Relations Between Heat Capacities - Derivation

Derivation

If an infinitesimal small amount of heat is supplied to a system in a quasistatic way then, according to the second law of thermodynamics, the entropy change of the system is given by:

Since

where C is the heat capacity, it follows that:

The heat capacity depends on how the external variables of the system are changed when the heat is supplied. If the only external variable of the system is the volume, then we can write:

From this follows:

Expressing dS in terms of dT and dP similarly as above leads to the expression:

One can find the above expression for by expressing dV in terms of dP and dT in the above expression for dS.

results in

and it follows:

Therefore,

The partial derivative can be rewritten in terms of variables that do not involve the entropy using a suitable Maxwell relation. These relations follow from the fundamental thermodynamic relation:

It follows from this that the differential of the Helmholtz free energy is:

This means that

and

The symmetry of second derivatives of F with respect to T and V then implies

allowing one to write:

The r.h.s. contains a derivative at constant volume, which can be difficult to measure. It can be rewritten as follows. In general,

Since the partial derivative is just the ratio of dP and dT for dV = 0, one can obtain this by putting dV = 0 in the above equation and solving for this ratio:

which yields the expression:

The expression for the ratio of the heat capacities can be obtained as follows:

The partial derivative in the numerator can be expressed as a ratio of partial derivatives of the pressure w.r.t. temperature and entropy. If in the relation

we put and solve for the ratio we obtain . Doing so gives:

One can similarly rewrite the partial derivative by expressing dV in terms of dS and dT, putting dV equal to zero and solving for the ratio . When one substitutes that expression in the heat capacity ratio expressed as the ratio of the partial derivatives of the entropy above, it follows:

Taking together the two derivatives at constant S:

Taking together the two derivatives at constant T:

From this one can write: