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[Thermo] - Maxwells Relations? Prove the Validity of the Equations

  1. Apr 18, 2008 #1
    1. The problem statement, all variables and given/known data


    We are working on some problems for class and we are given statements which I accept as valid but don't know how to prove they are valid. I believe I have to utilize the maxwell relations but the terms seem unfamiliar to me.


    2. Relevant equations

    (1)
    Partial
    (d^2f / ds^2)_T = T / Kv

    (2)
    Partial
    (d^2h / ds^2)_P = T / C_p

    (3)
    Partial
    (d^2u / ds^2)_v = T / C_v






    3. The attempt at a solution

    For the first equation, I know that the isothermal compressibiity K = -1/v partial(dv/dP)_T

    For the second equation I also know that C_p/T = partial(ds/dT)_P = 1/T * partial(dh/dT)_P and i need to take this knowledge to combine the equations but I don't see how I would be getting the square on the ds out of this. Obviously I'm missing something.

    C_v/T should be the same proof with u in place of h holding v constant.

    Thanks for any help.
     
  2. jcsd
  3. Apr 18, 2008 #2

    Mapes

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    Take a look at my recent post on Maxwell relations. To handle the second derivative, note that [itex]\left(\frac{\partial H}{\partial S}\right)_P[/itex] is also known as [itex]T[/itex]. So differentiate [itex]T[/itex] with respect to [itex]S[/itex] again at constant pressure. Does this help?
     
  4. Apr 21, 2008 #3
    Thank You, I think the answers were staring me in the face but in a different format and i didn't put 2 and 2 together until your response. I now have a solution where T=du/ds with v constant and T=dh/ds with p constant (correct?) and plugging these in to the equations cv=T(ds/dt)_v and cp=T(dt/ds)_p to get the desired proof. I think its right.

    I'm still struggling with the third one however so any assistance would be appreciated. How do you make the proper symbols appear?
     
  5. Apr 21, 2008 #4

    Mapes

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    The first relation can't be true, the units don't match up.
     
  6. Apr 21, 2008 #5
    Yes thats the one i'm struggling with sorry, the first one. The 2nd and 3rd I believe I have proven as said in the previous post.

    I completely screwed up the first one, the actual problem is as follows:

    (d^2f / dv^2)_T = 1 / Kv

    This is the one I am struggling with at the moment. Sorry I was trying too hard to convey the format of the problem that i got the variables wrong somehow.
     
  7. Apr 21, 2008 #6

    Mapes

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    OK, that makes a lot more sense. Try writing out the differential form of the Helmholtz free energy dF and taking the derivative of that twice with respect to V at constant T. The first derivative equals a well-known parameter.
     
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