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Thermal Physics Relation

  1. Feb 24, 2008 #1
    1. The problem statement, all variables and given/known data
    Derive the following equation

    2. Relevant equations
    [tex]TdS = C_{V} \left( \frac{\partial T}{\partial P} \right)_{V}dP + C_{P} \left( \frac{\partial T}{\partial V} \right)_{P}dV [/tex]

    3. The attempt at a solution

    [tex] dU = \delta Q - \delta W [/tex]

    [tex] \delta Q = TdS [/tex] for a closed system

    [tex] C_{P} = T \left( \frac{\partial S}{\partial T} \right)_{P} [/tex]

    [tex] C_{V} = T \left( \frac{\partial S}{\partial T} \right)_{V} [/tex]

    I am not sure where to go from here.
  2. jcsd
  3. Feb 24, 2008 #2


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    Homework Helper
    Gold Member

    I think the trick here is playing around with the relations.

    If you consider S as a function of p and v as independent variables, then
    [tex] dS = \left(\frac{\partial S}{\partial P}\right)_V dP + \left(\frac{\partial S}{\partial V}\right)_P dV[/tex]

    But, [tex]\frac{\partial S}{\partial P}_V = \left(\frac{\partial S}{\partial T}\right)_V \left(\frac{\partial T}{\partial P}\right)_V [/tex], and so on.
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