Virial coefficient

Derivation

The first step in obtaining a closed expression for virial coefficients is a cluster expansion of the grand canonical partition function

${\displaystyle \mathrm{\Xi }=\sum _n{\lambda }^n{Q}_n=e^{\left(pV\right)/\left(k_{\text{B}}T\right)}}$

Here ${\displaystyle p}$ is the pressure, ${\displaystyle V}$ is the volume of the vessel containing the particles, ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, ${\displaystyle T}$ is the absolute temperature, ${\displaystyle \lambda =\exp [\mu /(k_{\text{B}}T)]}$ is the fugacity, with ${\displaystyle \mu }$ the chemical potential. The quantity ${\displaystyle Q_n}$ is the canonical partition function of a subsystem of ${\displaystyle n}$ particles:

${\displaystyle Q_n=\mathrm{tr}[e^{-H(1,2,\dots ,n)/(k_{\text{B}}T)}].}$

Here ${\displaystyle H(1,2,\dots ,n)}$ is the Hamiltonian (energy operator) of a subsystem of ${\displaystyle n}$ particles. The Hamiltonian is a sum of the kinetic energies of the particles and the total ${\displaystyle n}$-particle potential energy (interaction energy). The latter includes pair interactions and possibly 3-body and higher-body interactions. The grand partition function ${\displaystyle \mathrm{\Xi }}$ can be expanded in a sum of contributions from one-body, two-body, etc. clusters. The virial expansion is obtained from this expansion by observing that ${\displaystyle \ln \mathrm{\Xi }}$ equals ${\displaystyle pV/(k_{B}T)}$. In this manner one derives

${\displaystyle B_2=V\left(\frac{1}{2}-\frac{Q_2}{Q_1^2}\right)}$

${\displaystyle B_3=V^2\left[\frac{2Q_2}{Q_1^2}(\frac{2Q_2}{Q_1^2}-1)-\frac{1}{3}(\frac{6Q_3}{Q_1^3}-1)\right]}$.

These are quantum-statistical expressions containing kinetic energies. Note that the one-particle partition function ${\displaystyle Q_1}$ contains only a kinetic energy term. In the classical limit ${\displaystyle \hslash =0}$ the kinetic energy operators commute with the potential operators and the kinetic energies in numerator and denominator cancel mutually. The trace (tr) becomes an integral over the configuration space. It follows that classical virial coefficients depend on the interactions between the particles only and are given as integrals over the particle coordinates.

The derivation of higher than ${\displaystyle B_3}$ virial coefficients becomes quickly a complex combinatorial problem. Making the classical approximation and neglecting non-additive interactions (if present), the combinatorics can be handled graphically as first shown by Joseph E. Mayer and Maria Goeppert-Mayer.

They introduced what is now known as the Mayer function:

${\displaystyle f(1,2)=\exp \left[-\frac{u(|{\overrightarrow{r}}_1-{\overrightarrow{r}}_2|)}{k_{B}T}\right]-1}$

and wrote the cluster expansion in terms of these functions. Here ${\displaystyle u(|{\overrightarrow{r}}_1-{\overrightarrow{r}}_2|)}$ is the interaction potential between particle 1 and 2 (which are assumed to be identical particles).