Finding chemical potential with given thermodynamic relation

[Mayan Fung]

Homework Statement

Suppose you are given the following relation among the entropy S, volume V , internal energy U, and number of particles N of a thermodynamic system, where A is a constant.:
$$ S = A(NVU)^{\frac 1 3} $$
Find the chemical potential μ(T,P)

Homework Equations

$$ \frac μ T = -(\frac{∂S}{∂N})_{U,V} $$
U = TS - PV + μN

The Attempt at a Solution

$$ \frac μ T = -(\frac{∂S}{∂N})_{U,V} = \frac 1 3 A N^{-\frac 2 3} (VU)^{\frac 1 3} $$
This only solved μ(U,V,N)
Since internal energy is a function of N,V,T : U(N,V,T)
$$ ∴ μ(U,V,N) → μ(T,\frac V N) → μ(T,P) $$
However, I don't know how to carry out the above transform in this case. I don't know U(N,V,T)

Thanks!

 

[DrDu]

Try to express p and T in terms of U, S and V, using dU=TdS-pdV.
Then set up G(T,p,N)=U-TS+pV.
Calculate mu from G.

 

[Mayan Fung]

Oh! Thanks, I didn't think of that before