How to calculate the Gibbs free energy out of a given Entropy? I'm stuck in a small calculation. I try to solve a small exercise.
You should calculate for a given entropy the Gibbs free energy, where the entropy is given as
$S\left(U,V\right) = \frac{4}{3}\left(\alpha V U^3\right)^{1/4} \qquad\dots\qquad U$ is the internal energy, $V$ is the volume and $\alpha > 0, \text{const}$
The exercise say something like "you get a surprisingly expression for the Gibbs free energy".

Basically I have to use the definition of the Gibbs free energy:
$$
\begin{equation}
G\left(T,p\right) = U\left(S,V\right) - T\cdot S + p \cdot V
\label{eqn:gibbs}
\end{equation}
$$
The question is now, how can we exchange the volume $V$ and the entropy $S$ in the equation above. My idea was to calculate $p=p\left(T,V\right)$ and transpose it to $V=V\left(p,T\right)$. The thing is, that I get for the pressure $p$ something like:
$p = \frac{\alpha}{3}T^4$, which is not $p = p\left(T,V\right)$ and it's not possible to find a relation for $V=V\left(p,T\right)$. I thing, I'm making some mistake. Can someone help me?
 A: You may also use $T=\left(\frac{\partial U}{\partial S}\right)_V=\left(\frac{\partial S}{\partial U}\right)_V^{-1}$.
A: Let me add a more general solution to this question.
The proposed fundamental equation for the entropy $S(U,V)$ is a homogeneous function of degree 1 of its variables (extensiveness of the entropy). As a consequence also the fundamental equation $U(S,V)$, obtained from the original fundamental equation is a homogeneous function of degree 1. Therefore, as a consequence of Euler's theorem:
$$
U=\left(\frac{\partial{U}}{\partial{S}}\right)_VS+\left(\frac{\partial{U}}{\partial{V}}\right)_SV=TS-pV
$$
The definition of $G$ as double Legendre transform of $U(S,V)$ with respect to its variables implies
$$
G=U-TS+pV
$$
and combining the two expressions, we get the general result that
$$
G=0
$$
as soon $U$ (or $S$) is a homogeneous function of ($S,V$) ( or ($U,V$)).
Let's notice that the same "zero" thermodynamic potential is obtained every time one evaluates the  Legendre transform of the internal energy with respect to all its extensive variables.
