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Derivation of Maxwell's relations

  1. Oct 21, 2012 #1
    In thermodynamics one of the maxwell relations is:
    [itex]
    \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V
    [/itex]

    When I try to derive it from [itex] dU = TdS - PdV [/itex] i get:
    [itex]
    T = \left( \frac{\partial U}{\partial S} \right)_V
    [/itex]
    [itex]
    P = -\left( \frac{\partial U}{\partial V} \right)_S
    [/itex]
    [itex]
    \left( \frac{\partial T}{\partial V} \right)_S = \frac{\partial}{\partial V}\left( \frac{\partial U}{\partial S} \right)_V = \frac{\partial}{\partial S}\left( \frac{\partial U}{\partial V}\right)_S = -\left( \frac{\partial P}{\partial S} \right)_V
    [/itex]
    I then multiply with [itex]\frac{\partial S}{\partial T}[/itex],
    [itex]
    \frac{\partial S}{\partial T} \left( \frac{\partial T}{\partial V} \right)_S = \frac{\partial S}{\partial T} \left( -\frac{\partial P}{\partial S} \right)_V
    [/itex]
    [itex]
    \left( \frac{\partial S}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial T} \right)_V
    [/itex]
    So, what am I doing wrong?
     
  2. jcsd
  3. Oct 21, 2012 #2
    I don't think you can derive that Maxwell relation from the internal energy differential. Try enthalpy, Gibbs free energy, or Helmholtz free energy instead.
     
  4. Oct 21, 2012 #3
    Helmholtz free energy specifically. The form he is trying to derive implies T & V are the parameters of the system.
     
  5. Oct 22, 2012 #4
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