Local Shape of the Vapor–Liquid Critical Point on the Thermodynamic Surface and the van der Waals Equation of State

Abstract

Differential geometry is a powerful tool to analyze the vapor–liquid critical point on the surface of the thermodynamic equation of state. The existence of usual condition of the critical point (∂p/∂V)T=0 requires the isothermal process, but the universality of the critical point is its independence of whatever process is taken, and so we can assume (∂p/∂T)V=0. The distinction between the critical point and other points on the surface leads us to further assume that the critical point is geometrically represented by zero Gaussian curvature. A slight extension of the van der Waals equation of state is to letting the two parameters a and b in it vary with temperature, which then satisfies both assumptions and reproduces its usual form when the temperature is approximately the critical one.

Geometrical Aspect of Compressibility Critical Exponent

“…For instance, it is well-known that both the van der Waals EoS and the general theory of the Landau theory of phase transitions predict the compressibility critical exponent γ = γ′ to be 1, and all known empirical EoS fail to exactly reproduce the experimental values γ = 1.2 ~1.3 ≳ γ′ = 1.1 ~1.2 [16,17]. Recently, we have conjectured that the Gaussian curvature of the local shape of the vaporliquid critical point is zero [18]. In the present study, we first prove that the conjecture is true and secondly report the construction of a fluid EoS, which has γ = γ′ = 3, which is quantitatively different from the experimental values but leads to the zero Gaussian curvature of the vapor-liquid critical point.…”