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How do I derive these results (multivariable calculus in thermodynamics context)?

  1. Nov 3, 2012 #1
    1. The problem statement, all variables and given/known data
    The thermodynamic state of a system may be defined by setting any 2 of the variables P, V and T to be constant because they are related by f(P,V,T)=0.

    Given a thermodynamic function K(V,T), prove that:

    (∂K/∂T)_P = (∂K/∂T)_V + [(∂K/∂V)_T]*[(∂V/∂T)_P]

    and that

    (∂K/∂P)_T = [(∂K/∂V)_T]*[∂V/∂P)_T]

    2. The attempt at a solution
    To be honest I have written down various differentials and tried to combine them in various ways to get these results but to no avail.

    Obviously an expression for dK would be useful:

    dK = [(∂K/∂V)_T]dV + [(∂K/∂T)_V]dT

    Since f(P,V,T) = 0, I can say that p is a function of V and T, and then:

    dP = [(∂P/∂V)_T]dV + [(∂P/∂T)_V]dT

    Similarly:

    dV = [(∂V/∂P)_T]dP + [(∂V/∂T)_P]dT
    dT = [(∂T/∂V)_P]dV + [(∂T/∂P)_V]dP

    But I cannot combine these to make the required results, and can't think of anything else I can write down.

    I did write that since P is a function of V and T and K is a function of V and T then K is a function of P -- but I've just thought this isn't actually (necessarily) correct?

    Any help/hints on this would be appreciated.
     
  2. jcsd
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