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Enthalpy, partial derivatives

  1. Apr 29, 2012 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    Demonstrate that [itex]C_{Y,N}=\left ( \frac{ \partial H}{\partial T } \right ) _{Y,N}[/itex] where H is the enthalpy and Y is an intensive variable.


    2. Relevant equations
    (1) [itex]C_{Y,N}=\frac{T}{N} \left ( \frac{ \partial S}{\partial T } \right ) _{Y,N}[/itex]
    (2) [itex]T= \left ( \frac{ \partial U}{\partial S } \right ) _{X,N}[/itex] where X is an extensive variable.

    3. The attempt at a solution
    Using (1) and (2) I reach that [itex]C_{Y,N}=T \left ( \frac{ \partial S}{\partial T } \right ) _{Y,N}+ P \left ( \frac{ \partial V}{\partial T } \right ) _{Y,N}[/itex]. I don't know how to proceed further, I'm really stuck here.
     
    Last edited: Apr 29, 2012
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  3. Apr 29, 2012 #2

    I like Serena

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    Which intensive variables are we talking about?
    What do you get if you pick the first one that springs to mind?
     
    Last edited by a moderator: Apr 29, 2012
  4. Apr 29, 2012 #3

    fluidistic

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    Usually the pressure, but it is not specified.
    [itex]C_{P,N}=\frac{T}{N} \left ( \frac{ \partial S}{\partial T } \right ) _{P,N}[/itex].
    Where [itex]\left ( \frac{ \partial S}{\partial T } \right ) _{P,N}=\left ( \frac{ \partial U}{\partial T } \right ) _{P,N}\left ( \frac{ \partial S}{\partial U } \right ) _{P,N}=\frac{1}{T} \left ( \frac{ \partial U}{\partial T } \right ) _{P,N}[/itex].
    Thus [itex]C_{P,N}= \frac{1}{N} \left ( \frac{\partial U }{\partial T } \right ) _{P,N } [/itex].
    Now I use the relation [itex]U=H-PV[/itex] to get [itex]\left ( \frac{ \partial U}{\partial T } \right ) _{P,N}=\left ( \frac{ \partial H}{\partial T } \right ) _{P,N} - \left [ \underbrace { \left ( \frac{ \partial P}{\partial T } \right ) _{P,N} V }_{=0} + P \left ( \frac{ \partial V}{\partial T } \right ) _{P,N} \right ][/itex].
    Therefore I'm left with [itex]C_{P,N}=\frac{1}{N} \left [ \left ( \frac{ \partial H}{\partial T } \right ) _{P,N} - P \left ( \frac{ \partial V}{\partial T } \right ) _{P,N} \right ][/itex].
     
    Last edited by a moderator: Apr 29, 2012
  5. Apr 29, 2012 #4

    I like Serena

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    Hmm, let's start with H=U+PV, or rather dH=TdS+VdP.

    When I take the partial derivative with respect to T, and with P,N constant, I almost get what you're looking for (typo?).
     
    Last edited: Apr 29, 2012
  6. Apr 29, 2012 #5

    fluidistic

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    Hmm I don't think there's a typo.
    Anyway you took the partial derivative of "dH"? I'm having some troubles to figure this out :)
     
  7. Apr 29, 2012 #6

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    dH=TdS+VdP
    So:
    $$\left({\partial H \over \partial T}\right)_{P,N}=\left({T\partial S + V\partial P \over \partial T}\right)_{P,N}$$

    Factor out and replace with ##C_{P,N}## where applicable...
     
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