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:

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

where

• $p$ is the pressure

• $V$ is the volume

• $N$ is the number of molecules

• $\rho \equiv \frac{N}{V}$ is the (number) density

• $B_k(T)$ is called the k-th virial coefficient

Virial coefficients

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

$B_2(T)=\frac{N_0}{2V}\int ....\int (1-e^{-\mathrm{\Phi }/k_{B}T})d{\tau }_{1}d{\tau }_2$

where $N_0$ 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

$B_3(T)=-\frac{1}{3V}\int \int \int f_{12}{f}_{13}{f}_{23}d{r}_{1}d{r}_{2}d{r}_3$

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)