# 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

Seeking the right Maxwell Relation

**Activity Highlights**

- This small group activity is designed to help students understand the origin of Maxwell relations.
- Students seek a Maxwell relation that will allow them to find a relationship between two given thermodynamic derivatives.
- 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

- (HardSphereFluid)
*What goes here?***Hard-sphere fluid**A hard-sphere fluid (or gas) is just like and ideal gas except that the gas (fluid) particles are not point particles, but instead have a finite volume. The following equations approximately describe the hard-sphere fluid.\[U=\frac{3}{2} N k_B T\]

\[\eta=\frac{N}{V} \frac{4 \pi}{3} R^3\]

\[\frac{F}{N k_B T}=\ln \left( \frac{N}{V} \left( \frac{3 h^2}{4 \pi m} \frac{N}{U} \right)^\frac32 \right) + \frac{4 \eta - 3 \eta^2}{\left( 1-\eta \right)^2}-\frac12\]

where $k_B$ is a constant, and you may consider the number of spheres $N$ as a constant for this problem. Similarly the mass $m$, the radius $R$, and Plank's constant $h$ are constants. In these equations $\eta$ is called the packing fraction, and represents the fraction of the volume that is occupied by hard spheres (which has no impact whatsoever on how you solve this problem).

Solve for \[S \equiv -\left( \frac{\partial F}{\partial T} \right) _V\]

Solve for \[S \equiv -\left( \frac{\partial F}{\partial V} \right) _T\]

Solve for \[\left( \frac{\partial U}{\partial V} \right) _T\]

Solve for \[\left( \frac{\partial U}{\partial V} \right) _S\]

- (SimpleCycleII)
*What goes here?*