Virial Coefficient - Derivation

Derivation

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

Here is the pressure, is the volume of the vessel containing the particles, is Boltzmann's constant, is the absolute temperature, with the chemical potential. The quantity is the canonical partition function of a subsystem of particles:

Here is the Hamiltonian (energy operator) of a subsystem of particles. The Hamiltonian is a sum of the kinetic energies of the particles and the total -particle potential energy (interaction energy). The latter includes pair interactions and possibly 3-body and higher-body interactions. The grand partition function 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 equals . In this manner one derives

.

These are quantum-statistical expressions containing kinetic energies. Note that the one-particle partition function contains only a kinetic energy term. In the classical limit 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 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:

and wrote the cluster expansion in terms of these functions. Here is the interaction between particle 1 and 2 (which are assumed to be identical particles).