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Thermodynamics, entropy representation problem

  1. Mar 25, 2012 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    The problem is taken from Callen's book (page 36).
    Find the three equations of state in the entropy representation for a system with the fundamental equation [itex]u =A\frac{s^{5/2}}{v^{1/2}}[/itex].
    Show by a diagram (drawn to abitrary scale) the dependence of temperature on volume for fixed pressure. Draw two such "isobars" corresponding to two values of the pressure and indicate which isobar corresponds to the higher pressure.

    2. Relevant equations
    [itex]dU=TdS-PdV+\mu dN[/itex].


    3. The attempt at a solution
    I've been looking in the book for the "entropy representation" and what I understood is that they ask for [itex]T(S,V,N)[/itex], [itex]P(S,V,N)[/itex] and [itex]\mu (S,V,N)[/itex]. Google didn't give me a better clue on the "entropy representation" either.
    So I've found out the 3 equations of state to be [itex]T=\frac{5AS^{3/2}}{2N^2V^{1/2}}[/itex], [itex]P=\frac{AS^{5/2}}{2N^2V^{3/2}}[/itex] and [itex]\mu =-\frac{AS^{5/2}}{N^3V^{1/2}}[/itex].
    What destroys me is the next question. They ask me to graph T(V) for a fixed P; as if they had asked me first to find [itex]T(V,P)[/itex] instead of [itex]T(S,V,N)[/itex]. Did I get the "entropy representation" wrong?
     
  2. jcsd
  3. Mar 26, 2012 #2

    fluidistic

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    Apparently what I did is wrong.
    The 3 equations of state are [itex]\left ( \frac{\partial S }{\partial U } \right ) _{V,N}=\frac{1}{T}[/itex], [itex]\left ( \frac{\partial S }{\partial N } \right ) _{V,U}=\frac{\mu}{T}[/itex] and [itex]\left ( \frac{\partial S }{\partial V } \right ) _{U,N}=\frac{-P}{T}[/itex]. I calculated them to be worth [itex]\frac{2N^{2/5}V^{1/5}}{5A^{2/5}U^{3/5}}[/itex], [itex]\frac{2U^{2/5}V^{1/5}}{5A^{2/5}N^{3/5}}[/itex] and [itex]\frac{N^{2/5}U^{2/5}}{5A^{2/5}V^{4/5}}[/itex] respectively.
    I am not sure how to do the diagram though.
     
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