# Proving the property of entrophy

## Homework Statement

$-\left ( \frac{\partial U}{\partial V} \right )_{S, N}$ is a definition of an imporant thermodynamic property,where S denote the entropy and the subscript 0 denotes reference state, so they must be constant. show what is this property. In your analysis, use the equation below.
$$\frac{U}{U_{0}} = (\frac{V}{V_{0}})^{-2/3}exp\left \{ \frac{2(S-S_0)}{3k_BN} \right \}$$

## Homework Equations

As I thought this was done by first-order ordinary derivative equation, I tried to differentiate it, but I was unable to do so. As this is about entrophy, I think U means current energy and V means volume.

## The Attempt at a Solution

I think this is done by first-order ordinary derivative, but I can't specify.

The way I understand it, this question is not really about entropy. You have to explain what $- (\frac{dU}{dV})_{S,N}$ stands for. So, if it has not already been defined, I guess you have to look at other thermodynamic identities and compare your result to them.

The differentiation should not be that difficult. Since S and N are held constant, you only need to differentiate the $(\frac{V}{V_0})^{-2/3}$ term. Use the power rule.

Entropy does not change, nor do the number of particles; What changes when the internal energy of a system changes due to volume? When you figure that out, you have explained what $- (\frac{dU}{dV})_{S,N}$ stands for.

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Then, what kind of property is it?

That seems to be your homework question.

Think about it. You have a closed system (ie its number of particles stay constant). Its entropy does not change either. However, you can change its volume. What happens if you make a canister of gas smaller? If you put your finger into the canister, how would it feel different now compared to before you made the canister smaller? Remember, temperature changes with entropy, and in our case entropy is constant.

Also, I think you should have access to more relevant equations. I am assuming you have a book on thermodynamics. Does it have a section on the internal energy of a system? Or, thermodynamic potentials?