Local Shape of the Vapor–Liquid Critical Point on the Thermodynamic Surface and the van der Waals Equation of State - Yu, J. S.; Zhou, X.; Chen, J. F.; Du, W. K.; Wang, X.; Liu, Q. H. (Frontiers in Physics, 2021)
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 ( 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 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.