1 mm2 in cross-section, approximately 104.9 cm in length at 0 °C, similar to the apparatus suggested by Siemens. In the electronics industry it is common to use the character R instead of the Ω symbol, thus, a 10 Ω resistor may be represented as 10R. Multiple drug resistance (MDR), multidrug resistance or multiresistance is antimicrobial resistance shown by a species of microorganism to at least one antimicrobial drug in three or more antimicrobial categories. Multiple resistance may refer to: * Multiple drug resistance ** including Antimicrobial resistance * The measured voltage will contain both the fundamental and third harmonic components (ω and 3ω respectively), because the Joule heating of the metal structure induces oscillations in its resistance with frequency 2ω due to the temperature coefficient of resistance (TCR) of the metal heater/sensor as stated in the following equation: :V=IR=I_0e^{i\omega t}\left (R_0+\frac{\partial R}{\partial T}\Delta T \right )=I_0e^{i\omega t}\left (R_0+C_0e^{i2\omega t} \right )=I_0R_0e^{i\omega t} + I_0C_0e^{i3\omega t}, where C0 is constant. In Mac OS, does the same. == See also == * Electronic color code * History of measurement * International Committee for Weights and Measures * Orders of magnitude (resistance) * Resistivity == Notes and references == == External links == * Scanned books of Georg Simon Ohm at the library of the University of Applied Sciences Nuernberg * Official SI brochure * NIST Special Publication 811 * History of the ohm at sizes.com * History of the electrical units. :\Omega = \dfrac{\text{V}}{\text{A}} = \dfrac{1}{\text{S}} = \dfrac{\text{W}}{\text{A}^2} = \dfrac{\text{V}^2}{\text{W}} = \dfrac{\text{s}}{\text{F}} = \dfrac{\text{H}}{\text{s}} = \dfrac{\text{J} {\cdot} \text{s}}{\text{C}^2} = \dfrac{\text{kg} {\cdot} \text{m}^2}{\text{s} {\cdot} \text{C}^2} = \dfrac{\text{J}}{\text{s} {\cdot} \text{A}^2}=\dfrac{\text{kg}{\cdot}\text{m}^2}{\text{s}^3 {\cdot} \text{A}^2} in which the following units appear: volt (V), ampere (A), siemens (S), watt (W), second (s), farad (F), henry (H), joule (J), coulomb (C), kilogram (kg), and meter (m). Since resistivity usually increases as defect prevalence increases, a large RRR is associated with a pure sample. Since the ohm belongs to a coherent system of units, when each of these quantities has its corresponding SI unit (watt for , ohm for , volt for and ampere for , which are related as in ) this formula remains valid numerically when these units are used (and thought of as being cancelled or omitted). == History == The rapid rise of electrotechnology in the last half of the 19th century created a demand for a rational, coherent, consistent, and international system of units for electrical quantities. Various artifact standards were proposed as the definition of the unit of resistance. The quantum Hall experiments are used to check the stability of working standards that have convenient values for comparison.R. Dzuiba and others, Stability of Double-Walled Maganin Resistors in NIST Special Publication Proceedings of SPIE, The Institute, 1988 pp. 63–64 Following the 2019 redefinition of the SI base units, in which the ampere and the kilogram were redefined in terms of fundamental constants, the ohm is now also defined in terms of these constants. == Symbol == The symbol Ω was suggested, because of the similar sound of ohm and omega, by William Henry Preece in 1867. Note that since it is a unitless ratio there is no difference between a residual resistivity and residual-resistance ratio. ==Background== Usually at "warm" temperatures the resistivity of a metal varies linearly with temperature. Advances in metrology allowed definitions to be formulated with a high degree of precision and repeatability. === Historical units of resistance === UnitGordon Wigan (trans. and ed.), Electrician's Pocket Book, Cassel and Company, London, 1884 Definition Value in B.A. ohms Remarks Absolute foot/second × 107 using imperial units 0.3048 considered obsolete even in 1884 Thomson's unit using imperial units 0.3202 , considered obsolete even in 1884 Jacobi copper unit A specified copper wire long weighing 0.6367 Used in 1850s Weber's absolute unit × 107 Based on the meter and the second 0.9191 Siemens mercury unit 1860.

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0.05882352941

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49

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The length and width of a rectangle are measured as $30 \mathrm{~cm}$ and $24 \mathrm{~cm}$, respectively, with an error in measurement of at most $0.1 \mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.

The big rectangle has width m and length T + 3m. For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 points, producing a larger error of approximately 9.54. thumb|A circle of radius 5 centered at the origin has area 25, approximately 78.54, but it contains 81 integer points, so the error in estimating its area by counting grid points is approximately 2.46. There are a number of variants of the problem, depending on the particularities of this generic formulation, in particular, depending on the measure of the "size", domain (type of obstacles), and the orientation of the rectangle. In particular, for the case of points within rectangle an optimal algorithm of time complexity \Theta(n \log n) is known. ===Domain: rectangle containing points=== A problem first discussed by Naamad, Lee and Hsu in 1983 is stated as follows: given a rectangle A containing n points, find a largest-area rectangle with sides parallel to those of A which lies within A and does not contain any of the given points. Each side of a maximal empty rectangle abuts an obstacle (otherwise the side may be shifted outwards, increasing the empty rectangle). E shows a maximal empty rectangle with arbitrary orientation In computational geometry, the largest empty rectangle problem, maximal empty rectangle problem or maximum empty rectangle problem, is the problem of finding a rectangle of maximal size to be placed among obstacles in the plane. In contrast, in rectangle packing (as in real-life packing problems) the sizes of the rectangles are given, but their locations are flexible. In this more general case, it is not clear if the problem is in NP, since it is much harder to verify a solution. == Packing different rectangles in a minimum-area rectangle == In this variant, the small rectangles can have varying lengths and widths, and their orientation is fixed (they cannot be rotated). This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. In the contexts of many algorithms for largest empty rectangles, "maximal empty rectangles" are candidate solutions to be considered by the algorithm, since it is easily proven that, e.g., a maximum-area empty rectangle is a maximal empty rectangle. ==Classification== In terms of size measure, the two most common cases are the largest-area empty rectangle and largest-perimeter empty rectangle. Rectangle packing is a packing problem where the objective is to determine whether a given set of small rectangles can be placed inside a given large polygon, such that no two small rectangles overlap. * Maximum disjoint set (or Maximum independent set) is a problem in which both the sizes and the locations of the input rectangles are fixed, and the goal is to select a largest sum of non-overlapping rectangles. Another major classification is whether the rectangle is sought among axis-oriented or arbitrarily oriented rectangles. ==Special cases == ===Maximum-area square=== The case when the sought rectangle is an axis-oriented square may be treated using Voronoi diagrams in L_1metrics for the corresponding obstacle set, similarly to the largest empty circle problem. Later in 1977, Jon Bentley considered a 2-dimensional analogue of this problem: given a collection of n rectangles, find the area of their union. Finding a largest square-packing is NP-hard; one may prove this by reducing from 3SAT. == Packing different rectangles in a given rectangle == In this variant, the small rectangles can have varying lengths and widths, and they should be packed in a given large rectangle. The light green rectangle would be suboptimal (non- maximal) solution. Probabilistic bounding analysis in the quantification of margins and uncertainties. The goal is to pack them in an enclosing rectangle of minimum area, with no boundaries on the enclosing rectangle's width or height. The Gauss circle problem concerns bounding this error more generally, as a function of the radius of the circle. Therefore, the packing must involve exactly m rows where each row contains rectangles with a total length of exactly T. A maximal empty rectangle is a rectangle which is not contained in another empty rectangle.

29.36

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B

The planet Mercury travels in an elliptical orbit with eccentricity 0.206 . Its minimum distance from the sun is $4.6 \times 10^7 \mathrm{~km}$. Find its maximum distance from the sun.

Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). thumb|An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. Conclusions: *The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (a\,\\!), *For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law). ==Energy== Under standard assumptions, the specific orbital energy (\epsilon) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form: :{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0 where: *v\, is the orbital speed of the orbiting body, *r\, is the distance of the orbiting body from the central body, *a\, is the length of the semi-major axis, *\mu\, is the standard gravitational parameter. The following chart of the perihelion and aphelion of the planets, dwarf planets and Halley's Comet demonstrates the variation of the eccentricity of their elliptical orbits. 250px|thumb|Animation of Orbit by eccentricity In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit. Note the almost-zero eccentricity of Earth and Venus compared to the enormous eccentricity of Halley's Comet and Eris. ==Radial elliptic trajectory== A radial trajectory can be a double line segment, which is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1. Mathematically, the distance between a central body and an orbiting body can be expressed as: r(\theta) = \frac{a(1-e^2)}{1+e\cos(\theta)} where: *r is the distance *a is the semi- major axis, which defines the size of the orbit *e is the eccentricity, which defines the shape of the orbit *\theta is the true anomaly, which is the angle between the current position of the orbiting object and the location in the orbit at which it is closest to the central body (called the periapsis). This is true for r being the closest / furthest distance so we get two simultaneous equations which we solve for E: :E = - G \frac{Mm}{r_1 + r_2} Since r_1 = a + a \epsilon and r_2 = a - a \epsilon, where epsilon is the eccentricity of the orbit, we finally have the stated result. ==Flight path angle== The flight path angle is the angle between the orbiting body's velocity vector (= the vector tangent to the instantaneous orbit) and the local horizontal. Kepler-1704b goes on a highly eccentric 2.7 year-long (988.88 days) orbit around its star as well as transiting. The extreme eccentricity yields a temperature difference of up to 700 K. == Star == The star, Kepler-1704, is a G2, 5745-kelvin star from Earth and the sun. Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 The total energy of the orbit is given by :E = - G \frac{M m}{2a}, where a is the semi major axis. ==== Derivation ==== Since gravity is a central force, the angular momentum is constant: :\dot{\mathbf{L}} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times F(r)\mathbf{\hat{r}} = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: :L = r p = r m v. thumb|right|Kepler's equation solutions for five different eccentricities between 0 and 1 In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. Alternatively, Kepler's equation can be solved numerically. This includes the radial elliptic orbit, with eccentricity equal to 1. Kepler-1704b is a super-Jupiter on a highly eccentric orbit around the star Kepler-1704. The planet's distance from its star varies from 0.16 to 3.9 AU. For 0 this is an ellipse with = a \cdot \sqrt{1-e^2}|}} For e = 1 this is a parabola with focal length \tfrac{p}{2} For e > 1 this is a hyperbola with = a \cdot \sqrt{e^2-1}|}} The following image illustrates a circle (grey), an ellipse (red), a parabola (green) and a hyperbola (blue) thumb|300px|A diagram of the various forms of the Kepler Orbit and their eccentricities. Conclusions: *For a given semi-major axis the specific orbital energy is independent of the eccentricity. The total energy of the orbit is given by :E = \frac{1}{2}m v^2 - G \frac{Mm}{r}. The standard Kepler equation is used for elliptic orbits (0 \le e < 1).

0.086

7

30.0

54.7

322

B

Use differentials to estimate the amount of tin in a closed tin can with diameter $8 \mathrm{~cm}$ and height $12 \mathrm{~cm}$ if the tin is $0.04 \mathrm{~cm}$ thick.

A standard size tin can holds roughly 400 g; though the weight can vary between 385 g and 425 g depending on the density of the contents. Estimates of tin production have historically varied with the market and mining technology. Most of the world's tin is produced from placer deposits, which can contain as little as 0.015% tin. The springs on the tinning machine can be set to different forces to give different thicknesses of tin. thumb|upright|An empty tin can A steel can, tin can, tin (especially in British English, Australian English, Canadian English and South African English), steel packaging, or can is a container for the distribution or storage of goods, made of thin metal. The US Occupational Safety and Health Administration (OSHA) set the permissible exposure limit for tin exposure in the workplace as 2 mg/m3 over an 8-hour workday. Tin melts at about , the lowest in group 14\. Recovery of tin through recycling is increasing rapidly. I Tall 3 × 4 16.70 0.813 No. 303 3 × 4 16.88 0.821 Fruits, Vegetables, Soups No. 303 Cylinder 3 × 5 21.86 1.060 No. 2 Vacuum 3 × 3 14.71 0.716 No. 2 3 × 4 20.55 1.000 Juices, Soups, Vegetables Jumbo 3 × 5 25.80 1.2537 No. 2 Cylinder 3 × 5 26.40 1.284 No. 1.25 4 × 2 13.81 0.672 No. 2.5 4 × 4 29.79 1.450 Fruits, Vegetables No. 3 Vacuum 4 × 3 23.90 1.162 No. 3 Cylinder 4 × 7 51.70 2.515 No. 5 5 × 5 59.10 2.8744 Fruit Juice, Soups No. 10 6 × 7 109.43 5.325 Fruits, Vegetables In parts of the world using the metric system, tins are made in 250, 500, 750 ml (millilitre) and 1 L (litre) sizes (250 ml is approximately 1 cup or 8 ounces). For instance, if a plate is desired the tin bar is cut to a length and width that is divisible by 14 and 20. Because of the higher specific gravity of tin dioxide, about 80% of mined tin is from secondary deposits found downstream from the primary lodes. The flat surfaces of rimmed cans are recessed from the edge of any rim (toward the middle of the can) by about the width of the rim; the inside diameter of a rim, adjacent to this recessed surface, is slightly smaller than the inside diameter of the rest of the can. (See BPA controversy#Chemical manufacturers reactions to bans.) ==See also== * Albion metal *Can collecting * Drink can * Oil can * Tin box * Tin can wall ==References== ===General references, further reading=== * Nicolas Appert * Guide to Tinplate * History of the Tin Can on About.com * Yam, K. L., Encyclopedia of Packaging Technology, John Wiley & Sons, 2009, * Soroka, W, Fundamentals of Packaging Technology, Institute of Packaging Professionals (IoPP), 2002, ==External links== *Steeluniversity Packaging Module *Steel industry fact sheet on food cans *Standard U.S. can sizes at GourmetSleuth Category:Containers Category:Packaging Category:Food storage containers Category:British inventions Category:1810 introductions Category:Food packaging Category:Steel Category:19th-century inventions Category:Metallic objects The Can Manufacturers Institute defines these sizes, expressing them in three-digit numbers, as measured in whole and sixteenths of an inch for the container's nominal outside dimensions: a 307 × 512 would thus measure 3 and 7/16" in diameter by 5 and 3/4" (12/16") in height. Depending on contents and available coatings, some canneries still use tin-free steel. The bar is then rolled and doubled over, with the number of times being doubled over dependent on how large the tin bar is and what the final thickness is. If the starting tin bar is then it must be at least finished on the fours, or doubled over twice, and if a thin gauge is required then it may be finished on the eights, or doubled over three times. The tin layer is usually applied by electroplating. ===Two-piece steel can design=== Most steel beverage cans are two-piece designs, made from 1) a disc re-formed into a cylinder with an integral end, double-seamed after filling and 2) a loose end to close it. The International Tin Association estimated that global refined tin consumption will grow 7.2 percent in 2021, after losing 1.6 percent in 2020 as the COVID-19 pandemic disrupted global manufacturing industries. ==Applications== thumb|right|World consumption of refined tin by end-use, 2006 In 2018, just under half of all tin produced was used in solder. A 2002 study showed that 99.5% of 1200 tested cans contained below the UK regulatory limit of 200 mg/kg of tin, an improvement over most previous studies largely attributed to the increased use of fully lacquered cans for acidic foods, and concluded that the results do not raise any long term food safety concerns for consumers. Because of the low toxicity of inorganic tin, tin-plated steel is widely used for food packaging as tin cans. File:Inside of a tin platted can.jpg|Inside of a tin can. ==Design and manufacture== ===Steel for can making=== The majority of steel used in packaging is tinplate, which is steel that has been coated with a thin layer of tin, whose functionality is required for the production process.

0.69

1.8763

16.0

24

0.000226

C

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A cable that weighs $2 \mathrm{~lb} / \mathrm{ft}$ is used to lift $800 \mathrm{~lb}$ of coal up a mine shaft $500 \mathrm{~ft}$ deep. Find the work done.

If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . The work of the net force is calculated as the product of its magnitude and the particle displacement. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Work is defined by: : \delta W = \mathbf{F}\cdot\mathbf{v}\delta t, Therefore :\frac{\partial W}{\partial t}=\mathbf{F_E} \cdot \,\mathbf{v} ==References== Category:Electromagnetism This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The work done is given by the dot product of the two vectors. To show that the external work done to move a point charge q+ from infinity to a distance r is: :W_{ext} = \frac{q_1q_2}{4\pi\varepsilon_0}\frac{1}{r} This could have been obtained equally by using the definition of W and integrating F with respect to r, which will prove the above relationship. This is easy to see mathematically, as reversing the boundaries of integration reverses the sign. === Uniform electric field === Where the electric field is constant (i.e. not a function of displacement, r), the work equation simplifies to: :W = Q (\mathbf{E} \cdot \, \mathbf{r})=\mathbf{F_E} \cdot \, \mathbf{r} or 'force times distance' (times the cosine of the angle between them). ==Electric power== The electric power is the rate of energy transferred in an electric circuit. Therefore, work need only be computed for the gravitational forces acting on the bodies. In physics, work is the energy transferred to or from an object via the application of force along a displacement. Notice that the work done by gravity depends only on the vertical movement of the object. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Rope access or industrial climbing or commercial climbing, is a form of work positioning, initially developed from techniques used in climbing and caving, which applies practical ropework to allow workers to access difficult-to-reach locations without the use of scaffolding, cradles or an aerial work platform. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. The work of the torque is calculated as \delta W = \mathbf{T} \cdot \boldsymbol{\omega} \, dt, where the is the power over the instant . If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. In general this integral requires that the path along which the velocity is defined, so the evaluation of work is said to be path dependent. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2.

524

0.33333333

650000.0

-1

57.2

C

A patient takes $150 \mathrm{mg}$ of a drug at the same time every day. Just before each tablet is taken, 5$\%$ of the drug remains in the body. What quantity of the drug is in the body after the third tablet?

Such drugs need only a low maintenance dose in order to keep the amount of the drug in the body at the appropriate level, but this also means that, without an initial higher dose, it would take a long time for the amount of the drug in the body to reach that level. == Calculating the maintenance dose == The required maintenance dose may be calculated as: :\mbox{MD} = \frac{C_p CL}{F } Where: : MD is the maintenance dose rate [mg/h] Cp = desired peak concentration of drug [mg/L] CL = clearance of drug in body [L/h] F = bioavailability For an intravenously administered drug, the bioavailability F will equal 1, since the drug is directly introduced to the bloodstream. If the pill removed is a half pill, then it is simply consumed and nothing is returned to the jar. ==Mathematical derivation== The problem becomes very easy to solve once a binary variable Xk defined as Xk = 1, if the kth half pill remains inside the jar after all the whole pills are removed. One half pill is consumed and the other one is returned to the jar. 150 Milligrams (, lit. The sales of generic drugs dominate in the prescription category at 64.5%. Continuing the maintenance dose for about 4 to 5 half-lives (t½) of the drug will approximate the steady state level. The prescription drugs sales historically took the biggest share of the market, capturing 61% of the market in 2016. The pill jar puzzle is a probability puzzle, which asks the expected value of the number of half-pills remaining when the last whole pill is popped from a jar initially containing whole pills and the way to proceed is by removing a pill from the bottle at random. In pharmacokinetics, a maintenance dose is the maintenance rate [mg/h] of drug administration equal to the rate of elimination at steady state. But for companies that approved between eight and 13 drugs over 10 years, the cost per drug went as high as $5.5 billion. 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). The pharmaceutical industry in Russia had a turnover of $16.5 billion in 2016, which was equal to 1.3 % of GDP and 19.9% of health spending. Thus, profits made from one drug need to cover the costs of previous "failed drugs". === Relationship === Overall, research and development expenses relating to a pharmaceutical drug amount to the billions. Therefore, an appropriate figure like $60 billion would be approximate sales figure that a pharmaceutical company like AstraZeneca would aim to generate to cover these costs and make a profit at the same time. Out of all pharmaceutical sales they constitute only 39.4%. If the patient requires an oral dose, bioavailability will be less than 1 (depending upon absorption, first pass metabolism etc.), requiring a larger loading dose. == See also == * Therapeutic index ==References== Category:Pharmacokinetics Xk = 1 if out of the n − k + 1 pills (n − k whole pills + kth half pill), the one half pill is removed at the very end. If the pill removed is a whole pill, it is broken into two half pills. This is not to be confused with dose regimen, which is a type of drug therapy in which the dose [mg] of a drug is given at a regular dosing interval on a repetitive basis. In an analysis of the drug development costs for 98 companies over a decade, the average cost per drug developed and approved by a single- drug company was $350 million. This is important in setting projected profit goals for a particular drug and thus, is one of the most necessary steps pharmaceutical companies take in pricing a particular drug. ==Research on costs== Tufts Center for the Study of Drug Development has published numerous studies estimating the cost of developing new pharmaceutical drugs. A 2022 study invalidated the common argument for high medication costs that research and development investments are reflected in and necessitate the treatment costs, finding no correlation for investments in drugs (for cases where transparency was sufficient) and their costs. == References == == Further reading == * * * * * Category:Drug pricing Category:Drug discovery

8.8

35

157.875

7.58

1.06

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A $360-\mathrm{lb}$ gorilla climbs a tree to a height of $20 \mathrm{~ft}$. Find the work done if the gorilla reaches that height in 5 seconds.

If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. Net work is typically calculated either from instantaneous power (muscle force x muscle velocity) or from the area enclosed by the work loop on a force vs. length plot. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Therefore, work need only be computed for the gravitational forces acting on the bodies. Therefore, the distance in feet down a 6% grade to reach the velocity is at least s = \frac{\Delta z}{0.06}= 8.3\frac{V^2}{g},\quad\text{or}\quad s=8.3\frac{88^2}{32.2}\approx 2000\mathrm{ft}. The work of the net force is calculated as the product of its magnitude and the particle displacement. Step 3) Obtain net work (a single number) by numerical integration of muscle power data. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. In physics, work is the energy transferred to or from an object via the application of force along a displacement. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge. === Work done by a variable force === Calculating the work as "force times straight path segment" would only apply in the most simple of circumstances, as noted above. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.Resnick, Robert, Halliday, David (1966), Physics, Section 1–3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 Work is the result of a force on a point that follows a curve , with a velocity , at each instant.

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Find the area of triangle $A B C$, correct to five decimal places, if $$ |A B|=10 \mathrm{~cm} \quad|B C|=3 \mathrm{~cm} \quad \angle A B C=107^{\circ} $$

Substituting this in the formula T=\tfrac12 bh derived above, the area of the triangle can be expressed as: :T = \tfrac12 ab\sin \gamma = \tfrac12 bc\sin \alpha = \tfrac12 ca\sin \beta (where α is the interior angle at A, β is the interior angle at B, \gamma is the interior angle at C and c is the line AB). In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. thumb|The area of the pink triangle is one-seventh of the area of the large triangle ABC. The reader is advised that several of the formulas in this source are not correct. gave a collection of over a hundred distinct area formulas for the triangle. thumb|Triangle with the area 6, a congruent number. thumb|upright=1.5|\begin{align}&\text{all inner angles} < 120^\circ : \\\ &\text{grey area} = 3 \Delta \leq \Delta_a+\Delta_b+\Delta_c \end{align} thumb|upright=1.5|\begin{align}&\text{one inner angle} \geq 120^\circ : \\\ &\text{grey area} = 3 \Delta \leq \Delta_c < \Delta_a+\Delta_b+\Delta_c \end{align} In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds: : a^2 + b^2 + c^2 \geq 4\sqrt{3}\, \Delta. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). Then a, b and c are the legs and hypotenuse of a right triangle with area n. In plane geometry, a triangle ABC contains a triangle having one-seventh of the area of ABC, which is formed as follows: the sides of this triangle lie on cevians p, q, r where :p connects A to a point on BC that is one-third the distance from B to C, :q connects B to a point on CA that is one-third the distance from C to A, :r connects C to a point on AB that is one-third the distance from A to B. :T = D^{2} \sqrt{S(S-\sin \alpha)(S-\sin \beta)(S-\sin \gamma)} where D is the diameter of the circumcircle: D=\tfrac{a}{\sin \alpha} = \tfrac{b}{\sin \beta} = \tfrac{c}{\sin \gamma}. ==Using vectors== The area of triangle ABC is half of the area of a parallelogram: :\tfrac12|\mathbf{AB}\times\mathbf{AC}|. The values (1201, 140, 1151, 1249) give (a, b, c) = (7/10, 120/7, 1201/70). 300px|thumb|A graphic derivation of the formula T=\frac{h}{2}b that avoids the usual procedure of doubling the area of the triangle and then halving it. The area can also be expressed asPathan, Alex, and Tony Collyer, "Area properties of triangles revisited," Mathematical Gazette 89, November 2005, 495–497. By Heron's formula: :T = \sqrt{s(s-a)(s-b)(s-c)} where s= \tfrac12(a+b+c) is the semiperimeter, or half of the triangle's perimeter. This can now can be shown by replicating area of the triangle three times within the equilateral triangles. The area of a triangle then falls out as the case of a polygon with three sides. Three other area bisectors are parallel to the triangle's sides. The area of triangle ABC can also be expressed in terms of dot products as follows: :\tfrac12 \sqrt{(\mathbf{AB} \cdot \mathbf{AB})(\mathbf{AC} \cdot \mathbf{AC}) -(\mathbf{AB} \cdot \mathbf{AC})^2} =\tfrac12 \sqrt{ |\mathbf{AB}|^2 |\mathbf{AC}|^2 -(\mathbf{AB} \cdot \mathbf{AC})^2}.\, In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to (x1,y1) and AC as (x2,y2), this can be rewritten as: :\tfrac12|x_1 y_2 - x_2 y_1|. ==Using coordinates== If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by and , then the area can be computed as times the absolute value of the determinant :T = \tfrac12\left|\det\begin{pmatrix}x_B & x_C \\\ y_B & y_C \end{pmatrix}\right| = \tfrac12 |x_B y_C - x_C y_B|. There can be one, two, or three of these for any given triangle. ==See also== *Area of a circle *Congruence of triangles ==References== Category:Area Category:Triangles The height of a triangle can be found through the application of trigonometry. The area of parallelogram ABDC is then :|\mathbf{AB}\times\mathbf{AC}|, which is the magnitude of the cross product of vectors AB and AC. However if one angle is greater or equal to 120^\circ it is possible to replicate the whole triangle three times within the largest equilateral triangle, so the sum of areas of all equilateral triangles stays greater than the threefold area of the triangle anyhow. == Further proofs == The proof of this inequality was set as a question in the International Mathematical Olympiad of 1961.

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Use Stokes' Theorem to evaluate $\int_C \mathbf{F} \cdot d \mathbf{r}$, where $\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+z x \mathbf{k}$, and $C$ is the triangle with vertices $(1,0,0),(0,1,0)$, and $(0,0,1)$, oriented counterclockwise as viewed from above.

The Kelvin–Stokes theorem: \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S abla \times \mathbf{F} \cdot \mathbf{\hat n} \, dS. thumb|right|An illustration of Stokes' theorem, with surface , its boundary and the normal vector . The classical Stokes' theorem relates the surface integral of the curl of a vector field over a surface \Sigma in Euclidean three-space to the line integral of the vector field over its boundary. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface. More explicitly, the equality says that \begin{align} &\iint_\Sigma \left(\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z} \right)\,\mathrm{d}y\, \mathrm{d}z +\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)\, \mathrm{d}z\, \mathrm{d}x +\left (\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\, \mathrm{d}x\, \mathrm{d}y\right) \\\ & = \oint_{\partial\Sigma} \Bigl(F_x\, \mathrm{d}x+F_y\, \mathrm{d}y+F_z\, \mathrm{d}z\Bigr). \end{align} The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. Nevertheless, it is possible to check that the conclusion of Stokes' theorem is still true. Start with the left side of Green's theorem: \oint_C (L\, dx + M\, dy) = \oint_C (L, M, 0) \cdot (dx, dy, dz) = \oint_C \mathbf{F} \cdot d\mathbf{r}. Stokes' theorem, also known as the Kelvin–Stokes theoremNagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12 (Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, \Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S abla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S} Here, the closed integration path is the boundary or perimeter of an open surface , whose infinitesimal element normal is oriented according to the right-hand rule. This classical case relates the surface integral of the curl of a vector field \textbf{F} over a surface (that is, the flux of \text{curl}\,\textbf{F}) in Euclidean three- space to the line integral of the vector field over the surface boundary. == Introduction == The second fundamental theorem of calculus states that the integral of a function f over the interval [a,b] can be calculated by finding an antiderivative F of f: \int_a^b f(x)\,dx = F(b) - F(a)\,. The expression inside the integral becomes abla \times \mathbf{F} \cdot \mathbf{\hat n} = \left[ \left(\frac{\partial 0}{\partial y} - \frac{\partial M}{\partial z}\right) \mathbf{i} + \left(\frac{\partial L}{\partial z} - \frac{\partial 0}{\partial x}\right) \mathbf{j} + \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) \mathbf{k} \right] \cdot \mathbf{k} = \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right). Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives: \oint_C L \,dx + M \,dy = \oint_{\partial D} \\! \omega = \int_D d\omega = \int_D \frac{\partial L}{\partial y} \,dy \wedge \,dx + \frac{\partial M}{\partial x} \,dx \wedge \,dy = \iint_D \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) \,dx \,dy. ==Relationship to the divergence theorem== Considering only two- dimensional vector fields, Green's theorem is equivalent to the two- dimensional version of the divergence theorem: :\iint_D\left( abla\cdot\mathbf{F}\right)dA=\oint_C \mathbf{F} \cdot \mathbf{\hat n} \, ds, where abla\cdot\mathbf{F} is the divergence on the two-dimensional vector field \mathbf{F}, and \mathbf{\hat n} is the outward- pointing unit normal vector on the boundary. Then \begin{align} 0 &= \int_\Omega \vec{ abla} \cdot \vec{c} f - \int_{\partial \Omega} \vec{n} \cdot \vec{c} f & \text{by the divergence theorem} \\\ &= \int_\Omega \vec{c} \cdot \vec{ abla} f - \int_{\partial \Omega} \vec{c} \cdot \vec{n} f \\\ &= \vec{c} \cdot \int_\Omega \vec{ abla} f - \vec{c} \cdot \int_{\partial \Omega} \vec{n} f \\\ &= \vec{c} \cdot \left( \int_\Omega \vec{ abla} f - \int_{\partial \Omega} \vec{n} f \right) \end{align} Since this holds for any \vec{c} (in particular, for every basis vector), the result follows. ==See also== *Chandrasekhar–Wentzel lemma ==Footnotes== ==References== ==Further reading== * * * * * * * * * * * ==External links== * * * Proof of the Divergence Theorem and Stokes' Theorem * Calculus 3 – Stokes Theorem from lamar.edu – an expository explanation Category:Differential topology Category:Differential forms Category:Duality theories Category:Integration on manifolds Category:Theorems in calculus Category:Theorems in differential geometry Stokes' theorem says that the integral of a differential form \omega over the boundary \partial\Omega of some orientable manifold \Omega is equal to the integral of its exterior derivative d\omega over the whole of \Omega, i.e., \int_{\partial \Omega} \omega = \int_\Omega d\omega\,. Stokes is a census-designated place in Pitt County, North Carolina, United States. Stokes' theorem is a special case of the generalized Stokes theorem. For Faraday's law, Stokes's theorem is applied to the electric field, \mathbf{E}: \oint_{\partial\Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{l}= \iint_\Sigma \mathbf{ abla}\times \mathbf{E} \cdot \mathrm{d} \mathbf{S} . Thus, by generalized Stokes's theorem, \oint_{\partial\Sigma}{\mathbf{F}\cdot\,\mathrm{d}\mathbf{\gamma}} =\oint_{\partial\Sigma}{\omega_{\mathbf{F}}} =\int_{\Sigma}{\mathrm{d}\omega_{\mathbf{F}}} =\int_{\Sigma}{\star\omega_{ abla\times\mathbf{F}}} =\iint_{\Sigma}{ abla\times\mathbf{F}\cdot\,\mathrm{d}\mathbf{\Sigma}} ==Applications== ===Irrotational fields=== In this section, we will discuss the irrotational field (lamellar vector field) based on Stokes's theorem. Stokes' theorem is a vast generalization of this theorem in the following sense. The circulation of a vector field around a closed curve is the line integral: \Gamma = \oint_{C}\mathbf{V} \cdot \mathrm d \mathbf{l}. With the above notation, if \mathbf{F} is any smooth vector field on \R^3, thenRobert Scheichl, lecture notes for University of Bath mathematics course \oint_{\partial\Sigma} \mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}} = \iint_{\Sigma} abla\times\mathbf{F}\, \cdot\, \mathrm{d}\mathbf{\Sigma}.

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A hawk flying at $15 \mathrm{~m} / \mathrm{s}$ at an altitude of $180 \mathrm{~m}$ accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation $$ y=180-\frac{x^2}{45} $$ until it hits the ground, where $y$ is its height above the ground and $x$ is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.

The short-tailed hawk hunts from a soaring flight, often at the borders between wooded and open areas. While following this path, the aircraft and its payload are in free fall at certain points of its flight path. The clutch of one or two eggs is incubated for around 37 days, beginning after the first egg is laid. ===Food and feeding=== The roadside hawk's diet consists mainly of insects, squamates, and small mammals, such as young common marmosets and similar small monkeys which are hunted quite often. The roadside hawk is the smallest hawk in the widespread genus Buteo; although Ridgway's hawk and the white-rumped hawk are scarcely larger. The short-tailed hawk (Buteo brachyurus) is an American bird of prey in the family Accipitridae, which also includes the eagles and Old World vultures. When approached on the nest, the adults will get airborne and observe the intruder from above, unlike related hawks, which usually wait much longer to flush and then launch a determined attack. ==Footnotes== ==References== * eNature (2007): White-tailed Hawk. In one case, 95% of a single hawk's prey selection was found to consist of red-winged blackbirds. The semiplumbeous hawk (Leucopternis semiplumbeus) is a species of bird of prey in the family Accipitridae. The white-tailed hawk (Geranoaetus albicaudatus) is a large bird of prey species found in tropical and subtropical environments of the Americas. ==Description== The white-tailed hawk is a large, stocky hawk. It has been reported up to 1600 m in altitude in one instance. == Population and research == Semiplumbeous hawks is currently categorized as "Least Concern" in terms of global threat level and is listed under CITES II, but was previously classified as "Near Threatened." Semiplumbeous Hawk Leucopternis semiplumbeus at Birdlife.org. In flight, the relatively long tail and disproportionately short wings of the roadside hawk are distinctive. The diet of the white-tailed hawk varies with its environment. It frequently soars, but does not hover. ==Distribution and habitat== The roadside hawk is common throughout its range: from Mexico through Central America to most of South America east of the Andes Cordillera. Like many Accipitridae, white- tailed hawks do not like to abandon a nest site, and nests built up over the years can thus reach sizes of up to three feet (1 m) across. "The dynamics of parabolic flight: flight characteristics and passenger percepts". griseocauda eating speckled racer, Belize The roadside hawk (Rupornis magnirostris) is a relatively small bird of prey found in the Americas. PDF fulltext * Hawk Conservancy Trust (HCT) (2008): White-tailed Hawk - Buteo albicaudatus. Breeding pairs of white- tailed hawks build nests out of freshly broken twigs, often of thorny plants, 5–15 ft (1.5–5 m) or more above the ground on top of a tree or yucca, preferably one growing in an elevated location giving good visibility from the nest. There are isolated records of short-tails preying on sharp-shinned hawks (Accipiter striatus) and American kestrels (Falco sparverius). Its natural habitat is subtropical or tropical moist lowland forests. ==Morphology== The semiplumbeous hawk is a small bird, averaging about in lengthHenderson, Carrol L. "Birds of Costa Rica." Space Hawk is a multidirectional shooter released by Mattel for its Intellivision console in 1982.

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The intensity of light with wavelength $\lambda$ traveling through a diffraction grating with $N$ slits at an angle $\theta$ is given by $I(\theta)=N^2 \sin ^2 k / k^2$, where $k=(\pi N d \sin \theta) / \lambda$ and $d$ is the distance between adjacent slits. A helium-neon laser with wavelength $\lambda=632.8 \times 10^{-9} \mathrm{~m}$ is emitting a narrow band of light, given by $-10^{-6}<\theta<10^{-6}$, through a grating with 10,000 slits spaced $10^{-4} \mathrm{~m}$ apart. Use the Midpoint Rule with $n=10$ to estimate the total light intensity $\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta$ emerging from the grating.

Defining the relative intensity I_\text{rel}as the intensity divided by the intensity of the undisturbed wavefront, the relative intensity for an extended circular source of diameter w can be expressed exactly using the following equation: I_\text{rel}(w) = J_0^2\left(\frac{w R \pi}{g \lambda}\right) + J_1^2\left(\frac{w R \pi}{g \lambda}\right) where J_0and J_1are the Bessel functions of the first kind. It is worth to notice that the parameter N depends on the excitation wavelength. == Impact of ambient pressure == The atmospheric pressure strongly influences the LIWE intensity. Only the width of the Arago spot intensity peak depends on the distances between source, circular object and screen, as well as the source's wavelength and the diameter of the circular object. The dependence of intensity on power is described by the formula: , where N is the number of near infrared photons absorbed for LIWE generation. So, the diffraction formula becomes U(P) = -\frac{i}{2\lambda} \frac{ae^{ikr_0}}{r_0} \int_{S} \frac{e^{iks}}{s} (1 + \cos\chi) dS, where the integral is done over the part of the wavefront at r0 which is the closest to the aperture in the diagram. If, however, we assume that the light from the source at each point in the aperture has a well-defined direction, which is the case if the distance between the source and the aperture is significantly greater than the wavelength, then we can write U_0(r) \approx a(r) e^{ikr}, where a(r) is the magnitude of the disturbance at the point r in the aperture. In order to derive the intensity behind the circular obstacle using this integral one assumes that the experimental parameters fulfill the requirements of the near-field diffraction regime (the size of the circular obstacle is large compared to the wavelength and small compared to the distances g = P0C and b = CP1). thumb|Photo of the Arago spot in a shadow of a 5.8 mm circular obstacle thumb|700px|Numerical simulation of the intensity of monochromatic light of wavelength λ = 0.5 µm behind a circular obstacle of radius . thumb|Formation of the Arago spot (select "WebM source" for good quality) thumb|Arago spot forming in the shadow In optics, the Arago spot, Poisson spot, or Fresnel spot"Although this phenomenon is often called Poisson's spot, Poisson probably was not happy to have seen it because it supported the wave model of light. Kirchhoff's diffraction formula (also called Fresnel–Kirchhoff diffraction formula) approximates light intensity and phase in optical diffraction: light fields in the boundary regions of shadows. The lateral intensity distribution on the screen has in fact the shape of a squared zeroth Bessel function of the first kind when close to the optical axis and using a plane wave source (point source at infinity): U(P_1, r) \propto J_0^2 \left(\frac{\pi r d}{\lambda b}\right) where * r is the distance of the point P1 on the screen from the optical axis * d is the diameter of circular object * λ is the wavelength * b is the distance between circular object and screen. This is not the case, and this is _one of the approximations_ used in deriving the Kirchhoff's diffraction formula.J.Z. Buchwald & C.-P. Yeang, "Kirchhoff's theory for optical diffraction, its predecessor and subsequent development: the resilience of an inconsistent theory" , Archive for History of Exact Sciences, vol.70, no.5 (Sep.2016), pp.463–511; .J. Saatsi & P. Vickers, "Miraculous success? The following images show the radial intensity distribution of the simulated Arago spot images above: 200px 200px 200px The red lines in these three graphs correspond to the simulated images above, and the green lines were computed by applying the corresponding parameters to the squared Bessel function given above. ===Finite source size and spatial coherence=== The main reason why the Arago spot is hard to observe in circular shadows from conventional light sources is that such light sources are bad approximations of point sources. The source intensity, which is the square of the field amplitude, is I_0 = \left|\frac{1}{g} A e^{\mathbf{i} k g}\right|^2 and the intensity at the screen I = \left| U(P_1) \right|^2. Generally when light of a certain wavelength falls on a subwavelength aperture, it is diffracted isotropically in all directions evenly, with minimal far-field transmission. Each image is 16 mm wide. ==Experimental aspects== ===Intensity and size=== For an ideal point source, the intensity of the Arago spot equals that of the undisturbed wave front. The on-axis intensity as a function of the distance b is hence given by: I = \frac{b^2}{b^2 + a^2} I_0. The white light emission intensity is exponentially dependent on excitation power density and pressure surrounding the samples. Laser-induced white emission (LIWE) is a broadband light in the visible spectral range. Going to polar coordinates then yields the integral for a circular object of radius a (see for example Born and Wolf): U(P_1) = - \frac{\mathbf{i}}{\lambda} \frac{A e^{\mathbf{i} k (g + b)}}{g b} 2\pi \int_a^\infty e^{\mathbf{i} k \frac{1}{2} \left(\frac{1}{g} + \frac{1}{b}\right) r^2} r \, dr. 300px|thumb|right|The on- axis intensity at the center of the shadow of a small circular obstacle converges to the unobstructed intensity. The approximation can be used to model light propagation in a wide range of configurations, either analytically or using numerical modelling. The adjacent Fresnel zone is approximately given by: \Delta r \approx \sqrt{r^2 + \lambda \frac{g b}{g + b}} - r. However, if random edge corrugation have amplitude comparable to or greater than the width of that adjacent Fresnel zone, the contributions from radial segments are no longer in phase and cancel each other reducing the Arago spot intensity.

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A model for the surface area of a human body is given by $S=0.1091 w^{0.425} h^{0.725}$, where $w$ is the weight (in pounds), $h$ is the height (in inches), and $S$ is measured in square feet. If the errors in measurement of $w$ and $h$ are at most $2 \%$, use differentials to estimate the maximum percentage error in the calculated surface area.

It is [4W (kg) + 7]/[90 + W (kg)].Costeff H, "A simple empirical formula for calculating approximate surface area in children.," The Growth of the Surface Area of the Human Body. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. In physiology and medicine, the body surface area (BSA) is the measured or calculated surface area of a human body. The surface area of a solid object is a measure of the total area that the surface of the object occupies. Thus, the surface area falls off steeply with increasing volume. == See also == * Perimeter length * Projected area * BET theory, technique for the measurement of the specific surface area of materials * Spherical area * Surface integral == References == * ==External links== *Surface Area Video at Thinkwell Category:Area The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. One of the subtleties of surface area, as compared to arc length of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. For different applications a minimal or maximal surface area may be desired. == In biology == The surface area of an organism is important in several considerations, such as regulation of body temperature and digestion. The Sail Area-Displacement ratio (SA/D) is a calculation used to express how much sail a boat carries relative to its weight. :\mathit{SA/D} = \frac{\mathit{Sail Area}(\text{ft}^2)} {[\mathit{Displacement}(\text{lb})/64]^{\frac{2}{3}}} = \frac{\mathit{Sail Area}(\text{m}^2)} {\mathit{Displacement}(\text{m}^3)^{\frac{2}{3}}} In the first equation, the denominator in pounds is divided by 64 to convert it to cubic feet (because 1 cubic foot of salt water weights 64 pounds). The below given formulas can be used to show that the surface area of a sphere and cylinder of the same radius and height are in the ratio 2 : 3, as follows. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. The most fundamental property of the surface area is its additivity: the area of the whole is the sum of the areas of the parts. Surface area is important in chemical kinetics. Increased surface area can also lead to biological problems. thumb|300px|Graphs of surface area, A against volume, V of the Platonic solids and a sphere, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. The resulting surface area to volume ratio is therefore . thumb|upright=1.3|Measurement of volume by displacement, (a) before and (b) after an object has been submerged. An increased surface area to volume ratio also means increased exposure to the environment. Thus the area of SD is obtained by integrating the length of the normal vector \vec{r}_u\times\vec{r}_v to the surface over the appropriate region D in the parametric uv plane. Let the radius be r and the height be h (which is 2r for the sphere). \begin{array}{rlll} \text{Sphere surface area} & = 4 \pi r^2 & & = (2 \pi r^2) \times 2 \\\ \text{Cylinder surface area} & = 2 \pi r (h + r) & = 2 \pi r (2r + r) & = (2 \pi r^2) \times 3 \end{array} The discovery of this ratio is credited to Archimedes. == In chemistry == thumb|Surface area of particles of different sizes.

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The temperature at the point $(x, y, z)$ in a substance with conductivity $K=6.5$ is $u(x, y, z)=2 y^2+2 z^2$. Find the rate of heat flow inward across the cylindrical surface $y^2+z^2=6$, $0 \leqslant x \leqslant 4$

Then the heat per unit volume u satisfies an equation : \frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q. In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. Thus the rate of heat flow into V is also given by the surface integral q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS where n(x) is the outward pointing normal vector at x. Rate of heat flow = - (heat transfer coefficient) * (area of the body) * (variation of the temperature) / (length of the material) The formula for the rate of heat flow is: :\frac{Q}{\Delta t} = -kA \frac{\Delta T}{\Delta x} where * Q is the net heat (energy) transfer, * \Delta t is the time taken, * \Delta T is the difference in temperature between the cold and hot sides, * \Delta x is the thickness of the material conducting heat (distance between hot and cold sides), * k is the thermal conductivity, and * A is the surface area of the surface emitting heat. Consider a fluid of uniform temperature T_o and velocity u_o impinging onto a stationary plate uniformly heated to a temperature T_s. This equation is also called the Churchill–Bernstein correlation. ==Heat transfer definition== :\overline{\mathrm{Nu}}_D \ = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\Pr^{1/3}}{\left[1 + (0.4/\Pr)^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad \Pr\mathrm{Re}_D \ge 0.2 where: * \overline{\mathrm{Nu}}_D is the surface averaged Nusselt number with characteristic length of diameter; * \mathrm{Re}_D\,\\! is the Reynolds number with the cylinder diameter as its characteristic length; * \Pr is the Prandtl number. This form is more general and particularly useful to recognize which property (e.g. cp or \rho) influences which term. :\rho c_p \frac{\partial T}{\partial t} - abla \cdot \left( k abla T \right) = \dot q_V where \dot q_V is the volumetric heat source. ===Three-dimensional problem=== In the special cases of propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is : \frac{\partial u}{\partial t} = \alpha abla^2 u = \alpha \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2 }\right) = \alpha \left( u_{xx} + u_{yy} + u_{zz} \right) where: * u = u(x, y, z, t) is temperature as a function of space and time; * \tfrac{\partial u}{\partial t} is the rate of change of temperature at a point over time; * u_{xx} , u_{yy} , and u_{zz} are the second spatial derivatives (thermal conductions) of temperature in the x , y , and z directions, respectively; * \alpha \equiv \tfrac{k}{c_p\rho} is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity k , the specific heat capacity c_p , and the mass density \rho . One then says that is a solution of the heat equation if :\frac{\partial u}{\partial t} = \alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right) in which is a positive coefficient called the thermal diffusivity of the medium. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The equation yields the surface averaged Nusselt number, which is used to determine the average convective heat transfer coefficient. Steady-state condition: :\frac{\partial u}{\partial t} = 0 The steady-state heat equation for a volume that contains a heat source (the inhomogeneous case), is the Poisson's equation: :-k abla^2 u = q where u is the temperature, k is the thermal conductivity and q is the rate of heat generation per unit volume. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The general solution of the heat equation on Rn is then obtained by a convolution, so that to solve the initial value problem with u(x, 0) = g(x), one has :u(\mathbf{x},t) = \int_{\R^n}\Phi(\mathbf{x}-\mathbf{y},t)g(\mathbf{y})d\mathbf{y}. That is, :\frac{\partial Q}{\partial t} = - \frac{\partial q}{\partial x} From the above equations it follows that :\frac{\partial u}{\partial t} \;=\; - \frac{1}{c \rho} \frac{\partial q}{\partial x} \;=\; - \frac{1}{c \rho} \frac{\partial}{\partial x} \left(-k \,\frac{\partial u}{\partial x} \right) \;=\; \frac{k}{c \rho} \frac{\partial^2 u}{\partial x^2} which is the heat equation in one dimension, with diffusivity coefficient :\alpha = \frac{k}{c\rho} This quantity is called the thermal diffusivity of the medium. ====Accounting for radiative loss==== An additional term may be introduced into the equation to account for radiative loss of heat. In the absence of heat energy generation, from external or internal sources, the rate of change in internal heat energy per unit volume in the material, \partial Q/\partial t, is proportional to the rate of change of its temperature, \partial u/\partial t. One further variation is that some of these solve the inhomogeneous equation :u_{t}=ku_{xx}+f. where f is some given function of x and t. ==== Homogeneous heat equation ==== ;Initial value problem on (−∞,∞) :\begin{cases} u_{t}=ku_{xx} & (x, t) \in \R \times (0, \infty) \\\ u(x,0)=g(x) & \text{Initial condition} \end{cases} :u(x,t) = \frac{1}{\sqrt{4\pi kt}} \int_{-\infty}^{\infty} \exp\left(-\frac{(x-y)^2}{4kt}\right)g(y)\,dy right|thumb|upright=2|Fundamental solution of the one-dimensional heat equation. * The time rate of heat flow into a region V is given by a time- dependent quantity qt(V). If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specify boundary conditions for u. The rate of heat flow is the amount of heat that is transferred per unit of time in some material, usually measured in watt (joules per second). * By the divergence theorem, the previous surface integral for heat flow into V can be transformed into the volume integral \begin{align} q_t(V) &= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS \\\ &= \int_{\partial V} \mathbf{A}(x) \cdot abla u (x) \cdot \mathbf{n}(x) \, dS \\\ &= \int_V \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t) \bigr)\,dx \end{align} * The time rate of temperature change at x is proportional to the heat flowing into an infinitesimal volume element, where the constant of proportionality is dependent on a constant κ \partial_t u(x,t) = \kappa(x) Q(x,t) Putting these equations together gives the general equation of heat flow: : \partial_t u(x,t) = \kappa(x) \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t)\bigr) Remarks. By Fourier's law for an isotropic medium, the rate of flow of heat energy per unit area through a surface is proportional to the negative temperature gradient across it: :\mathbf{q} = - k \, abla u where k is the thermal conductivity of the material, u=u(\mathbf{x},t) is the temperature, and \mathbf{q} = \mathbf{q}(\mathbf{x},t) is a vector field that represents the magnitude and direction of the heat flow at the point \mathbf{x} of space and time t. The temperature at the solid wall is T_s and gradually changes to T_o as one moves toward the free stream of the fluid.

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If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^2$. Find the velocity when $t=2$.

Since the derivative of the position with respect to time gives the change in position (in metres) divided by the change in time (in seconds), velocity is measured in metres per second (m/s). By combining this equation with the suvat equation , it is possible to relate the displacement and the average velocity by \boldsymbol{x} = \frac{(\boldsymbol{u} + \boldsymbol{v})}{2} t = \boldsymbol{\bar{v}}t. Velocity is the speed and the direction of motion of an object. From there, we can obtain an expression for velocity as the area under an acceleration vs. time graph. Average velocity can be calculated as: \boldsymbol{\bar{v}} = \frac{\Delta\boldsymbol{x}}{\Delta t} . Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. thumb|Forty Foot changing rooms and clubhouse kitchen, 2008 thumb|Sunrise at the Forty Foot, 2018 The Forty Foot () is a promontory on the southern tip of Dublin Bay at Sandycove, County Dublin, Ireland, from which people have been swimming in the Irish Sea all year round for some 250 years.as of 2008 * * ==Name== The name "Forty Foot" is somewhat obscure. The magnitude of the transverse velocity is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. It is also possible to derive an expression for the velocity independent of time, known as the Torricelli equation, as follows: v^{2} = \boldsymbol{v}\cdot\boldsymbol{v} = (\boldsymbol{u}+\boldsymbol{a}t) \cdot (\boldsymbol{u}+\boldsymbol{a}t) = u^{2} + 2t(\boldsymbol{a}\cdot\boldsymbol{u})+a^{2}t^{2} (2\boldsymbol{a})\cdot\boldsymbol{x} = (2\boldsymbol{a})\cdot(\boldsymbol{u}t + \tfrac{1}{2} \boldsymbol{a} t^2) = 2t (\boldsymbol{a} \cdot \boldsymbol{u}) + a^2 t^2 = v^{2} - u^{2} \therefore v^2 = u^2 + 2(\boldsymbol{a}\cdot\boldsymbol{x}) where etc. In calculus terms, the integral of the velocity function is the displacement function . In other words, only relative velocity can be calculated. ===Quantities that are dependent on velocity=== The kinetic energy of a moving object is dependent on its velocity and is given by the equation E_{\text{k}} = \tfrac{1}{2} m v^2 ignoring special relativity, where Ek is the kinetic energy and m is the mass. 20 Hrs. 40 Min.: The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. This makes "escape velocity" somewhat of a misnomer, as the more correct term would be "escape speed": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of the base body as long as it does not intersect with something in its path. ==Relative velocity== Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. In terms of a displacement-time ( vs. ) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line to the curve at any point, and the average velocity as the slope of the secant line between two points with coordinates equal to the boundaries of the time period for the average velocity. Hence, the car is considered to be undergoing an acceleration. ==Difference between speed and velocity== thumb|300px|Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a. The average velocity is always less than or equal to the average speed of an object. Rocket Racing Composite Corp. Acquires Velocity Aircraft, Parabolic Arc, 2008-04-14, accessed 2010-12-05. Velocity Aircraft Receives Purchase Order For 20 Velocity XL-5's, Space Fellowship, 2008-10-08, accessed 2010-12-11. As above, this is done using the concept of the integral: \boldsymbol{v} = \int \boldsymbol{a} \ dt . ====Constant acceleration==== In the special case of constant acceleration, velocity can be studied using the suvat equations. Escape velocity is the minimum speed a ballistic object needs to escape from a massive body such as Earth. Velocity is a physical vector quantity; both magnitude and direction are needed to define it.

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A woman walks due west on the deck of a ship at $3 \mathrm{mi} / \mathrm{h}$. The ship is moving north at a speed of $22 \mathrm{mi} / \mathrm{h}$. Find the speed of the woman relative to the surface of the water.

This is often called the hull speed and is a function of the length of the ship V=k\sqrt{L} where constant (k) should be taken as: 2.43 for velocity (V) in kn and length (L) in metres (m) or, 1.34 for velocity (V) in kn and length (L) in feet (ft). right|thumb|upright=1.3|Load lines, by showing how low a ship is sitting in the water, make it possible to determine displacement. A ship must be designed to move efficiently through the water with a minimum of external force. thumb|A close up quartering view of Grey Lady at dock MV Grey Lady is a high speed catamaran ferry operated by Hy-Line Cruises that travels on a route between Hyannis and Nantucket. Denise "Dee" Caffari MBE (born 23 January 1973) is a British sailor, and in 2006 became the first woman to sail single-handedly and non-stop around the world "the wrong way"; westward against the prevailing winds and currents. The relationship between the velocity of ships and that of the transverse waves can be found by equating the wave celerity and the ship's velocity. ==Propulsion== (Main article: Marine propulsion) Ships can be propelled by numerous sources of power: human, animal, or wind power (sails, kites, rotors and turbines), water currents, chemical or atomic fuels and stored electricity, pressure, heat or solar power supplying engines and motors. ==Gallery== Image:Archimedes principle.svg|A floating ship's displacement Fp and buoyancy Fa must be equal. Seawater (1,025 kg/m3) is more dense than fresh water (1,000 kg/m3);Turpin and McEwen, 1980. so a ship will ride higher in salt water than in fresh. The ship's hydrostatic tables show the corresponding volume displaced.George, 2005. p. 465. For a displacement vessel, that is the usual type of ship, three main types of resistance are considered: that due to wave-making, that due to the pressure of the moving water on the form, often not calculated or measured separately, and that due to friction of moving water on the wetted surface of the hull. As the term indicates, it is measured indirectly, using Archimedes' principle, by first calculating the volume of water displaced by the ship, then converting that value into weight. Ship's boats have always provided transport between the shore and other ships. The displacement or displacement tonnage of a ship is its weight. Grunnslep fra 1932.jpg|Principle of sea surveying with two boats, Norwegian Sea Survey, 1932. A ship's boat is a utility boat carried by a larger vessel. These waves were first studied by William Thomson, 1st Baron Kelvin, who found that regardless of the speed of the ship, they were always contained within the 39° wedge shape (19.5° on each side) following the ship. Froude had observed that when a ship or model was at its so-called Hull speed the wave pattern of the transverse waves (the waves along the hull) have a wavelength equal to the length of the waterline. The total (upward) force due to this buoyancy is equal to the (downward) weight of the displaced water. thumb|A ship for updating nautical charts. thumb|300px|HMS Thetis aground. The divergent waves do not cause much resistance against the ship's forward motion. The boundary layer undergoes shear at different rates extending from the hull surface until it reaches the field flow of the water. ==Wave-making resistance== (Main article: Wave-making resistance) A ship moving over the surface of undisturbed water sets up waves emanating mainly from the bow and stern of the ship.

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A $360-\mathrm{lb}$ gorilla climbs a tree to a height of $20 \mathrm{~ft}$. Find the work done if the gorilla reaches that height in 10 seconds.

If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. Therefore, the distance in feet down a 6% grade to reach the velocity is at least s = \frac{\Delta z}{0.06}= 8.3\frac{V^2}{g},\quad\text{or}\quad s=8.3\frac{88^2}{32.2}\approx 2000\mathrm{ft}. Net work is typically calculated either from instantaneous power (muscle force x muscle velocity) or from the area enclosed by the work loop on a force vs. length plot. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Step 3) Obtain net work (a single number) by numerical integration of muscle power data. Therefore, work need only be computed for the gravitational forces acting on the bodies. The work of the net force is calculated as the product of its magnitude and the particle displacement. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). In physics, work is the energy transferred to or from an object via the application of force along a displacement. If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge. === Work done by a variable force === Calculating the work as "force times straight path segment" would only apply in the most simple of circumstances, as noted above. Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.Resnick, Robert, Halliday, David (1966), Physics, Section 1–3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 Work is the result of a force on a point that follows a curve , with a velocity , at each instant. Notice that this formula uses the fact that the mass of the vehicle is . thumb|Lotus type 119B gravity racer at Lotus 60th celebration thumb|Gravity racing championship in Campos Novos, Santa Catarina, Brazil, 8 September 2010 ===Coasting down an inclined surface (gravity racing)=== Consider the case of a vehicle that starts at rest and coasts down an inclined surface (such as mountain road), the work–energy principle helps compute the minimum distance that the vehicle travels to reach a velocity , of say 60 mph (88 fps).

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A ball is thrown eastward into the air from the origin (in the direction of the positive $x$-axis). The initial velocity is $50 \mathrm{i}+80 \mathrm{k}$, with speed measured in feet per second. The spin of the ball results in a southward acceleration of $4 \mathrm{ft} / \mathrm{s}^2$, so the acceleration vector is $\mathbf{a}=-4 \mathbf{j}-32 \mathbf{k}$. What speed does the ball land?

The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. Ground speed is the horizontal speed of an aircraft relative to the Earth’s surface. The magnitude of the transverse velocity is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. Velocity is the speed and the direction of motion of an object. thumb|150px|A thrust ball bearing A thrust bearing is a particular type of rotary bearing. That is a consequence of the rotational symmetry of the system around the vertical axis. == Trajectory == thumb|250x250px|Trajectory of a spherical pendulum. The material acceleration at P is: \mathbf a_P = \frac{d \mathbf v_P}{dt} = \mathbf a_C + \boldsymbol \alpha \times (\mathbf r_P - \mathbf r_C) + \boldsymbol \omega \times (\mathbf v_P - \mathbf v_C) where \boldsymbol \alpha is the angular acceleration vector. thumb|upright=1.5|Spherical pendulum: angles and velocities. Hence, the car is considered to be undergoing an acceleration. ==Difference between speed and velocity== thumb|300px|Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a. In other words, only relative velocity can be calculated. ===Quantities that are dependent on velocity=== The kinetic energy of a moving object is dependent on its velocity and is given by the equation E_{\text{k}} = \tfrac{1}{2} m v^2 ignoring special relativity, where Ek is the kinetic energy and m is the mass. If an object A is moving with velocity vector v and an object B with velocity vector w, then the velocity of object A relative to object B is defined as the difference of the two velocity vectors: \boldsymbol{v}_{A\text{ relative to }B} = \boldsymbol{v} - \boldsymbol{w} Similarly, the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is: \boldsymbol{v}_{B\text{ relative to }A} = \boldsymbol{w} - \boldsymbol{v} Usually, the inertial frame chosen is that in which the latter of the two mentioned objects is in rest. ===Scalar velocities=== In the one-dimensional case,Basic principle the velocities are scalars and the equation is either: v_\text{rel} = v - (-w), if the two objects are moving in opposite directions, or: v_\text{rel} = v -(+w), if the two objects are moving in the same direction. ==Polar coordinates== In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good), and an angular velocity, which is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system). Velocity is a physical vector quantity; both magnitude and direction are needed to define it. The radial and angular velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. Speed, the scalar magnitude of a velocity vector, denotes only how fast an object is moving. The transverse velocity is the component of velocity along a circle centered at the origin. \boldsymbol{v}=\boldsymbol{v}_T+\boldsymbol{v}_R where *\boldsymbol{v}_T is the transverse velocity *\boldsymbol{v}_R is the radial velocity. This means that the outer race groove exerts less force inward against the ball as the bearing spins. His was the first modern ball-bearing design, with the ball running along a groove in the axle assembly. The magnitude of the radial velocity is the dot product of the velocity vector and the unit vector in the direction of the displacement. v_R = \frac{\boldsymbol{v} \cdot \boldsymbol{r}}{\left|\boldsymbol{r}\right|} where \boldsymbol{r} is displacement. This makes "escape velocity" somewhat of a misnomer, as the more correct term would be "escape speed": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of the base body as long as it does not intersect with something in its path. ==Relative velocity== Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. The linear velocity vector \mathbf v_P at P is expressed in terms of the velocity vector \mathbf v_C at C as: \mathbf v_P = \mathbf v_C + \boldsymbol \omega \times (\mathbf r_P - \mathbf r_C) where \boldsymbol \omega is the angular velocity vector. The purpose of a ball bearing is to reduce rotational friction and support radial and axial loads. In high speed applications, such as turbines, jet engines, and dentistry equipment, the centrifugal forces generated by the balls changes the contact angle at the inner and outer race.

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C

The demand function for a commodity is given by $$ p=2000-0.1 x-0.01 x^2 $$ Find the consumer surplus when the sales level is 100 .

McGraw-Hill 2005 To compute the inverse demand function, simply solve for P from the demand function. Specifying values for the non-price determinants, Prg = 4.00 and Y = 50, results in demand equation Q = 325 - P - 30(4) +1.4(50) or Q = 275 - P. A simple example of a demand equation is Qd = 325 - P - 30Prg \+ 1.4Y. To compute the inverse demand equation, simply solve for P from the demand equation.The form of the inverse linear demand equation is P = a/b - 1/bQ. Pe = 80 is the equilibrium price at which quantity demanded is equal to the quantity supplied. Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) × Q = 120Q - 0.5Q². Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) × Q = 120Q - 0.5Q². For example, if the demand equation is Q = 240 - 2P then the inverse demand equation would be P = 120 - .5Q, the right side of which is the inverse demand function.Samuelson, W & Marks, S. Managerial Economics 4th ed. p. 37. So 20 is the profit- maximizing quantity: to find the profit-maximizing price simply plug the value of Q into the inverse demand equation and solve for P. ==See also== *Supply and demand *Demand *Law of demand *Profit (economics) ==References== Category:Mathematical finance Category:Demand For example, if the demand function has the form Q = 240 - 2P then the inverse demand function would be P = 120 - .5Q.Samuelson & Marks, Managerial Economics 4th ed. (Wiley 2003) Note that although price is the dependent variable in the inverse demand function, it is still the case that the equation represents how the price determines the quantity demanded, not the reverse. ==Relation to marginal revenue== There is a close relationship between any inverse demand function for a linear demand equation and the marginal revenue function. *The x intercept of the marginal revenue function is one- half the x intercept of the inverse demand function. The value P in the inverse demand function is the highest price that could be charged and still generate the quantity demanded Q.Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London This is useful because economists typically place price (P) on the vertical axis and quantity (Q) on the horizontal axis in supply-and-demand diagrams, so it is the inverse demand function that depicts the graphed demand curve in the way the reader expects to see. The supply curve, shown in orange, intersects with the demand curve at price (Pe) = 80 and quantity (Qe)= 120. If good i is a Giffen good whose price increases while other goods' prices are held fixed (so that p_j'-p_j=0 \; \forall j eq i), the law of demand is clearly violated, as we have both p_i'-p_i>0 (as price increased) and q_i'-q_i>0 (as we consider a Giffen good), so that (p'-p)(x'-x)=(p_i'-p_i)(x_i'-x_i)>0. == Demand versus quantity demanded == It is very important to apprehend the difference between demand and quantity demanded as they are used to mean different things in the economic jargon. If income were to increase to 55, the new demand equation would be Q = 282 - P. Graphically, this change in a non-price determinant of demand would be reflected in an outward shift of the demand function caused by a change in the x-intercept. ==Demand curve== In economics the demand curve is the graphical representation of the relationship between the price and the quantity that consumers are willing to purchase. The mathematical relationship between the price of the substitute and the demand for the good in question is positive. The above equation, when plotted with quantity demanded (Q_x) on the x-axis and price (P_x) on the y-axis, gives the demand curve, which is also known as the demand schedule. The inverse demand equation, or price equation, treats price as a function f of quantity demanded: P = f(Q). The law of demand states that \frac{\partial f}{\partial P_x} < 0. The number of consumers in a market: The market demand for a good is obtained by adding individual demands of the present, as well as prospective consumers of a good at various possible prices. Demand vacuum in economics and marketing is the effect created by consumer demand on the supply chain. The formula to solve for the coefficient of price elasticity of demand is the percentage change in quantity demanded divided by the percentage change in Price.

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E

The linear density in a rod $8 \mathrm{~m}$ long is $12 / \sqrt{x+1} \mathrm{~kg} / \mathrm{m}$, where $x$ is measured in meters from one end of the rod. Find the average density of the rod.

upright=1.4|thumb|The deformation of a thin straight rod into a closed loop. Knowing the volume of the unit cell of a crystalline material and its formula weight (in daltons), the density can be calculated. In the United States until 1 January 2023, the rod was often defined as 16.5 US survey feet, or approximately 5.029 210 058 m. ==History== In England, the perch was officially discouraged in favour of the rod as early as the 15th century;Encyclopædia Britannica, English measure however, local customs maintained its use. The rod is useful as a unit of length because integer multiples of it can form one acre of square measure (area). * Determination of Density of Solid, instructions for performing classroom experiment. Standing at arm's length from the tree, estimate its average diameter by taking a note on the rod's markings. A cruising rod is a simple device used to quickly estimate the number of pieces of lumber yielded by a given piece of timber. In the US, the rod, along with the chain, furlong, and statute mile (as well as the survey inch and survey foot) were based on the pre-1959 values for United States customary units of linear measurement until 1 January 2023. Counting rods () are small bars, typically 3–14 cm (1" to 6") long, that were used by mathematicians for calculation in ancient East Asia. Mathematically, density is defined as mass divided by volume: \rho = \frac{m}{V} where ρ is the density, m is the mass, and V is the volume. In that case the density around any given location is determined by calculating the density of a small volume around that location. The rod as a survey measure was standardized by Edmund Gunter in England in 1607 as a quarter of a chain (of ), or long. ===In ancient cultures=== The perch as a lineal measure in Rome (also decempeda) was 10 Roman feet (2.96 metres), and in France varied from 10 feet (perche romanie) to 22 feet (perche d'arpent—apparently of "the range of an arrow"—about 220 feet). Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. Rod was also sometimes used as a unit of area to refer to a rood. * Gas density calculator Calculate density of a gas for as a function of temperature and pressure. See below for a list of some of the most common units of density. ===Homogeneous materials=== The density at all points of a homogeneous object equals its total mass divided by its total volume. The Railway Operating Division (ROD) ROD 2-8-0 is a type of 2-8-0 steam locomotive which was the standard heavy freight locomotive operated in Europe by the ROD during the First World War. ==ROD need for a standard locomotive== During the First World War the Railway Operating Division of the Royal Engineers requisitioned about 600 locomotives of various types from thirteen United Kingdom railway companies; the first arrived in France in late 1916. Rods can also be found on the older legal descriptions of tracts of land in the United States, following the "metes and bounds" method of land survey; as shown in this actual legal description of rural real estate: ==Area and volume== The terms pole, perch, rod and rood have been used as units of area, and perch is also used as a unit of volume. Bars of metal one rod long were used as standards of length when surveying land. The rod, perch, or pole (sometimes also lug) is a surveyor's tool and unit of length of various historical definitions. In some cases (for instance, in the United States oil and gas industry), density is loosely defined as its weight per unit volume, although this is scientifically inaccurate – this quantity is more specifically called specific weight. The length of the rod remains almost unchanged during the deformation, which indicates that the strain is small.

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A variable force of $5 x^{-2}$ pounds moves an object along a straight line when it is $x$ feet from the origin. Calculate the work done in moving the object from $x=1 \mathrm{~ft}$ to $x=10 \mathrm{~ft}$.

If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. This is illustrated in the figure to the right: The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative. In physics, work is the energy transferred to or from an object via the application of force along a displacement. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The work of the net force is calculated as the product of its magnitude and the particle displacement. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. Therefore, work need only be computed for the gravitational forces acting on the bodies. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. If force is changing, or if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point velocity is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge. === Work done by a variable force === Calculating the work as "force times straight path segment" would only apply in the most simple of circumstances, as noted above. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. Net work is typically calculated either from instantaneous power (muscle force x muscle velocity) or from the area enclosed by the work loop on a force vs. length plot. Work transfers energy from one place to another or one form to another. ===Derivation for a particle moving along a straight line=== In the case the resultant force is constant in both magnitude and direction, and parallel to the velocity of the particle, the particle is moving with constant acceleration a along a straight line. The work done is given by the dot product of the two vectors. If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction.

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C

One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it take for $80 \%$ of the population to become infected?

Now, the epidemic model is : \frac{\mathrm{d} x_i}{\mathrm{d}t}= F_i (x)-V_i(x), where V_i(x)= [V^-_i(x)-V^+_i(x)] In the above equations, F_i(x) represents the rate of appearance of new infections in compartment i . An epidemic (from Greek ἐπί epi "upon or above" and δῆμος demos "people") is the rapid spread of disease to a large number of hosts in a given population within a short period of time. * Epidemic – when this disease is found to infect a significantly larger number of people at the same time than is common at that time, and among that population, and may spread through one or several communities. In other words, the changes in population of a compartment can be calculated using only the history that was used to develop the model. ==Sub-exponential growth== A common explanation for the growth of epidemics holds that 1 person infects 2, those 2 infect 4 and so on and so on with the number of infected doubling every generation. This means that, on average, each infected person is infecting exactly one other person (any more and the number of people infected will grow sub-exponentially and there will be an epidemic, any less and the disease will die out). Classic epidemic models of disease transmission are described in Compartmental models in epidemiology. Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. There is another variation, both as regards the number of people affected and the number who die in successive epidemics: the severity of successive epidemics rises and falls over periods of five or ten years. ==Types== ===Common source outbreak=== In a common source outbreak epidemic, the affected individuals had an exposure to a common agent. This quantity determines whether the infection will increase sub- exponentially, die out, or remain constant: if R0 > 1, then each person on average infects more than one other person so the disease will spread; if R0 < 1, then each person infects fewer than one person on average so the disease will die out; and if R0 = 1, then each person will infect on average exactly one other person, so the disease will become endemic: it will move throughout the population but not increase or decrease. ==Endemic steady state== An infectious disease is said to be endemic when it can be sustained in a population without the need for external inputs. In epidemiology, the next-generation matrix is used to derive the basic reproduction number, for a compartmental model of the spread of infectious diseases. Thus this model of an epidemic leads to a curve that grows exponentially until it crashes to zero as all the population have been infected. i.e. no herd immunity and no peak and gradual decline as seen in reality. == Reproduction number == The basic reproduction number (denoted by R0) is a measure of how transferable a disease is. For example, in meningococcal infections, an attack rate in excess of 15 cases per 100,000 people for two consecutive weeks is considered an epidemic. Research topics include: * antigenic shift * epidemiological networks * evolution and spread of resistance * immuno- epidemiology * intra-host dynamics * Pandemic * pathogen population genetics * persistence of pathogens within hosts * phylodynamics * role and identification of infection reservoirs * role of host genetic factors * spatial epidemiology * statistical and mathematical tools and innovations * Strain (biology) structure and interactions * transmission, spread and control of infection * virulence ==Mathematics of mass vaccination== If the proportion of the population that is immune exceeds the herd immunity level for the disease, then the disease can no longer persist in the population and its transmission dies out. The earliest account of mathematical modelling of spread of disease was carried out in 1760 by Daniel Bernoulli. In epidemiology, particularly in the discussion of infectious disease dynamics (mathematical modeling of disease spread), the infectious period is the time interval during which a host (individual or patient) is infectious, i.e. capable of directly or indirectly transmitting pathogenic infectious agents or pathogens to another susceptible host. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic. Epidemics of infectious disease are generally caused by several factors including a change in the ecology of the host population (e.g., increased stress or increase in the density of a vector species), a genetic change in the pathogen reservoir or the introduction of an emerging pathogen to a host population (by movement of pathogen or host). It is the average number of people that a single infectious person will infect over the course of their infection. Statistical agent-level disease dissemination in small or large populations can be determined by stochastic methods. ===Deterministic=== When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are often used. thumb|upright=1.5|Example of an epidemic showing the number of new infections over time. A classic compartmental model in epidemiology is the SIR model, which may be used as a simple model for modelling epidemics. The Centers for Disease Control and Prevention defines epidemic broadly: "the occurrence of more cases of disease, injury, or other health condition than expected in a given area or among a specific group of persons during a particular period.

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Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths $3 \mathrm{~cm}$, $4 \mathrm{~cm}$, and $5 \mathrm{~cm}$

The volume of this tetrahedron is one-third the volume of the cube. Let be the volume of the tetrahedron; then :V=\frac{\sqrt{4 a^2 b^2 c^2-a^2 X^2-b^2 Y^2-c^2 Z^2+X Y Z}}{12} where :\begin{align}X&=b^2+c^2-x^2, \\\ Y&=a^2+c^2-y^2, \\\ Z&=a^2+b^2-z^2. \end{align} The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles. Let V be the volume of the tetrahedron. Since the four subtetrahedra fill the volume, we have V = \frac13A_1r+\frac13A_2r+\frac13A_3r+\frac13A_4r. ===Circumradius=== Denote the circumradius of a tetrahedron as R. Comparing this formula with that used to compute the volume of a parallelepiped, we conclude that the volume of a tetrahedron is equal to of the volume of any parallelepiped that shares three converging edges with it. thumb|3D model of regular tetrahedron. Given the distances between the vertices of a tetrahedron the volume can be computed using the Cayley–Menger determinant: :288 \cdot V^2 = \begin{vmatrix} 0 & 1 & 1 & 1 & 1 \\\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\\ 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\\ 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\\ 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0 \end{vmatrix} where the subscripts represent the vertices and d is the pairwise distance between them – i.e., the length of the edge connecting the two vertices. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. ===Coordinates for a regular tetrahedron=== The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, and two level edges: :\left(\pm 1, 0, -\frac{1}{\sqrt{2}}\right) \quad \mbox{and} \quad \left(0, \pm 1, \frac{1}{\sqrt{2}}\right) Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face parallel to the xy plane, the vertices are: v_1 = \left(\sqrt{\frac{8}{9}},0,-\frac{1}{3}\right) v_2 = \left(-\sqrt{\frac{2}{9}},\sqrt{\frac{2}{3}},-\frac{1}{3}\right) v_3 = \left(-\sqrt{\frac{2}{9}},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right) v_4 = (0,0,1) with the edge length of \sqrt{\frac{8}{3}}. Then another volume formula is given by :V = \frac {d |(\mathbf{a} \times \mathbf{(b-c)})| } {6}. ===Properties analogous to those of a triangle=== The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres. :V = \frac {abc} {6} \sqrt{1 + 2\cos{\alpha}\cos{\beta}\cos{\gamma}-\cos^2{\alpha}-\cos^2{\beta}-\cos^2{\gamma}} ==== Heron-type formula for the volume of a tetrahedron ==== right|thumb|240px|Six edge-lengths of Tetrahedron If , , , , , are lengths of edges of the tetrahedron (first three form a triangle; with opposite , opposite , opposite ), then : V = \frac{\sqrt {\,( - p + q + r + s)\,(p - q + r + s)\,(p + q - r + s)\,(p + q + r - s)}}{192\,u\,v\,w} where : \begin{align} p & = \sqrt {xYZ}, & q & = \sqrt {yZX}, & r & = \sqrt {zXY}, & s & = \sqrt {xyz}, \end{align} : \begin{align} X & = (w - U + v)\,(U + v + w), & x & = (U - v + w)\,(v - w + U), \\\ Y & = (u - V + w)\,(V + w + u), & y & = (V - w + u)\,(w - u + V), \\\ Z & = (v - W + u)\,(W + u + v), & z & = (W - u + v)\,(u - v + W). \end{align} ====Volume divider==== Any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron.Bottema, O. The three faces interior to the tetrahedron are: a right triangle with edges 1, \sqrt{\tfrac{3}{2}}, \sqrt{\tfrac{1}{2}}, a right triangle with edges \sqrt{\tfrac{1}{3}}, \sqrt{\tfrac{1}{2}}, \sqrt{\tfrac{1}{6}}, and a right triangle with edges \sqrt{\tfrac{4}{3}}, \sqrt{\tfrac{3}{2}}, \sqrt{\tfrac{1}{6}}. ===Space-filling tetrahedra=== A space-filling tetrahedron packs with directly congruent or enantiomorphous (mirror image) copies of itself to tile space. Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron edge arc arc dihedral dihedral 𝒍 2 109°28′16″ \pi - 2\text{𝜿} 70°31′44″ \pi - 2\text{𝟁} 𝟀 \sqrt{\tfrac{4}{3}} \approx 1.155 70°31′44″ 2\text{𝜿} 60° \tfrac{\pi}{3} 𝝓 1 54°44′8″ \tfrac{\pi}{2} - \text{𝜿} 60° \tfrac{\pi}{3} 𝟁 \sqrt{\tfrac{1}{3}} \approx 0.577 54°44′8″ \tfrac{\pi}{2} - \text{𝜿} 60° \tfrac{\pi}{3} _0R/l \sqrt{\tfrac{3}{2}} \approx 1.225 _1R/l \sqrt{\tfrac{1}{2}} \approx 0.707 _2R/l \sqrt{\tfrac{1}{6}} \approx 0.408 \text{𝜿} 35°15′52″ \tfrac{\text{arc sec }3}{2} If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths \sqrt{\tfrac{4}{3}}, 1, \sqrt{\tfrac{1}{3}} (the exterior right triangle face, the characteristic triangle 𝟀, 𝝓, 𝟁), plus \sqrt{\tfrac{3}{2}}, \sqrt{\tfrac{1}{2}}, \sqrt{\tfrac{1}{6}} (edges that are the characteristic radii of the regular tetrahedron). The law of cosines for this tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation: : \Delta_i^2 = \Delta_j^2 + \Delta_k^2 + \Delta_l^2 - 2(\Delta_j\Delta_k\cos\theta_{il} + \Delta_j\Delta_l \cos\theta_{ik} + \Delta_k\Delta_l \cos\theta_{ij}) === Interior point === Let P be any interior point of a tetrahedron of volume V for which the vertices are A, B, C, and D, and for which the areas of the opposite faces are Fa, Fb, Fc, and Fd. The exterior face is a 60-90-30 triangle which is one-sixth of a tetrahedron face. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube. (As a side-note: these two kinds of tetrahedron have the same volume.) This formula is obtained from dividing the tetrahedron into four tetrahedra whose points are the three points of one of the original faces and the incenter. The only two isometries are 1 and the rotation (12)(34), giving the group C2 isomorphic to the cyclic group, Z2. ==General properties== ===Volume=== The volume of a tetrahedron is given by the pyramid volume formula: :V = \frac13 A_0\,h \, where A0 is the area of the base and h is the height from the base to the apex. :Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1) :Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1) thumb|right|300px|Regular tetrahedron ABCD and its circumscribed sphere ===Angles and distances=== For a regular tetrahedron of edge length a: Face area A_0=\frac{\sqrt{3}}{4}a^2\, Surface areaCoxeter, Harold Scott MacDonald; Regular Polytopes, Methuen and Co., 1948, Table I(i) A=4\,A_0={\sqrt{3}}a^2\, Height of pyramidKöller, Jürgen, "Tetrahedron", Mathematische Basteleien, 2001 h=\frac{\sqrt{6}}{3}a=\sqrt{\frac23}\,a\, Centroid to vertex distance \frac34\,h = \frac{\sqrt{6}}{4}\,a = \sqrt{\frac{3}{8}}\,a\, Edge to opposite edge distance l=\frac{1}{\sqrt{2}}\,a\, Volume V=\frac13 A_0h =\frac{\sqrt{2}}{12}a^3=\frac{a^3}{6\sqrt{2}}\, Face-vertex-edge angle \arccos\left(\frac{1}{\sqrt{3}}\right) = \arctan\left(\sqrt{2}\right)\, (approx. 54.7356°) Face-edge-face angle, i.e., "dihedral angle" \arccos\left(\frac13\right) = \arctan\left(2\sqrt{2}\right)\, (approx. 70.5288°) Vertex-Center-Vertex angle, the angle between lines from the tetrahedron center to any two vertices. thumb|3D model of a truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. If all three pairs of opposite edges of a tetrahedron are perpendicular, then it is called an orthocentric tetrahedron. The three edges that meet at the right angle are called the legs and the perpendicular from the right angle to the base is called the altitude of the tetrahedron.

0.011

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The base of a solid is a circular disk with radius 3 . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.

The volume of a tricylinder is :V=8(2 - \sqrt{2}) r^3 and the surface area is :A=24(2 - \sqrt{2}) r^2. == More cylinders == With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is :V_4=12 \left( 2\sqrt{2} - \sqrt{6} \right) r^3 \, With six cylinders, with axes parallel to the diagonals of the faces of a cube, the volume is: :V_6=\frac{16}{3} \left( 3 + 2\sqrt{3} - 4\sqrt{2} \right) r^3 \, == See also == * Ungula == References == == External links == *A 3D model of Steinmetz solid in Google 3D Warehouse Category:Euclidean solid geometry It is well known that the relations of the volumes of a right circular cone, one half of a sphere and a right circular cylinder with same radii and heights are 1 : 2 : 3. The intersection of three cylinders with perpendicularly intersecting axes generates a surface of a solid with vertices where 3 edges meet and vertices where 4 edges meet. thumb|right|170px Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. The volume and the surface area of the curved triangles can be determined by similar considerations as it is done for the bicylinder above. right|thumb|236px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. The volume of the 8 pyramids is: \textstyle 8 \times \frac{1}{3} r^2 \times r = \frac{8}{3} r^3 , and then we can calculate that the bicylinder volume is \textstyle (2 r)^3 - \frac{8}{3} r^3 = \frac{16}{3} r^3 == Tricylinder == 450px|thumb|Generating the surface of a tricylinder: At first two cylinders (red, blue) are cut. The key for the determination of volume and surface area is the observation that the tricylinder can be resampled by the cube with the vertices where 3 edges meet (s. diagram) and 6 curved pyramids (the triangles are parts of cylinder surfaces). One simply must solve each equation for before one inserts them into the integration formula. ==See also== *Solid of revolution *Shell integration ==References== * * *Frank Ayres, Elliott Mendelson. Hence the area of this development is thumb|cloister vault :B = \int_{0}^{\pi r} r\sin\left(\frac{\xi}{r}\right) \mathrm{d}\xi = 2r^2 and the total surface area is: :A=8\cdot B=16r^2. === Alternate proof of the volume formula === Deriving the volume of a bicylinder (white) can be done by packing it in a cube (red). The concept of included angle is discussed at: * Congruence of triangles * Solution of triangles The Euler line of an isosceles triangle is perpendicular to the triangle's base. The altitudes of a triangle are perpendicular to their respective bases. thumb|Steinmetz solid (intersection of two cylinders) In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. File:Sphere volume derivation using bicylinder.jpg|Zu Chongzhi's method (similar to Cavalieri's principle) for calculating a sphere's volume includes calculating the volume of a bicylinder. For one half of a bicylinder a similar statement is true: * The relations of the volumes of the inscribed square pyramid (a=2r, h=r, V=\frac{4}{3}r^3), the half bicylinder (V=\frac{8}{3} r^3) and the surrounding squared cuboid ( a= 2r, h=r, V=4r^3) are 1 : 2 : 3. ==== Using Multivariable Calculus ==== Consider the equations of the cylinders: x^2+z^2=r^2 x^2+y^2=r^2 The volume will be given by: V = \iiint_V \mathrm{d}z\mathrm{d}y\mathrm{d}x With the limits of integration: -\sqrt{r^2-x^2} \leqslant z \leqslant \sqrt{r^2-x^2} -\sqrt{r^2-x^2} \leqslant y \leqslant \sqrt{r^2-x^2} -r \leqslant x \leqslant r Substituting, we have: V = \int_{-r}^{r}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \mathrm{d}z\mathrm{d}y\mathrm{d}x = 8r^3-\frac{8r^3}{3} = \frac{16r^3}{3} === Proof of the area formula === The surface area consists of two red and two blue cylindrical biangles. For example, the next figure shows the rotation along the -axis of the red "leaf" enclosed between the square- root and quadratic curves: thumb|Rotation about x-axis The volume of this solid is: :\pi\int_0^1\left(\left(\sqrt{x}\right)^2 - \left(x^2\right)^2 \right)\,dx\,. If is the value of a horizontal axis, then the volume equals :\pi\int_a^b\left(\left(h-R_\mathrm{O}(x)\right)^2 - \left(h-R_\mathrm{I}(x)\right)^2\right)\,dx\,. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. thumb|right|150px|Animated depiction of a bicylinder == Bicylinder == 300px|thumb|The generation of a bicylinder 180px|thumb|Calculating the volume of a bicylinder A bicylinder generated by two cylinders with radius r has the ;volume :V=\frac{16}{3} r^3 and the ;surface area :A=16 r^2. Wallis's Conical Edge with right|thumb|600px| Figure 2. File:Bicylinder and cube sections related by pyramids.png|Relationship of the area of a bicylinder section with a cube section The volume of the cube (red) minus the volume of the eight pyramids (blue) is the volume of the bicylinder (white). Wallis's Conical Edge with , In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt{a^2-b^2\cos^2u} where , and are constants.

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A projectile is fired with an initial speed of $200 \mathrm{~m} / \mathrm{s}$ and angle of elevation $60^{\circ}$. Find the speed at impact.

Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. Let tg be any time when the height of the projectile is equal to its initial value. : 0 = v t \sin \theta - \frac{1} {2} g t^2 By factoring: : t = 0 or : t = \frac{2 v \sin \theta} {g} but t = T = time of flight : T = \frac{2 v \sin \theta} {g} The first solution corresponds to when the projectile is first launched. Composites Science and Technology 64:35-54. . is as follows: V_b=\frac{\pi\,\Gamma\,\sqrt{\rho_t\,\sigma_e}\,D^2\,T}{4\,m} \left [1+\sqrt{1+\frac{8\,m}{\pi\,\Gamma^2\,\rho_t\,D^2\,T}}\, \right ] where *V_b\, is the ballistic limit *\Gamma\, is a projectile constant determined experimentally *\rho_t\, is the density of the laminate *\sigma_e\, is the static linear elastic compression limit *D\, is the diameter of the projectile *T\, is the thickness of the laminate *m\, is the mass of the projectile Additionally, the ballistic limit for small-caliber into homogeneous armor by TM5-855-1 is: V_1= 19.72 \left [ \frac{7800 d^3 \left [ \left ( \frac{e_h}{d} \right) \sec \theta \right ]^{1.6}}{W_T} \right ]^{0.5} where *V_1 is the ballistic limit velocity in fps *d is the caliber of the projectile, in inches *e_h is the thickness of the homogeneous armor (valid from BHN 360 - 440) in inches *\theta is the angle of obliquity *W_T is the weight of the projectile, in lbs == References == Category:Ballistics thumb|The aftermath of a hypervelocity impact, with a projectile the same size as the one that impacted for scale Hypervelocity is very high velocity, approximately over 3,000 meters per second (6,700 mph, 11,000 km/h, 10,000 ft/s, or Mach 8.8). The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. right|288px|thumb|Impact parameter and scattering angle In physics, the impact parameter is defined as the perpendicular distance between the path of a projectile and the center of a potential field created by an object that the projectile is approaching (see diagram). After several steps of algebraic manipulation : t = \frac {v \sin \theta} {g} \pm \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} The square root must be a positive number, and since the velocity and the sine of the launch angle can also be assumed to be positive, the solution with the greater time will occur when the positive of the plus or minus sign is used. Ballistic impact is a high velocity impact by a small mass object, analogous to runway debris or small arms fire. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. Thus, the solution is : t = \frac {v \sin \theta} {g} + \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} Solving for the range once again : d = \frac {v \cos \theta} {g} \left ( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2 g y_0} \right) To maximize the range at any height : \theta = \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} Checking the limit as y_0 approaches 0 : \lim_{y_0 \to 0} \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} = \frac {\pi} {4} ====Angle of impact==== The angle ψ at which the projectile lands is given by: : \tan \psi = \frac {-v_y(t_d)} {v_x(t_d)} = \frac {\sqrt { v^2 \sin^2 \theta + 2 g y_0 }} { v \cos \theta} For maximum range, this results in the following equation: : \tan^2 \psi = \frac { 2 g y_0 + v^2 } { v^2 } = C+1 Rewriting the original solution for θ, we get: : \tan^2 \theta = \frac { 1 - \cos^2 \theta } { \cos^2 \theta } = \frac { v^2 } { 2 g y_0 + v^2 } = \frac { 1 } { C + 1 } Multiplying with the equation for (tan ψ)^2 gives: : \tan^2 \psi \, \tan^2 \theta = \frac { 2 g y_0 + v^2 } { v^2 } \frac { v^2 } { 2 g y_0 + v^2 } = 1 Because of the trigonometric identity : \tan (\theta + \psi) = \frac { \tan \theta + \tan \psi } { 1 - \tan \theta \tan \psi } , this means that θ + ψ must be 90 degrees. == Actual projectile motion == In addition to air resistance, which slows a projectile and reduces its range, many other factors also have to be accounted for when actual projectile motion is considered. === Projectile characteristics === Generally speaking, a projectile with greater volume faces greater air resistance, reducing the range of the projectile. Here, the object that the projectile is approaching is a hard sphere with radius R. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. Mathematical equations of motion are used to analyze projectile trajectory. The ballistic limit or limit velocity is the velocity required for a particular projectile to reliably (at least 50% of the time) penetrate a particular piece of material. (And see Trajectory of a projectile.) (And see Trajectory of a projectile.) The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance. The impact parameter is related to the scattering angle byLandau L. D. and Lifshitz E. M. (1976) Mechanics, 3rd. ed., Pergamon Press. (hardcover) and (softcover). : \theta = \pi - 2b\int_{r_\text{min}}^\infty \frac{dr}{r^2\sqrt{1 - (b/r)^2 - 2U/(mv_\infty^2)}}, where is the velocity of the projectile when it is far from the center, and is its closest distance from the center. ==Scattering from a hard sphere== The simplest example illustrating the use of the impact parameter is in the case of scattering from a sphere. When b > R , the projectile misses the hard sphere.

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A force of $30 \mathrm{~N}$ is required to maintain a spring stretched from its natural length of $12 \mathrm{~cm}$ to a length of $15 \mathrm{~cm}$. How much work is done in stretching the spring from $12 \mathrm{~cm}$ to $20 \mathrm{~cm}$ ?

Let be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. The force an ideal spring would exert is exactly proportional to its extension or compression. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. If a spring has a rate of 100 then the spring would compress 1 inch (2.54 cm) with of load. == Types== Types of coil spring are: * Tension/extension coil springs, designed to resist stretching. The force is applied through the ends of the spring. When a conventional spring, without stiffness variability features, is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). The change in length may be expressed as \Delta L = \varepsilon L = \frac{F L}{A E}\,. === Spring energy === The potential energy stored in a spring is given by U_\mathrm{el}(x) = \tfrac 1 2 kx^2 which comes from adding up the energy it takes to incrementally compress the spring. The manufacture normally specifies the spring rate. Explaining the Power of Springing Bodies, London, 1678. as: ("as the extension, so the force" or "the extension is proportional to the force"). The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. : F is the resulting force vector – the magnitude and direction of the restoring force the spring exerts : k is the rate, spring constant or force constant of the spring, a constant that depends on the spring's material and construction. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils forces the spring to collapse or extend all the coils at the same rate when deformed. ===Simple harmonic motion=== Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like: :F = m a \quad \Rightarrow \quad -k x = m a. thumb|Hooke's law: the force is proportional to the extension In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its stiffness), and is small compared to the total possible deformation of the spring. Force of fully compressed spring : F_{max} = \frac{E d^4 (L-n d)}{16 (1+ u) (D-d)^3 n} \ where : E – Young's modulus : d – spring wire diameter : L – free length of spring : n – number of active windings : u – Poisson ratio : D – spring outer diameter ==Zero-length springs== thumb|left|120px|Simplified LaCoste suspension using a zero-length spring thumb|upright|Spring length L vs force F graph of ordinary (+), zero-length (0) and negative-length (−) springs with the same minimum length L0 and spring constant "Zero-length spring" is a term for a specially designed coil spring that would exert zero force if it had zero length; if there were no constraint due to the finite wire diameter of such a helical spring, it would have zero length in the unstretched condition. \, The mass of the spring is small in comparison to the mass of the attached mass and is ignored. The work can be split into two terms \delta W = \delta W_\mathrm{s} + \delta W_\mathrm{b} where is the work done by surface forces while is the work done by body forces. Coil springs can be made from various materials, including steel, brass, and bronze. == Spring rate == Spring rate is the measurement of how much a coil spring can hold until it compresses . To meet the demands of today's consumers, spring manufacturers must be able to produce springs in a wide range of sizes and shapes. A torsion spring's rate is in units of torque divided by angle, such as N·m/rad or ft·lbf/degree. A spring (made by winding a wire around a cylinder) is of two types: * Tension or extension springs are designed to become longer under load.

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Use Poiseuille's Law to calculate the rate of flow in a small human artery where we can take $\eta=0.027, R=0.008 \mathrm{~cm}$, $I=2 \mathrm{~cm}$, and $P=4000$ dynes $/ \mathrm{cm}^2$.

As , Poiseuille solution is recovered. ==Poiseuille flow in an annular section== thumb|Poiseuille flow in annular section If is the inner cylinder radii and is the outer cylinder radii, with constant applied pressure gradient between the two ends , the velocity distribution and the volume flux through the annular pipe are : \begin{align} u(r) &= \frac{G}{4\mu}\left(R_1^2-r^2\right) + \frac{G}{4\mu}\left(R_2^2-R_1^2\right) \frac{\ln r/R_1}{\ln R_2/R_1},\\\\[6pt] Q &= \frac{G \pi}{8\mu}\left[R_2^4-R_1^4- \frac{\left(R_2^2-R_1^2\right)^2}{\ln R_2/R_1}\right] . \end{align} When , , the original problem is recovered. ==Poiseuille flow in a pipe with an oscillating pressure gradient== Flow through pipes with an oscillating pressure gradient finds applications in blood flow through large arteries. Both Ohm's law and Poiseuille's law illustrate transport phenomena. == Medical applications – intravenous access and fluid delivery == The Hagen–Poiseuille equation is useful in determining the vascular resistance and hence flow rate of intravenous (IV) fluids that may be achieved using various sizes of peripheral and central cannulas. Hence the volumetric flow rate at the pipe outlet is given by : Q_2 =\frac{\pi R^4}{16 \mu L} \left( \frac{ p_1^2-p_2^2}{p_2}\right) = \frac{\pi R^4 \left( p_1-p_2\right)}{8 \mu L} \frac{\left( p_1+p_2\right)}{2 p_2}. However, the viscosity of blood will cause additional pressure drop along the direction of flow, which is proportional to length traveled (as per Poiseuille's law). The volumetric flow rate is usually expressed at the outlet pressure. It is also useful to understand that viscous fluids will flow slower (e.g. in blood transfusion). ==See also== * Couette flow * Darcy's law * Pulse * Wave * Hydraulic circuit ==Cited references== ==References== *. *. *. == External links == *Poiseuille's law for power-law non-Newtonian fluid *Poiseuille's law in a slightly tapered tube *Hagen–Poiseuille equation calculator Category:Equations of fluid dynamics Category:Mathematics in medicine Mass flow rate can be calculated by multiplying the volume flow rate by the mass density of the fluid, ρ. Finally, put this expression in the form of a differential equation, dropping the term quadratic in . : \frac{1}{\mu} \frac{\Delta p}{\Delta x} = \frac{\mathrm{d}^2 v}{\mathrm{d}r^2} + \frac{1}{r} \frac{\mathrm{d}v}{\mathrm{d}r} The above equation is the same as the one obtained from the Navier–Stokes equations and the derivation from here on follows as before. ===Startup of Poiseuille flow in a pipe=== When a constant pressure gradient is applied between two ends of a long pipe, the flow will not immediately obtain Poiseuille profile, rather it develops through time and reaches the Poiseuille profile at steady state. The ratio of length to radius of a pipe should be greater than one forty-eighth of the Reynolds number for the Hagen–Poiseuille law to be valid. Over a short section of the pipe, the gas flowing through the pipe can be assumed to be incompressible so that Poiseuille law can be used locally, :-\frac{\mathrm{d}p}{\mathrm{d}x} = \frac{8\mu Q}{\pi R^4} = \frac{8\mu Q_2p_2}{\pi p R^4} \quad \Rightarrow \quad -p\frac{\mathrm{d}p}{\mathrm{d}x} = \frac{8\mu Q_2p_2}{\pi R^4}. An artery (plural arteries) ()ἀρτηρία, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus is a blood vessel in humans and most animals that takes blood away from the heart to one or more parts of the body (tissues, lungs, brain etc.). The Navier–Stokes equations reduce to :\frac{\mathrm{d}^2 u}{\mathrm{d}y^2} = - \frac{G}{\mu} with no-slip condition on both walls :u(0)=0, \quad u(h)=0 Therefore, the velocity distribution and the volume flow rate per unit length are :u(y) = \frac{G}{2\mu} y(h-y), \quad Q = \frac{Gh^3}{12\mu}. ==Poiseuille flow through some non-circular cross-sections== Joseph Boussinesq derived the velocity profile and volume flow rate in 1868 for rectangular channel and tubes of equilateral triangular cross-section and for elliptical cross-section. Poiseuille's law was later in 1891 extended to turbulent flow by L. R. Wilberforce, based on Hagenbach's work. == Derivation == The Hagen–Poiseuille equation can be derived from the Navier–Stokes equations. In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. The volume flow rate is calculated by multiplying the flow velocity of the mass elements, v, by the cross-sectional vector area, A. Mass flow rate can also be calculated by: :\dot m = \rho \cdot \dot V = \rho \cdot \mathbf{v} \cdot \mathbf{A} = \mathbf{j}_{\rm m} \cdot \mathbf{A} where: *\dot V or Q = Volume flow rate, *ρ = mass density of the fluid, *v = Flow velocity of the mass elements, *A = cross-sectional vector area/surface, * jm = mass flux. Considering flow through porous media, a special quantity, superficial mass flow rate, can be introduced. The reason why Poiseuille's law leads to a different formula for the resistance is the difference between the fluid flow and the electric current. Mass flow rate can be used to calculate the energy flow rate of a fluid: :\dot{E}=\dot{m}e where: * e = unit mass energy of a system Energy flow rate has SI units of kilojoule per second or kilowatt. == See also == * Continuity equation * Fluid dynamics * Mass flow controller * Mass flow meter * Mass flux * Orifice plate * Standard cubic centimetres per minute * Thermal mass flow meter * Volumetric flow rate ==References== Category:Fluid dynamics Category:Temporal rates Category:Mass If we introduce a new dependent variable as :U = u +\frac{G}{4\mu}\left(y^2+z^2\right), then it is easy to see that the problem reduces to that integrating a Laplace equation :\frac{\partial^2 U}{\partial y^2}+\frac{\partial^2 U}{\partial z^2}=0 satisfying the condition :U = \frac{G}{4\mu}\left(y^2+z^2\right) on the wall. == Poiseuille's equation for an ideal isothermal gas== For a compressible fluid in a tube the volumetric flow rate and the axial velocity are not constant along the tube; but the mass flow rate is constant along the tube length. Since the net force acting on the fluid is equal to , where , i.e. , then from Poiseuille's law, it follows that :\Delta F = \frac{8 \mu LQ}{r^2}. Sometimes these equations are used to define the mass flow rate. If the pipe is too short, the Hagen–Poiseuille equation may result in unphysically high flow rates; the flow is bounded by Bernoulli's principle, under less restrictive conditions, by :\begin{align} \Delta p = \frac{1}{2} \rho \overline{v}_\text{max}^2 &= \frac{1}{2} \rho \left(\frac{Q_\text{max}}{\pi R^2}\right)^2 \\\\[6pt] \rightarrow \, \, \, Q_\max{} &= \pi R^2 \sqrt\frac{2 \Delta p}{\rho}, \end{align} because it is impossible to have negative (absolute) pressure (not to be confused with gauge pressure) in an incompressible flow. ==Relation to the Darcy–Weisbach equation== Normally, Hagen–Poiseuille flow implies not just the relation for the pressure drop, above, but also the full solution for the laminar flow profile, which is parabolic.

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In this problem, 3.00 mol of liquid mercury is transformed from an initial state characterized by $T_i=300 . \mathrm{K}$ and $P_i=1.00$ bar to a final state characterized by $T_f=600 . \mathrm{K}$ and $P_f=3.00$ bar. a. Calculate $\Delta S$ for this process; $\beta=1.81 \times 10^{-4} \mathrm{~K}^{-1}, \rho=13.54 \mathrm{~g} \mathrm{~cm}^{-3}$, and $C_{P, m}$ for $\mathrm{Hg}(l)=27.98 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$.

System A12 A21 Acetone(1)-Chloroform(2) -0.8404 -0.5610 Acetone(1)-Methanol(2) 0.6184 0.5788 Acetone(1)-Water(2) 2.0400 1.5461 Carbon tetrachloride(1)-Benzene (2) 0.0948 0.0922 Chloroform(1)-Methanol(2) 0.8320 1.7365 Ethanol(1)-Benzene(2) 1.8362 1.4717 Ethanol(1)-Water(2) 1.6022 0.7947 ==See also== * Van Laar equation ==Literature== ==External links== *Ternary systems Margules Category:Physical chemistry Category:Thermodynamic models When A>2 the system starts to demix in two liquids at 50/50 composition; i.e. plait point is at 50 mol%. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. Since: :A = \ln \gamma_1^\infty = \ln \gamma_2^\infty : \gamma_1^\infty = \gamma_2^\infty > \exp(2) \approx 7.38 For asymmetric binary systems, A12≠A21, the liquid-liquid separation always occurs for : :A_{21} + A_{12} > 4 Or equivalently: :\gamma_1^\infty \gamma_2^\infty > \exp(4) \approx 54.6 The plait point is not located at 50 mol%. ΔP (Delta P) is a mathematical term symbolizing a change (Δ) in pressure (P). ==Uses== *Young–Laplace equation ===Darcy–Weisbach equation=== Given that the head loss hf expresses the pressure loss Δp as the height of a column of fluid, :\Delta p = \rho \cdot g \cdot h_f where ρ is the density of the fluid. This yields, when applied only to the first term and using the Gibbs–Duhem equation,:Phase Equilibria in Chemical Engineering, Stanley M. Walas, (1985) p180 Butterworth Publ. : \left\\{\begin{matrix} \ln\ \gamma_1=[A_{12}+2(A_{21}-A_{12})x_1]x^2_2 \\\ \ln\ \gamma_2=[A_{21}+2(A_{12}-A_{21})x_2]x^2_1 \end{matrix}\right. {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup {{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 In chemical engineering the Margules Gibbs free energy model for liquid mixtures is better known as the Margules activity or activity coefficient model. thumb|The Mollier enthalpy–entropy diagram for water and steam. In an isenthalpic process, the enthalpy is constant. Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on "Thermal Conduction in Fluids" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. Calculating compliance on minute volume (VE: ΔV is always defined by tidal volume (VT), but ΔP is different for the measurement of dynamic vs. static compliance. ====Dynamic compliance (Cdyn)==== :C_{dyn} = \frac{V_T}\mathrm{PIP-PEEP} where PIP = peak inspiratory pressure (the maximum pressure during inspiration), and PEEP = positive end expiratory pressure. So the expansion process in a turbine can be easily calculated using the h–s chart when the process is considered to be ideal (which is the case normally when calculating enthalpies, entropies, etc. The enthalpy coordinate is skewed and the constant enthalpy lines are parallel and evenly spaced. ==See also== *Thermodynamic diagrams *Contour line *Phase diagram == References == Category:Thermodynamics Category:Entropy de:Wasserdampf#h-s- Diagramm The leading term X_1X_2 assures that the excess Gibbs energy becomes zero at x1=0 and x1=1. === Activity coefficient === The activity coefficient of component i is found by differentiation of the excess Gibbs energy towards xi. Although the model is old it has the characteristic feature to describe extrema in the activity coefficient, which modern models like NRTL and Wilson cannot. ==Equations== === Excess Gibbs free energy === Margules expressed the intensive excess Gibbs free energy of a binary liquid mixture as a power series of the mole fractions xi: : \frac{G^{ex}}{RT}=X_1 X_2 (A_{21} X_1 +A_{12} X_2) + X_1^2 X_2^2 (B_{21}X_1+ B_{12} X_2) + ... + X_1^m X_2^m (M_{21}X_1+ M_{12} X_2) In here the A, B are constants, which are derived from regressing experimental phase equilibria data. When A_{12}=A_{21}=A, which implies molecules of same molecular size but different polarity, the equations reduce to the one-parameter Margules activity model: : \left\\{\begin{matrix} \ln\ \gamma_1=Ax^2_2 \\\ \ln\ \gamma_2=Ax^2_1 \end{matrix}\right. The work done in a process on vapor cycles is represented by length of , so it can be measured directly, whereas in a T–s diagram it has to be computed using thermodynamic relationship between thermodynamic properties. The Margules activity model is a simple thermodynamic model for the excess Gibbs free energy of a liquid mixture introduced in 1895 by Max Margules.https://archive.org/details/sitzungsbericht10wiengoog After Lewis had introduced the concept of the activity coefficient, the model could be used to derive an expression for the activity coefficients \gamma_i of a compound i in a liquid, a measure for the deviation from ideal solubility, also known as Raoult's law. The Mollier diagram coordinates are enthalpy h and humidity ratio x. The parameters for a description at 20 °C are A12=0.6298 and A21=1.9522.

58.2

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3.54

358800

6.9

A

For an ensemble consisting of 1.00 moles of particles having two energy levels separated by $h v=1.00 \times 10^{-20} \mathrm{~J}$, at what temperature will the internal energy of this system equal $1.00 \mathrm{~kJ}$ ?

The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. The internal energy of an ideal gas is proportional to its mass (number of moles) n and to its temperature T : U = C_V n T, where C_V is the isochoric (at constant volume) molar heat capacity of the gas. The internal energy relative to the mass with unit J/kg is the specific internal energy. Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): U = U(n,T). The internal energy is an extensive function of the extensive variables S, V, and the amounts N_j, the internal energy may be written as a linearly homogeneous function of first degree: : U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots ) = \alpha U(S,V,N_{1},N_{2},\ldots), where \alpha is a factor describing the growth of the system. The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization.Crawford, F. H. (1963), pp. 106–107.Haase, R. (1971), pp. 24–28. The internal energy is the mean value of the system's total energy, i.e., the sum of all microstate energies, each weighted by its probability of occurrence: : U = \sum_{i=1}^N p_i \,E_i. At any temperature greater than absolute zero, microscopic potential energy and kinetic energy are constantly converted into one another, but the sum remains constant in an isolated system (cf. table). The differential internal energy may be written as :\mathrm{d} U = \frac{\partial U}{\partial S} \mathrm{d} S + \frac{\partial U}{\partial V} \mathrm{d} V + \sum_i\ \frac{\partial U}{\partial N_i} \mathrm{d} N_i\ = T \,\mathrm{d} S - P \,\mathrm{d} V + \sum_i\mu_i \mathrm{d} N_i, which shows (or defines) temperature T to be the partial derivative of U with respect to entropy S and pressure P to be the negative of the similar derivative with respect to volume V, : T = \frac{\partial U}{\partial S}, : P = -\frac{\partial U}{\partial V}, and where the coefficients \mu_{i} are the chemical potentials for the components of type i in the system. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39 Therefore, a convenient null reference point may be chosen for the internal energy. MJ/kg may refer to: * megajoules per kilogram * Specific kinetic energy * Heat of fusion * Heat of combustion In statistical mechanics, the internal energy of a body can be analyzed microscopically in terms of the kinetic energies of microscopic motion of the system's particles from translations, rotations, and vibrations, and of the potential energies associated with microscopic forces, including chemical bonds. The internal energy is an extensive property: it depends on the size of the system, or on the amount of substance it contains. kJ/kg may refer to: * kilojoules per kilogram * The SI derived units of specific energy * Specific Internal energy * Specific kinetic energy * Heat of fusion * Heat of combustion Furthermore, it relates the mean microscopic kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system. The internal energy of any gas (ideal or not) may be written as a function of the three extensive properties S, V, n (entropy, volume, mass). In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. In an ideal gas all of the extra energy results in a temperature increase, as it is stored solely as microscopic kinetic energy; such heating is said to be sensible. Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. The expression relating changes in internal energy to changes in temperature and volume is : \mathrm{d}U =C_{V} \, \mathrm{d}T +\left[T\left(\frac{\partial P}{\partial T}\right)_{V} - P\right] \mathrm{d}V. For a closed system, with transfers only as heat and work, the change in the internal energy is : \mathrm{d} U = \delta Q - \delta W, expressing the first law of thermodynamics. When a closed system receives energy as heat, this energy increases the internal energy.

8.8

8

7.0

3.8

449

E

The sedimentation coefficient of lysozyme $\left(\mathrm{M}=14,100 \mathrm{~g} \mathrm{~mol}^{-1}\right)$ in water at $20^{\circ} \mathrm{C}$ is $1.91 \times 10^{-13} \mathrm{~s}$ and the specific volume is $0.703 \mathrm{~cm}^3 \mathrm{~g}^{-1}$. The density of water at this temperature is $0.998 \mathrm{~g} \mathrm{~cm}^{-3}$ and $\eta=1.002 \mathrm{cP}$. Assuming lysozyme is spherical, what is the radius of this protein?

The commonly used molar ratio of lysozyme and mPEG-aldehyde is 1:6 or 1:6.67. The hydrodynamic radius of a macromolecule or colloid particle is R_{\rm hyd}. The protein and mPEG-aldehyde are dissolved using a sodium phosphate buffer with sodium cyanoborohydride, which acts as a reducing agent and conditions the aldehyde group of mPEG-aldehyde to have a strong affinity towards the lysine residue on the N-terminal of lysozyme. The CD spectra range from 189 - 260 nm with a pitch of 0.1 nm showed no significant change in the secondary structure of the intact and PEGylated lysozyme. === Enzymatic activity assay === ==== Glycol chitosan ==== Enzymatic activity of intact and PEGylated lysozyme can be evaluated using glycol chitosan by reacting 1 mL of 0.05% (w/v) glycol chitosan in 100 mM of pH 5.5 acetate buffer and 100 μL of the intact or PEGylated protein at 40 °C for 30 min and subsequently adding 2 mL of 0.5 M sodium carbonate with 1 μg of potassium ferricyanide. There is a negative correlation between molecular weight and the retention time of the PEGylated protein in the chromatogram; larger protein, or more PEGylated protein elutes first, and smaller protein, or intact protein the latest. == Characterization == === Identification === The most common analyses for identifying intact and PEGylated lysozyme can be achieved via size-exclusion chromatography (high-performance liquid chromatography or HPLC), SDS-PAGE and Matrix-assisted laser desorption/ionization (MALDI). === Conformation === The secondary structure of intact and PEGylated lysozyme can be characterized by circular dichroism (CD) spectroscopy. Previous works on lysozyme PEGylation showed various chromatographic schemes in order to purify PEGylated lysozyme, which included ion exchange chromatography, hydrophobic interaction chromatography, and size-exclusion chromatography (fast protein liquid chromatography), and proved its stable conformation via circular dichroism and improved thermal stability by enzymatic activity assays, SDS-PAGE, and size- exclusion chromatography (high-performance liquid chromatography). == Methodology == === PEGylation === The chemical modification of lysozyme by PEGylation involves the addition of methoxy-PEG-aldehyde (mPEG-aldehyde) with varying molecular sizes, ranging from 2 kDa to 40 kDa, to the protein. Due to the high pI of lysozyme (pI = 10.7), cation exchange chromatography is used. As the enzymatic activity to hydrolyze β-1,4- N-acetylglucosamine linkage was retained after PEGylation, there was no decay in the enzymatic activity by increasing the degree of PEGylation. ==== Micrococcus lysodeikticus ==== By the measurement of decrease in turbidity of M. lysodeikticus by incubating it with lysozyme, enzymatic activity can be evaluated. 7.5 μL of 0.1 - 1 mg/mL proteins is added to 200 μL of M. lysodeikticus at its optical density (OD) of 1.7 AU, and the mixture is measured at 450 nm periodically for reaction rate calculation. Inside the Vainshtein radius; see also :r_V = l_\text{P}\left( \frac{m_\text{P}^3M}{m^4_G} \right)^\frac{1}{5} :with Planck length l_\text{P} and Planck mass m_\text{P} the gravitational field around a body of mass M is the same in a theory where the graviton mass m_G is zero and where it's very small because the helicity 0 degree of freedom becomes effective on distance scales r \gg r_V. ==See also== * == References == Category:Quantum gravity In its full form, the Flory–Rehner equation is written as: : -\left[ \ln{\left(1 - u_2\right)}+ u_2+ \chi_1 u_2^2 \right] = \frac{V_1}{\bar{ u}M_c} \left(1-\frac{2M_c}{M}\right) \left( u_2^\frac{1}{3}-\frac{ u_2}{2}\right) where, \bar{ u} is the specific volume of the polymer, M is the primary molecular mass, and M_c is the average molecular mass between crosslinks or the network parameter. == Flory–Rehner theory == The Flory–Rehner theory gives the change of free energy upon swelling of the polymer gel similar to the Flory–Huggins solution theory: :\Delta F = \Delta F_\mathrm{mix} + \Delta F_\mathrm{elastic}. thumb|right| T4 lysozyme ribbon schematic (from PDB 1LZM) Brian W. Matthews is a biochemist and biophysicist educated at the University of Adelaide, contributor to x-ray crystallographic methodology at the University of Cambridge, and since 1970 at the University of Oregon as Professor of Physics and HHMI investigator in the Institute of Molecular Biology. Lysozyme has six lysine residues which are accessible for PEGylation reactions. Note that in biophysics, hydrodynamic radius refers to the Stokes radius, or commonly to the apparent Stokes radius obtained from size exclusion chromatography. Lysozyme PEGylation is the covalent attachment of Polyethylene glycol (PEG) to Lysozyme, which is one of the most widely investigated PEGylated proteins. The theoretical hydrodynamic radius R_{\rm hyd} arises in the study of the dynamic properties of polymers moving in a solvent. In the continuum limit, where the mean free path of the particle is negligible compared to a characteristic length scale of the particle, the hydrodynamic radius is defined as the radius that gives the same magnitude of the frictional force, \boldsymbol{F}_d as that of a sphere with that radius, i.e. :\boldsymbol{F}_d = 6\pi\mu R_{hyd}\boldsymbol{v} where \mu is the viscosity of the surrounding fluid, and \boldsymbol{v} is the velocity of the particle. He created hundreds of mutants of T4 lysozyme (making it the commonest structure in the PDB), determined their structure by x-ray crystallography and measured their melting temperatures. {{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 theoretical hydrodynamic radius R_{\rm hyd} was originally an estimate by John Gamble Kirkwood of the Stokes radius of a polymer, and some sources still use hydrodynamic radius as a synonym for the Stokes radius. Thus, the PEGylation of lysozyme, or lysozyme PEGylation, can be a good model system for the PEGylation of other proteins with enzymatic activities by showing the enhancement of its physical and thermal stability while retaining its activity. In polymer science Flory–Rehner equation is an equation that describes the mixing of polymer and liquid molecules as predicted by the equilibrium swelling theory of Flory and Rehner. The Flory–Rehner equation is written as: : -\left[ \ln{\left(1 - u_2\right)}+ u_2+ \chi_1 u_2^2 \right] = V_1 n \left( u_2^\frac{1}{3}-\frac{ u_2}{2}\right) where, u_2 is the volume fraction of polymer in the swollen mass, V_1 the molar volume of the solvent, n is the number of network chain segments bounded on both ends by crosslinks, and \chi_1 is the Flory solvent-polymer interaction term.

41.40

0.332

2.534324263

14

1.94

E

Determine the standard molar entropy of $\mathrm{Ne}$ and $\mathrm{Kr}$ under standard thermodynamic conditions.

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). 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. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. Factor (J K) Value Item 10 9.5699 J K Entropy equivalent of one bit of information, equal to k times ln(2) page 72 (page 5 of pdf) 10 1.381 Boltzmann Constant 10 5.74 J K Standard entropy of 1 mole of graphite 10 ≈ 10 J K Entropy of the Sun (given as ≈ 10 erg K in Bekenstein (1973)) 10 1.5 × 10 J K Entropy of a black hole of one solar mass (given as ≈ 10 erg K in Bekenstein (1973)) 10 4.3 × 10 J K One estimate of the theoretical maximum entropy of the universeCalculated: 3.1e104 * k = 3.1e104 * 1.381e-23 J/K = 4.3e81 J/K ==See also== *Orders of magnitude (data), relates to information entropy *Order of magnitude (terminology) ==References== Entropy Category:Entropy 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). More fundamentally, kT is the amount of heat required to increase the thermodynamic entropy of a system by k. Approximate values of kT at 298 K Units kT = J kT = pN⋅nm kT = cal kT = meV kT=-174 dBm/Hz kT/hc ≈ cm−1 kT/e = 25.7 mV RT = kT ⋅ NA = kJ⋅mol−1 RT = 0.593 kcal⋅mol−1 h/kT = 0.16 ps kT (also written as kBT) is the product of the Boltzmann constant, k (or kB), and the temperature, T. The following list shows different orders of magnitude of entropy. 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 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 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. Negative values for indicate that entropy decreases on forming the transition state, which often indicates an associative mechanism in which two reaction partners form a single activated complex.James H. Espenson Chemical Kinetics and Reaction Mechanisms (2nd ed., McGraw-Hill 2002), p.156-160 == Derivation == It is possible to obtain entropy of activation using Eyring equation. Because of this, the crystal is locked into a state with 2^N different corresponding microstates, giving a residual entropy of S=Nk\ln(2), rather than zero. Chemical equations make use of the standard molar entropy of reactants and products to find the standard entropy of reaction: :{\Delta S^\circ}_{rxn} = S^\circ_{products} - S^\circ_{reactants} The standard entropy of reaction helps determine whether the reaction will take place spontaneously. To calculate the conformational entropy, the possible conformations of the molecule may first be discretized into a finite number of states, usually characterized by unique combinations of certain structural parameters, each of which has been assigned an energy. It differs from kT only by a factor of the Avogadro constant, NA. 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 quantity \frac{dQ_{k}}{T} represents the ratio of a very small exchange of heat energy to the temperature . In chemical thermodynamics, conformational entropy is the entropy associated with the number of conformations of a molecule. 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 factor is needed because of the pressure dependence of the reaction rate. (bar·L)/(mol·K).Laidler, K.J. and Meiser J.H. Physical Chemistry (Benjamin/Cummings 1982) p.381-2 The value of provides clues about the molecularity of the rate determining step in a reaction, i.e. the number of molecules that enter this step.Laidler and Meiser p.365 Positive values suggest that entropy increases upon achieving the transition state, which often indicates a dissociative mechanism in which the activated complex is loosely bound and about to dissociate.

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Carbon-14 is a radioactive nucleus with a half-life of 5760 years. Living matter exchanges carbon with its surroundings (for example, through $\mathrm{CO}_2$ ) so that a constant level of ${ }^{14} \mathrm{C}$ is maintained, corresponding to 15.3 decay events per minute. Once living matter has died, carbon contained in the matter is not exchanged with the surroundings, and the amount of ${ }^{14} \mathrm{C}$ that remains in the dead material decreases with time due to radioactive decay. Consider a piece of fossilized wood that demonstrates $2.4{ }^{14} \mathrm{C}$ decay events per minute. How old is the wood?

The 774–775 carbon-14 spike is an observed increase of around 1.2% in the concentration of the radioactive carbon-14 isotope in tree rings dated to 774 or 775 CE, which is about 20 times higher than the normal year-to-year variation of radiocarbon in the atmosphere. The calculation of radiocarbon dates determines the age of an object containing organic material by using the properties of radiocarbon (also known as carbon-14), a radioactive isotope of carbon. Carbon-14 decays into nitrogen-14 () through beta decay. A calculation or (more accurately) a direct comparison of carbon-14 levels in a sample, with tree ring or cave-deposit carbon-14 levels of a known age, then gives the wood or animal sample age-since-formation. Carbon-14, C-14, or radiocarbon, is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. The presence of carbon-14 in the isotopic signature of a sample of carbonaceous material possibly indicates its contamination by biogenic sources or the decay of radioactive material in surrounding geologic strata. The following inventory of carbon-14 has been given: * Global inventory: ~8500 PBq (about 50 t) ** Atmosphere: 140 PBq (840 kg) ** Terrestrial materials: the balance * From nuclear testing (until 1990): 220 PBq (1.3 t) ===In fossil fuels=== Many man-made chemicals are derived from fossil fuels (such as petroleum or coal) in which is greatly depleted because the age of fossils far exceeds the half-life of . Because decays at a known rate, the proportion of radiocarbon can be used to determine how long it has been since a given sample stopped exchanging carbon – the older the sample, the less will be left. Contamination with old carbon, with no remaining , causes an error in the other direction, which does not depend on age—a sample that has been contaminated with 1% old carbon will appear to be about 80 years older than it really is, regardless of the date of the sample.Aitken (1990), pp. 85–86. As a tree grows, only the outermost tree ring exchanges carbon with its environment, so the age measured for a wood sample depends on where the sample is taken from. Accumulated dead organic matter, of both plants and animals, exceeds the mass of the biosphere by a factor of nearly 3, and since this matter is no longer exchanging carbon with its environment, it has a / ratio lower than that of the biosphere. ==Dating considerations== The variation in the / ratio in different parts of the carbon exchange reservoir means that a straightforward calculation of the age of a sample based on the amount of it contains will often give an incorrect result. If a sample that is in fact 17,000 years old is contaminated so that 1% of the sample is actually modern carbon, it will appear to be 600 years younger; for a sample that is 34,000 years old the same amount of contamination would cause an error of 4,000 years. Libby estimated that the radioactivity of exchangeable carbon-14 would be about 14 disintegrations per minute (dpm) per gram of pure carbon, and this is still used as the activity of the modern radiocarbon standard. If a sample that is 17,000 years old is contaminated so that 1% of the sample is modern carbon, it will appear to be 600 years younger; for a sample that is 34,000 years old, the same amount of contamination would cause an error of 4,000 years. Cosmogenic nuclides are also used as proxy data to characterize cosmic particle and solar activity of the distant past. ==Origin== ===Natural production in the atmosphere=== right|thumb| 1: Formation of carbon-14 2: Decay of carbon-14 3: The "equal" equation is for living organisms, and the unequal one is for dead organisms, in which the C-14 then decays (See 2). This means that radiocarbon dates on wood samples can be older than the date at which the tree was felled. The fossil fuel effect was eliminated from the standard value by measuring wood from 1890, and using the radioactive decay equations to determine what the activity would have been at the year of growth. If the benzene sample contains carbon that is about 5,730 years old (the half-life of ), then there will only be half as many decay events per minute, but the same error term of 80 years could be obtained by doubling the counting time to 500 minutes.Bowman, Radiocarbon Dating, pp. 38-39.Taylor, Radiocarbon Dating, p. 124. This resemblance is used in chemical and biological research, in a technique called carbon labeling: carbon-14 atoms can be used to replace nonradioactive carbon, in order to trace chemical and biochemical reactions involving carbon atoms from any given organic compound. ==Radioactive decay and detection== Carbon-14 goes through radioactive beta decay: : → + + + 156.5 keV By emitting an electron and an electron antineutrino, one of the neutrons in the carbon-14 atom decays to a proton and the carbon-14 (half-life of 5,730 ± 40 years) decays into the stable (non- radioactive) isotope nitrogen-14. Radiocarbon dating (also referred to as carbon dating or carbon-14 dating) is a method for determining the age of an object containing organic material by using the properties of radiocarbon, a radioactive isotope of carbon. Calculating radiocarbon ages also requires the value of the half-life for . Carbon-12 and carbon-13 are both stable, while carbon-14 is unstable and has a half-life of years.

4.86

432

0.15

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Carbon-14 is a radioactive nucleus with a half-life of 5760 years. Living matter exchanges carbon with its surroundings (for example, through $\mathrm{CO}_2$ ) so that a constant level of ${ }^{14} \mathrm{C}$ is maintained, corresponding to 15.3 decay events per minute. Once living matter has died, carbon contained in the matter is not exchanged with the surroundings, and the amount of ${ }^{14} \mathrm{C}$ that remains in the dead material decreases with time due to radioactive decay. Consider a piece of fossilized wood that demonstrates $2.4{ }^{14} \mathrm{C}$ decay events per minute. How old is the wood?

The 774–775 carbon-14 spike is an observed increase of around 1.2% in the concentration of the radioactive carbon-14 isotope in tree rings dated to 774 or 775 CE, which is about 20 times higher than the normal year-to-year variation of radiocarbon in the atmosphere. The calculation of radiocarbon dates determines the age of an object containing organic material by using the properties of radiocarbon (also known as carbon-14), a radioactive isotope of carbon. Carbon-14 decays into nitrogen-14 () through beta decay. A calculation or (more accurately) a direct comparison of carbon-14 levels in a sample, with tree ring or cave-deposit carbon-14 levels of a known age, then gives the wood or animal sample age-since-formation. Carbon-14, C-14, or radiocarbon, is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. The presence of carbon-14 in the isotopic signature of a sample of carbonaceous material possibly indicates its contamination by biogenic sources or the decay of radioactive material in surrounding geologic strata. The following inventory of carbon-14 has been given: * Global inventory: ~8500 PBq (about 50 t) ** Atmosphere: 140 PBq (840 kg) ** Terrestrial materials: the balance * From nuclear testing (until 1990): 220 PBq (1.3 t) ===In fossil fuels=== Many man-made chemicals are derived from fossil fuels (such as petroleum or coal) in which is greatly depleted because the age of fossils far exceeds the half-life of . As a tree grows, only the outermost tree ring exchanges carbon with its environment, so the age measured for a wood sample depends on where the sample is taken from. Because decays at a known rate, the proportion of radiocarbon can be used to determine how long it has been since a given sample stopped exchanging carbon – the older the sample, the less will be left. Accumulated dead organic matter, of both plants and animals, exceeds the mass of the biosphere by a factor of nearly 3, and since this matter is no longer exchanging carbon with its environment, it has a / ratio lower than that of the biosphere. ==Dating considerations== The variation in the / ratio in different parts of the carbon exchange reservoir means that a straightforward calculation of the age of a sample based on the amount of it contains will often give an incorrect result. This means that radiocarbon dates on wood samples can be older than the date at which the tree was felled. Cosmogenic nuclides are also used as proxy data to characterize cosmic particle and solar activity of the distant past. ==Origin== ===Natural production in the atmosphere=== right|thumb| 1: Formation of carbon-14 2: Decay of carbon-14 3: The "equal" equation is for living organisms, and the unequal one is for dead organisms, in which the C-14 then decays (See 2). Libby estimated that the radioactivity of exchangeable carbon-14 would be about 14 disintegrations per minute (dpm) per gram of pure carbon, and this is still used as the activity of the modern radiocarbon standard. Radiocarbon dating (also referred to as carbon dating or carbon-14 dating) is a method for determining the age of an object containing organic material by using the properties of radiocarbon, a radioactive isotope of carbon. If a sample that is 17,000 years old is contaminated so that 1% of the sample is modern carbon, it will appear to be 600 years younger; for a sample that is 34,000 years old, the same amount of contamination would cause an error of 4,000 years. This resemblance is used in chemical and biological research, in a technique called carbon labeling: carbon-14 atoms can be used to replace nonradioactive carbon, in order to trace chemical and biochemical reactions involving carbon atoms from any given organic compound. ==Radioactive decay and detection== Carbon-14 goes through radioactive beta decay: : → + + + 156.5 keV By emitting an electron and an electron antineutrino, one of the neutrons in the carbon-14 atom decays to a proton and the carbon-14 (half-life of 5,730 ± 40 years) decays into the stable (non- radioactive) isotope nitrogen-14. If the benzene sample contains carbon that is about 5,730 years old (the half-life of ), then there will only be half as many decay events per minute, but the same error term of 80 years could be obtained by doubling the counting time to 500 minutes.Bowman, Radiocarbon Dating, pp. 38-39.Taylor, Radiocarbon Dating, p. 124. Carbon-12 and carbon-13 are both stable, while carbon-14 is unstable and has a half-life of years. The fossil fuel effect was eliminated from the standard value by measuring wood from 1890, and using the radioactive decay equations to determine what the activity would have been at the year of growth. Calculating radiocarbon ages also requires the value of the half-life for . The old wood effect or old wood problem is a pitfall encountered in the archaeological technique of radiocarbon dating. The globally averaged production of carbon-14 for this event is . ==Hypotheses== Several possible causes of the event have been considered.

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135.36

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Determine the diffusion coefficient for Ar at $298 \mathrm{~K}$ and a pressure of $1.00 \mathrm{~atm}$.

thumb|Thermal diffusion coefficients vs. temperature, for air at normal pressure The Maxwell–Stefan diffusion (or Stefan–Maxwell diffusion) is a model for describing diffusion in multicomponent systems. 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. On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System). The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Rearranging yields: \ln k = \frac{-E_{\rm a}}{R}\left(\frac{1}{T}\right) + \ln A. "On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System)". Alternatively, the equation may be expressed as k = Ae^\frac{-E_{\rm a}}{k_{\rm B}T}, where * is the activation energy for the reaction (in the same units as kBT), * is the Boltzmann constant. Suction pressure is also called Diffusion Pressure Deficit. The Boltzmann–Matano method is used to convert the partial differential equation resulting from Fick's law of diffusion into a more easily solved ordinary differential equation, which can then be applied to calculate the diffusion coefficient as a function of concentration. DPD decreases with dilution of the solution. == History == The term diffusion pressure deficit (DPD) was coined by B.S. Meyer in 1938. We can then rearrange: :D(c^*) = - \frac{1}{2 t} \frac{\int^{c^*}_{c_R} x \mathrm{d}c}{(\mathrm{d}c/\mathrm{d}x)_{c=c^*}} Knowing the concentration profile c(x) at annealing time t, and assuming it is invertible as x(c), we can then calculate the diffusion coefficient for all concentrations between cR and cL. === The Matano interface === The last formula has one significant shortcoming: no information is given about the reference according to which x should be measured. The difference between diffusion pressure of pure solvent and solution is called diffusion pressure deficit (DPD). The diffusion equation is a parabolic partial differential equation. McGraw-Hill == External links == * Diffusion Calculator for Impurities & Dopants in Silicon * A tutorial on the theory behind and solution of the Diffusion Equation. The Diffusion Handbook: Applied Solutions for Engineers. This form is significantly easier to solve numerically, and one only needs to perform a back-substitution of t or x into the definition of ξ to find the value of the other variable. === The parabolic law === Observing the previous equation, a trivial solution is found for the case dc/dξ = 0, that is when concentration is constant over ξ. Stefan: Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 2te Abteilung a, 1871, 63, 63-124. for liquids. The Mathematics of Diffusion. If some solute is dissolved in solvent, its diffusion pressure decreases. The suction pressure, along with the suction temperature the wet bulb temperature of the discharge air are used to determine the correct refrigerant charge in a system. == Further reading == # The measurement of Diffusion Pressure Deficit in plants by the method of Vapour Equilibrium (By R. O. SLATYER, 1958) == References == Category:Diffusion This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time (x \propto \sqrt t), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the distance (t \propto x^2); the square term gives the name parabolic law.See an animation of the parabolic law. == Matano’s method == Chuijiro Matano applied Boltzmann's transformation to obtain a method to calculate diffusion coefficients as a function of concentration in metal alloys.

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Gas cylinders of $\mathrm{CO}_2$ are sold in terms of weight of $\mathrm{CO}_2$. A cylinder contains $50 \mathrm{lb}$ (22.7 $\mathrm{kg}$ ) of $\mathrm{CO}_2$. How long can this cylinder be used in an experiment that requires flowing $\mathrm{CO}_2$ at $293 \mathrm{~K}(\eta=146 \mu \mathrm{P})$ through a 1.00-m-long tube (diameter $\left.=0.75 \mathrm{~mm}\right)$ with an input pressure of $1.05 \mathrm{~atm}$ and output pressure of $1.00 \mathrm{~atm}$ ? The flow is measured at the tube output.

Fill the gas collecting tube with one of those fluids of given mass density and measure the overall mass, do the same with the second one giving the two mass values m_{fullgas}, m_{fullliquid}. Measure the overall mass m_{full} to calculate the mass of the fluid inside the tube m=m_{full}-m_{evactube} yielding the desired mass density \rho=\frac{m}{V}. ==Molar Mass from the Mass Density of a Gas== If the gas is a pure gaseous chemical substance (and not a mixture), with the mass density \rho=\frac{m}{V}, then using the ideal gas law permits to calculate the molar mass M of the gaseous chemical substance: :M = \frac{ m \cdot R \cdot T }{ p \cdot V}\, Where R represents the universal gas constant, T the absolute temperature at which the measurements took place. == References == ==External links== * Source of the notion "gas collecting tube", among others Category:Measuring instruments Category:Laboratory equipment Category:Laboratory glassware The gas collecting tube is weighed for a third and last time containing the liquid yielding the value m_{containing liquid}. The mass and volume of a displaced amount of gas are determined: At atmospheric pressure p, the gas collecting tube is filled with the gas to be investigated and the overall mass m_{full} is measured. ISBN 978-0-626-23561-1 == See also == * * * * * * * * * * * * * - a small, inexpensive, disposable metal gas cylinder for providing pneumatic power * * * * – a device for measuring leaf water potentials * * * or Knock-out drum * * * ==Notes== ==References== * A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed. * E.P. Popov, Engineering Mechanics of Solids, 1st ed. * Megyesy, Eugene F. "Pressure Vessel Handbook, 14th Edition." For example, the ASME Boiler and Pressure Vessel Code (BPVC) (UG-27) formulas are: Spherical shells: Thickness has to be less than 0.356 times inner radius :\sigma_\theta = \sigma_{\rm long} = \frac{p(r + 0.2t)}{2tE} Cylindrical shells: Thickness has to be less than 0.5 times inner radius :\sigma_\theta = \frac{p(r + 0.6t)}{tE} :\sigma_{\rm long} = \frac{p(r - 0.4t)}{2tE} where E is the joint efficiency, and all others variables as stated above. Consequently, for those two fluids, the definition of mass density can be rewritten: :\rho_{gas}=\frac{m_{fullgas}-m_{evactube}}{V}\, :\rho_{liquid}=\frac{m_{fullliquid}-m_{evactube}}{V} These two equations with two unknowns m_{evactube} and V can be solved by using elementary algebra: :V=\frac{m_{fullliquid}-m_{fullgas}}{\rho_{liquid}-\rho_{gas}}\, :m_{evactube}=\frac{\rho_{gas}\cdot m_{fullliquid}-\rho_{liquid} \cdot m_{fullgas}}{\rho_{liquid}-\rho_{gas}} (The relative error of the result significantly depends on the relative proportions of the given mass densities and the measured masses.) Each cylinder shall be painted externally in the colours corresponding to its gaseous contents. == Common sizes == The below are example cylinder sizes and do not constitute an industry standard. size Diameter × height, including 5.5 inches for valve and cap (inches) Nominal tare weight, including 4.5 lb for valve and cap (lb) Water capacity (lb) Internal volume, 70 °F (21 °C), 1atm Internal volume, 70 °F (21 °C), 1atm U.S. DOT specs size Diameter × height, including 5.5 inches for valve and cap (inches) Nominal tare weight, including 4.5 lb for valve and cap (lb) Water capacity (lb) (liters) (cu.ft) U.S. DOT specs 2HP 43.3 1.53 3AA3500 K 110 49.9 1.76 3AA2400 A 96 43.8 1.55 3AA2015 B 37.9 17.2 0.61 3AA2015 C 15.2 6.88 0.24 3AA2015 D 4.9 2.24 0.08 3AA2015 AL 64.8 29.5 1.04 3AL2015 BL 34.6 15.7 0.55 3AL2216 CL 13 5.9 0.21 3AL2216 XL 238 108 3.83 4BA240 SSB 41.6 18.9 0.67 3A1800 10S 8.3 3.8 0.13 3A1800 LB 1 0.44 0.016 3E1800 XF 60.9 2.15 8AL XG 278 126.3 4.46 4AA480 XM 120 54.3 1.92 3A480 XP 124 55.7 1.98 4BA300 QT (includes 4.5 inches for valve) (includes 1.5 lb for valve) 2.0 0.900 0.0318 4B-240ET LP5 47.7 21.68 0.76 4BW240 Medical E (excludes valve and cap) (excludes valve and cap) 9.9 4.5 0.16 3AA2015 (US DOT specs define material, making, and maximum pressure in psi. (The difference of masses of the nearly evacuated tube and the liquid-containing tube gives the mass (m_{liquid}=m_{containing liquid}-m_{sucked}) of the sucked-in liquid, that took the place of the extracted amount of gas.) Hydraulic (filled with water) testing pressure is usually 50% higher than the working pressure. ==== Vessel thread ==== Until 1990, high pressure cylinders were produced with conical (tapered) threads. For a cylinder with hemispherical ends, :M = 2 \pi R^2 (R + W) P {\rho \over \sigma}, where *R is the Radius (m) *W is the middle cylinder width only, and the overall width is W + 2R (m) ====Cylindrical vessel with semi-elliptical ends==== In a vessel with an aspect ratio of middle cylinder width to radius of 2:1, :M = 6 \pi R^3 P {\rho \over \sigma}. ====Gas storage==== In looking at the first equation, the factor PV, in SI units, is in units of (pressurization) energy. * ISO 11439: Compressed natural gas (CNG) cylinders. For gases that remain gaseous under ambient storage conditions, the upstream pressure gauge can be used to estimate how much gas is left in the cylinder according to pressure. thumb|A 20 lb () steel propane cylinder. upright=1.4|thumb|A welded steel pressure vessel constructed as a horizontal cylinder with domed ends. The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. Because (for a given pressure) the thickness of the walls scales with the radius of the tank, the mass of a tank (which scales as the length times radius times thickness of the wall for a cylindrical tank) scales with the volume of the gas held (which scales as length times radius squared). Further the volume of the gas is (4πr3)/3. The vapor pressure in the cylinder is a function of temperature. Now fill the gas collecting tube with the fluid to be investigated. A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. New York:McGraw-Hill, , pg 108 thumb|300px|Stress in the cylinder body of a pressure vessel.

3

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The vibrational frequency of $I_2$ is $208 \mathrm{~cm}^{-1}$. What is the probability of $I_2$ populating the $n=2$ vibrational level if the molecular temperature is $298 \mathrm{~K}$ ?

A molecular vibration is excited when the molecule absorbs energy, ΔE, corresponding to the vibration's frequency, ν, according to the relation ΔE = hν, where h is Planck's constant. The probability of resonance absorption is called the resonance factor \psi, and the sum of the two factors is p + \psi = 1. The excess energy of the excited vibrational mode is transferred to the kinetic modes in the same molecule or to the surrounding molecules. The vibration frequencies, νi, are obtained from the eigenvalues, λi, of the matrix product GF. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics For a diatomic molecule A−B, the vibrational frequency in s−1 is given by u = \frac{1}{2 \pi} \sqrt{k / \mu} , where k is the force constant in dyne/cm or erg/cm2 and μ is the reduced mass given by \frac{1}{\mu} = \frac{1}{m_A}+\frac{1}{m_B}. The vibrational temperature is used commonly when finding the vibrational partition function. Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the "effective area" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. Vibrational energy relaxation, or vibrational population relaxation, is a process in which the population distribution of molecules in quantum states of high energy level caused by an external perturbation returns to the Maxwell–Boltzmann distribution. Through this process, the initially excited vibrational mode moves to a vibrational state of a lower energy. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency u (in the harmonic oscillator approximation). Vibrations of polyatomic molecules are described in terms of normal modes, which are independent of each other, but each normal mode involves simultaneous vibrations of different parts of the molecule. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm−1 and wavelengths of approximately 30 to 3 µm. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. The normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular vibration. Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. A diatomic molecule has one normal mode of vibration, since it can only stretch or compress the single bond. The combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state. Intramolecular vibrational energy redistribution (IVR) is a process in which energy is redistributed between different quantum states of a vibrationally excited molecule, which is required by successful theories explaining unimolecular reaction rates such as RRKM theory. Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region. In addition, by the electronic transition, the molecule often moves to the vibrationally excited state of the electronic excited state.

0

0.082

'-2.5'

0.086

0.66666666666

D

The value of $\Delta G_f^{\circ}$ for $\mathrm{Fe}(g)$ is $370.7 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298.15 \mathrm{~K}$, and $\Delta H_f^{\circ}$ for $\mathrm{Fe}(g)$ is $416.3 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at the same temperature. Assuming that $\Delta H_f^{\circ}$ is constant in the interval $250-400 \mathrm{~K}$, calculate $\Delta G_f^{\circ}$ for $\mathrm{Fe}(g)$ at 400. K.

The heat equation can be efficiently solved numerically using the implicit Crank–Nicolson method of . 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 same effect can be achieved with a low temperature and a long holding time, or with a higher temperature and a short holding time. ==Formula== In the Hollomon–Jaffe parameter, this exchangeability of time and temperature can be described by the following formula: :H_p = \frac {(273.15 + T)}{1000} \cdot (C + \log(t)) This formula is not consistent concerning the units; the parameters must be entered in a certain manner. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. In either case, one uses the heat equation :c_t = D \Delta c, or :P_t = D \Delta P. T is in degrees Celsius. The other (trivial) solution is for all spatial temperature gradients to disappear as well, in which case the temperature become uniform in space, as well. The "diffusivity constant" is often not present in mathematical studies of the heat equation, while its value can be very important in engineering. Then, according to the chain rule, one has Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of and solutions of the heat equation with . The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The vibrational temperature is used commonly when finding the vibrational partition function. Thus instead of being for thermodynamic temperature (Kelvin - K), units of u should be J/L. Then the heat per unit volume u satisfies an equation : \frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q. The Hollomon–Jaffe parameter (HP), also generally known as the Larson–Miller parameter, describes the effect of a heat treatment at a temperature for a certain time. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units. Hence the heat capacity was measured within 2 nK below the transition in an experiment included in a Space Shuttle payload in 1992. thumb|250px|The plot of the specific heat capacity versus temperature. The temperature gradient is a dimensional quantity expressed in units of degrees (on a particular temperature scale) per unit length. This form is more general and particularly useful to recognize which property (e.g. cp or \rho) influences which term. :\rho c_p \frac{\partial T}{\partial t} - abla \cdot \left( k abla T \right) = \dot q_V where \dot q_V is the volumetric heat source. ===Three-dimensional problem=== In the special cases of propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is : \frac{\partial u}{\partial t} = \alpha abla^2 u = \alpha \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2 }\right) = \alpha \left( u_{xx} + u_{yy} + u_{zz} \right) where: * u = u(x, y, z, t) is temperature as a function of space and time; * \tfrac{\partial u}{\partial t} is the rate of change of temperature at a point over time; * u_{xx} , u_{yy} , and u_{zz} are the second spatial derivatives (thermal conductions) of temperature in the x , y , and z directions, respectively; * \alpha \equiv \tfrac{k}{c_p\rho} is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity k , the specific heat capacity c_p , and the mass density \rho . Holloman and Jaffe determined the value of C experimentally by plotting hardness versus tempering time for a series of tempering temperatures of interest and interpolating the data to obtain the time necessary to yield a number of different hardness values. These authors derived an expression for the temperature at the center of a sphere :\frac{T_C - T_S}{T_0 - T_S} =2 \sum_{n = 1}^{\infty} (-1)^{n+1} \exp\left({-\frac{n^2 \pi^2 \alpha t}{L^2}}\right) where is the initial temperature of the sphere and the temperature at the surface of the sphere, of radius . A temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location.

+10

1.61

2598960.0

92

355.1

E

The reactant 1,3-cyclohexadiene can be photochemically converted to cis-hexatriene. In an experiment, $2.5 \mathrm{mmol}$ of cyclohexadiene are converted to cis-hexatriene when irradiated with 100. W of 280. nm light for $27.0 \mathrm{~s}$. All of the light is absorbed by the sample. What is the overall quantum yield for this photochemical process?

A quantum yield of 1.0 (100%) describes a process where each photon absorbed results in a photon emitted. Since not all photons are absorbed productively, the typical quantum yield will be less than 1. The quantum yield of ANS is ~0.002 in aqueous buffer, but near 0.4 when bound to serum albumin. === Photochemical reactions === The quantum yield of a photochemical reaction describes the number of molecules undergoing a photochemical event per absorbed photon: \Phi=\frac{\rm \\#\ molecules\ undergoing\ reaction\ of\ interest}{\rm \\#\ photons\ absorbed\ by\ photoreactive\ substance} In a chemical photodegradation process, when a molecule dissociates after absorbing a light quantum, the quantum yield is the number of destroyed molecules divided by the number of photons absorbed by the system. In optical spectroscopy, the quantum yield is the probability that a given quantum state is formed from the system initially prepared in some other quantum state. Such effects can be studied with wavelength-tunable lasers and the resulting quantum yield data can help predict conversion and selectivity of photochemical reactions. For example, a singlet to triplet transition quantum yield is the fraction of molecules that, after being photoexcited into a singlet state, cross over to the triplet state. === Photosynthesis === Quantum yield is used in modeling photosynthesis: \Phi = \frac {\rm \mu mol\ CO_2 \ fixed} {\rm \mu mol\ photons \ absorbed} ==See also== *Quantum dot *Quantum efficiency == References == Category:Radiation Category:Spectroscopy Category:Photochemistry Photoexcitation is the production of an excited state of a quantum system by photon absorption. The quantum yield is then calculated by: \Phi = \Phi_\mathrm{R} \times \frac{\mathit{Int}}{\mathit{Int}_\mathrm{R}} \times \frac{1-10^{-A_\mathrm{R}}}{1-10^{-A}} \times \frac{{n}^2}{{n_\mathrm{R}}^2} where * is the quantum yield, * is the area under the emission peak (on a wavelength scale), * is absorbance (also called "optical density") at the excitation wavelength, * is the refractive index of the solvent. Quantum photoelectrochemistry in particular provides fundamental insight into basic light-harvesting and photoinduced electro-optical processes in several emerging solar energy conversion technologies for generation of both electricity (photovoltaics) and solar fuels.Ponseca Jr., Carlito S.; Chábera, Pavel; Uhlig, Jens; Persson, Petter; Sundström, Villy (August 2017). The absorption of photons with energy equal to or greater than the band gap of the semiconductor initiates photocatalytic reactions. Quantum photoelectrochemistry is the investigation of the quantum mechanical nature of photoelectrochemistry, the subfield of study within physical chemistry concerned with the interaction of light with electrochemical systems, typically through the application of quantum chemical calculations.Quantum Photoelectrochemistry - Theoretical Studies of Organic Adsorbates on Metal Oxide Surfaces, Petter Persson, Acta Univ. Upsaliensis., Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 544, 53 pp. Uppsala. . On the atomic and molecular scale photoexcitation is the photoelectrochemical process of electron excitation by photon absorption, when the energy of the photon is too low to cause photoionization. Fluorescence quantum yields are measured by comparison to a standard of known quantum yield. One example is the reaction of hydrogen with chlorine, in which as many as 106 molecules of hydrogen chloride can be formed per quantum of blue light absorbed.Laidler K.J., Chemical Kinetics (3rd ed., Harper & Row 1987) p.289 Quantum yields of photochemical reactions can be highly dependent on the structure, proximity and concentration of the reactive chromophores, the type of solvent environment as well as the wavelength of the incident light. Quantum photoelectrochemistry provides an expansion of quantum electrochemistry to processes involving also the interaction with light (photons). Key aspects of quantum photoelectrochemistry are calculations of optical excitations, photoinduced electron and energy transfer processes, excited state evolution, as well as interfacial charge separation and charge transport in nanoscale energy conversion systems.Multiscale Modelling of Interfacial Electron Transfer, Petter Persson, Chapter 3 in: Solar Energy Conversion – Dynamics of Electron and Excitation Transfer P. Piotrowiak (Ed.), RSC Energy and Environment Series (2013) thumbnail|Quantum photoelectrochemistry calculation of photoinduced interfacial electron transfer in a dye-sensitized solar cell. In particle physics, the quantum yield (denoted ) of a radiation-induced process is the number of times a specific event occurs per photon absorbed by the system. The term quantum efficiency (QE) may apply to incident photon to converted electron (IPCE) ratio of a photosensitive device, or it may refer to the TMR effect of a Magnetic Tunnel Junction. Thus, the fluorescence quantum yield is affected if the rate of any non-radiative pathway changes. Quantum yield is defined by the fraction of excited state fluorophores that decay through fluorescence: \Phi_f = \frac{k_f}{k_f + \sum k_\mathrm{nr}} where * is the fluorescence quantum yield, * is the rate constant for radiative relaxation (fluorescence), * is the rate constant for all non-radiative relaxation processes. This synthesis demonstrates that the thermal Diels–Alder reaction favors the undesired regioisomer, but the photoredox-catalyzed reaction gives the desired regioisomer in improved yield. 400px|frameless|center|Key photoredox cycloaddition in total synthesis of Heitziamide A === Photoredox organocatalysis === Organocatalysis is a subfield of catalysis that explores the potential of organic small molecules as catalysts, particularly for the enantioselective creation of chiral molecules. The photoexcitation causes the electrons in atoms to go to an excited state.

1.4

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In this problem, $2.50 \mathrm{~mol}$ of $\mathrm{CO}_2$ gas is transformed from an initial state characterized by $T_i=450 . \mathrm{K}$ and $P_i=1.35$ bar to a final state characterized by $T_f=800 . \mathrm{K}$ and $P_f=$ 3.45 bar. Using Equation (5.23), calculate $\Delta S$ for this process. Assume ideal gas behavior and use the ideal gas value for $\beta$. For $\mathrm{CO}_2$, $$ \frac{C_{P, m}}{\mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}}=18.86+7.937 \times 10^{-2} \frac{T}{\mathrm{~K}}-6.7834 \times 10^{-5} \frac{T^2}{\mathrm{~K}^2}+2.4426 \times 10^{-8} \frac{T^3}{\mathrm{~K}^3} $$

"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. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. 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. Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as : f = -kT \lim_{N \rightarrow \infty} \frac{1}{N}\log Z_N The magnetization M per site in the thermodynamic limit, depending on the external magnetic field H and temperature T is given by : M(T,H) \ \stackrel{\mathrm{def}}{=}\ \lim_{N \rightarrow \infty} \frac{1}{N} \left( \sum_i \sigma_i \right) = - \left( \frac{\partial f}{\partial H} \right)_T where \sigma_i is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively : \chi_T(T,H) = \left( \frac{\partial M}{\partial H} \right)_T and : c_H = -T \left( \frac{\partial^2 f}{\partial T^2} \right)_H. ==Definitions== The critical exponents \alpha, \alpha', \beta, \gamma, \gamma' and \delta are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows : M(t,0) \simeq (-t)^{\beta}\mbox{ for }t \uparrow 0 : M(0,H) \simeq |H|^{1/ \delta} \operatorname{sign}(H)\mbox{ for }H \rightarrow 0 : \chi_T(t,0) \simeq \begin{cases} (t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases} : c_H(t,0) \simeq \begin{cases} (t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases} where : t \ \stackrel{\mathrm{def}}{=}\ \frac{T-T_c}{T_c} measures the temperature relative to the critical point. ==Derivation== For the magnetic analogue of the Maxwell relations for the response functions, the relation : \chi_T (c_H -c_M) = T \left( \frac{\partial M}{\partial T} \right)_H^2 follows, and with thermodynamic stability requiring that c_H, c_M\mbox{ and }\chi_T \geq 0 , one has : c_H \geq \frac{T}{\chi_T} \left( \frac{\partial M}{\partial T} \right)_H^2 which, under the conditions H=0, t>0 and the definition of the critical exponents gives : (-t)^{-\alpha'} \geq \mathrm{constant}\cdot(-t)^{\gamma'}(-t)^{2(\beta-1)} which gives the Rushbrooke inequality : \alpha' + 2\beta + \gamma' \geq 2. The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on "Thermal Conduction in Fluids" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. \\! | 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. {{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 COSILAB is a software tool for solving complex chemical kinetics problems. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). Problems to be solved by COSILAB may involve thousands of reactions amongst hundreds of species for practically any mixture composition, pressure and temperature. For pressures above 1 bar, that means, above the normal boiling point, the typical errors are below 2%. Nowadays, statistical software, such as the R programming language, and functions available in many spreadsheet programs compute values of the t-distribution and its inverse without tables. ==See also== * F-distribution * Folded-t and half-t distributions * Hotelling's T-squared distribution * Multivariate Student distribution * Standard normal table (Z-distribution table) * t-statistic * Tau distribution, for internally studentized residuals * Wilks' lambda distribution * Wishart distribution * Modified half-normal distribution with the pdf on (0, \infty) is given as f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}}, where \Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\\\(1,0)\end{matrix};z \right) denotes the Fox–Wright Psi function. ==Notes== ==References== * * * * ==External links== * *Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term "Student's distribution") * First Students on page 112. If the flow velocity is negligible, the general equation of heat transfer reduces to the standard heat equation. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. With :T \sim t_ u and location-scale family transformation :X = \mu + \tau T we get :X \sim lst(\mu, \tau^2, u) The resulting distribution is also called the non- standardized Student's t-distribution. ===Density and first two moments=== The location-scale t distribution has a density defined by: :p(x\mid u,\mu,\tau) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u}\tau\,} \left(1+\frac{1}{ u} \left( \frac{ x-\mu } {\tau } \right)^2\right)^{-( u+1)/2} Equivalently, the density can be written in terms of \tau^2: :p(x\mid u, \mu, \tau^2) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u\tau^2}} \left(1+\frac{1}{ u}\frac{(x-\mu)^2}{{\tau}^2}\right)^{-( u+1)/2} Other properties of this version of the distribution are: :\begin{align} \operatorname{E}(X) &= \mu & \text{ for } u > 1 \\\ \operatorname{var}(X) &= \tau^2\frac{ u}{ u-2} & \text{ for } u > 2 \\\ \operatorname{mode}(X) &= \mu \end{align} ===Special cases=== * If X follows a location-scale t-distribution X \sim \mathrm{lst}\left(\mu, \tau^2, u\right) then for u \rightarrow \infty X is normally distributed X \sim \mathrm{N}\left(\mu, \tau^2\right) with mean \mu and variance \tau^2. In an ideal fluid, as described by the Euler equations, the conservation of energy is defined by the equation:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) \right] = 0 where h is the specific enthalpy. The Lee–Kesler method Lee B.I., Kesler M.G., This may also be written as :f(t) = \frac{1}{\sqrt{ u}\,\mathrm{B} (\frac{1}{2}, \frac{ u}{2})} \left(1+\frac{t^2} u \right)^{-( u+1)/2}, where B is the Beta function. Note that the t-distribution (red line) becomes closer to the normal distribution as u increases. ===Cumulative distribution function=== The cumulative distribution function (CDF) can be written in terms of I, the regularized incomplete beta function. It may also be extended to rotating, stratified flows, such as those encountered in geophysical fluid dynamics. == Derivation == === Extension of the ideal fluid energy equation === For a viscous, Newtonian fluid, the governing equations for mass conservation and momentum conservation are the continuity equation and the Navier-Stokes equations:\begin{aligned} {\partial \rho\over{\partial t}} &= - abla\cdot (\rho {\bf v}) \\\ \rho {D{\bf v}\over{Dt}} &= - abla p + abla \cdot \sigma \end{aligned}where p is the pressure and \sigma is the viscous stress tensor, with the components of the viscous stress tensor given by:\sigma_{ij} = \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right) + \zeta \delta_{ij} abla\cdot {\bf v} The energy of a unit volume of the fluid is the sum of the kinetic energy \rho v^{2}/2 \equiv \rho k and the internal energy \rho\varepsilon, where \varepsilon is the specific internal energy. In statistical mechanics, the Rushbrooke inequality relates the critical exponents of a magnetic system which exhibits a first-order phase transition in the thermodynamic limit for non-zero temperature T.

0

12

30.0

48.6

24

D

One mole of $\mathrm{CO}$ gas is transformed from an initial state characterized by $T_i=320 . \mathrm{K}$ and $V_i=80.0 \mathrm{~L}$ to a final state characterized by $T_f=650 . \mathrm{K}$ and $V_f=120.0 \mathrm{~L}$. Using Equation (5.22), calculate $\Delta S$ for this process. Use the ideal gas values for $\beta$ and $\kappa$. For CO, $$ \frac{C_{V, m}}{\mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}}=31.08-0.01452 \frac{T}{\mathrm{~K}}+3.1415 \times 10^{-5} \frac{T^2}{\mathrm{~K}^2}-1.4973 \times 10^{-8} \frac{T^3}{\mathrm{~K}^3} $$

"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. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? 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. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup {{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 In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on "Thermal Conduction in Fluids" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. Lange indirectly defines the values to be at a standard state pressure of "1 atm (101325 Pa)", although citing the same NBS and JANAF sources among others. K (? °C), ? K (? °C), ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? * J.D. Cox, DD., Wagman, and V.A. Medvedev, CODATA Key Values for Thermodynamics, Hemisphere Publishing Corp., * Values from CRC refer to "100 kPa (1 bar or 0.987 standard atmospheres)". J/(mol K) == Spectral data == UV- Vis Lambda-max ? nm Extinction coefficient ?

537

4.946

'-20.0'

-0.16

24.4

E

You are given the following reduction reactions and $E^{\circ}$ values: $$ \begin{array}{ll} \mathrm{Fe}^{3+}(a q)+\mathrm{e}^{-} \rightarrow \mathrm{Fe}^{2+}(a q) & E^{\circ}=+0.771 \mathrm{~V} \\ \mathrm{Fe}^{2+}(a q)+2 \mathrm{e}^{-} \rightarrow \mathrm{Fe}(s) & E^{\circ}=-0.447 \mathrm{~V} \end{array} $$ Calculate $E^{\circ}$ for the half-cell reaction $\mathrm{Fe}^{3+}(a q)+3 \mathrm{e}^{-} \rightarrow \mathrm{Fe}(s)$.

These changes can be represented in formulas by inserting appropriate electrons into each half reaction: :\begin{align} & \ce{Fe^2+ -> Fe^3+ + e-} \\\ & \ce{Cl2 + 2e- -> 2Cl-} \end{align} Given two half reactions it is possible, with knowledge of appropriate electrode potentials, to arrive at the complete (original) reaction the same way. The sum of these two half reactions is the oxidation–reduction reaction. ==Half-reaction balancing method== Consider the reaction below: :Cl2 + 2Fe^2+ -> 2Cl- + 2Fe^3+ The two elements involved, iron and chlorine, each change oxidation state; iron from +2 to +3, chlorine from 0 to −1\. For a half cell equation, conventionally written as a reduction reaction (i.e., electrons accepted by an oxidant on the left side): : a \, A + b \, B + h \, \ce{H+} + z \, e^{-} \quad \ce{<=>} \quad c \, C + d \, D The half-cell standard reduction potential E^{\ominus}_\text{red} is given by : E^{\ominus}_\text{red} (\text{volt}) = -\frac{\Delta G^\ominus}{zF} where \Delta G^\ominus is the standard Gibbs free energy change, is the number of electrons involved, and is the Faraday's constant. For a half cell equation, conventionally written as a reduction reaction (i.e., electrons accepted by an oxidant on the left side): : a \, A + b \, B + h \, \ce{H+} + z \, e^{-} \quad \ce{<=>} \quad c \, C + d \, D The half-cell standard reduction potential E^{\ominus}_\text{red} is given by : E^{\ominus}_\text{red} (\text{volt}) = -\frac{\Delta G^\ominus}{zF} where \Delta G^\ominus is the standard Gibbs free energy change, is the number of electrons involved, and is Faraday's constant. The determination of the formal reduction potential at pH = 7 for a given biochemical half-reaction requires thus to calculate it with the corresponding Nernst equation as a function of pH. The global reaction can thus be decomposed in half redox reactions as follows: :2 (Fe2+ → Fe3+ \+ e−) (oxidation of 2 iron(II) ions) :2 (H2O + e− → ½ H2 \+ OH−) (reduction of 2 water protons) to give: :2 Fe2+ \+ 2 H2O → 2 Fe3+ \+ H2 \+ 2 OH− Adding to this reaction one intact iron(II) ion for each two oxidized iron(II) ions leads to: :3 Fe2+ \+ 2 H2O → Fe2+ \+ 2 Fe3+ \+ H2 \+ 2 OH− Electroneutrality requires the iron cations on both sides of the equation to be counterbalanced by 6 hydroxyl anions (OH−): :3 Fe2+ \+ 6 OH− \+ 2 H2O → Fe2+ \+ 2 Fe3+ \+ H2 \+ 8 OH− :3 Fe(OH)2 \+ 2 H2O → Fe(OH)2 \+ 2 Fe(OH)3 \+ H2 For completing the main reaction, two companion reactions have still to be taken into account: The autoprotolysis of the hydroxyl anions; a proton exchange between two OH−, like in a classical acid–base reaction: :OH− \+ OH− → O2− \+ H2O :acid 1 + base 2 → base 1 + acid 2, or also, :2 OH− → O2− \+ H2O it is then possible to reorganize the global reaction as: :3 Fe(OH)2 \+ 2 H2O → (FeO + H2O) + (Fe2O3 \+ 3 H2O) + H2 :3 Fe(OH)2 \+ 2 H2O → FeO + Fe2O3 \+ 4 H2O + H2 :3 Fe(OH)2 → FeO + Fe2O3 \+ 2 H2O + H2 Considering then the formation reaction of iron(II,III) oxide: :Fe^{II}O + Fe^{III}2O3 -> Fe3O4 it is possible to write the balanced global reaction: :3 Fe(OH)2 → (FeO·Fe2O3) + 2 H2O + H2 in its final form, known as the Schikorr reaction: :3 Fe(OH)2 → Fe3O4 \+ 2 H2O + H2 == Occurrences == The Schikorr reaction can occur in the process of anaerobic corrosion of iron and carbon steel in various conditions. At pH = 7, \+ → P680 ~ +1.0 Half-reaction independent of pH as no is involved in the reaction ==See also== * Nernst equation * Electron bifurcation * Pourbaix diagram * Reduction potential ** Dependency of reduction potential on pH * Standard electrode potential * Standard reduction potential * Standard reduction potential (data page) * Standard state ==References== ==Bibliography== ;Electrochemistry * ;Bio-electrochemistry * * * * * * ;Microbiology * * Category:Biochemistry Standard reduction potentials for half-reactions important in biochemistry Category:Electrochemical potentials Category:Thermodynamics databases Category:Biochemistry databases The same also applies for the reduction potential of oxygen: : For , E^{\ominus}_\text{red} = 1.229 V, so, applying the Nernst equation for pH = 7 gives: : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 1.229 - \left(0.05916 \ \text{×} \ 7\right) = 0.815 \ V For obtaining the values of the reduction potential at pH = 7 for the redox reactions relevant for biological systems, the same kind of conversion exercise is done using the corresponding Nernst equation expressed as a function of pH. Is it simply: * E_h = E_\text{red} calculated at pH 7 (with or without corrections for the activity coefficients), * E^{\ominus '}_\text{red}, a formal standard reduction potential including the activity coefficients but no pH calculations, or, is it, * E^{\ominus '}_\text{red apparent at pH 7}, an apparent formal standard reduction potential at pH 7 in given conditions and also depending on the ratio \frac{h} {z} = \frac{\text{(number of involved protons)}} {\text{(number of exchanged electrons)}}. This immediately leads to the Nernst equation, which for an electrochemical half-cell is E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \ln Q_r=E^\ominus_\text{red} - \frac{RT}{zF} \ln\frac{a_\text{Red}}{a_\text{Ox}}. The values below are standard apparent reduction potentials for electro- biochemical half-reactions measured at 25 °C, 1 atmosphere and a pH of 7 in aqueous solution. The offset of -414 mV in E_\text{red} is the same for both reduction reactions because they share the same linear relationship as a function of pH and the slopes of their lines are the same. Taking into account the activity coefficients (\gamma) the Nernst equation becomes: E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \ln\left(\frac{\gamma_\text{Red}}{\gamma_\text{Ox}}\frac{C_\text{Red}}{C_\text{Ox}}\right) E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \left(\ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}} + \ln\frac{C_\text{Red}}{C_\text{Ox}}\right) E_\text{red} = \underbrace{\left(E^\ominus_\text{red} - \frac{RT}{zF} \ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}}\right)}_{E^{\ominus '}_\text{red}} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}} Where the first term including the activity coefficients (\gamma) is denoted E^{\ominus '}_\text{red} and called the formal standard reduction potential, so that E_\text{red} can be directly expressed as a function of E^{\ominus '}_\text{red} and the concentrations in the simplest form of the Nernst equation: E_\text{red}=E^{\ominus '}_\text{red} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}} ===Formal standard reduction potential=== When wishing to use simple concentrations in place of activities, but that the activity coefficients are far from unity and can no longer be neglected and are unknown or too difficult to determine, it can be convenient to introduce the notion of the "so-called" standard formal reduction potential (E^{\ominus '}_\text{red}) which is related to the standard reduction potential as follows: E^{\ominus '}_\text{red}=E^{\ominus}_\text{red}-\frac{RT}{zF}\ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}} So that the Nernst equation for the half-cell reaction can be correctly formally written in terms of concentrations as: E_\text{red}=E^{\ominus '}_\text{red} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}} and likewise for the full cell expression. Solving the Nernst equation for the half-reaction of reduction of two protons into hydrogen gas gives: : : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 0 - \left(0.05916 \ \text{×} \ 7\right) = -0.414 \ V In biochemistry and in biological fluids, at pH = 7, it is thus important to note that the reduction potential of the protons () into hydrogen gas is no longer zero as with the standard hydrogen electrode (SHE) at 1 M (pH = 0) in classical electrochemistry, but that E_\text{red} = -0.414\mathrm V versus the standard hydrogen electrode (SHE). In any given oxidation-reduction reaction, there are two half reactions—oxidation half reaction and reduction half reaction. For a complete electrochemical reaction (full cell), the equation can be written as E_\text{cell} = E^\ominus_\text{cell} - \frac{RT}{zF} \ln Q_r where: * is the half-cell reduction potential at the temperature of interest, * is the standard half-cell reduction potential, * is the cell potential (electromotive force) at the temperature of interest, * is the standard cell potential, * is the universal gas constant: , * is the temperature in kelvins, * is the number of electrons transferred in the cell reaction or half-reaction, * is the Faraday constant, the magnitude of charge (in coulombs) per mole of electrons: , * is the reaction quotient of the cell reaction, and * is the chemical activity for the relevant species, where is the activity of the reduced form and is the activity of the oxidized form. ===Thermal voltage=== At room temperature (25 °C), the thermal voltage V_T=\frac{RT}{F} is approximately 25.693 mV. In chemistry, a half reaction (or half-cell reaction) is either the oxidation or reduction reaction component of a redox reaction. So, : -zFE^\ominus_{cell} = -RT \ln{K} And therefore: : E^\ominus_{cell} = \frac{RT} {zF} \ln{K} Starting from the Nernst equation, one can also demonstrate the same relationship in the reverse way. The numerically simplified form of the Nernst equation is expressed as: :E_\text{red} = E^\ominus_\text{red} - \frac{0.059\ V}{z} \log_{10}\frac{a_\text{Red}}{a_\text{Ox}} Where E^\ominus_\text{red} is the standard reduction potential of the half-reaction expressed versus the standard reduction potential of hydrogen. This is represented in the following oxidation half reaction (note that the electrons are on the products side): :Zn_{(s)} -> Zn^2+ + 2e- At the Cu cathode, reduction takes place (electrons are accepted). Half reactions can be written to describe both the metal undergoing oxidation (known as the anode) and the metal undergoing reduction (known as the cathode). So, at pH = 7, E_\text{red} = -0.414 V for the reduction of protons.

-0.041

1.06

27.0

56

0

A

At $298.15 \mathrm{~K}, \Delta G_f^{\circ}(\mathrm{C}$, graphite $)=0$, and $\Delta G_f^{\circ}(\mathrm{C}$, diamond $)=2.90 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Therefore, graphite is the more stable solid phase at this temperature at $P=P^{\circ}=1$ bar. Given that the densities of graphite and diamond are 2.25 and $3.52 \mathrm{~kg} / \mathrm{L}$, respectively, at what pressure will graphite and diamond be in equilibrium at $298.15 \mathrm{~K}$ ?

At normal temperature and pressure, and , the stable phase of carbon is graphite, but diamond is metastable and its rate of conversion to graphite is negligible. At normal temperature and pressure, and , the stable phase of carbon is graphite, but diamond is metastable and its rate of conversion to graphite is negligible. Rapid conversion of graphite to diamond requires pressures well above the equilibrium line: at , a pressure of is needed. Rapid conversion of graphite to diamond requires pressures well above the equilibrium line: at , a pressure of is needed. ===Other properties=== thumb|upright=1.15|Molar volume against pressure at room temperature The acoustic and thermal properties of graphite are highly anisotropic, since phonons propagate quickly along the tightly bound planes, but are slower to travel from one plane to another. At surface air pressure (one atmosphere), diamonds are not as stable as graphite, and so the decay of diamond is thermodynamically favorable (δH = ). A similar proportion of diamonds comes from the lower mantle at depths between 660 and 800 km. Diamond is thermodynamically stable at high pressures and temperatures, with the phase transition from graphite occurring at greater temperatures as the pressure increases. The equilibrium pressure varies linearly with temperature, between at and at (the diamond/graphite/liquid triple point). It is chemically inert, not reacting with most corrosive substances, and has excellent biological compatibility. ===Thermodynamics=== The equilibrium pressure and temperature conditions for a transition between graphite and diamond are well established theoretically and experimentally. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, but diamond is metastable and converts to it at a negligible rate under those conditions. File:Graphite ambient STM.jpg|Scanning tunneling microscope image of graphite surface File:Graphite-layers- side-3D-balls.png|Side view of ABA layer stacking File:Graphite-layers- top-3D-balls.png|Plane view of layer stacking File:Graphite-unit- cell-3D-balls.png|Alpha graphite's unit cell ===Thermodynamics=== The equilibrium pressure and temperature conditions for a transition between graphite and diamond is well established theoretically and experimentally. However, at temperatures above about , diamond rapidly converts to graphite. However, at temperatures above about , diamond rapidly converts to graphite. Above the graphite- diamond-liquid carbon triple point, the melting point of diamond increases slowly with increasing pressure; but at pressures of hundreds of GPa, it decreases. At high pressure (~) diamond can be heated up to , and a report published in 2009 suggests that diamond can withstand temperatures of and above. Thus, graphite is much softer than diamond. The toughness of natural diamond has been measured as 7.5–10 MPa·m1/2. However, owing to a very large kinetic energy barrier, diamonds are metastable; they will not decay into graphite under normal conditions. ==See also== *Chemical vapor deposition of diamond *Crystallographic defects in diamond *Nitrogen-vacancy center *Synthetic diamond ==References== ==Further reading== *Pagel-Theisen, Verena. (2001). The pressure changes linearly between at and at (the diamond/graphite/liquid triple point). Sufficiently small diamonds can form in the cold of space because their lower surface energy makes them more stable than graphite. Diamonds are carbon crystals that form under high temperatures and extreme pressures such as deep within the Earth. Much higher pressures may be possible with nanocrystalline diamonds. ====Elasticity and tensile strength==== Usually, attempting to deform bulk diamond crystal by tension or bending results in brittle fracture. Research results published in an article in the scientific journal Nature Physics in 2010 suggest that at ultrahigh pressures and temperatures (about 10 million atmospheres or 1 TPa and 50,000 °C) diamond melts into a metallic fluid.

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Imagine tossing a coin 50 times. What are the probabilities of observing heads 25 times (i.e., 25 successful experiments)?

Find the probability that exactly two of the tosses result in heads. ===Solution=== For this experiment, let a heads be defined as a success and a tails as a failure. A fair coin has the probability of success 0.5 by definition. For comparison, we could define an event to occur when "at least one 'heads'" is flipped in the experiment - that is, when the outcome contains at least one 'heads'. Because the coin is assumed to be fair, the probability of success is p = \tfrac{1}{2}. Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely. Cambridge University Press, New York (NY), 1995, p.67-68 ==Example: tossing coins== Consider the simple experiment where a fair coin is tossed four times. For example, if one were to toss the same coin one hundred times and record each result, each toss would be considered a trial within the experiment composed of all hundred tosses. ==Mathematical description== A random experiment is described or modeled by a mathematical construct known as a probability space. However, there are experiments that are not easily described by a set of equally likely outcomes-- for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely. ==See also== * * * * * ==References== ==External links== * Category:Experiment (probability theory) In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. In probability theory, an outcome is a possible result of an experiment or trial. The probability function P is defined in such a way that, if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would approach agreement with the values P assigns them. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis. ==Experiments and trials== Random experiments are often conducted repeatedly, so that the collective results may be subjected to statistical analysis. The probability of exactly k successes in the experiment B(n,p) is given by: :P(k)={n \choose k} p^k q^{n-k} where {n \choose k} is a binomial coefficient. For the experiment where we flip a coin twice, the four possible outcomes that make up our sample space are (H, T), (T, H), (T, T) and (H, H), where "H" represents a "heads", and "T" represents a "tails". Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number n of statistically independent Bernoulli trials, each with a probability of success p, and counts the number of successes. As a simple experiment, we may flip a coin twice. For example, when tossing an ordinary coin, one typically assumes that the outcomes "head" and "tail" are equally likely to occur. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial. In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. When an experiment is conducted, one (and only one) outcome results-- although this outcome may be included in any number of events, all of which would be said to have occurred on that trial. The probability of the outcome of an experiment is never negative, although a quasiprobability distribution allows a negative probability, or quasiprobability for some events.

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In a rotational spectrum of $\operatorname{HBr}\left(B=8.46 \mathrm{~cm}^{-1}\right)$, the maximum intensity transition in the R-branch corresponds to the $J=4$ to 5 transition. At what temperature was the spectrum obtained?

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 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 Finally, rotational energy states describe semi-rigid rotation of the entire molecule and produce transition wavelengths in the far infrared and microwave regions (about 100-10,000 μm in wavelength). At high frequencies (short wavelengths) and low temperatures, the conversion must proceed through Planck's law. In agreement with this estimate, vibrational spectra show transitions in the near infrared (about 1 - 5 μm). The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. Typically, the largest fluctuations of the primordial CMB temperature occur on angular scales of about 1°. 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. Approximately half the time, this cascade will include the n = 3 to n = 2 transition and the atom will emit H-alpha light. For the radiation of a helium–neon laser with a power of 1 mW, a frequency spread Δf = 1 GHz, an output aperture of 1 mm, and a beam dispersion half-angle of 0.56 mrad, the brightness temperature would be . 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. 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 spectroscopy, a Soret peak or Soret band is an intense peak in the blue wavelength region of the visible spectrum. The "Cold Spot" is approximately 70 µK (0.00007 K) colder than the average CMB temperature (approximately 2.7 K), whereas the root mean square of typical temperature variations is only 18 µK.After the dipole anisotropy, which is due to the Doppler shift of the microwave background radiation due to our peculiar velocity relative to the comoving cosmic rest frame, has been subtracted out. In this case, the brightness temperature is simply a measure of the intensity of the radiation as it would be measured at the origin of that radiation. Since the emissivity is a value between 0 and 1, the real temperature will be greater than or equal to the brightness temperature. This combination will pass only a narrow (<0.1 nm) range of wavelengths of light centred on the H-alpha emission line. Also in . , vacuum (nm) 2 121.57 3 102.57 4 97.254 5 94.974 6 93.780 ∞ 91.175 Source: ===Balmer series ( = 2)=== 757px|thumb|center|The four visible hydrogen emission spectrum lines in the Balmer series. In the first year of data recorded by the Wilkinson Microwave Anisotropy Probe (WMAP), a region of sky in the constellation Eridanus was found to be cooler than the surrounding area. In particular, it is the temperature at which a black body would have to be in order to duplicate the observed intensity of a grey body object at a frequency u. For a grey body the spectral radiance is a portion of the black body radiance, determined by the emissivity \epsilon. Four of the Balmer lines are in the technically "visible" part of the spectrum, with wavelengths longer than 400 nm and shorter than 700 nm.

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Given that the work function for sodium metal is $2.28 \mathrm{eV}$, what is the threshold frequency $v_0$ for sodium?

Using the equations given above one can then translate the electron energy E into the threshold energy T. 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. 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. Therefore, datasheets will specify threshold voltage according to a specified measurable amount of current (commonly 250 μA or 1 mA). 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. Although an analytical description of the displacement is not possible, the "sudden approximation" gives fairly good approximations of the threshold displacement energies at least in covalent materials and low-index crystal directions An example molecular dynamics simulation of a threshold displacement event is available in 100_20eV.avi. The threshold energy should not be confused with the threshold displacement energy, which is the minimum energy needed to permanently displace an atom in a crystal to produce a crystal defect in radiation material science. ==Example of pion creation== Consider the collision of a mobile proton with a stationary proton so that a {\pi}^0 meson is produced: p^+ + p^+ \to p^+ + p^+ + \pi^0 We can calculate the minimum energy that the moving proton must have in order to create a pion. Thus, the thinner the oxide thickness, the lower the threshold voltage. The threshold limit value (TLV) is believed to be a level to which a worker can be exposed per shift in the worktime without adverse effects. If the desired result is to produce a third particle then the threshold energy is greater than or equal to the rest energy of the desired particle. Looking above, that the threshold voltage does not have a direct relationship but is not independent of the effects. The threshold voltage, commonly abbreviated as Vth or VGS(th), of a field-effect transistor (FET) is the minimum gate-to- source voltage (VGS) that is needed to create a conducting path between the source and drain terminals. Hence theoretical study of the threshold displacement energy is conventionally carried out using either classical or quantum mechanical molecular dynamics computer simulations. Such simulations have given significant qualitative insights into the threshold displacement energy, but the quantitative results should be viewed with caution. In particle physics, the threshold energy for production of a particle is the minimum kinetic energy that must be imparted to one of a pair of particles in order for their collision to produce a given result. Sodium vanadate can refer to: * Sodium metavanadate (sodium trioxovanadate(V)), NaVO3 * Sodium orthovanadate (sodium tetraoxovanadate(V)), Na3VO4 * Sodium decavanadate, Na6V10O28 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 . Using the body formulas above, V_{TN} is directly proportional to \gamma, and t_{OX}, which is the parameter for oxide thickness. This is the fundamental ("primary damage") threshold displacement energy, and also the one usually simulated by molecular dynamics computer simulations. Accordingly, the term threshold voltage does not readily apply to turning such devices on, but is used instead to denote the voltage level at which the channel is wide enough to allow electrons to flow easily. The initial stage A. of defect creation, until all excess kinetic energy has dissipated in the lattice and it is back to its initial temperature T0, takes < 5 ps. In most cases, since momentum is also conserved, the threshold energy is significantly greater than the rest energy of the desired particle.

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Calculate the de Broglie wavelength of an electron traveling at $1.00 \%$ of the speed of light.

Free electron propagation (in vacuum) can be accurately described as a de Broglie matter wave with a wavelength inversely proportional to its longitudinal (possibly relativistic) momentum. 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. The Compton wavelength for this particle is the wavelength of a photon of the same energy. The CODATA 2018 value for the Compton wavelength of the electron is .CODATA 2018 value for Compton wavelength for the electron from NIST. The standard Compton wavelength of a particle is given by \lambda = \frac{h}{m c}, while its frequency is given by f = \frac{m c^2}{h}, where is the Planck constant, is the particle's proper mass, and is the speed of light. Fortunately as long as the electromagnetic field traversed by the electron changes only slowly compared with this wavelength (see typical values in matter wave#Applications of matter waves), Kirchhoff's diffraction formula applies. 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. Thus the uncertainty in position must be greater than half of the reduced Compton wavelength . ==Relationship to other constants== Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength for the electron {2\pi}\simeq 386~\textrm{fm})}} and the electromagnetic fine-structure constant The Bohr radius is related to the Compton wavelength by: a_0 = \frac{1}{\alpha}\left(\frac{\lambda_\text{e}}{2\pi}\right) = \frac{\bar{\lambda}_\text{e}}{\alpha} \simeq 137\times\bar{\lambda}_\text{e}\simeq 5.29\times 10^4~\textrm{fm} The classical electron radius is about 3 times larger than the proton radius, and is written: r_\text{e} = \alpha\left(\frac{\lambda_\text{e}}{2\pi}\right) = \alpha\bar{\lambda}_\text{e} \simeq\frac{\bar{\lambda}_\text{e}}{137}\simeq 2.82~\textrm{fm} The Rydberg constant, having dimensions of linear wavenumber, is written: \frac{1}{R_\infty}=\frac{2\lambda_\text{e}}{\alpha^2} \simeq 91.1~\textrm{nm} \frac{1}{2\pi R_\infty} = \frac{2}{\alpha^2}\left(\frac{\lambda_\text{e}}{2\pi}\right) = 2 \frac{\bar{\lambda}_\text{e}}{\alpha^2} \simeq 14.5~\textrm{nm} This yields the sequence: r_{\text{e}} = \alpha \bar{\lambda}_{\text{e}} = \alpha^2 a_0 = \alpha^3 \frac{1}{4\pi R_\infty}. The Compton wavelength is a quantum mechanical property of a particle, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). Thus, \textstyle \frac{1}{\lambda}=\frac{E}{hc} (in this formula the h represents Planck's constant). The electron particle trajectory formula matches the formula for geometrical optics with a suitable electron-optical index of refraction. 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. For photons of frequency , energy is given by E = h f = \frac{h c}{\lambda} = m c^2, which yields the Compton wavelength formula if solved for . ==Limitation on measurement== The Compton wavelength expresses a fundamental limitation on measuring the position of a particle, taking into account quantum mechanics and special relativity. 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). In this case, the Compton wavelength is equal to the square root of the quantum metric, a metric describing the quantum space: \sqrt{g_{kk}}=\lambda_c ==See also== * de Broglie wavelength * Planck–Einstein relation ==References== ==External links== * Length Scales in Physics: the Compton Wavelength Category:Atomic physics Category:Foundational quantum physics de:Compton-Effekt#Compton-Wellenlänge The reduced Compton wavelength appears in the relativistic Klein–Gordon equation for a free particle: \mathbf{ abla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \left(\frac{m c}{\hbar} \right)^2 \psi. 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. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. 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. The Planck mass and length are defined by: m_{\rm P} = \sqrt{\hbar c/G} l_{\rm P} = \sqrt{\hbar G /c^3}. ==Geometrical interpretation== A geometrical origin of the Compton wavelength has been demonstrated using semiclassical equations describing the motion of a wavepacket. As stressed by Niels Bohr, expressing results in terms of wavenumber, not wavelength, was the key to Rydberg's discovery.

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Show that $u(\theta, \phi)=Y_1^1(\theta, \phi)$ given in Example E-2 satisfies the equation $\nabla^2 u=\frac{c}{r^2} u$, where $c$ is a constant. What is the value of $c$ ?

For each angle \varphi the parameter :u = u(\varphi,m)=\int_0^\varphi r(\theta,m) \, d\theta (the incomplete elliptic integral of the first kind) is computed. In particular, :\theta =2\operatorname{am}\left(\frac{t\sqrt{2c}}{2},2\right)\rightarrow \frac{\mathrm d^2 \theta}{\mathrm dt^2}+c\sin \theta=0. The following table summarizes the expressions for all Jacobi elliptic functions pq(u,m) in the variables (x,y,r) and (φ,dn) with r = \sqrt{x^2+y^2} Jacobi elliptic functions pq[u,m] as functions of {x,y,r} and {φ,dn} q c s n d p c 1 x/y=\cot(\varphi) x/r=\cos(\varphi) x=\cos(\varphi)/\operatorname{dn} s y/x=\tan(\varphi) 1 y/r=\sin(\varphi) y=\sin(\varphi)/\operatorname{dn} n r/x=\sec(\varphi) r/y=\csc(\varphi) 1 r=1/\operatorname{dn} d 1/x=\sec(\varphi)\operatorname{dn} 1/y=\csc(\varphi)\operatorname{dn} 1/r=\operatorname{dn} 1 ==Definition in terms of Jacobi theta functions== ===Jacobi theta function description=== Equivalently, Jacobi's elliptic functions can be defined in terms of his theta functions. Note that when \varphi=\pi/2, that u then equals the quarter period K. ==Definition as trigonometry: the Jacobi ellipse== \cos \varphi, \sin \varphi are defined on the unit circle, with radius r = 1 and angle \varphi = arc length of the unit circle measured from the positive x-axis. For the x and y value of the point P with u and parameter m we get, after inserting the relation: :r(\varphi,m) = \frac 1 {\operatorname{dn}(u,m)} into: x = r(\varphi,m) \cos (\varphi), y = r(\varphi,m) \sin (\varphi) that: : x = \frac{\operatorname{cn}(u,m)} {\operatorname{dn}(u,m)},\quad y = \frac{\operatorname{sn}(u,m)} {\operatorname{dn}(u,m)}. In short: : \operatorname{cn}(u,m) = \frac{x}{r(\varphi,m)}, \quad \operatorname{sn}(u,m) = \frac{y}{r(\varphi,m)}, \quad \operatorname{dn}(u,m) = \frac{1}{r(\varphi,m)}. The quantity u[\varphi,k]=u(\varphi,k^2) is related to the incomplete elliptic integral of the second kind (with modulus k) by :u[\varphi,k]=\frac{1}{\sqrt{1-k^2}}\left(\frac{1+\sqrt{1-k^2}}{2}\operatorname{E}\left(\varphi+\arctan\left(\sqrt{1-k^2}\tan \varphi\right),\frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}}\right)-\operatorname{E}(\varphi,k)+\frac{k^2 \sin\varphi\cos\varphi}{2\sqrt{1-k^2 \sin ^2\varphi}}\right), and therefore is related to the arc length of an ellipse. Therefore, \frac{\partial {\mathbf r}}{\partial u} = \frac{\partial{s}}{\partial u} \mathbf u where is the arc length parameter. Let : \begin{align} & x^2 + \frac{y^2}{b^2} = 1, \quad b > 1, \\\ & m = 1 - \frac{1}{b^2}, \quad 0 < m < 1, \\\ & x = r \cos \varphi, \quad y = r \sin \varphi \end{align} then: : r( \varphi,m) = \frac{1} {\sqrt {1-m \sin^2 \varphi}}\, . Elliptic functions are functions of two variables. Equivalent potential temperature, commonly referred to as theta-e \left( \theta_e \right), is a quantity that is conserved during changes to an air parcel's pressure (that is, during vertical motions in the atmosphere), even if water vapor condenses during that pressure change. Then the familiar relations from the unit circle: : x' = \cos \varphi, \quad y' = \sin \varphi read for the ellipse: :x' = \operatorname{cn}(u,m),\quad y' = \operatorname{sn}(u,m). The first variable might be given in terms of the amplitude \varphi, or more commonly, in terms of u given below. In the above, the value m is a free parameter, usually taken to be real such that 0\leq m \leq 1, and so the elliptic functions can be thought of as being given by two variables, u and the parameter m. Let P=(x,y)=(r \cos\varphi, r\sin\varphi) be a point on the ellipse, and let P'=(x',y')=(\cos\varphi,\sin\varphi) be the point where the unit circle intersects the line between P and the origin O. The \varphi that satisfies :u=\int_0^\varphi \frac{\mathrm d\theta} {\sqrt {1-m \sin^2 \theta}} is called the Jacobi amplitude: :\operatorname{am}(u,m)=\varphi. Similarly, Jacobi elliptic functions are defined on the unit ellipse, with a = 1\. 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. This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. == Notes == * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): ** The polar angle is denoted by \theta \in [0, \pi]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question. Then divide on both sides by \mathrm d u_i to get: \frac{\partial s}{\partial u_i} \hat{\mathbf u}_i = \sum_{j} \frac{\partial s}{\partial v_j} \frac{\partial v_j}{\partial u_i} \hat{\mathbf v}_j. == See also == * Del * Orthogonal coordinates * Curvilinear coordinates * Vector fields in cylindrical and spherical coordinates == References == == External links == * Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates. Therefore, taking the partial derivative of this relation with respect to pressure yields: \left({\partial h \over \partial p}\right)_{S, \, \sigma} = \frac{1}{\rho} By integrating this equation, the potential enthalpy h^0 is defined as the enthalpy at a reference pressure p_r: h^0(S, \, \theta, \, p_r) = h(S, \, \theta, \, p) - \int^p_{p_r} \frac{1}{\rho(S, \, \theta, \, p')} dp' Here the enthalpy and density are defined in terms of the three state variables: salinity, potential temperature and pressure. === Conversion to conservative temperature === Conservative temperature \Theta is defined to be directly proportional to potential enthalpy. Damon Evans (born November 24, 1949) is an American actor best known as the second of two actors who portrayed Lionel Jefferson on the CBS sitcom The Jeffersons.

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The wave function $\Psi_2(1,2)$ given by Equation 9.39 is not normalized as it stands. Determine the normalization constant of $\Psi_2(1,2)$ given that the "1s" parts are normalized.

Note that if the probability density function is a function of various parameters, so too will be its normalizing constant. thumb|250px|Mexican hat In mathematics and numerical analysis, the Ricker wavelet :\psi(t) = \frac{2}{\sqrt{3\sigma}\pi^{1/4}} \left(1 - \left(\frac{t}{\sigma}\right)^2 \right) e^{-\frac{t^2}{2\sigma^2}} is the negative normalized second derivative of a Gaussian function, i.e., up to scale and normalization, the second Hermite function. A normalized 1s Slater-type function is a function which is used in the descriptions of atoms and in a broader way in the description of atoms in molecules. The constant by which one multiplies a polynomial so its value at 1 is a normalizing constant. In this case, the reciprocal of the value :P(D)=\sum_i P(D|H_i)P(H_i) \; is the normalizing constant.Feller, 1968, p. 124. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one. ==Definition== In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function.Continuous Distributions at University of Alabama.Feller, 1968, p. Thus, \mathbf E_{1s} = \left\langle \left(\frac{\zeta^3}{\pi} \right)^{0.50} e^{-\zeta r} \right|\left. -\left(\frac{\zeta^3}{\pi} \right)^{0.50}e^{-\zeta r}\left[\frac{-2r\zeta+r^2\zeta^2}{2r^2}\right]\right\rangle+\langle\psi_{1s}| - \frac{\mathbf Z}{r}|\psi_{1s}\rangle \mathbf E_{1s} = \frac{\zeta^2}{2}-\zeta \mathbf Z. It has the form :\psi_{1s}(\zeta, \mathbf{r - R}) = \left(\frac{\zeta^3}{\pi}\right)^{1 \over 2} \, e^{-\zeta |\mathbf{r - R}|}. The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. \\! | 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. Second normal form (2NF) is a normal form used in database normalization. 2NF was originally defined by E. F. Codd in 1971.Codd, E. F. And constant \frac{1}{\sqrt{2\pi}} is the normalizing constant of function p(x). In that context, the normalizing constant is called the partition function. ==Bayes' theorem== Bayes' theorem says that the posterior probability measure is proportional to the product of the prior probability measure and the likelihood function. For this reason { u} is also known as the normality parameter. The parametrised normalizing constant for the Boltzmann distribution plays a central role in statistical mechanics. Using the expression for Slater orbital, \mathbf \psi_{1s} = \left (\frac{\zeta^3}{\pi} \right ) ^{0.50}e^{-\zeta r} the integrals can be exactly solved. The normalization and the parameter χ have been obtained from data. is the probability distribution of the reconstructed invariant mass of a decayed particle candidate in continuum background. == Definition == The probability density function (pdf) of the ARGUS distribution is: : f(x; \chi, c ) = \frac{\chi^3}{\sqrt{2\pi}\,\Psi(\chi)} \cdot \frac{x}{c^2} \sqrt{1-\frac{x^2}{c^2}} \exp\bigg\\{ -\frac12 \chi^2\Big(1-\frac{x^2}{c^2}\Big) \bigg\\}, for 0 \leq x < c. *A tutorial on the first 3 normal forms by Fred Coulson *Description of the database normalization basics by Microsoft 2NF de:Normalisierung (Datenbank)#Zweite Normalform (2NF) Orthonormal functions are normalized such that \langle f_i , \, f_j \rangle = \, \delta_{i,j} with respect to some inner product . For concreteness, there are many methods of estimating the normalizing constant for practical purposes. The constant is used to establish the hyperbolic functions cosh and sinh from the lengths of the adjacent and opposite sides of a hyperbolic triangle. ==See also== *Normalization (statistics) ==Notes== ==References== *Continuous Distributions at Department of Mathematical Sciences: University of Alabama in Huntsville * Category:Theory of probability distributions Category:1 (number) It is also known as the Marr wavelet for David Marr.http://www2.isye.gatech.edu/~brani/isyebayes/bank/handout20.pdf : \psi(x,y) = \frac{1}{\pi\sigma^4}\left(1-\frac{1}{2} \left(\frac{x^2+y^2}{\sigma^2}\right)\right) e^{-\frac{x^2+y^2}{2\sigma^2}} thumb|3D view of 2D Mexican hat wavelet The multidimensional generalization of this wavelet is called the Laplacian of Gaussian function.

0.70710678

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A

Find the bonding and antibonding Hückel molecular orbitals for ethene.

A particular molecular orbital may be bonding with respect to some adjacent pairs of atoms and antibonding with respect to other pairs. Bonding and antibonding orbitals form when atoms combine into molecules. thumb|right|150px|H2 1sσ* antibonding molecular orbital In theoretical chemistry, an antibonding orbital is a type of molecular orbital that weakens the chemical bond between two atoms and helps to raise the energy of the molecule relative to the separated atoms. A molecular orbital becomes antibonding when there is less electron density between the two nuclei than there would be if there were no bonding interaction at all. If the bonding orbitals are filled, then any additional electrons will occupy antibonding orbitals. The four electrons occupy one bonding orbital at lower energy, and one antibonding orbital at higher energy than the atomic orbitals. Antibonding orbitals are also important for explaining chemical reactions in terms of molecular orbital theory. Antibonding orbitals are often labelled with an asterisk (*) on molecular orbital diagrams. Roald Hoffmann and Kenichi Fukui shared the 1981 Nobel Prize in Chemistry for their work and further development of qualitative molecular orbital explanations for chemical reactions. ==See also== *Bonding molecular orbital *Valence and conduction bands *Valence bond theory *Molecular orbital theory *Conjugated system ==References== ==Further reading== * Orchin, M. Jaffe, H.H. (1967) The Importance of Antibonding Orbitals. Similarly benzene with six carbon atoms has three bonding pi orbitals and three antibonding pi orbitals. When a molecular orbital changes sign (from positive to negative) at a nodal plane between two atoms, it is said to be antibonding with respect to those atoms. right|400px|the first four radialenes are alicyclic organic compounds containing n cross-conjugated exocyclic double bonds. There are two bonding pi orbitals which are occupied in the ground state: π1 is bonding between all carbons, while π2 is bonding between C1 and C2 and between C3 and C4, and antibonding between C2 and C3. If the bonding interactions outnumber the antibonding interactions, the MO is said to be bonding, whereas, if the antibonding interactions outnumber the bonding interactions, the molecular orbital is said to be antibonding. Since the antibonding orbital is more antibonding than the bonding orbital is bonding, the molecule has a higher energy than two separated helium atoms, and it is therefore unstable. ==Polyatomic molecules== thumb|right|200px|Butadiene pi molecular orbitals. There are also antibonding pi orbitals with two and three antibonding interactions as shown in the diagram; these are vacant in the ground state, but may be occupied in excited states. The higher-energy orbital is the antibonding orbital, which is less stable and opposes bonding if it is occupied. PLATO (Package for Linear-combination of ATomic Orbitals) is a suite of programs for electronic structure calculations. But-2-ene () is an acyclic alkene with four carbon atoms. The double bonds are commonly alkene groups but those with a carbonyl (C=O) group are also called radialenes. This overlap leads to the formation of a bonding molecular orbital with three nodal planes which contain the internuclear axis and go through both atoms. Since each carbon atom contributes one electron to the π-system of benzene, there are six pi electrons which fill the three lowest-energy pi molecular orbitals (the bonding pi orbitals).

0.396

0.70710678

655.0

0.24995

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B

Using the explicit formulas for the Hermite polynomials given in Table 5.3, show that a $0 \rightarrow 1$ vibrational transition is allowed and that a $0 \rightarrow 2$ transition is forbidden.

From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html The first Hermite numbers are: :H_0 = 1\, :H_1 = 0\, :H_2 = -2\, :H_3 = 0\, :H_4 = +12\, :H_5 = 0\, :H_6 = -120\, :H_7 = 0\, :H_8 = +1680\, :H_9 =0\, :H_{10} = -30240\, ==Recursion relations== Are obtained from recursion relations of Hermitian polynomials for x = 0: :H_{n} = -2(n-1)H_{n-2}.\,\\! } H_{n-2m}(x). ===Generating function=== The Hermite polynomials are given by the exponential generating function \begin{align} e^{xt - \frac12 t^2} &= \sum_{n=0}^\infty \mathit{He}_n(x) \frac{t^n}{n!}, \\\ e^{2xt - t^2} &= \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}. \end{align} This equality is valid for all complex values of and , and can be obtained by writing the Taylor expansion at of the entire function (in the physicist's case). H_{n-m}(x) &&= 2^m m! \binom{n}{m} H_{n-m}(x). \end{align} It follows that the Hermite polynomials also satisfy the recurrence relation \begin{align} \mathit{He}_{n+1}(x) &= x\mathit{He}_n(x) - n\mathit{He}_{n-1}(x), \\\ H_{n+1}(x) &= 2xH_n(x) - 2nH_{n-1}(x). \end{align} These last relations, together with the initial polynomials and , can be used in practice to compute the polynomials quickly. The corresponding expressions for the physicist's Hermite polynomials follow directly by properly scaling this: x^n = \frac{n!}{2^n} \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{1}{m!(n-2m)! Since H0 = 1 and H1 = 0 one can construct a closed formula for Hn: :H_n = \begin{cases} 0, & \mbox{if }n\mbox{ is odd} \\\ (-1)^{n/2} 2^{n/2} (n-1)!! , & \mbox{if }n\mbox{ is even} \end{cases} where (n - 1)!! = 1 × 3 × ... × (n - 1). ==Usage== From the generating function of Hermitian polynomials it follows that :\exp (-t^2 + 2tx) = \sum_{n=0}^\infty H_n (x) \frac {t^n}{n!}\,\\! In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. ==Definition== The polynomials are given in terms of basic hypergeometric functions by :H_n(x|q)=e^{in\theta}{}_2\phi_0\left[\begin{matrix} q^{-n},0\\\ -\end{matrix} ;q,q^n e^{-2i\theta}\right],\quad x=\cos\,\theta. ==Recurrence and difference relations== : 2x H_n(x\mid q) = H_{n+1} (x\mid q) + (1-q^n) H_{n-1} (x\mid q) with the initial conditions : H_0 (x\mid q) =1, H_{-1} (x\mid q) = 0 From the above, one can easily calculate: : \begin{align} H_0 (x\mid q) & = 1 \\\ H_1 (x\mid q) & = 2x \\\ H_2 (x\mid q) & = 4x^2 - (1-q) \\\ H_3 (x\mid q) & = 8x^3 - 2x(2-q-q^2) \\\ H_4 (x\mid q) & = 16x^4 - 4x^2(3-q-q^2-q^3) + (1-q-q^3+q^4) \end{align} ==Generating function== : \sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1} {\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty} where \textstyle x=\cos \theta. ==References== * * * * Category:Orthogonal polynomials Category:Q-analogs Category:Special hypergeometric functions Hermite polynomials were defined by Pierre-Simon Laplace in 1810, Collected in Œuvres complètes VII. though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. These numbers may also be expressed as a special value of the Hermite polynomials: T(n) = \frac{\mathit{He}_n(i)}{i^n}. === Completeness relation === The Christoffel–Darboux formula for Hermite polynomials reads \sum_{k=0}^n \frac{H_k(x) H_k(y)}{k!2^k} = \frac{1}{n!2^{n+1}}\,\frac{H_n(y) H_{n+1}(x) - H_n(x) H_{n+1}(y)}{x - y}. The Hermite functions satisfy the differential equation \psi_n(x) + \left(2n + 1 - x^2\right) \psi_n(x) = 0. Since they are an Appell sequence, they are a fortiori a Sheffer sequence. ==Contour-integral representation== From the generating- function representation above, we see that the Hermite polynomials have a representation in terms of a contour integral, as \begin{align} \mathit{He}_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{tx-\frac{t^2}{2}}}{t^{n+1}}\,dt, \\\ H_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{2tx-t^2}}{t^{n+1}}\,dt, \end{align} with the contour encircling the origin. ==Generalizations== The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}, which has expected value 0 and variance 1. One can also derive the (physicist's) generating function by using Cauchy's integral formula to write the Hermite polynomials as H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} = (-1)^n e^{x^2} \frac{n!}{2\pi i} \oint_\gamma \frac{e^{-z^2}}{(z-x)^{n+1}} \,dz. In terms of the probabilist's polynomials this translates to He_n(0) = \begin{cases} 0 & \text{for odd }n, \\\ (-1)^\frac{n}{2} (n-1)!! & \text{for even }n. \end{cases} ==Relations to other functions== ===Laguerre polynomials=== The Hermite polynomials can be expressed as a special case of the Laguerre polynomials: \begin{align} H_{2n}(x) &= (-4)^n n! In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. The polynomials are sometimes denoted by , especially in probability theory, because \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} is the probability density function for the normal distribution with expected value 0 and standard deviation 1. thumb|right|390px|The first six probabilist's Hermite polynomials thumb|right|390px|The first six (physicist's) Hermite polynomials * The first eleven probabilist's Hermite polynomials are: \begin{align} \mathit{He}_0(x) &= 1, \\\ \mathit{He}_1(x) &= x, \\\ \mathit{He}_2(x) &= x^2 - 1, \\\ \mathit{He}_3(x) &= x^3 - 3x, \\\ \mathit{He}_4(x) &= x^4 - 6x^2 + 3, \\\ \mathit{He}_5(x) &= x^5 - 10x^3 + 15x, \\\ \mathit{He}_6(x) &= x^6 - 15x^4 + 45x^2 - 15, \\\ \mathit{He}_7(x) &= x^7 - 21x^5 + 105x^3 - 105x, \\\ \mathit{He}_8(x) &= x^8 - 28x^6 + 210x^4 - 420x^2 + 105, \\\ \mathit{He}_9(x) &= x^9 - 36x^7 + 378x^5 - 1260x^3 + 945x, \\\ \mathit{He}_{10}(x) &= x^{10} - 45x^8 + 630x^6 - 3150x^4 + 4725x^2 - 945. \end{align} * The first eleven physicist's Hermite polynomials are: \begin{align} H_0(x) &= 1, \\\ H_1(x) &= 2x, \\\ H_2(x) &= 4x^2 - 2, \\\ H_3(x) &= 8x^3 - 12x, \\\ H_4(x) &= 16x^4 - 48x^2 + 12, \\\ H_5(x) &= 32x^5 - 160x^3 + 120x, \\\ H_6(x) &= 64x^6 - 480x^4 + 720x^2 - 120, \\\ H_7(x) &= 128x^7 - 1344x^5 + 3360x^3 - 1680x, \\\ H_8(x) &= 256x^8 - 3584x^6 + 13440x^4 - 13440x^2 + 1680, \\\ H_9(x) &= 512x^9 - 9216x^7 + 48384x^5 - 80640x^3 + 30240x, \\\ H_{10}(x) &= 1024x^{10} - 23040x^8 + 161280x^6 - 403200x^4 + 302400x^2 - 30240. \end{align} ==Properties== The th-order Hermite polynomial is a polynomial of degree . The Hermite polynomial is then represented as H_n(x) = (-1)^n e^{x^2} \frac {d^n}{dx^n} \left( \frac {1}{2\sqrt{\pi}} \int e^{isx - \frac{s^2}{4}} \,ds \right) = (-1)^n e^{x^2}\frac{1}{2\sqrt{\pi}} \int (is)^n e^{isx - \frac{s^2}{4}} \,ds. A special case of the cross-sequence identity then says that \sum_{k=0}^n \binom{n}{k} \mathit{He}_k^{[\alpha]}(x) \mathit{He}_{n-k}^{[-\alpha]}(y) = \mathit{He}_n^{[0]}(x + y) = (x + y)^n. ==Applications== ===Hermite functions=== One can define the Hermite functions (often called Hermite- Gaussian functions) from the physicist's polynomials: \psi_n(x) = \left (2^n n! \sqrt{\pi} \right )^{-\frac12} e^{-\frac{x^2}{2}} H_n(x) = (-1)^n \left (2^n n! \sqrt{\pi} \right)^{-\frac12} e^{\frac{x^2}{2}}\frac{d^n}{dx^n} e^{-x^2}. In mathematics, the discrete q-Hermite polynomials are two closely related families hn(x;q) and ĥn(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. hn(x;q) is also called discrete q-Hermite I polynomials and ĥn(x;q) is also called discrete q-Hermite II polynomials. ==Definition== The discrete q-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-Salam–Carlitz polynomials by :\displaystyle h_n(x;q)=q^{\binom{n}{2}}{}_2\phi_1(q^{-n},x^{-1};0;q,-qx) = x^n{}_2\phi_0(q^{-n},q^{-n+1};;q^2,q^{2n-1}/x^2) = U_n^{(-1)}(x;q) :\displaystyle \hat h_n(x;q)=i^{-n}q^{-\binom{n}{2}}{}_2\phi_0(q^{-n},ix;;q,-q^n) = x^n{}_2\phi_1(q^{-n},q^{-n+1};0;q^2,-q^{2}/x^2) = i^{-n}V_n^{(-1)}(ix;q) and are related by :h_n(ix;q^{-1}) = i^n\hat h_n(x;q) ==References== * * * * * * Category:Orthogonal polynomials Category:Q-analogs Category:Special hypergeometric functions Similarly, \begin{align} H_{2n}(x) &= (-1)^n \frac{(2n)!}{n!} \,_1F_1\big(-n, \tfrac12; x^2\big), \\\ H_{2n+1}(x) &= (-1)^n \frac{(2n+1)!}{n!}\,2x \,_1F_1\big(-n, \tfrac32; x^2\big), \end{align} where is Kummer's confluent hypergeometric function. ==Differential-operator representation== The probabilist's Hermite polynomials satisfy the identity \mathit{He}_n(x) = e^{-\frac{D^2}{2}}x^n, where represents differentiation with respect to , and the exponential is interpreted by expanding it as a power series. In mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. ==Definition== The polynomials are given in terms of basic hypergeometric functions. Reference gives a formal power series: :H_n (x) = (H+2x)^n\,\\! where formally the n-th power of H, Hn, is the n-th Hermite number, Hn. The Hermite functions are closely related to the Whittaker function : D_n(z) = \left(n! \sqrt{\pi}\right)^{\frac12} \psi_n\left(\frac{z}{\sqrt 2}\right) = (-1)^n e^\frac{z^2}{4} \frac{d^n}{dz^n} e^\frac{-z^2}{2} and thereby to other parabolic cylinder functions.

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To a good approximation, the microwave spectrum of $\mathrm{H}^{35} \mathrm{Cl}$ consists of a series of equally spaced lines, separated by $6.26 \times 10^{11} \mathrm{~Hz}$. Calculate the bond length of $\mathrm{H}^{35} \mathrm{Cl}$.

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 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. Bond lengths are measured in the solid phase by means of X-ray diffraction, or approximated in the gas phase by microwave spectroscopy. 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. Unusually long bond lengths do exist. The carbon–carbon (C–C) bond length in diamond is 154 pm. 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 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. Bond lengths are given in picometers. 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. Longest C-C bond within the cyclobutabenzene category is 174 pm based on X-ray crystallography. The W band of the microwave part of the electromagnetic spectrum ranges from 75 to 110 GHz, wavelength ≈2.7–4 mm. In molecular geometry, bond length or bond distance is defined as the average distance between nuclei of two bonded atoms in a molecule. The smallest theoretical C–C single bond obtained in this study is 131 pm for a hypothetical tetrahedrane derivative. It is generally considered the average length for a carbon–carbon single bond, but is also the largest bond length that exists for ordinary carbon covalent bonds. 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). Structural formula 104x104px 100x100px 100x100px 107x107px Name Fluoromethane Chloromethane Bromomethane Iodomethane Melting point −137,8 °C −97,4 °C −93,7 °C −66 °C Boiling point −78,4 °C −23,8 °C 4,0 °C 42 °C Space-filling model 90x90px 110x110px 120x120px 130x130px The monohalomethanes are organic compounds in which a hydrogen atom in methane is replaced by a halogen. The 21 cm L/35 were a family of German naval artillery developed in the years before World War I and used in limited numbers. In a bond between two identical atoms, half the bond distance is equal to the covalent radius. 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 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.

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-31.95

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A

The unit of energy in atomic units is given by $$ 1 E_{\mathrm{h}}=\frac{m_{\mathrm{e}} e^4}{16 \pi^2 \epsilon_0^2 \hbar^2} $$ Express $1 E_{\mathrm{h}}$ in units of joules (J), kilojoules per mole $\left(\mathrm{kJ} \cdot \mathrm{mol}^{-1}\right)$, wave numbers $\left(\mathrm{cm}^{-1}\right)$, and electron volts $(\mathrm{eV})$.

Other units sometimes used to describe reaction energetics are kilocalories per mole (kcal·mol−1), electron volts per particle (eV), and wavenumbers in inverse centimeters (cm−1). 1 kJ·mol−1 is approximately equal to 1.04 eV per particle, 0.239 kcal·mol−1, or 83.6 cm−1. In slightly more fundamental terms, is equal to 1 newton metre and, in terms of SI base units :1\ \mathrm{J} = 1\ \mathrm{kg} \left( \frac{\mathrm{m}}{\mathrm{s}} \right ) ^ 2 = 1\ \frac{\mathrm{kg} \cdot \mathrm{m}^2}{\mathrm{s}^2} An energy unit that is used in atomic physics, particle physics and high energy physics is the electronvolt (eV). The electron relative atomic mass is an adjusted parameter in the CODATA set of fundamental physical constants, while the electron rest mass in kilograms is calculated from the values of the Planck constant, the fine-structure constant and the Rydberg constant, as detailed above. == Relationship to other physical constants == The electron mass is used to calculate the Avogadro constant : :N_{\rm A} = \frac{M_{\rm u} A_{\rm r}({\rm e})}{m_{\rm e}} = \frac{M_{\rm u} A_{\rm r}({\rm e})c\alpha^2}{2R_\infty h} . This list compares various energies in joules (J), organized by order of magnitude. ==Below 1 J== List of orders of magnitude for energy Factor (joules) SI prefix Value Item 10−34 6.626×10−34J Photon energy of a photon with a frequency of 1 hertz. 10−33 2×10−33J Average kinetic energy of translational motion of a molecule at the lowest temperature reached, 100 picokelvins Calculated: KE ≈ (3/2) × T × 1.38 = (3/2) × 1 × 1.38 ≈ 2.07J 10−30 quecto- (qJ) 10−28 6.6×10−28J Energy of a typical AM radio photon (1 MHz) (4×10−9 eV)Calculated: E = hν = 6.626J-s × 1 Hz = 6.6J. This very small amount of energy is often expressed in terms of an even smaller unit such as the kJ·mol−1, because of the typical order of magnitude for energy changes in chemical processes. In spectroscopy the unit cm−1 ≈ is used to represent energy since energy is inversely proportional to wavelength from the equation E = h u = h c/\lambda . In the European Union, food energy labeling in joules is mandatory, often with calories as supplementary information. ==Atom physics and chemistry== In physics and chemistry, it is common to measure energy on the atomic scale in the non-SI, but convenient, units electronvolts (eV). 1 eV is equivalent to the kinetic energy acquired by an electron in passing through a potential difference of 1 volt in a vacuum. The electron rest mass can be calculated from the Rydberg constant and the fine-structure constant obtained through spectroscopic measurements. In eV: 6.6J / 1.6J/eV = 4.1 eV. 10−27 ronto- (rJ) 10−24 yocto- (yJ) 1.6×10−24J Energy of a typical microwave oven photon (2.45 GHz) (1×10−5 eV)Calculated: E = hν = 6.626J-s × 2.45 Hz = 1.62J. At room temperature (25 °C, or 298.15 K) 1 kJ·mol−1 is approximately equal to 0.4034 k_B T. == References == Category:SI derived units Hence it is also related to the atomic mass constant : :m_{\rm u} = \frac{M_{\rm u}}{N_{\rm A}} = \frac{m_{\rm e}}{A_{\rm r}({\rm e})} = \frac{2R_\infty h}{A_{\rm r}({\rm e})c\alpha^2} , where * is the molar mass constant (defined in SI); * is a directly measured quantity, the relative atomic mass of the electron. In eV: 13J / 6.022 molecules/mol / 1.6 eV/J = 0.13 eV. 10−20 4.5×10−20J Upper bound of the mass–energy of a neutrino in particle physics (0.28 eV)Calculated: 0.28 eV × 1.6J/eV = 4.5J 10−19 1.6×10−19J ≈1 electronvolt (eV) 10−19 3–5×10−19J Energy range of photons in visible light (≈1.6–3.1 eV)Calculated: E = hc/λ. In SI units, one kilocalorie per mole is equal to 4.184 kilojoules per mole (kJ/mol), which comes to approximately joules per molecule, or about 0.043 eV per molecule. Energy is defined via work, so the SI unit of energy is the same as the unit of work - the joule (J), named in honour of James Prescott Joule and his experiments on the mechanical equivalent of heat. For example, the Gibbs free energy of a compound in the area of thermochemistry is often quantified in units of kilojoules per mole (symbol: kJ·mol−1 or kJ/mol), with 1 kilojoule = 1000 joules. Historically Rydberg units have been used. ==Spectroscopy== In spectroscopy and related fields it is common to measure energy levels in units of reciprocal centimetres. These units (cm−1) are strictly speaking not energy units but units proportional to energies, with \ hc\sim 2\cdot 10^{-23}\ \mathrm{J}\ \mathrm{cm} being the proportionality constant. ==Explosions== A gram of TNT releases upon explosion. 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. Constant Values Units kg Da MeV/c2 J MeV In particle physics, the electron mass (symbol: ) is the mass of a stationary electron, also known as the invariant mass of the electron. Note that is defined in terms of , and not the other way round, and so the name "electron mass in atomic mass units" for involves a circular definition (at least in terms of practical measurements). The joule per mole (symbol: J·mol−1 or J/mol) is the unit of energy per amount of substance in the International System of Units (SI), such that energy is measured in joules, and the amount of substance is measured in moles.

92

15.425

27.211

22.2

311875200

C

Calculate the probability that a particle in a one-dimensional box of length $a$ is found between 0 and $a / 2$.

Finally, the unknown constant A may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. The particle in a box model is one of the very few problems in quantum mechanics which can be solved analytically, without approximations. The potential energy in this model is given as V(x) = \begin{cases} 0, & x_c-\tfrac{L}{2} < x where L is the length of the box, xc is the location of the center of the box and x is the position of the particle within the box. The kinetic energy of a particle is given by E = p^2/(2m), and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above. ==Higher-dimensional boxes== ===(Hyper)rectangular walls=== thumb|320px|right|The wavefunction of a 2D well with nx=4 and ny=4 If a particle is trapped in a two-dimensional box, it may freely move in the x and y-directions, between barriers separated by lengths L_x and L_y respectively. However, the particle in a box may only have certain, discrete energy levels. For the particle in a box, the probability density for finding the particle at a given position depends upon its state, and is given by P_n(x,t) = \begin{cases} \frac{2}{L} \sin^2\left(k_n \left(x-x_c+\tfrac{L}{2}\right)\right), & x_c-\frac{L}{2} < x < x_c+\frac{L}{2},\\\ 0, & \text{otherwise,} \end{cases} Thus, for any value of n greater than one, there are regions within the box for which P(x)=0, indicating that spatial nodes exist at which the particle cannot be found. The boxcar function can be expressed in terms of the uniform distribution as \operatorname{boxcar}(x)= (b-a)A\,f(a,b;x) = A(H(x-a) - H(x-b)), where is the uniform distribution of x for the interval and H(x) is the Heaviside step function. This can be explained in terms of the uncertainty principle, which states that the product of the uncertainties in the position and momentum of a particle is limited by \Delta x\Delta p \geq \frac{\hbar}{2} It can be shown that the uncertainty in the position of the particle is proportional to the width of the box.Davies, p. 15 Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box. 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. Using a similar approach to that of the one-dimensional box, it can be shown that the wavefunctions and energies for a centered box are given respectively by \psi_{n_x,n_y} = \psi_{n_x}(x,t,L_x)\psi_{n_y}(y,t,L_y), E_{n_x,n_y} = \frac{\hbar^2 k_{n_x,n_y}^2}{2m}, where the two-dimensional wavevector is given by \mathbf{k}_{n_x,n_y} = k_{n_x}\mathbf{\hat{x}} + k_{n_y}\mathbf{\hat{y}} = \frac{n_x \pi }{L_x} \mathbf{\hat{x}} + \frac{n_y \pi }{L_y} \mathbf{\hat{y}}. For a centered box, the position wave function may be written including the length of the box as \psi_n(x,t,L). For a shifted box (xc = L/2), the solution is particularly simple. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end.Davies, p.4 The walls of a one-dimensional box may be seen as regions of space with an infinitely large potential energy. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. This can be seen in the following equation, where m^*_e and m^*_h are the effective masses of the electron and hole, r is radius of the dot, and h is Planck's constant: \Delta E(r) = E_{\text{gap}}+\left ( \frac{h^2}{8r^2} \right ) \left( \frac{1}{m^*_e}+\frac{1}{m^*_h} \right) Hence, the energy gap of the quantum dot is inversely proportional to the square of the “length of the box,” i.e. the radius of the quantum dot. The Box is a 2007 American crime film starring Gabrielle Union, A.J. Buckley, RZA, Giancarlo Esposito, Jason Winston George, Brett Donowho and written and directed by A.J. Kparr. ==Plot== A disgraced former LAPD cop leads a home invasion in search of millions in stolen money. In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems. == One-dimensional solution == thumb|right|The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential. Shown is the shifted well, with x_c = L/2 The simplest form of the particle in a box model considers a one-dimensional system. In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wavefunction as P(x) = |\psi(x)|^2. If we set the origin of coordinates to the center of the box, we can rewrite the spacial part of the wave function succinctly as: \psi_n (x) = \begin{cases} \sqrt{\frac{2}{L}} \sin(k_nx) \quad{} \text{for } n \text{ even} \\\ \sqrt{\frac{2}{L}} \cos(k_nx) \quad{} \text{for } n \text{ odd}. \end{cases} === Momentum wave function === The momentum wavefunction is proportional to the Fourier transform of the position wavefunction. Mathematically, \int_0^L \left\vert \psi(x) \right\vert^2 dx = 1 (The particle must be somewhere) It follows that \left| A \right| = \sqrt{\frac{2 }{L}}.

0.5

1.7

6.6

0.87

2283.63

A

What happens to you if you eat watermelon seeds?

Strange as it seems, seedless watermelon plants are grown from seeds. Seedless cucumbers are an example of vegetative parthenocarpy, seedless watermelon is an example of stenospermocarpy as they are immature seeds (aborted ones). Eating before going to sleep, which triggers an increase in the body's metabolism and brain activity, can be a potential stimulus for nightmares. Eating amygdalin will cause it to release cyanide in the human body, and may lead to cyanide poisoning. In addition, contact your doctor immediately if the diarrhea symptoms remain severe or if your child exhibits any symptoms of dehydration, such as dry mouth, lack of urination, listlessness, and rapid heart rate. ==Alternatives== An alternative to the CRAM diet is the BRAT diet, which consists of bananas, rice, applesauce, and toast, the BRATT diet, which consists of bananas, rice, applesauce, toast, and tea, and the BRATTY diet which consists of bananas, rice, applesauce, toast, tea, and yogurt. == References == == External links == *CRAM Diet - Fitness website Category:Diets thumbnail|Square watermelon from Japan Square or cube watermelons are watermelons grown into the shape of a cube. right|thumb|Seedless watermelon In botany and horticulture, parthenocarpy is the natural or artificially induced production of fruit without fertilisation of ovules, which makes the fruit seedless. Amygdalin (from Ancient Greek: "almond") is a naturally occurring chemical compound found in many plants, most notably in the seeds (kernels) of apricots, bitter almonds, apples, peaches, cherries and plums, and in the roots of manioc. Stenospermocarpy may also produce apparently seedless fruit, but the seeds are actually aborted while they are still small. Amygdalin is contained in stone fruit kernels, such as almonds, apricot (14 g/kg), peach (6.8 g/kg), and plum (4–17.5 g/kg depending on variety), and also in the seeds of the apple (3 g/kg). Watermelon rind preserves are made by boiling chunks of watermelon rind with sugar and other ingredients. For one method of isolating amygdalin, the stones are removed from the fruit and cracked to obtain the kernels, which are dried in the sun or in ovens. A nightmare, also known as a bad dream, Retrieved 11 July 2016. is an unpleasant dream that can cause a strong emotional response from the mind, typically fear but also despair, anxiety, disgust or sadness. The CRAM diet (cereal, rice, applesauce, and milk) is a short term dietary treatment for diarrhea and gastroenteritis. The European Food Safety Agency's Panel on Contaminants in the Food Chain has studied the potential toxicity of the amygdalin in apricot kernels. If bad thoughts in a dream are more prominent than good thoughts, the dream may proceed to be a nightmare. Since the advent of the cube watermelon, other watermelon shapes have been introduced, such as hearts and pyramids. The dream may contain situations of discomfort, psychological or physical terror, or panic. There is a view, possibly featured in the story A Christmas Carol, that eating cheese before sleep can cause nightmares, but there is little scientific evidence for this. He presented the watermelons in a gallery in Ginza, Tokyo.Melonen nun im Quadrat, Arbeiter- Zeitung, 1978-08-21, p. 5 (Austrian Newspaper, German) He also applied for and received a patent in the United States.Molding process for a natural fruit of a fruit-tree or vegetable, a molding frame therefor and molded above fruit. The CRAM diet has more complete protein and fat content than the BRAT diet. ==Recent research== The use of cereals, rice and milk as a stop-gap eating plan for stomach upset, has been validated as a more effective remedy to manage diarrhea than BRAT by recent research in hospitals in South America and Asia. Typical dreams or recurrent dreams may also have some of these topics. ==Cause== Scientific research shows that nightmares may have many causes.

You have bad dreams

You fall unconscious

You get indigestion

You get sick

The watermelon seeds pass through your digestive system

E

Where did fortune cookies originate?

Several publications make the claim that fortune cookies are derived from tsujiura senbei. ==References== Category:Japanese stationery Category:Japanese words and phrases They make traditional fortune cookies, as well as chocolate flavored fortune cookies, almond cookies, and other sweets. The classic peanut butter blossom cookie can be easily adapted for different occasions. ==Overview== The cookie originated in Gibsonburg, Ohio, as an entry into the 1957 Pillsbury Bake-Off contest. thumbnail|Baking Japanese fortune cookies, Tsujiura Senbei in the Edo period (1603-1868). from a book written in 1878 are notes used in Japan in conjunction with rice crackers called senbei in a similar way to fortune cookies. The peanut butter blossom cookie originated in 1957, is made with a peanut butter cookie dough, and is topped with a piece of chocolate candy. thumb|right|Golden Gate Fortune Cookie Factory The Golden Gate Fortune Cookie Factory () is a fortune cookie company with its main entrance off Ross Alley, between Jackson Street and Washington Street in the Chinatown neighborhood of San Francisco, California in the United States. Fancy Fortune Cookies is a company, founded by Mike L. Fry in 1988, that produces custom fortune cookies. ==History== thumb|Mike L. Fry In late 1987, former circus clown and children's show host Mike Fry got the idea to produce custom fortune cookies, and left his job on the children's television series Happy's Place to start the company. The company also makes "fortuneless" cookies. Honolulu Cookie Company is a food and gift producer and retailer based in Honolulu, Hawaii. The company continued to innovate, introducing giant fortune cookies in 1999. The company created an online retail site in 1995, making it the first on-line seller of flavored and gourmet fortune cookies. The cookie company was opened in 1962. The Honolulu Cookie Company began opening stores in Guam in 2016, and during the same year the pumpkin flavor was introduced. The original cookie recipe can be found on the back of the Hershey's Kisses bag, and in the 9th Pillsbury Bake-Off Contest cookbook Authors of dessert recipe books, cooking blogs and websites have since created their own variations on the cookie. == History == ===Invention=== Freda Strasel Smith of Gibsonburg, Ohio, created the cookie by substituting chocolate chips out for Hershey's Kisses in a batch of peanut butter cookie dough. The appearance of Cookies proved to be nostalgic, as Cookies and Shine had debuted near the same time. The pineapple shape was chosen because of its symbolism of hospitality associated with Hawaii. == History == In 1998, Honolulu Cookie Company introduced its first cookies to the wholesale market. In 2011 they began selling caramel- dipped fortune cookies, another innovation because they don't require special shipping like chocolate-dipped cookies do. ==In the Media== Fry's success with Fancy Fortune Cookies has led to him being featured in books and articles. Cookies were a music group in Hong Kong created by Chan Chi Kwong under the giant record label EMI Hong Kong and later managed by Paco Wong of then Gold Label Entertainment (now known as Gold Typhoon). ==History== The group was formed with nine members in 2002 and was considered an emulation of the Japanese group Morning Musume.That same year the group released their debut album Happy Birthday.KKBOX: Cookies - Happy Birthday This helped the group become one of the best selling new groups of 2002 in Hong Kong.Hong Kong Record Sales Award 2002 However, in 2003, the management contract of Cookies was passed over to Gold Label Entertainment. Lee, she mentions Fancy Fortune Cookies, and Mike Fry’s invention of the gourmet fortune cookie as part of the history of the fortune cookie. The peanut butter blossom cookie went through to the final round of the competition held in Beverly Hills, California, and finished in third place. ===Later history=== The peanut butter blossom cookie has become a recognized dessert across the US, largely due to Pillsbury and the Hershey Company capitalizing on the popularity of the cookie after the contest by using the recipe to promote their own brands. The cookie is considered a snack or dessert and is often served at events or during holidays in the United States. 何綺玲婚宴 傅穎沒獲邀 * Gloria Chan (陳素瑩) Chan had left the entertainment industry after the downsizing of Cookies to Mini Cookies in 2003.

Fortune cookies originated in Kyoto

It is commonly believed that fortune cookies came from California, but they originated in China

The precise origin of fortune cookies is unclear

Fortune cookies originated in Los Angeles

Fortune cookies originated in China

C

Why do veins appear blue?

Normally, oxygenated blood appears red and deoxygenated blood has more of a blue appearance. The blood reaching the extremities is not oxygen-rich and when viewed through the skin a combination of factors can lead to the appearance of a blue color. In these defects, some of the blood that is pumped to the body has not been oxygenated by the lungs and therefore will appear more blue. In babies with low levels of oxygen or mixing of oxygenated and deoxygenated blood, the blood can have a blue or purple color, causing cyanosis. == Signs and symptoms == The main sign of blue baby syndrome is cyanosis. Blue Veins may refer to: *"Blue Veins", a song by The Raconteurs on their 2006 album Broken Boy Soldiers *Blue Veins (Pakistan), a women's health advocacy group *Blue Veins (TV series), a 2016 Hong Kong vampire series ==See also== *Blue Vein, a hamlet in Wiltshire, England thumbnail|right|Blood circulation: Red = oxygenated (arteries), Blue = deoxygenated (veins) Oxygen saturation is the fraction of oxygen-saturated hemoglobin relative to total hemoglobin (unsaturated + saturated) in the blood. Blue pigments are natural or synthetic materials, usually made from minerals and insoluble with water, used to make the blue colors in painting and other arts. Blue baby syndrome can refer to conditions that cause cyanosis, or blueness of the skin, in babies as a result of low oxygen levels in the blood. Oxygenated and deoxygenated hemoglobin differ in absorption of light of different wavelengths. The blue baby syndrome or cyanosis occurs when absolute amount of deoxygenated hemoglobin >3g/dL which is typically reflected with an O2 saturation of <85%. Cyanosis is the change of body tissue color to a bluish-purple hue as a result of having decreased amounts of oxygen bound to the hemoglobin in the red blood cells of the capillary bed. Since the late 18th and 19th century, blue pigments are largely synthetic, manufactured in laboratories and factories. == Ultramarine == Ultramarine was historically the most prestigious and expensive of blue pigments. Air Force blue colours are a variety of colours that are mostly various tones of the colour azure, the purest tones of which are identified as being the colour of the sky on a clear day. However, they are actually dark tones of azure, not blue. Egyptian blue is responsible for the blue colour seen very commonly in Egyptian faience. This is possible because methylene blue oxidizes NADPH, which in turn can convert methemoglobin back to hemoglobin. == Epidemiology == Out of all the babies born with congenital heart defects, about 25% have cyanosis as a result. These air force blue colours are used by these various air forces for colour identification. ==RAF blue== Air Force blue, more specifically Air Force blue (RAF) or RAF blue, is a medium shade of the colour azure. Some air force blue colours, notably the air force blue colour used by the United States Air Force and the colour used by the US Air Force Academy, may look like they are tones of blue instead of azure. It is derived from the color cyan, which comes from cyanós (κυανός), the Greek word for blue. A Dictionary of Color New York: McGraw Hill p. 206 Synthetic ultramarine was widely appreciated by the French impressionists, and Vincent van Gogh used both French ultramarine and cobalt blue for his painting The Starry Night (1889).Yonghui Zhao, Roy S. Berns, Lawrence A. Taplin, James Coddington, An Investigation of Multispectral Imaging for the Mapping of Pigments in Paintings, in Proc. SPIE 6810, Computer Image Analysis in the Study of Art, 681007 (29 February 2008) File:Ultramarinepigment.jpg|Synthetic ("French") ultramarine File:Van Gogh - Starry Night - Google Art Project.jpg|The Starry Night (1899) Vincent van Gogh == Cobalt blue == Cobalt blue is a synthetic blue pigment was invented in 1803 as a rival to ultramarine. In contrast, peripheral cyanosis typically has a blueish discoloration over the extremities. Although these are the most common causes of cyanosis, there are other potential factors that can cause a blue tint to a baby's skin or mucous membranes.

Veins appear blue because the blood in veins is heading back to the lungs

Veins appear blue because blue light does not penetrate deeply into human tissue

Veins appear blue because deoxygenated blood is blue

Veins appear blue due to the deoxygenated blood

People think that veins appear blue due to how blue and red light penetrate human tissue, but it's actually because deoxygenated blood is blue

B

What is the spiciest part of a chili pepper?

While capsaicin is present at some level in every part of the pepper, the chemical has its highest concentration in and around the seeds within chilies. While chili peppers are (to varying degrees) pungent or "spicy", there are other varieties of capsicum such as bell peppers, which generally provide additional sweetness and flavor to a meal rather than "heat". Chilies are sometimes used whole or in large slices, by roasting, or other means of blistering or charring the skin, so as not to entirely cook the flesh beneath. This is advantageous to the plant, as chili pepper seeds consumed by birds pass through the digestive tract and can germinate later, whereas mammals have molar teeth which destroy such seeds and prevent them from germinating. thumb|upright=1.3|Young chili plants Chili peppers (also chile, chile pepper, chilli pepper, or chilli), from Nahuatl chīlli (), are varieties of the berry- fruit of plants from the genus Capsicum, which are members of the nightshade family Solanaceae, cultivated for their pungency. Though there are only a few commonly used species, there are many cultivars and methods of preparing chili peppers that have different names for culinary use. Chili peppers are eaten by birds living in the chili peppers' natural range, possibly contributing to seed dispersal and evolution of the protective capsaicin in chili peppers, as a bird in flight can spread the seeds further away from the parent plant after they pass through its digestive system than any land or tree dwelling mammal could do so under the same circumstances, thus reducing competition for resources. == Nutritional value == Red hot chili peppers are 88% water, 9% carbohydrates, 2% protein, and 0.4% fat (table). Most popular pepper varieties are seen as falling into one of these categories or a cross between them. == Intensity == The substances that give chili peppers their pungency (spicy heat) when ingested or applied topically are capsaicin (8-methyl-N-vanillyl-6-nonenamide) and several related chemicals, collectively called capsaicinoids. Capsaicin and related compounds known as capsaicinoids are the substances giving chili peppers their intensity when ingested or applied topically. Chili peppers are widely used in many cuisines as a spice to add "heat" to dishes. Peppers are commonly broken down into two groupings: bell peppers (UK: sweet peppers) and hot peppers. Consuming hot peppers may cause stomach pain, hyperventilation, sweating, vomiting, and symptoms possibly requiring hospitalization. ==Gallery== File:Leiden University Library - Seikei Zusetsu vol. 25, page 019 - 蕃椒 - Capsicum annuum L., 1804.jpg|Illustration from the Japanese agricultural encyclopedia Seikei Zusetsu (1804) File:Habanero closeup edit2.jpg|The habanero pepper File:Buds and flowers of chili plants.jpg|Buds and flowers File:Chili pepper.jpg|Immature chilies in the field File:Shan Hills, Myanmar, Red chili pepper plant.jpg|Ripe chilies in the field, Myanmar File:Black pearl cultivar.jpg|The Black Pearl cultivar File:Cubanelle Peppers.jpg|Cubanelle peppers File:Chilli paper bd.jpg|Ripe chili pepper with seeds File:HotPeppersinMarket.jpg|Scotch bonnet chili peppers in a Caribbean market File:Chillies drying in Kathmandu.jpg|Chili peppers drying in Kathmandu, Nepal File:Mujer chiles 3.JPG|Removing seeds and pith from dried chilies in San Pedro Atocpan, Mexico File:2014 Dried chilli flakes.jpg|Dried chili pepper flakes and fresh chilies File:Chili pepper 01.JPG|Chili pepper dip in a traditional restaurant in Amman, Jordan File:Phrik haeng.jpg|Dried Thai bird's eye chilies File:Aesthetic green Chillies.JPG|Green chilies File:Andhra Chillies.jpg|Guntur chilli drying in the sun, Andhra Pradesh, India File:Sundried chilli.jpg|Sundried chili at Imogiri, Yogyakarta, Indonesia File:Red chili peppers Mesilla NM.jpg|New Mexico chiles dried on the plant in Mesilla, New Mexico File:A bottle of chili pepper wine.jpg|Chili pepper wine from Virginia File:Ristras Drying.jpg|alt=Chili peppers drying in hanging ristras|Ristras of chili peppers drying in Arizona File:White flower of chili paper plant.jpg|White flower of chili paper at night File:Pimientos choriceros.jpg|Choricero peppers File:Chilli pickle in a plate 2.jpg|Pickled chili in India File:Capsicum -Chili - Peperoncino - Il Viagra Calabrese - Calabria - Italy - July 17th 2013 - 02.jpg|Peperoncino chili in Tropea, Italy, with a sign saying il viagra calabrese ("the Calabrian viagra") File:Chili peppers cultivated in Myanmar.jpg|Chili peppers cultivated in Myanmar File:Inle Lake, Dried red chili (chilli) pepper, Capsicum annuum, Myanmar.jpg|Dried chili pepper flakes, Myanmar File:Green-chillies.jpg|Green Chillies from North India. Capsaicin (8-methyl-N-vanillyl-6-nonenamide) ( or ) is an active component of chili peppers, which are plants belonging to the genus Capsicum. Though almost all other Solanaceous crops have toxins in their leaves, chili peppers do not. Although they are structurally similar to capsaicin, the substance that causes pungency in hot peppers, they largely lack that characteristic. Despite the fact that chilies within the Capsicum genus are found throughout the world, the capsaicin found within them all exhibit similar properties that serve as defensive and adaptive features. In a 100 gram reference amount, chili peppers supply 40 calories, and are a rich source of vitamin C and vitamin B6 (table). == Spelling and usage == The three primary spellings are chili, chile and chilli, all of which are recognized by dictionaries. Peppers increased the quantity of capsaicin in proportion to the damage caused by fungal predation on the plant's seeds. ===Common peppers=== thumb|Red Bhut Jolokia and green bird's eye chilies A wide range of intensity is found in commonly used peppers: Bell pepper 0 SHU Fresno, jalapeño 3,500–10,000 SHU Cayenne 30,000–50,000 SHU Piri piri 50,000–100,000 SHU Habanero, Scotch bonnet, bird's eye 100,000–350,000 SHU === Notable hot chili peppers === The top 8 world's hottest chili peppers (by country) are: Country Type Hotness Pepper X 3.18M SHU(*) Dragon's Breath 2.48M SHU(*) Carolina Reaper 2.2M SHU Trinidad moruga scorpion 2.0M SHU(*) Ghost pepper (Bhut jolokia) 1.58M SHU Trinidad Scorpion Butch T 1.463M SHU Naga Viper 1.4M SHU Infinity chili 1.2M SHU NOTE: SHU claims marked with an asterisk (*) have not been confirmed by Guinness World Records. == Uses == === Culinary === thumb|Smoke-dried chipotle Chili pepper pods are technically berries. Chilies are present in many cuisines. Many fresh chilies such as poblano have a tough outer skin that does not break down on cooking. After the Columbian Exchange, many cultivars of chili pepper spread around the world, used for both food and traditional medicine. However, in scientific speech, it refers specifically to the "hot" or "spicy" quality of chili peppers.

People think that the spiciest part of a chili pepper is the fruit, but it's actually the seeds

The spiciest part of a chili pepper is the stem

The spiciest part of a chili pepper is the flesh

The spiciest part of a chili pepper is the seeds

The spiciest part of a chili pepper is the placenta

E

How long should you wait before filing a missing person report?

A missing person is a person who has disappeared for usually unknown reasons Missing Person(s) or Missing People may also refer to: * Missing persons (Pakistan), referring to the enforced disappearances in Pakistan * Missing Persons (band), an American rock band * Missing Persons (TV series), an American TV series (1993–94) * The First 48: Missing Persons, documentary episode of the TV show The First 48 * The Missing Person, a 2009 American film * Missing Person (novel) (French: Rue des Boutiques Obscures), a 1978 French novel by Patrick Modiano * Missing People, UK charity * The Missing People, 1940 film ==See also== * Mising people, a South Asian ethnic group * International Commission on Missing Persons ** List of missing people organizations * "Of Missing Persons", short story by Jack Finney * Missing Persons Unit, Australian TV documentary * Bureau of Missing Persons, American comedic film If an adult unreported missing person is located in such an instance, the police are not obligated to inform the family of the missing person's whereabouts. Unreported missing (also known as missing missing) describes persons who cannot be found, yet have not been or cannot be reported as missing persons to law enforcement, specifically the National Crime Information Center database of missing persons in the United States. Reported Missing is a British documentary television series broadcast on BBC One (2017–present). Missing is a British crime drama television series, based on the 2000 novel Missing by crime author Karin Alvtegen, which was first broadcast on STV on 2 November 2008. Another reason missing persons may not be formally listed as missing is that those over the age of majority can be "voluntarily missing." Reported Missing may refer to: * Reported Missing!, a 1937 American thriller film * Reported Missing (1922 film), an American silent comedy film * Reported Missing (TV series), a British documentary television series Missing later received its official UK premiere on UKTV Drama, being broadcast in the original two-part format of 90-minutes per episode. ==Cast== * Joanne Froggatt as Sybil Foster * Dean Andrews as Mark Lanser * Gregor Fisher as DS Doug Duvall * Ralph Ineson as DCI John Carter * Mhairi Morrison as DC Mairi Wilson * Pip Torrens as Derek Mailer * Phyllis Logan as Karen Foster * Christopher Fulford as Dr. Webster ==Episode list== ==References== ==External links== * Category:2000s British drama television series Category:2000s British crime television series Category:2000s British television miniseries Category:2008 British television series debuts Category:2008 British television series endings Category:English-language television shows Category:ITV television dramas Category:Serial drama television series Category:Television series by STV Studios Category:Television shows based on British novels The term applies whether the missing person is a child or an adult. ==Reasons== According to Outpost for Hope, people can become unreported missing for a variety of reasons, including: * the person may be estranged from family or friends; * law enforcement may not take a "missing person" report; * the person may be in the country illegally; * the person may be an unknown dependent child of unreported missing adults or teens; * the person might be the victim of an undiscovered crime; or * the person may be homeless. This is a list of episodes of Disappeared, a television program broadcast on the Investigation Discovery network that documents missing persons cases. Missing had previously been screened in the United States in August 2006, and was distributed by Koch. For persons subsequently found alive or deceased, names of subjects are linked to reliable news sources regarding their disappearance and discovery. ===Season 1 (2009–2010)=== ===Season 2 (2010)=== ===Season 3 (2011)=== ===Season 4 (2011–2012)=== ===Season 5 (2012)=== ===Season 6 (2012–2013)=== ===Specials (2014–2015)=== ===Season 7 (2016)=== === Season 8 (2017) === ===Season 9 (2018)=== ===Season 10 (2022)=== ==See also== *List of people who disappeared mysteriously ==Notes== ==References== ==External links== *Official website *Disappeared at TV Guide Category:Lists of American non-fiction television series episodes Category:Television series about missing people Missing was released on Region 1 DVD on 5 September 2006. ==Plot== Sybil Foster (Joanne Froggatt) escapes from a psychiatric hospital, only to find herself the prime suspect in a series of murders. Reported Missing has eleven episodes over three series. In the majority of US states, when foster children go missing their name is not publicly released and, with a few exceptions, child protective services representatives will not speak to the public—in some states, biological parents are not even permitted to contact the news media about their fostered child going missing. Missing was first broadcast on STV on 2 and 9 November 2008, after being filmed in 2004. The series follows various police departments and associated organisations in the United Kingdom, in search for missing people. There have also been cases of missing persons reports being accidentally purged from databases, leading to those persons becoming unreported missing. ==Risks== Being unreported puts a person at a higher risk of exploitation than a person who is reported missing. A new series began in 2022 following a four-year absence. ==Series overview== Season Episodes Season premiere Season finale 1 13 2 13 3 13 4 14 5 17 6 15 Specials 4 7 10 8 13 9 13 10 8 ==Episodes== *Note: Episode subjects who are still actively missing are linked to their profiles at The Charley Project, a missing persons database. The police investigation, led by DS Doug Duvall (Gregor Fisher) unearths skeletons from her past, and a link to a prominent political figure who may hold the key to unlock the secrets of the past – and open the door to solving the mysteries of the present. ==Production== Missing reunites the BAFTA award-winning duo behind BBC2 crime drama The Cops – writer Jimmy Gardner and executive producer Eric Coulter, and is based on the best-selling novel, Missing, by Karin Alvtegen. *Jason Callahan, discovered in 1995, not reported missing until 2015, as his family did not know what jurisdiction to report his disappearance to and thought he may have gone to live on his own and not wanted contact. Of the approximately 2,340 unreported missing foster children in the US (in 2002) who are not accounted for, it is estimated that one in five (468) will be the victim of a homicide. ==Notable cases== * Barbara Precht, discovered in 2006 and was not identified until late 2014.

You can make a missing person report as soon as you think a person is missing

Wait for the person to return

As anyone knows from watching television shows, you should wait 24 hours to file a missing person report

You should wait 48 hours before filing a missing person report

Do not file a missing person report

A