# 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, ${\displaystyle Z}$, 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

${\displaystyle {\frac {pV}{Nk_{B}T}}=Z=1+\sum _{k=2}^{\infty }B_{k}(T)\rho ^{k-1}}$.

where

• ${\displaystyle p}$ is the pressure
• ${\displaystyle V}$ is the volume
• ${\displaystyle N}$ is the number of molecules
• ${\displaystyle T}$ is the temperature
• ${\displaystyle k_{B}}$ is the Boltzmann constant
• ${\displaystyle \rho \equiv {\frac {N}{V}}}$ is the (number) density
• ${\displaystyle B_{k}\left(T\right)}$ is called the k-th virial coefficient

## Virial coefficients

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

${\displaystyle B_{2}(T)={\frac {N_{A}}{2V}}\int ....\int (1-e^{-\Phi /k_{B}T})~d\tau _{1}d\tau _{2}}$

where ${\displaystyle N_{A}}$ is Avogadros number and ${\displaystyle d\tau _{1}}$ and ${\displaystyle d\tau _{2}}$ are volume elements of two different molecules in configuration space.

One can write the third virial coefficient as

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