How Can You Prove That PV Equals E in Thermodynamics?

[dumbperson]

Homework Statement

Show that PV = E

Homework Equations

$$E= \int^\infty_0 D(\epsilon)n_{FD}(\epsilon) \epsilon \cdot d\epsilon$$
$$n_{FD}=\frac{1}{(1+ e^{-(\alpha +\beta \epsilon_k)})}$$

$$\psi(\alpha ,\beta, V) =\beta PV =\sum_\vec{k} \ln{(1+e^{-(\alpha +\beta \epsilon_k)}) }$$

and in an earlier problem I found that

$$D= \frac{A\cdot m \cdot (2s+1)}{2\pi \hbar^2 }$$

The Attempt at a Solution

I think I'm supposed to wirte $$ \frac{\psi}{\beta }$$ as an integral over $$\epsilon$$ and then compare it to the integral for E that I gave, but I have no clue on how to do this. I also have no idea on how to get rid of the logarithm.

 

[Simon Bridge]

If $PV=E$, then, off what you wrote above: $$\frac{1}{\beta}\sum_\vec{k} \ln{(1+e^{-(\alpha +\beta \epsilon_k)}) }=\int^\infty_0 D(\epsilon)n_{FD}(\epsilon) \epsilon \cdot d\epsilon$$ ... so you need to play around with this expression a bit to see if you can make LHS look like the RHS.

i.e. can you take the derivative of both sides?