# Maxwell Relations

## Prerequisites

For this lesson, students should:

• Know what a state function is and what makes it a state function.
• Know the first and second thermodynamic laws as well as the thermodynamic identity.
• Be comfortable with manipulating mixed partial derivatives.
• Understand the symmetry of mixed partials.

## In-class Content

### Lecture: Maxwell Relations (?? minutes)

##### Lecture notes from Dr. Roundy's 2014 course website:

In the Interlude, we learned that mixed partial derivatives are the same, regardless of the order in which we take the derivative, so $$\left(\frac{\partial \left(\frac{\partial f}{\partial x}\right)_y}{\partial y}\right)_x=\left(\frac{\partial \left(\frac{\partial f}{\partial y}\right)_x}{\partial x}\right)_y$$ $$\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}$$ In the Interlude we found a Maxwell relation from the energy conservation law: $$dU = F_1dx_1 + F_2dx_2$$ $$\left(\frac{\partial \left(\frac{\partial U}{\partial x_1}\right)_{x_2}}{\partial x_2}\right)_{x_1}=\left(\frac{\partial \left(\frac{\partial U}{\partial x_2}\right)_{x_1}}{\partial x_1}\right)_{x_2}$$ $$\left(\frac{\partial F_1}{\partial x_2}\right)_{x_1}=\left(\frac{\partial F_2}{\partial x_1}\right)_{x_2}$$ As you know, in thermodynamics, partial derivatives are often physically measurable quantities. In such a case, their derivatives are also be measurable quantities that we often care about.

In your groups, consider mixed partial derivatives of the thermodynamic potential assigned to you, to derive a Maxwell relation. [GROUP]

### Activity: Seeking the right Maxwell Relation

Activity Highlights

1. This small group activity is designed to help students understand the origin of Maxwell relations.
2. Students seek a Maxwell relation that will allow them to find a relationship between two given thermodynamic derivatives.
3. The compare-and-contrast wrap-up discussion addresses how to decide on a thermodynamic potential that is likely to generate a helpful Maxwell relation.

## Homework for Energy and Entropy

1. (HardSphereFluid) What goes here?

2. (SimpleCycleII) What goes here?

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