(55d) A Careful Look at the Development of the Virial Equation of State

The virial equation of state (VEOS) has the distinction of being the only general-purpose thermodynamic model that has a direct, unambiguous connection to the intermolecular interactions, such that the coefficients of the VEOS may be computed rigorously from a given, arbitrarily complex molecular model. Practical application of the VEOS is limited however, due to failure of the series to converge for higher densities and condensed phases. It is interesting to consider the origins of this failure, and the manner in which the series represented by the grand-canonical partition function, which applies at all densities, is transformed into the VEOS, which does not. We identify a flaw in the textbook derivation of the series which, while not rendering the VEOS incorrect, does suggest how it can become inaccurate at high densities. Attempts to remedy this issue are described.