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# 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 was introduced in 1901 by Heike Kamerlingh Onnes as a generalization of the ideal gas law. He wrote for a gas containing N atoms or molecules,

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

### Additional recommended knowledge

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

Writing β = (kBT) − 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 Bi(T) are characteristic of the interactions between the particles in the system and in general depend on the temperature T.