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Show that (du/dv)t=T(dp/dT)v-p - please explain!

  1. May 29, 2016 #1
    1. The problem statement, all variables and given/known data
    Show that (du/dv)T = T(dp/dt)v - p

    2. Relevant equations
    Using Tds = du + pdv and a Maxwell relation

    3. The attempt at a solution
    I've solved the problem, but I'm not entirely sure my method is correct.

    Tds = du + pdv ---> du = Tds - Pdv

    - Using dF=(dF/dx)ydx +(dF/dy)xdy
    du=(du/dT)v+(du/dv)Tdv

    - Therefore Tds - Pdv = (du/dT)v+(du/dv)Tdv

    - Divide by dv:
    (du/dT)vdT/dv + (du/dv)T = T(ds/dv)T - p

    Now, to get the right answer, this term:

    (du/dT)vdT/dv

    must equal zero, but I'm not sure why - please can somebody explain?


    Then you simply insert Maxwell relation -(ds/dv)T = -(dp/dT)v
    and rearrange to get the correct answer.

    Many thanks for any help!
     
  2. jcsd
  3. May 29, 2016 #2
    You should have started out by substituting $$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV$$
     
  4. May 29, 2016 #3
    Thanks for your help. I still get to a similar problem unfortunately. I get to here:

    (ds/dT)vdT/dv + (ds/dv)T = (du/dv)T1/T + p/T

    How do I get ride of the (ds/dT)vdT/dv term?

    Many thanks!
     
  5. May 29, 2016 #4
    T(ds/dT)vdT + T(ds/dv)TdV = (du/dv)TdV+(du/dT)VdT + pdV
    Collect factors of dV and dT.
     
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