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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 behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law. It can also be derived from kinetic theory, as was achieved (apparently independently) by August Krönig in 1856 and Rudolf Clausius in 1857.

The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation is:

${\displaystyle pV=nRT\,}$

where p is the absolute pressure of the gas; V is the volume of the gas; n is the amount of substance of the gas, usually measured in moles; R is the gas constant (which is 8.314472 J·K·mol in SI units); and T is the absolute temperature.

Since it neglects both molecular size and intermolecular attractions, the ideal gas law is most accurate for monatomic gases at high temperatures and low pressures. The neglect of molecular size becomes less important for larger volumes, i.e., for lower pressures. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy i.e., with increasing temperatures. More sophisticated equations of state, such as the van der Waals equation, allow deviations from ideality caused by molecular size and intermolecular forces to be taken into account.

Alternative Forms

As the amount of substance could be given in mass instead of moles, sometimes an alternative form of the ideal gas law is useful. The number of moles (n) is equal to the mass (m) divided by the molar mass (M):

${\displaystyle n={\frac {m}{M}}}$

By replacing ${\displaystyle n\,}$, we get:

${\displaystyle \ pV={\frac {m}{M}}RT}$

from where

${\displaystyle \ p=\rho {\frac {R}{M}}T}$.

This form of the ideal gas law is very useful because it links pressure, density ρ = m/V, and temperature in a unique formula independent from the quantity of the considered gas.

In statistical mechanics the following molecular equation is derived from first principles:

${\displaystyle \ pV=NkT.}$

Here k is the Boltzmann constant, and N is the actual number of molecules, in contrast 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.

From here we can notice that for an average particle mass of μ times the atomic mass constant m_u (i.e., the mass is μ u)

${\displaystyle N={\frac {m}{\mu m_{\mathrm {u} }}}}$

and since ρ = m/V, we find that the ideal gas law can be rewritten as:

${\displaystyle p={\frac {1}{V}}{\frac {m}{\mu m_{\mathrm {u} }}}kT={\frac {k}{\mu m_{\mathrm {u} }}}\rho T.}$

Calculations

The table below essentially simplifies the ideal gas equation for a particular processes, thus making this 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 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.

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).

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.

a. In an isentropic process, system entropy (Q) is constant. Under these conditions, p₁V₁ = p₂V₂, 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 (N₂) and oxygen (O₂), (and air, which is 99% diatomic). Also γ is typically 1.6 for monatomic gases like the noble gases helium (He), and argon (Ar). In internal combustion engines γ varies between 1.35 and 1.15, depending on constitution gases and temperature.

Derivations

Empirical

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

${\displaystyle {\frac {pV}{T}}=C}$

where C is a constant which is directly proportional to the amount of gas, n (Avogadro's law). The proportionality factor is the universal gas constant, R, i.e. C = nR.

Hence the ideal gas law

${\displaystyle pV=nRT\,}$

Theoretical

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.

From statistical mechanics

Let q = (q_x, q_y, q_z) and p = (p_x, p_y, p_z) 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:
${\displaystyle {\begin{aligned}\langle \mathbf {q} \cdot \mathbf {F} \rangle &={\Bigl \langle }q_{x}{\frac {dp_{x}}{dt}}{\Bigr \rangle }+{\Bigl \langle }q_{y}{\frac {dp_{y}}{dt}}{\Bigr \rangle }+{\Bigl \langle }q_{z}{\frac {dp_{z}}{dt}}{\Bigr \rangle }\\&=-{\Bigl \langle }q_{x}{\frac {\partial H}{\partial q_{x}}}{\Bigr \rangle }-{\Bigl \langle }q_{y}{\frac {\partial H}{\partial q_{y}}}{\Bigr \rangle }-{\Bigl \langle }q_{z}{\frac {\partial H}{\partial q_{z}}}{\Bigr \rangle }=-3k_{B}T,\end{aligned}}}$
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

${\displaystyle 3Nk_{B}T=-{\biggl \langle }\sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}{\biggr \rangle }.}$

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

${\displaystyle -{\biggl \langle }\sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}{\biggr \rangle }=p\oint _{\mathrm {surface} }\mathbf {q} \cdot d\mathbf {S} ,}$

where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is

${\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {q} ={\frac {\partial q_{x}}{\partial q_{x}}}+{\frac {\partial q_{y}}{\partial q_{y}}}+{\frac {\partial q_{z}}{\partial q_{z}}}=3,}$

the divergence theorem implies that

${\displaystyle p\oint _{\mathrm {surface} }\mathbf {q} \cdot d\mathbf {S} =p\int _{\mathrm {volume} }\left({\boldsymbol {\nabla }}\cdot \mathbf {q} \right)dV=3pV,}$

where dV is an infinitesimal volume within the container and V is the total volume of the container.

Putting these equalities together yields

${\displaystyle 3Nk_{B}T=-{\biggl \langle }\sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}{\biggr \rangle }=3pV,}$

which immediately implies the ideal gas law for N particles:

${\displaystyle pV=Nk_{B}T=nRT,\,}$

where n = N/N_A is the number of moles of gas and R = N_Ak_B is the gas constant.

The readers are referred to the comprehensive article 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, but without using the equipartition theorem, is provided.

See also

• Combined gas law
• Van der Waals equation
• Boltzmann's constant
• Configuration integral

References

1. Clapeyron, E. (1834), "Mémoire sur la puissance motrice de la chaleur", Journal de l'École Polytechnique, XIV: 153–90. Facsimile at the Bibliothèque nationale de France (pp. 153–90).
2. Krönig, A. (1856), "Grundzüge einer Theorie der Gase", Annalen der Physik, 99: 315–22, doi:10.1002/andp.18561751008. Facsimile at the Bibliothèque nationale de France (pp. 315–22).
3. Clausius, R. (1857), "Ueber die Art der Bewegung, welche wir Wärme nennen", Annalen der Physik und Chemie, 100: 353–79, doi:10.1002/andp.18571760302. Facsimile at the Bibliothèque nationale de France (pp. 353–79).
4. R is also sometimes expressed as Error in {{val}}: parameter 2 cannot use e notation.

Further reading

• Davis and Masten Principles of Environmental Engineering and Science, McGraw-Hill Companies, Inc. New York (2002) ISBN 0-07-235053-9
• Website giving credit to Benoît Paul Émile Clapeyron, (1799-1864) in 1834

v t e Statistical mechanics
Theory	Principle of maximum entropy ergodic theory
Statistical thermodynamics	Ensembles partition functions equations of state thermodynamic potential: U H F G Maxwell relations
Models	Ferromagnetism models Ising Potts Heisenberg percolation Particles with force field depletion force Lennard-Jones potential
Mathematical approaches	Boltzmann equation H-theorem Vlasov equation BBGKY hierarchy stochastic process mean-field theory and conformal field theory
Critical phenomena	Phase transition Critical exponents correlation length size scaling
Entropy	Boltzmann Shannon Tsallis Rényi von Neumann
Applications	Statistical field theory elementary particle superfluidity Condensed matter physics Complex system chaos information theory Boltzmann machine

• Gas laws