# 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?

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).

1. Solve for $S \equiv -\left( \frac{\partial F}{\partial T} \right) _V$

2. Solve for $S \equiv -\left( \frac{\partial F}{\partial V} \right) _T$

3. Solve for $\left( \frac{\partial U}{\partial V} \right) _T$

4. Solve for $\left( \frac{\partial U}{\partial V} \right) _S$

2. (SimpleCycleII) What goes here?

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