Explore BrainMass

Derivation of thermodynamic identities

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.


Solution Preview

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.