Bitangent planes of surfaces and applications to thermodynamics

The classical van der Waals equation, applied to one or two mixing fluids, and the Helmholtz (free) energy function \(A\) yield, for fixed temperature \(T\), a curve in the plane \(\mathbb{R}^2\) (one fluid) or a surface in 3-space \(\mathbb{R}^3\) (binary fluid). A line tangent to this curve in two places (bitangent line), or a set of planes tangent to this surface in two places (bitangent planes) have a thermodynamic significance which is well documented in the classical literature. Points of contact of bitangent planes trace `binodal curves' on the surface in \(\mathbb{R}^3\). The study of these bitangents is also classical, starting with D.J. Korteweg and J.D. van der Waals at the end of the \(19^{\rm th}\) century, but continuing into modern times. In this paper we give a summary of the thermodynamic background and of other mathematical investigations and then present a new mathematical approach which classifies a wide range of situations in \(\mathbb{R}^3\) where bitangents occur. In particular, we are able to justify many of the details in diagrams of binodal curves observed by Korteweg and others, using techniques from singularity theory.