The application of statistical mechanics to the study of fluids over the past fifty years † or so has progressed through a series of problems of gradually increasing difficulty. The first and most elementary calculations were for the thermodynamic functions (heat capacities, entropies, free energies, etc.) of perfect gases. These properties are related to the molecular energy levels, which for perfect gases can be determined theoretically (by quantum calculations) or experimentally (by spectroscopic methods, for example). For simple molecules (CO2 , CH4 , etc.) the energy levels, and hence the thermodynamic properties, can be determined with great accuracy, and even for quite complex organic molecules it is now possible to obtain thermodynamic properties with satisfactory accuracy. With the advent of digital computers it became possible to calculate thermodynamic properties for a wide variety of substances and temperatures, and several useful tabulations of perfect gas properties now exist. Having successfully treated the perfect gas, it was natural to consider gases of moderate density, where intermolecular forces begin to have an effect, by expanding the thermodynamic functions in a power series (or virial series) in density. Although the mathematical basis for a theoretical treatment of this series was laid by Ursell in 1927, it was not exploited until ten years later, when Mayer re-examined the problem. Since that time a great deal of effort has been put into evaluating the virial coefficients that appear in the series for a variety of intermolecular force models. As the expressions for the virial coefficients are exact, they provide a very useful means of checking such force models by comparison of calculated and experimental coefficients. While the theory of dilute gases at equilibrium is essentially complete, this is far from being the case for all dense gases and liquids. The virial series cannot be applied directly to liquids. As an alternative to the ‘dense gas’ approach to liquids, there were early attempts to treat liquids as disordered solids by using cell or lattice theories; these were popular from the mid-1930s until the early 1960s.
Thermodynamic properties of the trigonometric Rosen–Morse potential and applications to a quantum gas of mesons
