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Derive heat capacity at constant pressure

  1. Jan 27, 2009 #1
    1. The problem statement, all variables and given/known data

    For temperatures [tex]T >> T_C[/tex] (critical temperature) derive the heat capacity at constant pressure [tex]C_P[/tex] from van der Waals equation.

    2. Relevant equations

    Critical temperature:
    [tex]T_C = \frac{2N(V - Nb)^2}{kV^2}[/tex]

    [tex]T_C[/tex] is derived from the fact that it exist at the point in which
    [tex](\frac{\mathrm{d}P}{\mathrm{d}V})_T = 0[/tex] but I'm pretty certain that this is not needed in this derivation.

    Van der Waals equation:
    [tex](P + aN^2/V^2)(V - Nb) = NkT[/tex]

    Heat capacity:
    [tex]C_P = (\frac{\mathrm{d}U}{\mathrm{d}T})_P + P(\frac{\mathrm{d}V}{\mathrm{d}T})_P[/tex]

    3. The attempt at a solution

    I've tried to get the point in this. The first term in the equation of [tex]C_P[/tex] is easy, but the problem is the second term.

    If I try to solve for the [tex]V[/tex] in the van der Waals equation I get a long equation set to derive. And I don't think this is what is the point of the exercise.

    Is there another approach or something to simplify the van der Waals equation before the derivation?
  2. jcsd
  3. Jan 28, 2009 #2


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    Staff Emeritus
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    Homework Helper

    It looks like you get a cubic equation in V to solve, which I agree is probably not what you're expected to do.

    Just taking an educated guess here, but it's probably the case that
    V >> Nb​
    P >> aN2/V2
    It may be a reasonable approximation to replace the "V2" term with whatever the ideal gas equation gives for V, since it appears as part of a term that is small to begin with.
  4. Dec 9, 2011 #3
    You're using the wrong equation for hear capacity, you should use Cp = dh/dt

    where h is enthaply
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