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| {{Thermodynamics|cTopic=[[Thermodynamic equations|Equations]]}}
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| [[Image:Ideal gas isotherms.svg|thumb|right|[[Isothermal process|Isotherm]]s of an [[ideal gas]]. The curved lines represent the relationship between [[pressure]] (on the vertical, ''y''-axis) and [[Volume (thermodynamics)|volume]] (on the horizontal, ''x''-axis) for an ideal gas at different [[temperature]]s: lines which are further away from the [[origin (mathematics)|origin]] (that is, lines that are nearer to the top right-hand corner of the diagram) represent higher temperatures.]]
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| The '''ideal gas law''' is the [[equation of state]] of a hypothetical [[ideal gas]]. It is a good approximation to the behaviour of many [[gas]]es under many conditions, although it has several limitations. It was first stated by [[Benoît Paul Émile Clapeyron|Émile Clapeyron]] in 1834 as a combination of [[Boyle's law]] and [[Charles' law]].<ref>
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| {{Cite journal
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| | author = Clapeyron, E. | authorlink = Benoît Paul Émile Clapeyron
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| | year = 1834
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| | title = Mémoire sur la puissance motrice de la chaleur
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| | journal = [[Journal de l'École Polytechnique]]
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| | volume = XIV | pages = 153–90
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| }} {{Fr icon}} [http://gallica.bnf.fr/ark:/12148/bpt6k4336791/f157.table Facsimile at the Bibliothèque nationale de France (pp. 153–90).]</ref> The ideal gas law is often introduced in its common form:
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| :<math>PV=nRT\,</math>
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|
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|
| where ''P'' is the absolute [[pressure]] of the gas, ''V'' is the [[volume]] of the gas, ''n'' is the [[amount of substance]] of gas (measured in [[Mole (unit)|moles]]), ''T'' is the absolute [[temperature]] of the gas and ''R'' is the ideal, or universal, [[Ideal gas constant|gas constant]].
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| It can also be derived from [[kinetic theory]], as was achieved (apparently independently) by [[August Krönig]] in 1856<ref>
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| {{Cite journal
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| | author = Krönig, A. | authorlink = August Krönig
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| | year = 1856
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| | title = Grundzüge einer Theorie der Gase
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| | journal = [[Annalen der Physik|Annalen der Physik und Chemie]]
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| | volume = 99 | pages = 315–22
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| | doi = 10.1002/andp.18561751008
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| |bibcode = 1856AnP...175..315K
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| | issue = 10 }} {{De icon}} [http://gallica.bnf.fr/ark:/12148/bpt6k15184h/f327.table Facsimile at the Bibliothèque nationale de France (pp. 315–22).]</ref> and [[Rudolf Clausius]] in 1857.<ref>
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| {{Cite journal
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| | author = Clausius, R. | authorlink = Rudolf Clausius
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| | year = 1857
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| | title = Ueber die Art der Bewegung, welche wir Wärme nennen
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| | journal = [[Annalen der Physik und Chemie]]
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| | volume = 176 | pages = 353–79
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| | doi = 10.1002/andp.18571760302
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| |bibcode = 1857AnP...176..353C
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| | issue = 3 }} {{De icon}} [http://gallica.bnf.fr/ark:/12148/bpt6k15185v/f371.table Facsimile at the Bibliothèque nationale de France (pp. 353–79).]</ref> The universal gas constant was discovered and first introduced into the ideal gas law instead of a large number of specific gas constants by [[Dmitri Mendeleev]] in 1874.<ref>
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| {{Cite journal
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| | author = Mendeleev, D. I. | authorlink = Dmitri Mendeleev
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| | year = 1874
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| | title = О сжимаемости газов (On the compressibility of gases)
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| | journal = [[Russian Journal of Chemical Society and the Physical Society]]
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| | volume = 6 | pages = 309–352 }} {{Ru icon}} (From the Laboratory of the University of St. Petersburg).</ref><ref>
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| {{Cite journal
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| | author = Mendeleev, D. I. | authorlink = Dmitri Mendeleev
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| | year = 1875
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| | title = Об упругости газов (On the elasticity of gases) }} {{Ru icon}} [http://gallica.bnf.fr/ark:/12148/bpt6k95208b.r=mendeleev.langEN Facsimile at the Bibliothèque nationale de France]</ref><ref>
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| {{Cite journal
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| | author = Mendeleef D. | authorlink = Dmitri Mendeleev
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| | year = 1877
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| | title = Researches on Mariotte's Law
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| | journal = [[Nature (journal)|Nature]]
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| | volume = 15 (388) | pages = 498–500 |bibcode = 1877Natur..15..498D |doi = 10.1038/015498a0 }} doi: 10.1038/015498a0.</ref>
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| ==Equation==
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| The [[state function|state]] of an amount of [[gas]] is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms. The temperature used in the equation of state is an absolute temperature: in the SI system of units, [[kelvin]].<ref name="Equation of State">{{Cite web|url=http://www.grc.nasa.gov/WWW/K-12/airplane/eqstat.html|title=Equation of State}}</ref>
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| ===Common form===
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| The most frequently introduced form is
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| :<math>PV=nRT\,</math>
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| where ''P'' is the [[pressure]] of the gas, ''V'' is the volume of the gas, ''n'' is the [[amount of substance]] of gas (also known as number of moles), ''T'' is the temperature of the gas and ''R'' is the ideal, or universal, [[Ideal gas constant|gas constant]], equal to the product of the [[Boltzmann constant]] and the [[Avogadro constant]].
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| In [[SI units]], ''P'' is measured in [[Pascal (unit)|pascals]], ''V'' is measured in [[cubic metre]]s, ''n'' is measured in [[Mole (unit)|moles]], and ''T'' in kelvins (273.15 kelvins = 0.00 degrees Celsius). R has the value 8.314 [[joule|J]]·[[kelvin|K]]<sup>−1</sup>·[[Mole (unit)|mol]]<sup>−1</sup> or 0.08206 L·atm·mol<sup>−1</sup>·K<sup>−1</sup> if using pressure in [[Atmosphere (unit)|standard atmospheres]] (atm) instead of pascals, and volume in [[liter]]s instead of cubic metres.
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| ===Molar form===
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| How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount (''n'') (in moles) is equal to the mass (''m'') (in grams) divided by the [[molar mass]] (''M'') (in grams per mole):
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| : <math> n = {\frac{m}{M}} </math>
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| By replacing ''n'' with ''m / M'', and subsequently introducing [[density]] ''ρ'' = ''m''/''V'', we get:
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| : <math>\ PV = \frac{m}{M}RT </math>
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| : <math>\ P = \rho \frac{R}{M}T </math>
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| Defining the [[Gas constant#Specific gas constant|specific gas constant]] ''R''<sub>specific</sub> as the ratio ''R''/''M'',
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| : <math>\ P = \rho R_{\rm specific}T </math>
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| This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas. Alternatively, the law may be written in terms of the [[specific volume]] ''v'', the reciprocal of density, as
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| : <math>\ Pv = R_{\rm specific}T </math>
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| It is common, especially in engineering applications, to represent the '''specific''' gas constant by the symbol ''R''. In such cases, the '''universal''' gas constant is usually given a different symbol such as <u style="font-style:italic; text-decoration:overline">R</u> to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to.<ref>Moran and Shapiro, ''Fundamentals of Engineering Thermodynamics'', Wiley, 4th Ed, 2000</ref>
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| ===Statistical mechanics===
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| In [[statistical mechanics]] the following molecular equation is derived from first principles:
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| : <math>\ PV = NkT</math>
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| where ''P'' is the absolute [[pressure]] of the gas measured in [[Atmosphere (unit)|Pascals]]; ''V'' is the [[Volume (thermodynamics)|volume]] (in this equation the volume is expressed in meters cubed, as pascal times cubic meter equals one Joule); ''N'' is the number of particles in the gas; ''k'' is the [[Boltzmann constant]] relating temperature and energy; and ''T'' is the [[absolute temperature]].
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| The ''actual number'' of molecules contrasts to the other formulation, which uses ''n'', the ''number of moles''. This relation implies that ''Nk'' = ''nR'', and the consistency of this result with experiment is a good check on the principles of statistical mechanics.
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| From this we can notice that for an average particle mass of ''μ'' times the [[atomic mass constant]] ''m''<sub>u</sub> (i.e., the mass is ''μ'' [[atomic mass unit|u]])
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| :<math> N = \frac{m}{\mu m_\mathrm{u}} </math> | |
| and since ''ρ'' = ''m''/''V'', we find that the ideal gas law can be rewritten as:
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| :<math> P = \frac{1}{V}\frac{m}{\mu m_\mathrm{u}} kT = \frac{k}{\mu m_\mathrm{u}} \rho T .</math>
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| In SI units, ''P'' is measured in [[Pascal (unit)|pascals]]; ''V'' in cubic metres; ''N'' is a dimensionless number; and ''T'' in kelvins.
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| ''k'' has the value 1.38·10<sup>−23</sup> [[joule|J]]·[[kelvin|K]]<sup>−1</sup> in [[SI unit]]s.
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| == Applications to thermodynamic processes == | |
| The table below essentially simplifies the ideal gas equation for a particular processes, thus making this equation easier to solve using numerical methods.
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| A [[thermodynamic process]] is defined as a system that moves from state 1 to state 2, where the state number is denoted by subscript. As shown in the first column of the table, basic thermodynamic processes are defined such that one of the gas properties (''P'', ''V'', ''T'', or ''S'') is constant throughout the process.
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| For a given thermodynamics process, in order to specify the extent of a particular process, one of the properties ratios (listed under the column labeled "known ratio") must be specified (either directly or indirectly). Also, the property for which the ratio is known must be distinct from the property held constant in the previous column (otherwise the ratio would be unity, and not enough information would be available to simplify the gas law equation).
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| In the final three columns, the properties (''P'', ''V'', or ''T'') at state 2 can be calculated from the properties at state 1 using the equations listed.
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| {| class="wikitable"
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| |-
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| ! Process
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| ! Constant
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| ! Known ratio
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| ! P<sub>2</sub>
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| ! V<sub>2</sub>
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| ! T<sub>2</sub>
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| |-
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| |rowspan="2"|[[Isobaric process]]
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| |rowspan="2"| <center>Pressure</center>
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| | <center>V<sub>2</sub>/V<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>
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| | V<sub>2</sub> = V<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)
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| | T<sub>2</sub> = T<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)
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| |-
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| | <center>T<sub>2</sub>/T<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>
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| | V<sub>2</sub> = V<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
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| | T<sub>2</sub> = T<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
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| |-
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| |rowspan="2"| [[Isochoric process]]<br>(Isovolumetric process)<br>(Isometric process)
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| |rowspan="2"| <center>Volume</center>
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| | <center>P<sub>2</sub>/P<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>(P<sub>2</sub>/P<sub>1</sub>)
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| | V<sub>2</sub> = V<sub>1</sub>
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| | T<sub>2</sub> = T<sub>1</sub>(P<sub>2</sub>/P<sub>1</sub>)
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| |-
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| | <center>T<sub>2</sub>/T<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
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| | V<sub>2</sub> = V<sub>1</sub>
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| | T<sub>2</sub> = T<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
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| |-
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| |rowspan="2"| [[Isothermal process]]
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| |rowspan="2"| <center> Temperature </center>
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| | <center>P<sub>2</sub>/P<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>(P<sub>2</sub>/P<sub>1</sub>)
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| | V<sub>2</sub> = V<sub>1</sub>/(P<sub>2</sub>/P<sub>1</sub>)
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| | T<sub>2</sub> = T<sub>1</sub>
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| |-
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| | <center>V<sub>2</sub>/V<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>/(V<sub>2</sub>/V<sub>1</sub>)
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| | V<sub>2</sub> = V<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)
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| | T<sub>2</sub> = T<sub>1</sub>
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| |-
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| |rowspan="3"| [[Isentropic process]]<br>(Reversible [[adiabatic process]])
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| |rowspan="3"| <center>[[Entropy]]{{ref_label|A|a|none}}</center>
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| | <center>P<sub>2</sub>/P<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>(P<sub>2</sub>/P<sub>1</sub>)
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| | V<sub>2</sub> = V<sub>1</sub>(P<sub>2</sub>/P<sub>1</sub>)<sup>(−1/γ)</sup>
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| | T<sub>2</sub> = T<sub>1</sub>(P<sub>2</sub>/P<sub>1</sub>)<sup>(γ − 1/γ)</sup>
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| |-
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| | <center>V<sub>2</sub>/V<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)<sup>−γ</sup>
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| | V<sub>2</sub> = V<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)
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| | T<sub>2</sub> = T<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)<sup>(1 − γ)</sup>
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| |-
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| | <center>T<sub>2</sub>/T<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)<sup>γ/(γ − 1)</sup>
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| | V<sub>2</sub> = V<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)<sup>1/(1 − γ) </sup>
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| | T<sub>2</sub> = T<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
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| |-
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| |rowspan="3"| [[Polytropic process]]<br>
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| |rowspan="3"| <center>P V<sup>n</sup></center>
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| | <center>P<sub>2</sub>/P<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>(P<sub>2</sub>/P<sub>1</sub>)
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| | V<sub>2</sub> = V<sub>1</sub>(P<sub>2</sub>/P<sub>1</sub>)<sup>(-1/n)</sup>
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| | T<sub>2</sub> = T<sub>1</sub>(P<sub>2</sub>/P<sub>1</sub>)<sup>(n - 1/n)</sup>
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| |-
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| | <center>V<sub>2</sub>/V<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)<sup>−n</sup>
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| | V<sub>2</sub> = V<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)
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| | T<sub>2</sub> = T<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)<sup>(1−n)</sup>
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| |-
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| | <center>T<sub>2</sub>/T<sub>1</sub></center>
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| | P<sub>2</sub> = P<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)<sup>n/(n − 1)</sup>
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| | V<sub>2</sub> = V<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)<sup>1/(1 − n) </sup>
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| | T<sub>2</sub> = T<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
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| |}
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| {{Note_label|A|a|none}} '''a.''' In an isentropic process, system entropy (''S'') is constant. Under these conditions, ''P''<sub>1</sub> ''V''<sub>1</sub><sup>''γ''</sup> = ''P''<sub>2</sub> ''V''<sub>2</sub><sup>''γ''</sup>, where ''γ'' is defined as the [[heat capacity ratio]], which is constant for an ideal gas. The value used for ''γ'' is typically 1.4 for diatomic gases like [[nitrogen]] (N<sub>2</sub>) and [[oxygen]] (O<sub>2</sub>), (and air, which is 99% diatomic). Also ''γ'' is typically 1.6 for monatomic gases like the [[noble gas]]es [[helium]] (He), and [[argon]] (Ar). In internal combustion engines ''γ'' varies between 1.35 and 1.15, depending on constitution gases and temperature.
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| ==Deviations from ideal behavior of real gases==
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| The equation of state given here applies only to an ideal gas, or as an approximation to a real gas that behaves sufficiently like an ideal gas. There are in fact many different forms of the equation of state. Since the ideal gas law neglects both molecular size and intermolecular attractions, it is most accurate for [[monatomic]] gases at high temperatures and low pressures. The neglect of molecular size becomes less important for lower densities, i.e. for larger volumes at lower pressures, because the average distance between adjacent molecules becomes much larger than the molecular size. The relative importance of intermolecular attractions diminishes with increasing [[thermal energy|thermal kinetic energy]], i.e., with increasing temperatures. More detailed ''[[Equation of state|equations of state]]'', such as the [[van der Waals equation]], account for deviations from ideality caused by molecular size and intermolecular forces. | |
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| A [[residual property (physics)|residual property]] is defined as the difference between a [[real gas]] property and an ideal gas property, both considered at the same pressure, temperature, and composition.
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| == Derivations ==
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| ===Empirical===
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| The ideal gas law can be derived from combining two empirical [[gas laws]]: the [[combined gas law]] and [[Avogadro's law]]. The combined gas law states that
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| :<math>\frac{PV}{T}= C </math>
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| where ''C'' is a constant which is directly proportional to the amount of gas, ''n'' ([[Avogadro's law]]). The proportionality factor is the [[gas constant|universal gas constant]], ''R'', i.e. ''C'' = ''nR''.
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| Hence the ideal gas law
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| :<math> PV = nRT \,</math>
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| ===Theoretical===
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| ====Kinetic theory====
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| {{Main|kinetic theory}}
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| The ideal gas law can also be derived from [[first principles]] using the [[kinetic theory of gases]], in which several simplifying assumptions are made, chief among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.
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| ====Statistical mechanics====
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| {{Main|Statistical mechanics}}
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| Let '''q''' = (''q''<sub>x</sub>, ''q''<sub>y</sub>, ''q''<sub>z</sub>) and '''p''' = (''p''<sub>x</sub>, ''p''<sub>y</sub>, ''p''<sub>z</sub>) denote the position vector and momentum vector of a particle of an ideal gas, respectively. Let '''F''' denote the net force on that particle. Then the time average momentum of the particle is:<br>
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| : <math>
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| \begin{align}
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| \langle \mathbf{q} \cdot \mathbf{F} \rangle &= \Bigl\langle q_{x} \frac{dp_{x}}{dt} \Bigr\rangle +
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| \Bigl\langle q_{y} \frac{dp_{y}}{dt} \Bigr\rangle +
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| \Bigl\langle q_{z} \frac{dp_{z}}{dt} \Bigr\rangle\\
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| &=-\Bigl\langle q_{x} \frac{\partial H}{\partial q_x} \Bigr\rangle -
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| \Bigl\langle q_{y} \frac{\partial H}{\partial q_y} \Bigr\rangle -
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| \Bigl\langle q_{z} \frac{\partial H}{\partial q_z} \Bigr\rangle = -3k_{B} T,
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| \end{align}
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| </math>
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| where the first equality is [[Newton's second law]], and the second line uses [[Hamilton's equations]] and the [[equipartition theorem]]. Summing over a system of ''N'' particles yields
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| :<math>
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| 3Nk_{B} T = - \biggl\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \biggr\rangle.
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| </math>
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| By [[Newton's third law]] and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure ''P'' of the gas. Hence
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| :<math>
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| -\biggl\langle\sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k}\biggr\rangle = P \oint_{\mathrm{surface}} \mathbf{q} \cdot d\mathbf{S},
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| </math>
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| | |
| where d'''S''' is the infinitesimal area element along the walls of the container. Since the [[divergence]] of the position vector '''q''' is
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| :<math>
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| \nabla \cdot \mathbf{q} =
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| \frac{\partial q_{x}}{\partial q_{x}} +
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| \frac{\partial q_{y}}{\partial q_{y}} +
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| \frac{\partial q_{z}}{\partial q_{z}} = 3,
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| </math>
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| the [[divergence theorem]] implies that
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| :<math>P \oint_{\mathrm{surface}} \mathbf{q} \cdot d\mathbf{S} = P \int_{\mathrm{volume}} \left( \nabla \cdot \mathbf{q} \right) dV = 3PV,
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| </math>
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| where ''dV'' is an infinitesimal volume within the container and ''V'' is the total volume of the container.
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| Putting these equalities together yields
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| :<math>
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| 3Nk_{B} T = -\biggl\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \biggr\rangle = 3PV,
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| </math>
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| which immediately implies the ideal gas law for ''N'' particles:
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| :<math>
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| PV = Nk_{B} T = nRT,\,
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| </math>
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| where ''n'' = ''N''/''N''<sub>A</sub> is the number of [[mole (unit)|moles]] of gas and ''R'' = ''N''<sub>A</sub>''k''<sub>B</sub> is the [[gas constant]].
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| ==See also==
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| * [[Combined gas law]]
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| * [[Van der Waals equation]]
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| * [[Boltzmann constant]]
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| * [[Configuration integral]]
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| * [[Dynamic pressure]]
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| * [[Internal energy]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| *Davis and Masten ''Principles of Environmental Engineering and Science'', McGraw-Hill Companies, Inc. New York (2002) ISBN 0-07-235053-9
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| *[http://www.gearseds.com/curriculum/learn/lesson.php?id=23&chapterid=5 Website giving credit to [[Benoît Paul Émile Clapeyron]], (1799–1864) in 1834]
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| == External links ==
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| * [http://www.webqc.org/ideal_gas_law.html Ideal Gas Law Calculator]
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| * [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)] where an alternative statistical mechanics derivation of the ideal-gas law, using the relationship between the [[Helmholtz free energy]] and the [[partition function (statistical mechanics)|partition function]], but without using the equipartition theorem, is provided.
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| {{Statistical mechanics topics}}
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| {{Diving medicine, physiology and physics}}
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| {{DEFAULTSORT:Ideal Gas Law}}
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| [[Category:Gas laws]]
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| [[Category:Ideal gas]]
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| [[Category:Equations of state]]
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| {{Link GA|zh}}
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