Equation of state for all regimes of a fluid: From gas to liquid

Abstract

The study of Mayer's cluster expansion (CE) for the partition function demonstrates a possible way to resolve the problem of the CE non-physical behavior at condensed states of fluids. In particular, a general equation of state is derived for finite closed systems of interacting particles, where the pressure is expressed directly in terms of the density (or system volume) and temperature-volume dependent reducible cluster integrals. Although its accuracy is now greatly affected by the limited character of the existing data on the reducible cluster integrals and, especially, the absence of any information on their density dependence, a number of simple approximations indicate the qualitative adequacy of this equation in various regimes of a fluid: from gaseous to liquid states (including the transition region).

Approximation of Cluster Integrals for Various Lattice-Gas Models

“…Such a behavior confirms once more the correctness of all conclusions concerning the convergence radius of VESA and SVESA, the divergence character of those equations, and the physical meaning of the quantities and , which were obtained analytically in [14] and [9][10][11][12]24]. Fur thermore, the very accuracy of the results obtained on the basis of a rather crude approximation of the medium-order cluster integrals (the low-order reducible integrals are calculated on the basis of the exact information about the irreducible integrals of the corresponding low orders, and the asymptotics of the integrals of very high orders must also be determined rather precisely by the convergence radius 0 ) may indirectly confirm conclusions made in works [13,15,18,20] about the real influence of cluster integrals with various orders on the partition function behavior.…”

“…New equations derived in terms of irreducible cluster integrals (virial coefficients) [9][10][11][12][13] made it possible to establish the applicability limits of the well-known virial expansion for pressure in powers of density (the virial equation of state, VES [6]) and to obtain a general theoretical criterion that exactly determines the saturation point for various systems of interacting particles. The study of the virial expansions for pressure and density in powers of activity with reducible cluster integrals [14,15,18,19] (the virial equation of state in terms of the activity, VESA [6,13]) completely confirmed those results (obtained in terms of irreducible integrals) and even demonstrated a potential possibility to determine the boiling point in the framework of the Mayer cluster approach [15].…”

“…For a constant set of reducible integrals (determined in the infinite volume), the density has an infinite singularity (an essential discontinuity). In work [15], it was shown that a true finite density jump to the boiling point, which would correspond to the actual physical picture of the first-order phase transition, can still be obtained in the framework of the Mayer cluster expansion, but only if the dependence of the reducible high-order integrals corresponding to macroscopic particle clusters on the real system volume (real integration limits) is taken into account.…”

“…Unfortunately, this example is only applicable to the description of the phase transition itself (the Lee-Yang solution [3] only determines the condensation parameters -pressure, activity, and density at the saturation and boiling points -for a two-dimensional lattice gas with a square-well potential) and provides no accurate information about the behavior of the system in the gaseous or liquid state. Recent researches on the basis of the Mayer cluster expansion of a partition function [6][7][8] significantly advanced the statistical theory of nonideal gases in general [9][10][11][12][13][14][15] and the theoretical description of the lattice-gas behavior in particular [16][17][18]. New equations derived in terms of irreducible cluster integrals (virial coefficients) [9][10][11][12][13] made it possible to establish the applicability limits of the well-known virial expansion for pressure in powers of density (the virial equation of state, VES [6]) and to obtain a general theoretical criterion that exactly determines the saturation point for various systems of interacting particles.…”

Adiabatic lapse rate of nonideal gases: The role of molecular interactions and vibrations

“…where (8a) is valid only at gaseous regimes up to the saturation point [19][20][21] and the B k+1 are the so-called virial coefficients, which generically are sums of the cluster integrals involving the (k + 1)-particle interactions [22]. They are, by construction, functions that can only depend on the temperature T [13] (this statement becomes invalid at the vicinity of the boiling point [23]). In (8b), M, k B , and h are the mass of the molecule, the Boltzmann constant, and the Planck constant, respectively.…”