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## Helmholtz Free Energy Expressions: Instructor's Guide

### Main Ideas

- Total differentials
- Partial differentials
- The Helmholtz Free Energy
- Thermodynamic relations

### Students' Task

*Estimated Time: 5 minutes*

Students are asked to compare the mathematical and physical expressions for the Helmholtz Free Energy and write down on their small white boards any equivalent terms in the expressions.

### Prerequisite Knowledge

- Familiarity with total differentials
- Familiarity with partial derivatives and their interpretations
- Familiarity with Helmholtz Free Energy and its derivation via Legendre transform

### Props/Equipment

- Small Whiteboards with markers

### Activity: Introduction

Before asking these small white board questions, a similar activity can be done in a lecture with the entire class relating terms in the internal energy expressions

$$ dU=\left(\frac{\partial U}{\partial V}\right)_{S}dV + \left(\frac{\partial U}{\partial S}\right)_{V} dS $$

and

$$ dU=pdV - TdS \; \; .$$

Then, the class can be asked the small white board questions:

- SWBQ: “Write down the equivalent physical and mathematical statements for the Helmholtz free energy equation.”
- SWBQ: “Compare the math statement with the physics statement and find any corresponding relationships.”

### Activity: Student Conversations

### Activity: Wrap-up

- After the students have had time to write their answers, the instructor may collect white boards and discuss the results of several students if they wish.
- If a similar example was performed with the internal energy before asking the white board questions, most students will find the relations rather quickly. For those that are still struggling to find the equivalent terms in the mathematical and physical expressions, write down the mathematical and physical expressions one on top of the other and circle the equivalent terms to emphasize where the equivalent terms are found in the expressions.
- To make the solution to the small white board questions clear, it helps to write the solutions on the board, where

$$ p=-\left(\frac{\partial F}{\partial V}\right)_T \; \; \; \; \; \; and \; \; \; \; \; \; \; S=-\left(\frac{\partial F}{\partial T}\right)_V \; \; .$$

- For students that had a difficult time remembering what the Helmholtz Free Energy expressions were, restate to the class that you can always re-derive the physical expression using Legendre transforms. Otherwise, students can just memorize any minus signs that appear in the expression, memorize the extensive variables, and use conjugate pairs to find the intensive variable.