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 Kammerlingh Onnes. 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

See Ref. 3.

References

1. H. Kammerlingh Onnes "", Communications from the Physical Laboratory Leiden 71 (1901)
2. James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics 7 pp. 195-229 (1940)
3. J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics 5 pp. 841-847 (1964)