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Proving a thermodynamic relationship

  1. Nov 30, 2016 #1
    1. The problem statement, all variables and given/known data
    Prove that ##TdS = C_vdT + \alpha T / \kappa dV##

    2. Relevant equations
    ##T dS = dU - pdV##
    ##\alpha = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right )_P##
    ##\kappa = -\frac{1}{v}\left(\frac{\partial v}{\partial P}\right)_T##

    3. The attempt at a solution

    The ##C_vdT## part is quite easy since for a constant volume process ##dU = C_vdT## but I can't seem to figure out how to get the second part of the expression. After multiplying by forms of 1 I end up with $$-pdV = \frac{\alpha\left(\frac{\partial v}{\partial P}\right)_T}{\kappa \left(\frac{\partial v}{\partial T}\right)_P}PdV$$, now using the cyclical rule here doesn't seem logical since that would introduce a negative so it seems like I need to replace the pressure P with something although I'm not sure what relation I can use to do that.
     
  2. jcsd
  3. Dec 3, 2016 #2

    Mapes

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    You've probably figured it out over the past few days, but for one thing, you've got a sign problem: ##T\,dS=dU+p\,dV## because ##p## is compressive stress.
     
  4. Dec 4, 2016 #3
    Your mistake is that dU is not equal to ##C_vdT##. That is only correct for an ideal gas. In general, $$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV=\frac{C_vdT}{T}+\left(\frac{\partial S}{\partial V}\right)_TdV$$
    From one of the Maxwell relationships, $$\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial P}{\partial T}\right)_V$$Therefore, $$dS=\frac{C_vdT}{T}+\left(\frac{\partial P}{\partial T}\right)_VdV$$So, $$TdS=C_vdT+T\left(\frac{\partial P}{\partial T}\right)_VdV$$
     
  5. Dec 4, 2016 #4

    Mapes

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    At constant volume, ##dU=C_V\,dT## for all materials, as Potatochip911 noted.
     
  6. Dec 4, 2016 #5
    But the problem statement does not say anything about constant volume. In fact, it explicitly indicates that the volume is not considered constant.
     
    Last edited: Dec 4, 2016
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