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## Introducing Legendre Transforms: Instructor's Guide

### Main Ideas

• Total differentials
• Internal energy
• Thermodynamic Identity

### Students' Task

Estimated Time: 25 minutes

1. Students are placed into small groups and first perform the total differential of the expression $F=U-TS$.
2. Then, by adding terms of $TS$ and $pV$ to the internal energy, the small groups must find expressions that are only a function of $S$, $p$, and $N$ or $T$, $p$, and $N$.

### Prerequisite Knowledge

• Familiarity with total differentials
• Basic understanding of the thermodynamic identity is helpful

### Activity: Introduction

This activity can be introduced after a lecture or activity that focuses on performing total differentials. Students should be placed into small groups. The instructor should write

$$F=U-TS$$

on the board and ask the class to perform the total differential of the expression. After the small groups have completed this, each group should be asked to add or subtract terms of $TS$ and/or $pV$ to acquire a particular variable dependence. Each group can be given either the $S$, $p$, and $N$ or $T$, $p$, and $N$ variable dependence and then move on to the other if they finish early.

### Activity: Student Conversations

The math in this activity is typically within the students' capabilities, and the main questions are why questions.

Students will ask, “What is the enthalpy?” (and the others). Ideally, this activity would either be preceded or followed by the Melting ice lab , which would enable you to discuss enthalpy as the state function that changes by equal amounts for the ice and water in that experiment.

### Activity: Wrap-up

• After the small groups have finished finding the expressions with the variable dependence detailed in the Student Task section, the class should be told that the operation they performed by adding and subtracting conjugate pairs from the internal energy is a Legendre transform. The students should be informed that this operation is a Legendre transform because the variable dependence changes from $V$ and $S$ (extensive variables for the internal energy) to different variables depending on what terms are added or subtracted from the internal energy expressions.
• If the variable dependences $S$, $p$, and $N$ or $T$, $p$, and $N$ were chosen to be found, write down the final total differentials on the board and inform them that they are called the Enthalpy and the Gibbs Free Energy. Also, note that the first total differential performed gives the Helmholtz Free Energy.
• Since some students will not have seen these expressions in several years, and since many students have never derived the expressions using total differentials, express the importance of these results to the class (this would be a good time to mention Maxwell Relations and see how familiar the class is with them).

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