Ideal gas law

Values of R	Units
8.314472	J·K-1·mol-1
0.082057	L·atm·K-1·mol-1
8.205745 × 10-5	m3·atm·K-1·mol-1
8.314472	L·kPa·K-1·mol-1
8.314472	m3·Pa·K-1·mol-1
62.36367	L·mmHg·K-1·mol-1
62.36367	L·Torr·K-1·mol-1
83.14472	L·mbar·K-1·mol-1
10.7316	ft3·psi· °R-1·lb-mol-1
0.73024	ft3·atm·°R-1·lb-mol-1

The ideal gas law is the equation of state of an ideal gas (also known as perfect gas). As an equation of state it relates the absolute pressure p of an ideal gas to its absolute temperature T. Further parameters that enter the equation are the volume V of the container holding the gas and the number of moles n in the container. The equation is,

$pV=nRT,\,$

where R is the molar gas constant defined as the product of the Boltzmann constant k_B and Avogadro's constant N_A,

$R\equiv N_{\mathrm {A} }k_{\mathrm {B} }.$

Currently, the most accurate value of R is: 8.314472 ± 0.000015 J·K^-1·mol^-1.

The law applies to hypothetical gases that consist of atoms or molecules that do not interact, i.e., that move through the container independently of one another. The law is a useful approximation for calculating temperatures, volumes, pressures or number of moles for many gases over a wide range of temperatures and pressures, as long as the temperatures and pressures are far from the values where condensation or sublimation occurs.

The ideal gas law is the combination of Boyle's law (given in 1662 and stating that pressure is inversely proportional to volume) and Gay-Lussac's law (given in 1808 and stating that pressure is proportional to temperature). Gay-Lussac's law was found a few decennia before Gay-Lussac's publication by Charles. In some countries the ideal gas law is known as the Boyle-Gay-Lussac law.

Real gases deviate from ideal gas behavior because of the intermolecular attractive and repulsive forces. The deviation is especially significant at low temperatures or high pressures, i.e., close to condensation. There are many equations of state available for use with real gases, the simplest of which is the van der Waals equation.

Statistical mechanics derivation

The statistical mechanics derivation of the ideal gas law, now given, provides the most precise insight into the microscopic conditions that a gas must satisfy in order to be called an ideal gas. In the derivation below it will be assumed that the molecules constituting the gas are practically independent systems, each pursuing its own motion. We must also assume, somewhat contradictorily, that exchange of energy between molecules occasionally takes place, so that the system can achieve a thermal equilibrium. This occasional exchange of energy can proceed via collisions with the walls, through interaction with a radiation field, or sporadic molecule-molecule collisions. This energy exchange is not explicitly included in the following formalism.

We recall from statistical mechanics that the canonical partition function is a function of N ≡ nN_A, V, and T and is defined by

$Q(N,V,T)=\sum _{I}e^{-{\mathcal {E}}_{I}/(k_{\mathrm {B} }T)}$

where ${\mathcal {E}}_{I}$ is the I-th energy of the total gas (energy of all N molecules). We further recall that according to statistical mechanics the absolute pressure is obtained from the partition function by

$p=k_{\mathrm {B} }T\left({\frac {\partial \ln Q}{\partial V}}\right)_{N,T}.$

The only approximation that will be made is that the energies ${\mathcal {E}}_{I}$ are taken to be sums of one-molecule energies $\varepsilon _{i}$. These one-molecule energies are those of a single molecule moving by itself in the vessel. We write

${\mathcal {E}}_{I}=\varepsilon _{i_{1}}+\varepsilon _{i_{2}}+\cdots$

The total partition function Q will factorize into one-molecule partition functions q given by,

$q(N,V,T)=\sum _{i}e^{-\varepsilon _{i}/(k_{\mathrm {B} }T)}.$

From the additivity of the molecular energies follows (assuming that the gas consists of one type of molecules only),

$Q={\frac {1}{N!}}q^{N}$

The appearance of the factorial N! is a consequence of the molecules being non-distinguishable; this factor is of no importance to the equation of state, but contributes to the entropy of the gas. Now,

$p=k_{\mathrm {B} }T\left({\frac {\partial \ln Q}{\partial V}}\right)=k_{\mathrm {B} }T\left({\frac {\partial (N\ln q-\ln N!)}{\partial V}}\right)=Nk_{\mathrm {B} }T\left({\frac {\partial \ln q}{\partial V}}\right).$

The molecular energy $\varepsilon _{i}$ can be exactly separated as

$\varepsilon _{i}=\varepsilon _{i}^{\mathrm {transl} }+\varepsilon _{i}^{\mathrm {internal} }\quad \Longrightarrow \quad q=q^{\mathrm {transl} }\;q^{\mathrm {internal} },$

where $\varepsilon _{i}^{\mathrm {transl} }$ is the translational energy of the center of mass of the molecule and $\varepsilon _{i}^{\mathrm {internal} }$ is the internal (rotation, vibration, electronic, nuclear) energy of the molecule. The internal energy of the molecule does not depend on the volume V, but the translational does depend on V, hence

$p=Nk_{\mathrm {B} }T\;{\frac {\partial \ln q^{\mathrm {transl} }}{\partial V}}.$

The problem of one molecule moving in a box of volume V is one of the few problems in quantum mechanics that can be solved analytically. That is, the energies $\varepsilon _{i}^{\mathrm {transl} }$ are known exactly. To a very good approximation one may replace the sum appearing in q^transl by an integral, finding

$q^{\mathrm {transl} }={\frac {V}{\Lambda ^{3}}}\quad {\hbox{with}}\quad \Lambda =\left({\frac {h^{2}}{2\pi Mk_{\mathrm {B} }T}}\right)^{1/2}.$

Note that the "thermal de Broglie wavelength" Λ does not depend on the volume V, so that

$p=Nk_{\mathrm {B} }T\left({\frac {\partial (\ln V-3\ln \Lambda )}{\partial V}}\right)={\frac {Nk_{\mathrm {B} }T}{V}}.$

Using that N = nN_A and N_Ak_B = R, we have proved the ideal gas law. In this derivation neither collisions nor sizes of molecules play a role; the only assumption made is that a single molecules moves in the vessel unhindered by the other molecules.

Background

The gas laws were developed in the 1660's, starting with Boyle's law, derived by Robert Boyle. Boyle's law states that the volume of a sample of gas at a given temperature varies inversely with the applied pressure, or V = constant / p (at a fixed temperature and amount of gas). Jacques Alexandre César Charles' experiments with hot-air balloons, and additional contributions by John Dalton (1801) and Joseph Louis Gay-Lussac (1808) showed that a sample of gas, at a fixed pressure, increases in volume linearly with the temperature, or V/T is constant. Extrapolations of volume/temperature data for many gases, to a volume of zero, all cross at about −273 °C, which is defined as absolute zero. Since real gases would liquefy before reaching this temperature, this temperature region remains a theoretical minimum.

In 1811 Amedeo Avogadro re-interpreted Gay-Lussac's law of combining volumes to state Avogadro's law: equal volumes of any two gases at the same temperature and pressure contain the same number of molecules.