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## Thermodynamic Partials: Instructor's Guide

### Main Ideas

- Partial derivatives
- Symmetry of mixed partials
- Maxwell relations
- Internal Energy
- Gibbs free energy
- Enthalpy
- Helmholtz free energy

### Students' Task

*Estimated Time: 20 minutes *

Students are given two thermodynamic partials and asked to find how the two partials are related.

### Prerequisite Knowledge

- Familiarity with the first thermodynamic law, the thermodynamic identity, and the internal energy expressions is useful.
- Understanding of the symmetry of mixed partials.
- An ability to interpret and manipulate partial derivatives.

### Props/Equipment

- Tabletop Whiteboard with markers

### Activity: Introduction

Before entering this activity, a lecture or activity that helps recall the symmetry of mixed partials is useful. If students have been struggling with the symmetry of mixed partials, write on the board

$$ \frac{\partial ^{2} f}{\partial y \partial x}=\frac{\partial ^{2} f}{\partial x \partial y} $$

as a starting point for this activity. Then, each small group is given two partials that are each equivalent to two different variables (typically, to focus on Maxwell relations, these expressions can be found from looking at the mathematical and physical expressions for different internal energy expressions) and asked to find how the partials are related. If a group is struggling to find any relations, give them the hint that they can look at the mathematical and physical expressions of different internal energy equations for more information on the partials. If a group finishes early, have them practice the same calculation with a different set of partial derivatives.

The derivatives to be assigned to groups are:

- $\left(\frac{\partial T}{\partial p}\right)_V$ and $\left(\frac{\partial S}{\partial V}\right)_T$
- $\left(\frac{\partial T}{\partial p}\right)_S$ and $\left(\frac{\partial S}{\partial V}\right)_p$
- $\left(\frac{\partial T}{\partial V}\right)_p$ and $\left(\frac{\partial p}{\partial S}\right)_T$

### Activity: Student Conversations

- Groups typically struggle to figure out which thermodynamic potential to use for their derivatives.
- In addition, some of the derivatives have been inverted, and groups seldom consider turning them upside down.
- Overall, students get more practice using the differentials of thermodynamic potentials, and finding Maxwell relations.

### Activity: Wrap-up

Have each group report what the relation was between the partial derivatives. As the groups present their relations, write them on the board so that everyone can copy them down. Tell the students that these expressions are called the Maxwell relations, and that each of them is found (as they calculated) using the different internal energy expressions. If you choose, you can also write on the board what Maxwell relation pairs with what internal energy expression.

In the previous Name the Experiment activities, the students may have received a partial derivative with an entropy term in it; be sure to note that the Maxwell relations are useful for measuring a challenging partial by instead measuring it's equivalent Maxwell relation partial.