# Virial expansion

The classical virial expansion expresses the pressure of a many-particle system in equilibrium as a power series in the density. The virial expansion, introduced in 1901 by Heike Kamerlingh Onnes, is a generalization of the ideal gas law. He wrote that for a gas containing $N$ atoms or molecules,

$\frac{p}{k_BT} = \rho + B_2(T) \rho^2 +B_3(T) \rho^3+ \cdots,$

where $p$ is the pressure, $k_B$ is the Boltzmann constant, $T$ is the absolute temperature, and $\rho \equiv N/V$ is the number density of the gas. Note that for a gas containing a fraction $n$ of $N_A$ (Avogadro's number) molecules, truncation of the virial expansion after the first term leads to $pV = n N_A k_B T = nRT$, which is the ideal gas law.

Writing $\beta=(k_{B}T)^{-1}$, the virial expansion can be written in closed form as

$\frac{\beta p}{\rho}=1+\sum_{i=1}^{\infty}B_{i+1}(T)\rho^{i}$.

The virial coefficients $B_i(T)$ are characteristic of the interactions between the particles in the system and in general depend on the temperature $T$.