# Applying Maxwell Relations to Experiments

## 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.

## Homework for Energy and Entropy

1. (IsothermalAdiabaticCompressibility) What goes here?

The isothermal compressibility is defined as $$K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T}$$ $K_T$ is be found by measuring the fractional change in volume when the the pressure is slightly changed with the temperature held constant. In contrast, the adiabatic compressibility is defined as $$K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S}$$ and is measured by making a slight change in pressure without allowing for any heat transfer. This is the compressibility, for instance, that would directly affect the speed of sound. Show that $$\frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}}$$ Where the heat capacities at constant pressure and volume are given by \begin{align} C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\ C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V} \end{align}

2. (NonIdealGas) What goes here?

The equation of state of a gas that departs from ideality can be approximated by $p=\frac{NkT}{V}\left(1+\frac{NB_{2}(T)}{V}\right)$ where $B_{2}(T)$ is called the second virial coefficient which increases monotonically with temperature. Find $\left( \frac{\partial U}{\partial V}\right)_{T}$ and determine its sign.

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