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.