Abstract
An extension of Mayer’s cluster theory beyond the gas spinodal density is discussed. Volume dependence of Mayer’s reducible cluster integrals bj(T,V) is examined and included in the discussion. It is concluded that Mayer’s first equation for the canonical partition function has a built in uniform density assumption and will give a van der Waals loop if correct bj(T,V) are used. The virial expansion is then meaningful in the liquid state as can be shown by the cumulant expansion method. An approximate but complete set of virial coefficients βk for simple fluid is constructed by supplementation of the known β1, β2, β3, β4, for the Lennard-Jones 12-6 potential with the β5, β6, ..., βN−1 given by Carnahan and Starling for the hard sphere potential. The virial series based on this set of βk gives remarkable agreement with the pressure isotherms obtained from Monte Carlo simulation of the Lennard-Jones fluid. To obtain the correct isotherms for a Lennard-Jones fluid not constrained to density homogeneity the uniform density canonical ensemble partition functions Q(T,V,N) obtained by integration of this virial series are used to construct a grand canonical ensemble partition function ∑N≥0λNQ(T,V,N). The isotherms obtained from these agree with the Monte Carlo results at densities corresponding to the homogeneous single phases of gas and of liquid, but display the proper constant pressure at the intermediate overall-average system densities appropriate to gas–liquid coexistence.
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