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 a perfect gas) that relates its absolute pressure p to its absolute temperature T. Further parameters that enter the equation are the volume V of the container holding the gas and the amount n (mol) of gas contained in there. The law reads

$pV=nRT\,$

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 ideal gases which are hypothetical gases that consist of molecules that do not interact, i.e., that move through the container independently of one another. In contrast to what is sometimes stated (see, e.g., Ref.) an ideal gas does not necessarily consist of point particles without internal structure, but may consist of polyatomic molecules with internal rotational, vibrational, and electronic degrees of freedom. The ideal gas law describes the motion of the centers of mass of the molecules and, indeed, mass centers may be seen as structureless point masses. However, for other properties of ideal gases, such as entropy, the internal structure may play a role.

The ideal gas law is a useful approximation for calculating temperatures, volumes, pressures or number of moles for many gases over a wide range of values, as long as the temperatures and pressures are far from the values where condensation or sublimation occurs.

Real gases deviate from ideal gas behavior because the intermolecular attractive and repulsive forces cause the motions of the molecules to be correlated. 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.

Historic background

The ideal gas law was initialized by Robert Boyle who formulated in 1662 Boyle's law, which 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). In 1699, Guillaume Amontons formulated what is now known as Amontons' law, p = constant / T.

At the end of the 18th century and the beginning of the 19th century, Jacques Alexandre César Charles' experimented (around 1780) 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, i.e. V / T is constant. Because Boyle and Gay-Lussac published their findings, the ideal gas law is known in some countries as the Boyle-Gay-Lussac law.

Extrapolation of the volume/temperature relationship of ideal and many real gases to zero volume crosses the T-axis at about −273 °C. This temperature is defined as the absolute zero temperature. Since any real gas would liquefy before reaching it, 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.

With hindsight, Boyle-Gay-Lussac's law, Amontons' law and Avogadro's law all turned out to be special cases of the ideal gas law.

Statistical mechanics derivation

The statistical mechanics derivation of the ideal gas law 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 is assumed that the molecules constituting the gas are practically independent systems, each pursuing its own motion. On the other hand, it is assumed 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 equilibrium 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). From quantum mechanics follows that a gas in a finite-size container has discrete (countable) energies; I is the discrete index labeling the different energies. Further we recall that according to statistical mechanics the Helmoltz free energy is given by $A=-k_{\mathrm {B} }T\,\ln Q$. The following expression for the absolute pressure p will be the starting point in the derivation

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

The only approximation (but a very drastic one) that must be made is that the energies ${\mathcal {E}}_{I}$ are sums of one-molecule energies $\varepsilon _{i}$. These one-molecule energies are those of a single molecule moving by itself in the vessel. This approximation, which is encountered in many branches of physics, is known as the the independent particle approximation. Thus

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

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

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

Now,

$Q=\sum _{i_{1},i_{2},\ldots }e^{-(\varepsilon _{i_{1}}+\varepsilon _{i_{2}}+\cdots )/(k_{\mathrm {B} }T)}=\sum _{i_{1}}e^{-\varepsilon _{i_{1}}/(k_{\mathrm {B} }T)}\sum _{i_{2}}e^{-\varepsilon _{i_{2}}/(k_{\mathrm {B} }T)}\cdots =q^{N}.$

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

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

where the factorial 1/N! must be inserted to avoid overcounting: the molecules are indistinguishable. This overcounting correction is of no consequence to the equation of state, but contributes to the entropy of the gas. The factorization of Q would not involve any approximation if (i) the molecules would not interact and if (ii) every molecule had the whole volume V of the container to its disposal, or in other words, if the molecules themselves had zero volume.

Now,

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

where we used the rules ln(a/b) = lna - lnb and lnaⁿ = n lna.

It follows from both classical mechanics and quantum mechanics that 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 (rotational, vibrational, electronic) energy of the molecule. This factorization of the one-molecule partition function into a translational and an internal factor proceeds in the same way as the factorization of the N-molecule partition function Q into one-molecule partition functions.

The internal energy of the molecule does not depend on the volume V (this is an exact result), but the translational energy does, hence

$p=Nk_{\mathrm {B} }T\left({\frac {\partial \ln q^{\mathrm {transl} }}{\partial V}}+{\frac {\partial \ln q^{\mathrm {internal} }}{\partial V}}\right)=Nk_{\mathrm {B} }T\;{\frac {\partial \ln q^{\mathrm {transl} }}{\partial V}}$

The determination of the translational energy 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 $q^{\mathrm {transl} }$ by an integral, finding

$q^{\mathrm {transl} }\equiv \sum _{i}e^{-\varepsilon _{i}^{\mathrm {transl} }/(k_{\mathrm {B} }T)}={\frac {V}{\Lambda ^{3}}}\quad {\hbox{with}}\quad \Lambda =\left({\frac {h^{2}}{2\pi Mk_{\mathrm {B} }T}}\right)^{1/2}$

where h is Planck's constant and M is the total mass of the molecule. Note that the "thermal de Broglie wavelength" Λ does not depend on the volume V, so that

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

Here we applied that

${\frac {\partial \ln V}{\partial V}}={\frac {1}{V}}\quad {\hbox{and}}\quad {\frac {\partial \ln \Lambda }{\partial V}}=0$

Using that N = nN_A and N_Ak_B = R, we have

$p\,V=nN_{\mathrm {A} }\,k_{\mathrm {B} }\,T=nR\,T$

and that completes the proof of the ideal gas law. In this derivation neither collisions nor sizes of molecules play a role; the only assumptions made are that a single molecule moves in the vessel unhindered by the other molecules and that there is sufficient, negligible, direct or indirect molecular interaction to obtain thermal equilibrium.