Deriving Internal Energy from Volume with Constant N: Thermodynamics Proof

[LeT374]

Hi all, I have a small question about a proof.

Question:
Under control variables T, V, and N, derive an expression to relate internal energy as a function of volume. Assume that N is constant throughout.

Thoughts:
Starting with dU = TdS - PdV + udN.
Cancel out dN --> dU = TdS - PdV
Divide by dV --> (dU/dV) = (TdS/dV) - P
In my answer key,
It jumps from the above equation to (dU/dV) = (TdP/dT) - P
I don't understand why dS/dV was replaced by dP/dT, how was that relationship derived?

Thanks.

 

[siddharth]

Maxwell's relations.

You know that the Helmholtz free energy dA is
dA = -SdT - PdV
Since dA is an exact differential, dS/dV=dP/dT. In fact, you can get a similar relationship between the properties for each of the four fundamental equations.

Here's more on Maxwell's relations
http://chsfpc5.chem.ncsu.edu/~franzen/CH431/lecture/lec_13_maxwell.htm"

 

[LeT374]

Ah, next chapter in class. Thanks!