where ρ is the density of the substance under the applicable conditions.
The corresponding expression for the ratio of specific heat capacities remains the same since the thermodynamic system size-dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. Thus:
The difference relation allows one to obtain the heat capacity for solids at constant volume which is not readily measured in terms of quantities that are more easily measured. The ratio relation allows one to express the isentropic compressibility in terms of the heat capacity ratio.
If an infinitesimally small amount of heat is supplied to a system in a reversible 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.
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:
The following partial derivatives are obtained from the above equation of state:
The following simple expressions are obtained for thermal expansion coefficient :
and for isothermal compressibility :
One can now calculate for ideal gases from the previously obtained general formula:
Substituting from the ideal gas equation gives finally:
where n = number of moles of gas in the thermodynamic system under consideration and R = universal gas constant. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows:
This result would be consistent if the specific difference were derived directly from the general expression for .