Pressure of Solid Homework: Show P = -dΦ₀/dV + γU/V

[qbslug]

Homework Statement

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

[tex]\Phi_0(V) [/tex] 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.
[tex]U[/tex] is the internal energy arsing from the vibrations of the atoms.

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

Homework Equations

[tex]E = \Phi_0 + \sum_i(n_i + 1/2) \hbar \omega_i [/tex]

[tex]U = \sum_i \hbar\omega_i + \sum_i \frac {\hbar\omega_i}{e^{\frac{\hbar\omega_i}{k T}}-1}[/tex]

The Attempt at a Solution

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

 

[NateD]

We have U = E - \Phi_0, so we can substitute this into the first equation to getP = -\frac{\partial\Phi_0}{\partial V} + \gamma \frac{E - \Phi_0}{V}Since \Phi_0 is a function of V and E is a function of \omega_i which in turn is a function of V, then I can sayP = -\frac{\partial\Phi_0}{\partial V} + \gamma \frac{\frac{\partial E}{\partial \omega_i} \frac{\partial \omega_i}{\partial V}}{V}I am not sure how to proceed from here.