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Homework Help: I need help deriving this Maxwell relation

  1. Dec 17, 2013 #1
    1. The problem statement, all variables and given/known data
    By considering small changes in enthalpy,
    and using the central equation, derive the Maxwell relation

    [tex]\left (\frac{\partial S}{\partial V} \right )_{T}= \left (\frac{\partial p}{\partial T} \right )_{V}[/tex]

    2. Relevant equations

    3. The attempt at a solution
    So the way I attempted this was to get an expression for dH,

    Honestly I don't know the reason that pV splits up into pdV and Vdp but just know that it does that so I'd appreciate it if someone could input on that?

    Then I used the fact that,
    [tex]dH=\left ( \frac{\partial H}{\partial S} \right )_{p}dS+\left ( \frac{\partial H}{\partial p} \right )_{S}dp[/tex]
    and that
    [tex]\frac{\partial^{2} H}{\partial S\partial p}=\frac{\partial^{2} H}{\partial p\partial S}[/tex]

    This basically yields me with the relation,

    [tex]\left (\frac{\partial T}{\partial p} \right )_{S}= \left (\frac{\partial V}{\partial S} \right )_{p}[/tex]

    So I know this is a correct relation, however it isn't the relation that the question requires. I'm not really sure where to go from here, is there a way to rearrange the equation I ended up with? Or was this a Red Herring and I should have gone about it a different way?

    Thank you for your time. I really appreciate it!
  2. jcsd
  3. Dec 17, 2013 #2


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    Since, as independent variables in the partial derivatives you have [itex]T[/itex] and [/itex]V[/itex], you should use the corresponding thermodynamic potential, for which these variables are "natural". This is the free energy rather than the enthalpy, i.e.,
    Write down the first Law, i.e., evaluate [itex]\mathrm{d} F=\ldots[/itex] and use the commutativity of the 2nd derivatives as in your example with the enthalpy.
  4. Dec 17, 2013 #3
    Sorry I may have been a little vague. I understand what you're saying but the question specifically asks to consider Enthalpy and use the equation,
  5. Dec 17, 2013 #4
    From the central equation and the definition of F, it follows that dF = -SdT-PdV. This should give you the required maxwell relation. The equation for the enthalpy is consistent with all this too. So, who knows what the question means.

  6. Dec 18, 2013 #5
    That is just the chain rule from simple calculus. d(pV) = pdV+Vdp

    For the rest of it, you did indeed prove one of the Maxwell relations, although I am not sure, how to do it for the one you had to find, using the enthalpy. An idea would be to try using:

    U = -T^2 (∂(F/T)/∂T)V

    in the equation for the enthalpy, or some other way of expressing U, that allows you to introduce F
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