Thermodynamic derivation involving heat capacities

mcdonkdik

I have the answer to this question but I'm finding it hard making sense of it...

Q5) Dervive a relationship relating C_p-C_v to the isothermal compressibility (∂p/∂V)_T and the coefficient of thermal expansion (∂V/∂T)_p. Hint: consider the intensive entropy S as a function of T and V.

So I've started with S(T, V):

dS = (∂S/∂T)dT + (∂S/∂V)dV

Apparently we take the partial derivative wrt T while holding p(pressure) constant.. then we use a Maxwell relation to remove the partial derivative containing S. Then we use the triple product rule for something.

We end up with:

C_p - C_v = -T(∂p/∂V)_T(∂V/∂T)²_p



I'd really appreciate it if someone could give me a thorough explanation of how to do this.

Many thanks!

Mapes

Hi mcdonkdik, welcome to PF. Nobody's going to solve your problem for you, but if you work through the recommended steps (which constitute the entire solution already!) and show where you get hung up, you'll likely get helpful comments.

mcdonkdik

It's ok, I'm being dumb. This post can be deleted!

Thnx