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

Q. H. Liu

Published 2021-01-26Version 1

The equation of state for a real gas can be treated as a two-dimensional surface in the three-dimensional Cartesian coordinates, and the local shapes of the surface are completely classified in to three types:\ elliptic, hyperbolic and parabolic. The principles of thermodynamics excludes the situation the vapor-liquid critical point is geometrically an elliptic point. Thus a reasonable thermodynamic equation of state must be capable of accounting for the fact that the critical point must be geometrically either the hyperbolic point or the parabolic point, corresponding to the negative and zero Gaussian curvature, respectively. With the van der Waals parameters $a$ and $b$ being extended to be of temperature dependence, we first demonstrate that the extended\ van der Waals equation of state suffices to contain both cases, and secondly propose an experimental scheme to fix these two parameters $a$ and $b$ via measuring two response functions at the critical point.