Local shape of the vapor-liquid critical point on the thermodynamic surface and the van der Waals equation of state

TL;DR

This paper uses differential geometry to analyze the vapor-liquid critical point on the thermodynamic surface, proposing a geometric representation and extending the van der Waals equation with temperature-dependent parameters.

Contribution

It introduces a geometric approach to characterize the critical point and extends the van der Waals model with temperature-dependent parameters.

Findings

Critical point represented by zero Gaussian curvature.

Extended van der Waals equation with temperature-dependent parameters.

Reproduces classical form near critical temperature.

Abstract

Differential geometry is 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 $\left( \partial p/\partial V\right) _{T}=0$ requires the isothermal process, but the universality of the critical point is its independence of whatever process is taken, and so we can assume $\left( \partial p/\partial T\right) _{V}=0$. 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 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.