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Helmholtz Energy Proof (thermodynamics)

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Homework Statement


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

Homework Equations


S=([itex]\frac{∂F}{∂T}[/itex])V

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

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
 

Answers and Replies

  • #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
 
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  • #3
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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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