Thermodynamics, Helmholtz free energy, Legendre transformation

[SoggyBottoms]

Homework Statement

The Helmholtz free energy of a certain system is given by $F(T,V) = -\frac{VT^2}{3}$. Calculate the energy U(S,V) with a Legendre transformation.

Homework Equations

F = U - TS
$S = -\left(\frac{\partial F}{\partial T}\right)_V$

The Attempt at a Solution

We have $U = -\frac{VT^2}{3} + TS$. S is given by $S = -\left(\frac{\partial F}{\partial T}\right)_V = -\frac{2}{3}VT$. Then:

$U = -\frac{VT^2}{3} - \frac{2}{3}VT^2 = -VT^2$

Now I didn't end up with a function U that depends on S and V, but on V and T instead. Should I somehow describe T in terms of S instead? If so, how can I do that?

 

[ehild]

Homework Equations

F = U - TS
$S = -\left(\frac{\partial F}{\partial T}\right)_V$

The Attempt at a Solution

We have $U = -\frac{VT^2}{3} + TS$. S is given by $S = -\left(\frac{\partial F}{\partial T}\right)_V = -\frac{2}{3}VT$. Then:

Check the sign of S: it is 2/3 VT .
Having this relation between T, V and S, express T as function of S and V and substitute into the expression for U.

ehild