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_{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 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 can also be applied to aqueous ionic solutions, as shown by Harold Friedman.

See also