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!