Deriving the Relationship between Pressure and Energy at Constant Entropy

[spaghetti3451]

Hi,

I need to prove that $$\left(\frac{\partial U}{\partial V}\right)_{S} = \sum n_{j} \frac{\partial \epsilon_{j}}{\partial V}$$.

 

[spaghetti3451]

In the bracket on the LHS, you have partial dU by dV. In the denominator on the RHS, you have partial dV.

This is my reasoning.

$$U = \sum n_{j} \epsilon_{j}$$.

We need to take the partial derivative of U at constant S.

At constant S and variable V, the number of particles at each energy level does not change but their energy levels may change. (I don't understand why this should be the case.) This means that $$n_{j}$$ does not vary, but $$\epsilon_{j}$$ does. Therefore, the derivative of $$\epsilon_{j}$$ is taken with respect to V?