The virial expansion re-visited: A new interpretation

A central topic in statistical mechanics, a research area Professor Hao Bailin devoted himself throughout the years, is the study of the state of matter. For a gas system the study begins with an understanding of the equation of state relating the pressure P, volume V and the temperature T. In 1901 Onnes introduced the virial expansion P/kT = rho + B-2 rho 2 + B-3 rho 3 + B-4 rho 4 + ... as an empirical formula expressing P in a power series of particle density rho = N/V, where N is the number of particles. A first-principle understanding of the virial expansion was provided years later by the advent of the Mayer cluster expansion in statistical mechanics in the 1930s. However, following Onnes the virial expansion has since been generally regarded as an expansion in density. Here we re-visit the virial expansion using the Mayer expansion, and show that the virial expansion should be considered as an expansion in specific volume, the ratio of the effective volume of a gas molecule and its allotted mean volume. This consideration is illustrated in the case of the hard sphere gas.