Gibbs solution of the van der Waals–Maxwell problem and universality of the liquid–gas coexistence curve

Abstract

The conventional statements of the scaling theory are: (i) the real fluids belong to the lattice gas universality class; (ii) the coexistence curves of real fluids, as well as of the van der Waals (vdW) model do not possess any apparent symmetry and therefore corrections to the asymptotic power laws and asymmetry must be taken into account in the extended critical region. We demonstrate that scaling and classical basic models, applied to the extended critical region have a common line of symmetry—the continuation of the critical isochore ρ_c by the line of the rectilinear diameter ρ_c = (ρ_l + ρ_g)/2 if a new parameter for the distance from the critical point is used instead of temperature. This is the reduced difference of molar entropies: x = (s_g − s_l)/2R which can be obtained from the measurable latent heat r_s(T)–data along the coexistence curve. The parametric solution of the van der Waals–Maxwell problem proposed by Gibbs in terms of an unspecified parameter demonstrates similar symmetry if the parameter is identified as x. Nearly ideal linearity between the reduced densities ρ_l/ρ_c, ρ_g/ρ_c, (ρ_l − ρ_g)/ρ_c and parameter x has been found for a set of well-studied fluids: Ar, C₂H₄, CO₂, H₂O. The symmetry of the vdW-model differs from that of real fluids by the value of critical slopes for ρ_l,g(x)-functions expressed in terms of dimensionless variables. The slope is close to ±1/2 for real fluids, ±2/3 for the vdW-model, and ±∞ for the lattice gas model. We conclude that the symmetry in real fluids is much more similar to the vdW-model than to the lattice gas model. Therefore, to achieve an adequate description of real fluids in the extended critical region (ρ = ρ_c ± 0,3ρ_c, x ≤ 0,5), a combination of background (vdW-like) and scaling terms should be taken into account up to the critical point. With the introduction of constant rescaling factor for each above-named fluid a novel coexistence curve model can be obtained providing a high level of prediction accuracy on the basis of the parametric solution of the vdW–Maxwell problem.