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Homework Help: Helmholtz Energy Proof (thermodynamics)

  1. Sep 19, 2013 #1
    1. The problem statement, all variables and given/known data
    Define the Helmholtz free energy as F=E-TS.
    Show that the internal energy E=-T2[itex]\frac{∂}{∂T}[/itex]([itex]\frac{F}{T}[/itex])V

    2. Relevant equations
    S=([itex]\frac{∂F}{∂T}[/itex])V

    Perhaps [itex]\beta[/itex]=[itex]\frac{1}{\tau}[/itex]
    and [itex]\tau[/itex]=kBT

    3. The attempt at a solution
    E = F+TS
    E = F+T([itex]\frac{∂F}{∂T}[/itex])V
    .
    .
    . (some steps to final equation)
    .
    .
    E=-T2[itex]\frac{∂}{∂T}[/itex]([itex]\frac{F}{T}[/itex])V


    Any help/hints would be greatly appreciated. My partial derivatives are a bit rusty. Thanks
     
  2. jcsd
  3. Sep 19, 2013 #2

    TSny

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    Note [itex]\frac{∂}{∂T}[/itex]([itex]\frac{F}{T}[/itex])V = [itex]\frac{∂}{∂T}[/itex]([itex]\frac{1}{T} \cdot F)[/itex]V and use the product rule to write out the partial derivative.

    Also, check to see if there's a sign error in your equation S=([itex]\frac{∂F}{∂T}[/itex])V
     
  4. Sep 21, 2013 #3
    Okay! I got it..

    So

    E=-T2[-[itex]\frac{1}{T^2}[/itex]F + [itex]\frac{∂F}{∂T}[/itex][itex]\frac{1}{T}[/itex]]

    E= F + ([itex]\frac{∂F}{∂T}[/itex])(-T)
    E= F+(-S)(-T)
    E= F+TS

    F=E-TS

    Thank you!!
     
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