Finding Pressure, Temperature, and Chemical Potential for a Non-Ideal Gas System

[Selveste]

Homework Statement

A generalized $TdS$-equation for systems of several types of "work-parts" and varying number of particles in multiple components, is given by

$$dU = TdS + \sum_{i}y_idX_i+\sum_{\alpha =1}^{c}\mu_\alpha dN_{\alpha}$$

Thus, its natural to regard the internal energy $U$ (an extensive property), as a function of the extensive variables $U, S, {X_i}, {N_{\alpha}}.$ Here $U_\alpha$ is the chemical potential for component $\alpha$, and $N_\alpha$ is the number of particles in component $\alpha$ of the system (a number that can vary by $dN_\alpha \neq 0$). Thus we have

$$U = U(S, X_i, N_\alpha)$$

Because $(U, S, X_i, N_\alpha)$ are all extensive properties, we have the following homogeneity condition

$$U(\lambda S, \lambda {X_i}, \lambda {N_\alpha}) = \lambda U(S, {X_i}, {N_\alpha})$$

Homework Equations

My question regards a special case of this, namely a one-component gass system (not an ideal gass!) with the following internal energy

$$U = U(S, V, N) = \frac{aS^3}{NV}$$

where $a$ is a a constant with dimension $K^3m^3/J^2$.

Problem: find the pressure $p$, the temperature $T$ and the chemical potential $\mu$ of this gas expressed by $(S, V, N)$. And then find the heat capacities at constant volume $C_V$ and pressure $C_p$, expressed by $(N, T, V )$ and $(N, T, p)$, respectively.

The Attempt at a Solution

The $TdS$-equation becomes

$$TdS = dU + pdV - \mu dN = C_vdT + \left[\left(\frac{\partial U}{\partial V}\right)_T + p\right]dV - \mu dN$$

But here I am completely at a loss.

 

[Charles Link]

I can give you one answer. This whole problem, I think, is not very difficult. $p=- (\frac{\partial{U}}{\partial{V}})_{S,N}$. Now apply this to the function $U$ that they gave you.