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

• Published in: arXiv.org 2021-05
• Main Authors: Yu, J S, Zhou, X, Chen, J F, Du, W K, Wang, X, Liu, Q H
• Format: Article
• Language: English
• Subjects: Cartesian coordinates; Cases (containers); Critical point; Equations of state; Mathematical analysis; Parameters; Real gases; Response functions; Temperature dependence
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2101.10618

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.