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Thermodynamics - Maxwell Reactions

  1. Sep 2, 2008 #1
    We have been asked the following: "For fixed compositions, show that T = (partial U/partial S)_V = (partial H/partial S)_P". There are 7 other equations after this one, but I think once I know how to solve the first one I will be ok with the rest.

    We have not been provided with any relevant equations. My knowledge of partial differential equations is a bit rusty, but I have been doing some reading today to brush up. I initially tried to differentiate H = U + PV and substitute that in, but I ended up with an extra Pdv/S, which with constant P I am not sure how to cancel out. I suspect it has something to do with C_v = T(ds/dt)_v and C_p = T(dt/ds)_p but am not sure how to apply them or how to convert from equations at constant volume to equations at constant pressure.

    I have ordered Classical Thermodynamics by Van Ness and Abbott at the recommendation of our professor, but it won't arrive until next week. The homework isn't due in for another 2 weeks, but if anyone could help me get started, I would really appreciate it. Maybe just some basic pointers so that I had an idea of what path to follow.

    Thanks in advance.
  2. jcsd
  3. Sep 2, 2008 #2


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  4. Sep 2, 2008 #3


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    For most thermodynamics problems of these sorts, you can start with one of these basic equations and go from there:

    dU = T dS - P dV
    dH = T dS + V dP
    dA = -S dT - P dV
    dG = -S dT + V dP

    For example, with the first, you can start with the differential for U. Since you want the partial derivative where V is constant, dV = 0. You should be able to take it from there.
  5. Sep 2, 2008 #4


    dU = T dS - PdV
    As V is constant, dV = 0. Hence, - PdV = 0
    dU = T dS
    dU/dS = T

    dH = T dS - V dP
    As P is constant, dP = 0. Hence, - V dP = 0
    dH = T dS
    dH/dS = T

    T = dU/dS @ constant V = dH/dS @ constant P


    Also, when you use "d" are you referring to partial differential or ordinary differential? If the former, does it matter that I need to find the partial differential?
  6. Sep 3, 2008 #5


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    The four equations have the ordinary differential. However, by fixing P or V, you convert the ordinary differential into a partial differential (since a partial derivative is an ordinary derivative with one or more of the other variables treated as constant).
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