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 $\tau$=k_BT

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

 

[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

 

[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!