# 2. Foundations of Classical Thermodynamics

The following little article shows how classical equations describing ideal gas mixtures and ideal solutions emerge in incredible detail from simple verbal postulates that seems to say very little at all. Unlike the other thermodynamics articles listed below, this one requires very little in the way of mathematics preparation. This article is recommended for people who like magic.

When I taught thermodynamics in the past, I was dissatisﬁed with arguments in support of Gibbs’s phase rule. (Gibbs seemed to have some reservations about his own arguments.) Then I came across an explicitly mathematical paper by Walter Noll that, for me, was beautiful and satisfying. The paper below is a variant of Noll’s, using weaker assumptions and, I think, a simpler argument.

One of the central problems in the foundations of classical thermodynamics is, loosely speaking, proof of the existence of entropy and thermodynamic temperature scales from more fundamental assertions about, for example, heat. And then one wants to examine properties of the derived notions of entropy and thermodynamic temperature. The classical 19th century arguments are brilliant, but they are also somewhat mysterious, typically relying on putative properties of engines, and they also raise questions about the full range of their applicability.

I am very proud of work with my friend Rick Lavine, in which we addressed these issues, using mathematical tools (e.g. the Hahn-Banach Theorem) that were not available to the pioneers. Unfortunately, the intersection of the class of people who are are familiar with these tools and the class of people who are interested in the thermodynamic questions is practically empty. At least in part, I think it is for this reason that the work didn’t receive much attention. It was gratifying to notice, however, that L. Craig Evans at the University of California, Berkeley, recently resurrected ideas from the papers below in his online notes, *Entropy and Partial Di**ff**erential Equations*.

The paper just above had its genesis in some handwritten notes prepared in 1978 for James Serrin. Those notes were less ambitious in their goals and are far more succinct. (They also relied on the Hahn-Banach Theorem). Some readers might find them helpul. A scanned copy is available here: