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 in 1885 by Thiesen [1] and extensively studied by Heike Kamerlingh Onnes [2] [3], and mathematically by Ursell [4]. One has

$\frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}$.

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 behaviour

$B_{2}(T)= \frac{N_A}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2$

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

$B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23} dr_1 dr_2 dr_3$