Editing Ideal gas law

{{Short description|Equation of the state of a hypothetical ideal gas}}
{{Thermodynamics|cTopic=[[Thermodynamic|Equations]]}}
[[Image:Ideal gas isotherms.svg|thumb|right|[[Isothermal process|Isotherms]] of an [[ideal gas]] for different temperatures. The curved lines are [[Hyperbola#Hyperbola with equation y = A/x|rectangular hyperbolae]] of the form y = a/x. They represent the relationship between [[pressure]] (on the vertical axis) and [[Volume (thermodynamics)|volume]] (on the horizontal axis) for an ideal gas at different [[temperature]]s: lines that are farther away from the [[origin (mathematics)|origin]] (that is, lines that are nearer to the top right-hand corner of the diagram) correspond to higher temperatures.]]

The '''ideal gas law''', also called the '''general gas equation''', is the [[equation of state]] of a hypothetical [[ideal gas]]. It is a good approximation of the behavior of many [[gas]]es under many conditions, although it has several limitations. It was first stated by [[Benoît Paul Émile Clapeyron]] in 1834 as a combination of the empirical [[Boyle's law]], [[Charles's law]], [[Avogadro's law]], and [[Gay-Lussac's law]].<ref>
{{Cite journal
 | author = Clapeyron, E. | author-link = Benoît Paul Émile Clapeyron
 | year = 1835
 | title = Mémoire sur la puissance motrice de la chaleur
 | journal = [[Journal de l'École Polytechnique]]
 | volume = XIV | pages = 153–90
|language=fr}} [http://gallica.bnf.fr/ark:/12148/bpt6k4336791/f157.table Facsimile at the Bibliothèque nationale de France (pp.&nbsp;153–90)].</ref> The ideal gas law is often written in an empirical form:

<math display="block">pV = nRT</math>
where <math>p</math>, <math>V</math> and <math>T</math> are the [[pressure]], [[volume]] and [[Thermodynamic temperature|temperature]] respectively; <math>n</math> is the [[amount of substance]]; and <math>R</math> is the [[ideal gas constant]].
It can also be derived from the microscopic [[kinetic theory of gases|kinetic theory]], as was achieved (apparently independently) by [[August Krönig]] in 1856<ref>
{{Cite journal
 | author = Krönig, A. | author-link = August Krönig
 | year = 1856
 | title = Grundzüge einer Theorie der Gase
 | journal = [[Annalen der Physik|Annalen der Physik und Chemie]]
 | volume = 99 | pages = 315–22
 | doi = 10.1002/andp.18561751008
|bibcode = 1856AnP...175..315K
 | issue = 10 |language=de| url = https://zenodo.org/record/1423642
 }} [http://gallica.bnf.fr/ark:/12148/bpt6k15184h/f327.table Facsimile at the Bibliothèque nationale de France (pp.&nbsp;315–22)].</ref> and [[Rudolf Clausius]] in 1857.<ref>
{{Cite journal
 | author = Clausius, R. | author-link = Rudolf Clausius
 | year = 1857
 | title = Ueber die Art der Bewegung, welche wir Wärme nennen
 | journal = [[Annalen der Physik und Chemie]]
 | volume = 176 | pages = 353–79
 | doi = 10.1002/andp.18571760302
|bibcode = 1857AnP...176..353C
 | issue = 3 |language=de| url = https://zenodo.org/record/1423644
 }} [http://gallica.bnf.fr/ark:/12148/bpt6k15185v/f371.table Facsimile at the Bibliothèque nationale de France (pp.&nbsp;353–79)].</ref>

==Equation==

[[File:Ideal Gas Law.jpg|thumb|Molecular collisions within a closed container (a propane tank) are shown (right). The arrows represent the random motions and collisions of these molecules. The pressure and temperature of the gas are directly proportional: As temperature increases, the pressure of the propane gas increases by the same factor. A simple consequence of this proportionality is that on a hot summer day, the propane tank pressure will be elevated, and thus propane tanks must be rated to withstand such increases in pressure.]]

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: the appropriate [[SI unit]] is the [[kelvin]].<ref name="Equation of State">{{Cite web|url=http://www.grc.nasa.gov/WWW/K-12/airplane/eqstat.html|title=Equation of State|access-date=2010-08-29|archive-url=https://web.archive.org/web/20140823003747/http://www.grc.nasa.gov/WWW/k-12/airplane/eqstat.html|archive-date=2014-08-23|url-status=dead}}</ref>

===Common forms===

The most frequently introduced forms are:<math display="block">pV = nRT = n k_\text{B} N_\text{A} T = N k_\text{B} T </math>where:
* <math>p</math> is the absolute [[pressure]] of the gas,
* <math>V</math> is the [[volume]] of the gas,
* <math>n</math> is the [[amount of substance]] of gas (also known as number of moles),
* <math>R</math> is the ideal, or universal, [[gas constant]], equal to the product of the [[Boltzmann constant]] and the [[Avogadro constant]],
* <math>k_\text{B}</math> is the [[Boltzmann constant]],
* '''''<math>N_{A}</math>''''' is the [[Avogadro constant]],
* <math>T</math> is the [[Thermodynamic temperature|absolute temperature]] of the gas,
* <math>N</math> is the number of particles (usually atoms or molecules) of the gas.

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 [[Kelvin (unit)|kelvins]] (the [[William Thomson, 1st Baron Kelvin|Kelvin]] scale is a shifted [[Celsius|Celsius scale]], where 0 K = −273.15&nbsp;°C, the [[Absolute zero|lowest possible temperature]]). ''R'' has for value 8.314 [[joule|J]]/([[Mole (unit)|mol]]·[[Kelvin|K]]) = 1.989 ≈ 2 [[calorie|cal]]/(mol·K), or 0.0821 L⋅[[Atmosphere (unit)|atm]]/(mol⋅K).

===Molar form===
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 total mass of the gas (''m'') (in kilograms) divided by the [[molar mass]], ''M'' (in kilograms per mole):
: <math> n = \frac{m}{M}. </math>
By replacing ''n'' with ''m''/''M'' and subsequently introducing [[density]] ''ρ'' = ''m''/''V'', we get:
: <math> pV = \frac{m}{M} RT </math>
: <math> p = \frac{m}{V} \frac{RT}{M} </math>
: <math> p = \rho \frac{R}{M} T </math>
Defining the [[Gas constant#Specific gas constant|specific gas constant]] ''R''<sub>specific</sub> as the ratio ''R''/''M'',
: <math> p = \rho R_\text{specific}T </math>
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
: <math> pv = R_\text{specific}T. </math>

It is common, especially in engineering and meteorological 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 <math>\bar R</math> or <math>R^*</math> 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 used.<ref>{{cite book |last1=Moran |last2=Shapiro |title=Fundamentals of Engineering Thermodynamics |publisher=Wiley |edition=4th |year=2000 |isbn=0-471-31713-6 }}</ref>

=== Statistical mechanics ===
In [[statistical mechanics]], the following molecular equation (i.e. the ideal gas law in its theoretical form) is derived from first principles:
: <math> p = nk_\text{B}T, </math>
where {{math|''p''}} is the absolute [[pressure]] of the gas, {{math|''n''}} is the [[number density]] of the molecules (given by the ratio {{math|1=''n'' = ''N''/''V''}}, in contrast to the previous formulation in which {{math|''n''}} is the ''number of moles''), {{math|''T''}} is the [[absolute temperature]], and {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]] relating temperature and energy, given by:
: <math> k_\text{B} = \frac{R}{N_\text{A}} </math>
where {{math|''N''<sub>A</sub>}} is the [[Avogadro constant]].
The form can be furtherly simplified by defining the kinetic energy corresponding to the temperature:
: <math> T := k_\text{B}T, </math>
so the ideal gas law is more simply expressed as:
: <math> p = n \, T, </math>

From this we notice that for a gas of mass {{math|''m''}}, with an average particle mass of {{math|''μ''}} times the [[atomic mass constant]], {{math|''m''<sub>u</sub>}}, (i.e., the mass is {{math|''μ''}}&nbsp;[[Dalton (unit)|Da]]) the number of molecules will be given by
: <math> N = \frac{m}{\mu m_\text{u}}, </math>
and since {{math|1=''ρ'' = ''m''/''V'' = ''nμm''<sub>u</sub>}}, we find that the ideal gas law can be rewritten as
: <math> p = \frac{1}{V}\frac{m}{\mu m_\text{u}} k_\text{B} T = \frac{k_\text{B}}{\mu m_\text{u}} \rho T. </math>

In SI units, {{math|''p''}} is measured in [[Pascal (unit)|pascals]], {{math|''V''}} in cubic metres, {{math|''T''}} in kelvins, and {{math|1=''k''<sub>B</sub> = {{physconst|k|ref=no|round=2}}}} in [[SI unit]]s.

=== Combined gas law ===
Combining the laws of Charles, Boyle, and Gay-Lussac gives the '''combined gas law''', which can take the same functional form as the ideal gas law. This form does not specify the number of moles, and the ratio of <math>PV</math> to <math>T</math> is simply taken as a constant:<ref>{{cite book |last1=Raymond |first1=Kenneth W. |title=General, organic, and biological chemistry : an integrated approach |publisher=John Wiley & Sons |isbn=9780470504765 |page=186 |edition=3rd |year=2010|url=https://books.google.com/books?id=iIltMoHUtJUC&pg=PA186 |access-date=29 January 2019}}</ref>
: <math>\frac{PV}{T}=k,</math>
where <math>P</math> is the [[pressure]] of the gas, <math>V</math> is the [[volume]] of the gas, <math>T</math> is the [[Thermodynamic temperature|absolute temperature]] of the gas, and <math>k</math> is a constant. More commonly, when comparing the same substance under two different sets of conditions, the law is written as:
: <math> \frac{P_1 V_1}{T_1}= \frac{P_2 V_2}{T_2}. </math>

== Energy associated with a gas ==
According to the assumptions of the kinetic theory of ideal gases, one can consider that there are no intermolecular attractions between the molecules, or atoms, of an ideal gas. In other words, its [[potential energy]] is zero. Hence, all the energy possessed by the gas is the kinetic energy of the molecules, or atoms, of the gas.
: <math>E=\frac{3}{2} n RT </math>

This corresponds to the kinetic energy of ''n'' moles of a [[monatomic|monoatomic]] gas having 3 [[Degrees of freedom (physics and chemistry)|degrees of freedom]]; ''x'', ''y'', ''z''. The table here below gives this relationship for different amounts of a monoatomic gas.

{| class="wikitable"
!Energy of a monoatomic gas
!Mathematical expression
|-
|Energy associated with one mole
|<math>E=\frac{3}{2}RT</math>
|-
|Energy associated with one gram
|<math>E=\frac{3}{2}rT</math>
|-
|Energy associated with one atom
|<math>E=\frac{3}{2}k_{\rm B}T</math>
|}

== Applications to thermodynamic processes ==
The table below essentially simplifies the ideal gas equation for a particular process, making the equation easier to solve using numerical methods.

A [[thermodynamic process]] is defined as a system that moves from state 1 to state 2, where the state number is denoted by a 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'', ''S'', or ''H'') is constant throughout the process.

For a given thermodynamic process, in order to specify the extent of a particular process, one of the properties ratios (which are 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).

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.

{| class="wikitable"
|-
! Process
! Constant
! Known ratio or delta
! p<sub>2</sub>
! V<sub>2</sub>
! T<sub>2</sub>
|-
|rowspan="2"|[[Isobaric process]]
|rowspan="2" style="text-align:center;" |Pressure
|style="text-align:center;"| V<sub>2</sub>/V<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>
| V<sub>2</sub> = V<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)
| T<sub>2</sub> = T<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)
|-
|style="text-align:center;"| T<sub>2</sub>/T<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>
| V<sub>2</sub> = V<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
| T<sub>2</sub> = T<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
|-
|rowspan="2"| [[Isochoric process]]<br>(Isovolumetric process)<br>(Isometric process)
|rowspan="2" style="text-align:center;" |Volume
|style="text-align:center;"| p<sub>2</sub>/p<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>(p<sub>2</sub>/p<sub>1</sub>)
| V<sub>2</sub> = V<sub>1</sub>
| T<sub>2</sub> = T<sub>1</sub>(p<sub>2</sub>/p<sub>1</sub>)
|-
|style="text-align:center;"| T<sub>2</sub>/T<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
| V<sub>2</sub> = V<sub>1</sub>
| T<sub>2</sub> = T<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
|-
|rowspan="2"| [[Isothermal process]]
|rowspan="2" style="text-align:center;" |&nbsp;Temperature&nbsp;
|style="text-align:center;"| p<sub>2</sub>/p<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>(p<sub>2</sub>/p<sub>1</sub>)
| V<sub>2</sub> = V<sub>1</sub>(p<sub>1</sub>/p<sub>2</sub>)
| T<sub>2</sub> = T<sub>1</sub>
|-
|style="text-align:center;"| V<sub>2</sub>/V<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>(V<sub>1</sub>/V<sub>2</sub>)
| V<sub>2</sub> = V<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)
| T<sub>2</sub> = T<sub>1</sub>
|-
|rowspan="3"| [[Isentropic process]]<br>(Reversible [[adiabatic process]])
|rowspan="3"| {{center|[[Entropy]]{{ref_label|A|a|none}}}}
|style="text-align:center;"| p<sub>2</sub>/p<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>(p<sub>2</sub>/p<sub>1</sub>)
| V<sub>2</sub> = V<sub>1</sub>(p<sub>2</sub>/p<sub>1</sub>)<sup>(−1/γ)</sup>
| T<sub>2</sub> = T<sub>1</sub>(p<sub>2</sub>/p<sub>1</sub>)<sup>(γ − 1)/γ</sup>
|-
|style="text-align:center;"| V<sub>2</sub>/V<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)<sup>−γ</sup>
| V<sub>2</sub> = V<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)
| T<sub>2</sub> = T<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)<sup>(1 − γ)</sup>
|-
|style="text-align:center;"| T<sub>2</sub>/T<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)<sup>γ/(γ − 1)</sup>
| V<sub>2</sub> = V<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)<sup>1/(1 − γ) </sup>
| T<sub>2</sub> = T<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
|-
|rowspan="3"| [[Polytropic process]]
|rowspan="3" style="text-align:center;" |P V<sup>''n''</sup>
|style="text-align:center;"| p<sub>2</sub>/p<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>(p<sub>2</sub>/p<sub>1</sub>)
| V<sub>2</sub> = V<sub>1</sub>(p<sub>2</sub>/p<sub>1</sub>)<sup>(−1/''n'')</sup>
| T<sub>2</sub> = T<sub>1</sub>(p<sub>2</sub>/p<sub>1</sub>)<sup>(''n'' − 1)/''n''</sup>
|-
|style="text-align:center;"| V<sub>2</sub>/V<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)<sup>−''n''</sup>
| V<sub>2</sub> = V<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)
| T<sub>2</sub> = T<sub>1</sub>(V<sub>2</sub>/V<sub>1</sub>)<sup>(1 − ''n'')</sup>
|-
|style="text-align:center;"| T<sub>2</sub>/T<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)<sup>''n''/(''n'' − 1)</sup>
| V<sub>2</sub> = V<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)<sup>1/(1 − ''n'') </sup>
| T<sub>2</sub> = T<sub>1</sub>(T<sub>2</sub>/T<sub>1</sub>)
|-
|rowspan="2"| [[Isenthalpic process]]<br>(Irreversible [[adiabatic process]])
|rowspan="2"| {{center|[[Enthalpy]]{{ref_label|B|b|none}}}}
|style="text-align:center;"| p<sub>2</sub> − p<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub> + (p<sub>2</sub> − p<sub>1</sub>)
|
| T<sub>2</sub> = T<sub>1</sub> + μ<sub>JT</sub>(p<sub>2</sub> − p<sub>1</sub>)
|-
|style="text-align:center;"| T<sub>2</sub> − T<sub>1</sub>
| p<sub>2</sub> = p<sub>1</sub> + (T<sub>2</sub> − T<sub>1</sub>)/μ<sub>JT</sub>
|
| T<sub>2</sub> = T<sub>1</sub> + (T<sub>2</sub> − T<sub>1</sub>)
|}

{{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 a calorifically [[perfect 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 mono atomic 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.

{{Note_label|B|b|none}} '''b.''' In an isenthalpic process, system [[enthalpy]] (''H'') is constant. In the case of [[free expansion]] for an ideal gas, there are no molecular interactions, and the temperature remains constant.  For real gases, the molecules do interact via attraction or repulsion depending on temperature and pressure, and heating or cooling does occur.  This is known as the [[Joule–Thomson effect]].  For reference, the Joule–Thomson coefficient μ<sub>JT</sub> for air at room temperature and sea level is 0.22&nbsp;°C/[[bar (unit)|bar]].<ref>{{Cite journal| pmc=1084398|title= The Joule-Thomson Effect in Air|journal= Proceedings of the National Academy of Sciences of the United States of America|volume= 12|issue= 1|pages= 55–58 |author= J. R. Roebuck|year= 1926|bibcode= 1926PNAS...12...55R|doi= 10.1073/pnas.12.1.55|pmid=16576959|doi-access= free}}</ref>

== Deviations from ideal behavior of real gases ==
The equation of state given here (''PV'' = ''nRT'') 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 [[Molecule#Molecular size|molecular size]] and intermolecular attractions, it is most accurate for [[monatomic]] gases at high temperatures and low pressures. The 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.

== Derivations ==

=== Empirical ===
The empirical laws that led to the derivation of the ideal gas law were discovered with experiments that changed only 2 state variables of the gas and kept every other one constant.

All the possible gas laws that could have been discovered with this kind of setup are:
* [[Boyle's law]] ({{EquationRef|1|Equation 1}}) <math display="block">PV = C_1 \quad \text{or} \quad P_1 V_1 = P_2 V_2 </math>
* [[Charles's law]] ({{EquationRef|2|Equation 2}}) <math display="block">\frac{V}{T} = C_2 \quad \text{or} \quad \frac{V_1}{T_1} = \frac{V_2}{T_2} </math>
* [[Avogadro's law]] ({{EquationRef|3|Equation 3}}) <math display="block">\frac{V}{N}=C_3 \quad \text{or} \quad \frac{V_1}{N_1}=\frac{V_2}{N_2} </math>
* [[Gay-Lussac's law]] ({{EquationRef|4|Equation 4}}) <math display="block">\frac{P}{T}=C_4 \quad \text{or} \quad \frac{P_1}{T_1}=\frac{P_2}{T_2} </math>
* {{EquationRef|5|Equation 5}} <math display="block">NT = C_5 \quad \text{or} \quad N_1 T_1 = N_2 T_2 </math>
* {{EquationRef|6|Equation 6}} <math display="block">\frac{P}{N} = C_6 \quad \text{or} \quad \frac{P_1}{N_1}=\frac{P_2}{N_2} </math>
{{Ideal gas law relationships.svg}}
where ''P'' stands for [[pressure]], ''V'' for [[volume]], ''N'' for number of particles in the gas and ''T'' for [[temperature]]; where <math>C_1, C_2, C_3, C_4, C_5, C_6 </math> are constants in this context because of each equation requiring only the parameters explicitly noted in them changing.

To derive the ideal gas law one does not need to know all 6 formulas, one can just know 3 and with those derive the rest or just one more to be able to get the ideal gas law, which needs 4.

Since each formula only holds when only the state variables involved in said formula change while the others (which are a property of the gas but are not explicitly noted in said formula) remain constant, we cannot simply use algebra and directly combine them all. This is why:  Boyle did his experiments while keeping ''N'' and ''T'' constant and this must be taken into account (in this same way, every experiment kept some parameter as constant and this must be taken into account for the derivation).

Keeping this in mind, to carry the derivation on correctly, one must imagine the [[gas]] being altered by one process at a time (as it was done in the experiments). The derivation using 4 formulas can look like this:

at first the gas has parameters <math>P_1, V_1, N_1, T_1 </math>

Say, starting to change only pressure and volume, according to [[Boyle's law]] ({{EquationNote|1|Equation 1}}), then:
{{NumBlk||<math display="block">P_1 V_1 = P_2 V_2 </math>|{{EquationRef|7}}}}
After this process, the gas has parameters <math>P_2,V_2,N_1,T_1 </math>

Using then equation ({{EquationNote|5}}) to change the number of particles in the gas and the temperature,
{{NumBlk||<math display="block">N_1 T_1 = N_2 T_2 </math>|{{EquationRef|8}}}}
After this process, the gas has parameters <math>P_2,V_2,N_2,T_2 </math>

Using then equation ({{EquationNote|6}}) to change the pressure and the number of particles,
{{NumBlk||<math display="block">\frac{P_2}{N_2} = \frac{P_3}{N_3} </math>|{{EquationRef|9}}}}
After this process, the gas has parameters <math>P_3,V_2,N_3,T_2 </math>

Using then [[Charles's law]] (equation 2) to change the volume and temperature of the gas,
{{NumBlk||<math display="block">\frac{V_2}{T_2} = \frac{V_3}{T_3} </math>|{{EquationRef|10}}}}
After this process, the gas has parameters <math>P_3,V_3,N_3,T_3 </math>

Using simple algebra on equations ({{EquationNote|7}}), ({{EquationNote|8}}), ({{EquationNote|9}}) and ({{EquationNote|10}}) yields the result:
<math display="block">\frac{P_1 V_1}{N_1 T_1} = \frac{P_3 V_3}{N_3 T_3} </math> or <math display="block">\frac{PV}{NT} = k_\text{B} ,</math> where <math> k_\text{B} </math> stands for the [[Boltzmann constant]].

Another equivalent result, using the fact that <math>nR = N k_\text{B} </math>, where ''n'' is the number of [[Mole (unit)|moles]] in the gas and ''R'' is the [[Gas constant|universal gas constant]], is:
<math display="block">PV = nRT, </math> which is known as the ideal gas law.

If three of the six equations are known, it may be possible to derive the remaining three using the same method. However, because each formula has two variables, this is possible only for certain groups of three. For example, if you were to have equations ({{EquationNote|1}}), ({{EquationNote|2}}) and ({{EquationNote|4}}) you would not be able to get any more because combining any two of them will only give you the third. However, if you had equations ({{EquationNote|1}}), ({{EquationNote|2}}) and ({{EquationNote|3}}) you would be able to get all six equations because combining ({{EquationNote|1}}) and ({{EquationNote|2}}) will yield ({{EquationNote|4}}), then ({{EquationNote|1}}) and ({{EquationNote|3}}) will yield ({{EquationNote|6}}), then ({{EquationNote|4}}) and ({{EquationNote|6}}) will yield ({{EquationNote|5}}), as well as would the combination of ({{EquationNote|2}}) and ({{EquationNote|3}}) as is explained in the following visual relation:

[[File:Relationship between the 6 gas laws .jpg|center|thumb|Relationship between the six gas laws]]
where the numbers represent the gas laws numbered above.

If you were to use the same method used above on 2 of the 3 laws on the vertices of one triangle that has a "O" inside it, you would get the third.

For example:

Change only pressure and volume first:
{{NumBlk||<math display="block">P_1 V_1 = P_2 V_2</math>|{{EquationRef|1'}}}}

then only volume and temperature:
{{NumBlk||<math display="block">\frac{V_2}{T_1} = \frac{V_3}{T_2}</math>|{{EquationRef|2'}}}}

then as we can choose any value for <math>V_3</math>, if we set <math>V_1 = V_3</math>, equation ({{EquationNote|2'}}) becomes:
{{NumBlk||<math display="block">\frac{V_2}{T_1} = \frac{V_1}{T_2}</math>|{{EquationRef|3'}}}}

combining equations ({{EquationNote|1'}}) and ({{EquationNote|3'}}) yields <math>\frac{P_1}{T_1} = \frac{P_2}{T_2}</math>, which is equation ({{EquationNote|4}}), of which we had no prior knowledge until this derivation.

=== Theoretical ===

==== Kinetic theory ====
{{Main|Kinetic theory of gases}}

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.

First we show that the fundamental assumptions of the kinetic theory of gases imply that
: <math>P = \frac{1}{3}nmv_{\text{rms}}^2.</math>

Consider a container in the <math>xyz</math> Cartesian coordinate system. For simplicity, we assume that a third of the molecules moves parallel to the <math>x</math>-axis, a third moves parallel to the <math>y</math>-axis and a third moves parallel to the <math>z</math>-axis. If all molecules move with the same velocity <math>v</math>, denote the corresponding pressure by <math>P_0</math>. We choose an area <math>S</math> on a wall of the container, perpendicular to the <math>x</math>-axis. When time <math>t</math> elapses, all molecules in the volume <math>vtS</math> moving in the positive direction of the <math>x</math>-axis will hit the area. There are <math>NvtS</math> molecules in a part of volume <math>vtS</math> of the container, but only one sixth (i.e. a half of a third) of them moves in the positive direction of the <math>x</math>-axis. Therefore, the number of molecules <math>N'</math> that will hit the area <math>S</math> when the time <math>t</math> elapses is <math>NvtS/6</math>.

When a molecule bounces off the wall of the container, it changes its momentum <math>\mathbf{p}_1</math> to <math>\mathbf{p}_2=-\mathbf{p}_1</math>. Hence the magnitude of change of the momentum of one molecule is <math>|\mathbf{p}_2-\mathbf{p}_1|=2mv</math>. The magnitude of the change of momentum of all molecules that bounce off the area <math>S</math> when time <math>t</math> elapses is then <math>|\Delta \mathbf{p}|=2mvN'/V=NtSmv^2/(3V)=ntSmv^2/3</math>. From <math>F=|\Delta \mathbf{p}|/t</math> and <math>P_0=F/S</math> we get

: <math>P_0=\frac{1}{3}nm v^2.</math>

We considered a situation where all molecules move with the same velocity <math>v</math>. Now we consider a situation where they can move with different velocities, so we apply an "averaging transformation" to the above equation, effectively replacing <math>P_0</math> by a new pressure <math>P</math> and <math>v^2</math> by the arithmetic mean of all squares of all velocities of the molecules, i.e. by <math>v_{\text{rms}}^2.</math> Therefore
: <math>P=\frac{1}{3}nm v_{\text{rms}}^2</math>
which gives the desired formula.

Using the [[Maxwell–Boltzmann distribution]], the fraction of molecules that have a speed in the range <math>v</math> to <math>v + dv</math> is <math>f(v) \, dv</math>, where
: <math>f(v) = 4\pi \left(\frac{m}{2\pi k_{\rm B}T}\right)^{\!\frac{3}{2}}v^2 e^{-\frac{mv^2}{2k_{\rm B}T}}</math>
and <math>k</math> denotes the Boltzmann constant. The root-mean-square speed can be calculated by
: <math>v_{\text{rms}}^2 = \int_0^\infty v^2 f(v) \, dv = 4\pi \left(\frac{m}{2\pi k_{\rm B}T}\right)^{\frac{3}{2}}\int_0^\infty v^4 e^{-\frac{mv^2}{2k_{\rm B}T}} \, dv.</math>

Using the integration formula
: <math>\int_0^\infty x^{2n}e^{-\frac{x^2}{a^2}} \, dx = \sqrt{\pi} \, \frac{(2n)!}{n!}\left(\frac{a}{2}\right)^{2n+1},\quad n\in\mathbb{N},\,a\in\mathbb{R}^+,</math>
it follows that
: <math>v_{\text{rms}}^2 = 4\pi\left(\frac{m}{2\pi k_{\rm B}T}\right)^{\!\frac{3}{2}}\sqrt{\pi} \, \frac{4!}{2!}\left(\frac{\sqrt{\frac{2k_{\rm B}T}{m}}}{2}\right)^{\!5} = \frac{3k_{\rm B}T}{m},</math>
from which we get the ideal gas law:
: <math>P = \frac{1}{3} nm\left(\frac{3k_{\rm B}T}{m}\right) = nk_{\rm B}T.</math>

==== Statistical mechanics ====
{{Main|Statistical mechanics}}

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 (two times) the time-averaged kinetic energy of the particle is:
: <math>\begin{align}
\langle \mathbf{q} \cdot \mathbf{F} \rangle
&= \left\langle q_{x} \frac{dp_{x}}{dt} \right\rangle +
\left\langle q_{y} \frac{dp_{y}}{dt} \right\rangle +
\left\langle q_{z} \frac{dp_{z}}{dt} \right\rangle\\
&=-\left\langle q_{x} \frac{\partial H}{\partial q_x} \right\rangle -
\left\langle q_{y} \frac{\partial H}{\partial q_y} \right\rangle -
\left\langle q_{z} \frac{\partial H}{\partial q_z} \right\rangle = -3k_\text{B} T,
\end{align}</math>
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
: <math>3Nk_{\rm B} T = - \left\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \right\rangle.</math>

By [[Newton's third law]] and the ideal gas assumption, the net force of the system is the force applied by the walls of the container, and this force is given by the pressure ''P'' of the gas. Hence
: <math>-\left\langle\sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k}\right\rangle = P \oint_{\text{surface}} \mathbf{q} \cdot d\mathbf{S},</math>
where d'''S''' is the infinitesimal area element along the walls of the container. Since the [[divergence]] of the position vector '''q''' is
: <math>
\nabla \cdot \mathbf{q} =
\frac{\partial q_{x}}{\partial q_{x}} +
\frac{\partial q_{y}}{\partial q_{y}} +
\frac{\partial q_{z}}{\partial q_{z}} = 3,
</math>
the [[divergence theorem]] implies that
: <math>P \oint_{\text{surface}} \mathbf{q} \cdot d\mathbf{S}
= P \int_{\text{volume}} \left( \nabla \cdot \mathbf{q} \right) dV
= 3PV,</math>
where ''dV'' is an infinitesimal volume within the container and ''V'' is the total volume of the container.

Putting these equalities together yields
: <math>3 N k_\text{B} T = -\left\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \right\rangle = 3PV,</math>
which immediately implies the ideal gas law for ''N'' particles:
: <math>PV = Nk_{\rm B} T = nRT,</math>
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]].

== Other dimensions ==
For a ''d''-dimensional system, the ideal gas pressure is:<ref>{{cite journal |last1=Khotimah |first1=Siti Nurul |last2=Viridi | first2=Sparisoma | date=2011-06-07 | title=Partition function of 1-, 2-, and 3-D monatomic ideal gas: A simple and comprehensive review |journal=Jurnal Pengajaran Fisika Sekolah Menengah |volume=2 |issue=2 |pages=15–18 |arxiv=1106.1273 |bibcode=2011arXiv1106.1273N}}</ref>
: <math>P^{(d)} = \frac{N k_{\rm B} T}{L^d}, </math>
where <math>L^d</math> is the volume of the ''d''-dimensional domain in which the gas exists. The dimensions of the pressure changes with dimensionality.

== See also ==
{{Portal|Physics}}
* {{annotated link|Boltzmann constant}}
* {{annotated link|Partition function (statistical mechanics)|Configuration integral}}
* {{annotated link|Dynamic pressure}}
* [[Gas laws]]
* {{annotated link|Internal energy}}
* {{annotated link|Van der Waals equation}}

== References ==
{{Reflist}}

== Further reading ==
* {{cite book |last1=Davis |last2=Masten |title=Principles of Environmental Engineering and Science |publisher=McGraw-Hill |location=New York |year=2002 |isbn=0-07-235053-9 }}

== External links ==
* {{cite web |archive-url=https://web.archive.org/web/20070705093306/http://www.gearseds.com/curriculum/learn/lesson.php?id=23&chapterid=5 |archive-date=July 5, 2007 |url=http://www.gearseds.com/curriculum/learn/lesson.php?id=23&chapterid=5 |title=Website giving credit to Benoît Paul Émile Clapeyron, (1799–1864) in 1834}}
* [https://web.archive.org/web/20120428193950/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. Vu-Quoc, L., [https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)], 2008. this wiki site is down; see [https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 this article in the web archive on 2012 April 28].
* [http://www.nigelhewitt.co.uk/diving/maths/vdw.html Gas equations in detail]

{{Mole concepts}}
{{Statistical mechanics topics}}
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{{DEFAULTSORT:Ideal Gas Law}}
[[Category:Gas laws]]
[[Category:Ideal gas]]
[[Category:Equations of state]]
[[Category:1834 introductions]]