# Helmholtz Energy Proof (thermodynamics)

musicure

## Homework Statement

Define the Helmholtz free energy as F=E-TS.
Show that the internal energy E=-T2$\frac{∂}{∂T}$($\frac{F}{T}$)V

## Homework Equations

S=($\frac{∂F}{∂T}$)V

Perhaps $\beta$=$\frac{1}{\tau}$
and $\tau$=kBT

## The Attempt at a Solution

E = F+TS
E = F+T($\frac{∂F}{∂T}$)V
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. (some steps to final equation)
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E=-T2$\frac{∂}{∂T}$($\frac{F}{T}$)V

Any help/hints would be greatly appreciated. My partial derivatives are a bit rusty. Thanks

Homework Helper
Gold Member
Note $\frac{∂}{∂T}$($\frac{F}{T}$)V = $\frac{∂}{∂T}$($\frac{1}{T} \cdot F)$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=($\frac{∂F}{∂T}$)V

1 person
musicure
Okay! I got it..

So

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

E= F + ($\frac{∂F}{∂T}$)(-T)
E= F+(-S)(-T)
E= F+TS

F=E-TS

Thank you!!