Thermodynamic identities with sunray diagrams

TL;DR

This paper introduces the sunray diagram, a novel visual tool based on symplectic geometry, to simplify and clarify the derivation and interpretation of thermodynamic identities for students.

Contribution

The paper presents a new diagrammatic method called the sunray diagram that intuitively visualizes thermodynamic derivatives and connects them to the symplectic structure of thermodynamics.

Findings

Sunray diagrams enable intuitive manipulation of partial derivatives.

The method provides a physical interpretation on P-V and T-S planes.

It enhances pedagogical approaches in thermodynamics education.

Abstract

One of the hurdles in teaching undergraduate thermodynamics is a plethora of complicated partial derivative identities. Students suffer from difficulties in deriving, justifying, or interpreting the identities, misconceptions about partial derivatives, and a lack of in-depth understanding of the meanings of identities. We propose a novel diagrammatic method for the calculus of differentials and partial derivatives called the "sunray diagram" that resolves the difficulties above. The sunray diagram technique relates a partial derivative with "arrow sliding," which enables an aesthetic and intuitive manipulation of partial derivative expressions in the form of successive arrow slidings. Furthermore, the sunray diagram is more than an ad hoc or abstract machinery but is based on the symplectic structure of thermodynamics; the sunray diagram admits a direct physical interpretation on the P-V (or T-S) plane. We elaborate on such physical semantics of the sunray diagram by taking Maxwell's approach to the geometry of thermodynamic structures--reinterpreted in terms of differential geometry--as a reference point. We anticipate that our discussion introduces the geometry of thermodynamics to learners and enriches the graphical pedagogy in physics education.