Thermodynamic derivation involving heat capacities

AI Thread Summary
The discussion revolves around deriving a relationship between heat capacities (Cp and Cv), isothermal compressibility, and the coefficient of thermal expansion. The initial approach involves using the intensive entropy S as a function of temperature and volume, leading to the differential form dS. The process includes taking partial derivatives while holding pressure constant and applying Maxwell relations, ultimately resulting in the equation Cp - Cv = -T(∂p/∂V)T(∂V/∂T)²p. Participants encourage the original poster to work through the steps and seek clarification on specific points rather than expecting a complete solution. The thread highlights the importance of understanding the derivation process in thermodynamics.
mcdonkdik
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I have the answer to this question but I'm finding it hard making sense of it...

Q5) Dervive a relationship relating Cp-Cv 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:

Cp - Cv = -T(∂p/∂V)T(∂V/∂T)2p



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

Many thanks!
 
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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.
 
Mapes said:
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.

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

Thnx
 

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