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Thermodynamics Proof

  1. Sep 20, 2006 #1
    Hi all, I have a small question about a proof.

    Question:
    Under control variables T, V, and N, derive an expression to relate internal energy as a function of volume. Assume that N is constant throughout.

    Thoughts:
    Starting with dU = TdS - PdV + udN.
    Cancel out dN --> dU = TdS - PdV
    Divide by dV --> (dU/dV) = (TdS/dV) - P
    In my answer key,
    It jumps from the above equation to (dU/dV) = (TdP/dT) - P
    I don't understand why dS/dV was replaced by dP/dT, how was that relationship derived?

    Thanks.
     
  2. jcsd
  3. Sep 20, 2006 #2

    siddharth

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    Homework Helper
    Gold Member

    Maxwell's relations.

    You know that the Helmholtz free energy dA is
    dA = -SdT - PdV
    Since dA is an exact differential, dS/dV=dP/dT. In fact, you can get a similar relationship between the properties for each of the four fundamental equations.

    Here's more on Maxwell's relations
    http://chsfpc5.chem.ncsu.edu/~franzen/CH431/lecture/lec_13_maxwell.htm
     
  4. Sep 20, 2006 #3
    Ah, next chapter in class. Thanks!
     
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