Evidence for a first-order phase transition at the divergence region of activity expansions

Abstract

On the example of a lattice-gas model, a convincing confirmation is obtained for the direct relationship between the condensation phenomenon and divergent behavior of the virial expansions for pressure and density in powers of activity. The present study analytically proves the pressure equality for the low-density and high-density virial expansions in powers of density (in terms of irreducible cluster integrals or virial coefficients) exactly at the symmetrical points, where their isothermal bulk modulus vanishes, as well as for the corresponding expansions in powers of activity (in terms of reducible cluster integrals) at the same points (the points of their divergence). For lattice-gas models of arbitrary geometry and dimensions, a simple and general expression is derived for the phase-transition activity (the convergence radius of activity expansions) that, in particular, exactly matches the well-known phase-transition activity of the Lee -Yang model. In addition, the study demonstrates that Mayer's expansion with the constant (volume independent) cluster integrals remains correct up to the condensation beginning, and the actual density-dependence may be taken into account for the high-order integrals only in more dense regimes beyond the saturation point.

Approximation of Cluster Integrals for Various Lattice-Gas Models

“…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].…”

“…the symmetry of its binodal, the gas-liquid coexistence curve) and additionally confirmed [19,20] the results presented above for the saturation point. Furthermore, it has been proved very recently that the pressure and activity are identical at the saturation and boiling points for both pairs of symmetric equations (VESA-SVESA and VES-SVES) [14]. Besides that, a general expression was obtained for the phase transition activity.…”

“…As a rule, those integrals are defined in an infinite integration volume, which makes them independent of the real system volume or density [6,13]. Actually, this simplification has to be correct only for microscopic particle clusters ( ≪ ) and, as was recently shown [14], remains applicable only to the saturation point of dry vapor. However, even with this simplification, the determination of highorder reducible integrals still remains a very complicated technical problem.…”

“…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.…”

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

“…In these regimes, the neglecting of the volume dependence for high-order cluster integrals (and, even, the neglecting of those integrals themselves) does not lead to any loses of accuracy at the thermodynamic limit (N → ∞; V → ∞). This fact has recently obtained a strict confirmation [15] for various lattice-gas models and, in particular, the Lee -Yang model [16].…”