# Helmholtz Energy Proof (thermodynamics)

1. Sep 19, 2013

### musicure

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$\frac{∂}{∂T}$($\frac{F}{T}$)V

2. Relevant equations
S=($\frac{∂F}{∂T}$)V

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

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

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

2. Sep 19, 2013

### TSny

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

3. Sep 21, 2013

### 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!!