# Pressure of a Solid

## Homework Statement

Show that the pressure of a solid is given by
$$P = -\frac{\partial\Phi_0}{\partial V} + \gamma \frac{U}{V}$$

$$\Phi_0(V)$$ is the potential energy of the solid when all atoms are at rest in their equilibrium positions and V is the volume of the solid.
$$U$$ is the internal energy arsing from the vibrations of the atoms.

$$\gamma = -\frac{\partial ln\omega}{\partial ln V} \approx 1/3$$ is the Gruneisen parameter
$$\omega_i (V)$$ where (i = 1,2,...,3N-6) are the normal frequencies of vibration

## Homework Equations

$$E = \Phi_0 + \sum_i(n_i + 1/2) \hbar \omega_i$$

$$U = \sum_i \hbar\omega_i + \sum_i \frac {\hbar\omega_i}{e^{\frac{\hbar\omega_i}{k T}}-1}$$

## The Attempt at a Solution

I guess I need to find how the frequencies w depend on volume?