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Homework Help: Thermodynamics, Helmholtz free energy, Legendre transformation

  1. Jan 4, 2012 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

    F = U - TS
    [itex]S = -\left(\frac{\partial F}{\partial T}\right)_V[/itex]

    3. The attempt at a solution

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

    [itex]U = -\frac{VT^2}{3} - \frac{2}{3}VT^2 = -VT^2 [/itex]

    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?
  2. jcsd
  3. Jan 5, 2012 #2


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    Homework Helper

    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.

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