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On the example of a lattice-gas model, a convincing confirmation is obtainedfor the direct relationship between the condensation phenomenon and divergentbehavior of the virial expansions for pressure and density in powers ofactivity. The present study analytically proves the pressure equality for thelow-density and high-density virial expansions in powers of density (in termsof irreducible cluster integrals or virial coefficients) exactly at thesymmetrical points, where their isothermal bulk modulus vanishes, as well asfor the corresponding expansions in powers of activity (in terms of reduciblecluster integrals) at the same points (the points of their divergence). Forlattice-gas models of arbitrary geometry and dimensions, a simple and generalexpression is derived for the phase-transition activity (the convergence radiusof activity expansions) that, in particular, exactly matches the well-knownphase-transition activity of the Lee -- Yang model. In addition, the studydemonstrates that Mayer's expansion with the constant (volume independent)cluster integrals remains correct up to the condensation beginning, and theactual density-dependence may be taken into account for the high-orderintegrals only in more dense regimes beyond the saturation point.

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