Virial expansion

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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,

${\frac {p}{k_{B}T}}=\rho +B_{2}(T)\rho ^{2}+B_{3}(T)\rho ^{3}+\cdots ,$

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 $pV=nN_{A}k_{B}T=nRT$ , 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 $T$ .

Virial expansion

See also