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Derivation of thermodynamic identities

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I need the solution to #10 on the attached file.

beta=(1/V)(dV/dT)_p=coefficient of expansion
k=-(1/V)(dV/dp)_T= compressibility
Cv and Cp are just heat capacities.

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I'm going to derive different but equivalent expressions for the firsty two partial derivatives. That they are in fact equal follows from the identity:

C_p - C_v = V T beta^2/kappa (1)

the derivation of which is given in almost all thermodynamics textbooks.

To simplify (dT/dV)_S consider rewriting this as partial derivatives of S w.r.t. T and V:

(dT/dV)_S = - (dS/dV)_T / (dS/dT)_V (2)

To see this, express dS in terms of dT and dV:

dS = (dS/dT)_V dT + (dS/dV)_T dV

The ratio of infinitesimal dT and dV at constant S then follows from setting dS zero here and solving for the ratio. This is by definition the desired partial derivative.

The heat capacity at some constant quantity X is T (dS/dT)_X. We can thus write (2) as:

(dT/dV)_S = - (dS/dV)_T / (dS/dT)_V = - T (dS/dV)_T / C_V (3)

We can simplify this further by using a Maxwell relation involving (dS/dV)_T. Starting from the fundamental thermodynamic ...

Solution Summary

We explain how to obtain the expressions for the partial derivatives. The hear capacities are given.