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?