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 compresiblity 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

• $\rho \equiv {\frac {N}{V}}$ 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

$B_{2}(T)={\frac {N_{0}}{2V}}\int ....\int (1-e^{-u/kT})~d\tau _{1}d\tau _{2}$

where $N_{0}$ is Avogadros number and $d\tau _{1}$ and $d\tau _{2}$ are volume elements of two different molecules in configuration space. The integration is to be performed over all available phase-space; that is, over the volume of the containing vessel. For the special case where the molecules posses spherical symmetry, so that $u$ depends not on orientation, but only on the separation $r$ of a pair of molecules, the equation can be simplified to

$B_{2}(T)=-{\frac {1}{2}}\int _{0}^{\infty }\left(\langle \exp \left(-{\frac {u(r)}{k_{B}T}}\right)\rangle -1\right)4\pi r^{2}dr$

Using the Mayer f-function

$f_{ij}=f(r_{ij})=\exp \left(-{\frac {u(r)}{k_{B}T}}\right)-1$

one can write the third virial coefficient more compactly as

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

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)