Evidence for the 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.