Virial equation of state

The virial equation of state is used to describe the behavior of diluted gases. It is usually written as an expansion of the compressibility factor, $Z$, in terms of either the density or the pressure. Such an expansion was first introduced by Heike Kamerlingh Onnes in 1901 (Ref. 1 and 2). In the first case:

${\frac {pV}{Nk_{B}T}}=Z=1+\sum _{k=2}^{\infty }B_{k}(T)\rho ^{k-1}$.

where

• $p$ is the pressure
• $V$ is the volume
• $N$ is the number of molecules
• $T$ is the temperature
• $k_{B}$ is the Boltzmann constant
• $\rho \equiv {\frac {N}{V}}$ is the (number) density
• $B_{k}\left(T\right)$ is called the k-th virial coefficient

Virial coefficients

The second virial coefficient represents the initial departure from ideal-gas behavior

$B_{2}(T)={\frac {N_{A}}{2V}}\int ....\int (1-e^{-\Phi /k_{B}T})~d\tau _{1}d\tau _{2}$

where $N_{A}$ is Avogadros number and $d\tau _{1}$ and $d\tau _{2}$ are volume elements of two different molecules in configuration space.

One can write the third virial coefficient as

$B_{3}(T)=-{\frac {1}{3V}}\int \int \int f_{12}f_{13}f_{23}dr_{1}dr_{2}dr_{3}$

where f is the Mayer f-function (see also: Cluster integrals). See also:

• M. S. Wertheim "Fluids of hard convex molecules III. The third virial coefficient", Molecular Physics 89 pp. 1005-1017 (1996)

Convergence

For a commentary on the convergence of the virial equation of state see Ref 4 and section 3 of Ref. 5.

References

1. H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden 71 pp. 3-25 (1901)
2. H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 4 pp. 125-147 (1902)
3. James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics 7 pp. 195-229 (1940)
4. J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics 5 pp. 841-847 (1964)
5. A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter 20 283102 (2008)