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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!

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# Thermodynamic derivation involving heat capacities

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