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Maxwell relations Thermodynamics

  1. Nov 29, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that: [itex] (\frac{∂T} {∂V})_S,_n=-(\frac {∂P} {∂S})_V,_n [/itex]

    2. Relevant equations
    [itex] dU=TdS-PdV+μdn [/itex]

    3. The attempt at a solution
    [itex] \frac {∂} {∂S} (\frac{∂U} {∂V})_S,_n=-(\frac {∂P} {∂S})_V,_n [/itex]

    [itex] \frac {∂} {∂V} (\frac{∂U} {∂S})_V,_n=(\frac{∂T} {∂V})_S,_n [/itex]

    I tried to isolate T and P, but I get stuck:
    [itex] TdS=dU+PdV [/itex]

    [itex] -PdV=dU-TdS [/itex]

    How can I demonstrate that they are equal?
     
  2. jcsd
  3. Nov 29, 2014 #2
  4. Nov 29, 2014 #3
    It looks like you already had it in your first two equations of Attempt at a Solution. You are aware that 2nd partial derivatives with respect to two variables are interchangeable (commutative), correct?

    Chet
     
  5. Nov 30, 2014 #4
    Yes I see that, but I ment to obtain [itex] −(\frac {∂P} {∂S})_V,_n [/itex] from [itex] (\frac {∂T} {∂V})_S,_n [/itex] or viceversa, demonstrating Schwartz relation in some way.
     
  6. Nov 30, 2014 #5
    Isn't that what your first two equations under Attempt demonstrate?

    Chet
     
  7. Nov 30, 2014 #6
    If I say they are equal, I am assuming that Schwartz relation is valid in this case, or that U is a continuos function. I want to demonstrate why Schwartz relation is valid.
     
  8. Nov 30, 2014 #7
    What's wrong with assuming that U is a continuous function for a single-component single-phase material?

    Chet
     
  9. Nov 30, 2014 #8
    Nothing, but in an exam I would have to reason why I assume that U is continuous, so if I could go from [itex] -\left( {\frac{\partial P} {\partial S}}\right)_{V,n} [/itex] to [itex] \left( {\frac{\partial T} {\partial V}}\right)_{S,n} [/itex] by using derivatives and their properties, the problem would be solved I think.
     
  10. Nov 30, 2014 #9
    Why would you think it's not continuous?

    Chet
     
  11. Nov 30, 2014 #10
    I think it's continuous, I know from theory, but it's not a given in the problem, so if i use it, I have to say why. As I don't know how to justify that U is continuous, I think that, if I can demonstrate that Schwartz relation is valid, then, I can avoid justifying its continuity.
     
  12. Nov 30, 2014 #11
    I don't know what to say. Are you a mathematician or a physicist? If the latter, why would you think that internal energy is not a continuous function of entropy and volume?

    Chet
     
  13. Nov 30, 2014 #12
    I'm studying Physics, and I know it is a continuous function of entropy and volume, but one thing is knowing and another is proving. I need to prove this knowledge.
     
  14. Nov 30, 2014 #13
    It's observed experimentally. Does that count as proof? Otherwise you need to start looking into statistical thermo.

    Chet
     
  15. Nov 30, 2014 #14
    Mmm statiscal thermo is a subject I will study next year, so if there's no other proof I suppose experimental proof is what I need.Thank you.
     
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