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Thermodynamics Proof : Cv (non-ideal gas) - Cv (ideal gas)

  1. Sep 17, 2015 #1
    Can someone please help me with the following proof ....I'm stuck and not sure if I'm even on the right path.

    Prove that upload_2015-9-17_14-52-7.png


    What I've done so far;
    if U = f(T,V)
    dU = (∂U/∂T)v dT + (∂U/∂V)t dV

    Cv (non ideal) = (∂U/∂T)v

    Using dU = TdS - PdV and Maxwell relation (∂S/∂V)t =(∂P/∂T)v,
    (∂U/∂V)t = T(∂P/∂T)v - P

    So;
    dU = CvdT + [ T(∂P/∂T)v - P ]dv

    I'm basically stuck here, tried different ways forward from here but I can't seem to arrive at the correct answer. Any help would be
     
  2. jcsd
  3. Sep 17, 2015 #2
    This is a homework problem, so I am moving it to a homework forum.

    You want to find U(T+dT, V) - U(T,V), where V is a small enough volume so that ideal gas behavior does not to apply. What you do is start at T,V and determine the isothermal change in U when you go from T,V to very large volume (infinite). This puts you in the ideal gas region. Then you take the change in U from T and infinite volume to T + dT and infinite volume. This is just the ideal gas heat capacity times dT. Then you go isothermally from T+dT and infinite volume to T+dT and finite volume V.

    Chet
     
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