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. 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

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

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

$B_2(T)=-\frac{1}{2}\underset{0}{\overset{\mathrm{\infty }}{\int }}(⟨\exp (-\frac{u(r)}{k_{B}T})⟩-1)4\pi r^{2}dr$

Using the Mayer f-function

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

one can write the third virial coefficient more compactly as

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

References

1. James A Beattie and Walter H Stockmayer "Equations of state",Reports on Progress in Physics 7 pp. 195-229 (1940)