Thermodynamics of Gas–Liquid Criticality: Rigidity Symmetry on Gibbs Density Surface

Abstract

The thermodynamic state function rigidity, defined simply as $$(\mathrm{d}p/\mathrm{d}\rho )_{T}$$ , where p is the pressure, $$\rho $$ is the density and T is the temperature, is the work required to reversibly increase the density of a fluid. Along any isotherm, rigidity $$(\omega )$$ decreases with density for a gas phase and increases with density for a liquid. Thermodynamics, therefore, can define a distinction between gas and liquid. For any one-phase system, rigidity is everywhere positive, in any two-phase region $$\omega =0$$ . For temperatures above critical coexistence, the rigidity has a constant value in the mesophase that separates the percolation loci, which bound the limits of existence of liquid and gas phases in the supercritical region. The law of rectilinear diameters extends in the supercritical region as a defining line of the colloid-like inversion between gas-in-liquid and liquid-in-gas. Every equilibrium state of gas phase has a corresponding isothermal state on the liquid phase with the same rigidity. We illustrate this symmetry between gas and liquid empirically using literature $$\rho (p,T)$$ equations-of-state for some real fluids, notably carbon dioxide, water and steam, and argon. At the molecular level, the symmetry can be explained by a correspondence between statistical properties of available holes in a liquid and sites of molecular clusters in the gas with equivalent number density fluctuations for complementary states of gas and liquid.