In thermodynamics, the heat capacity at constant volume,
, are extensive properties that have the magnitude of energy divided by temperature.
The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): Here
is the isothermal compressibility (the inverse of the bulk modulus): and
is the isentropic compressibility: A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is: 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.
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.
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.
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.
Doing so gives: One can similarly rewrite the partial derivative
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: This is a derivation to obtain an expression for
where The ideal gas equation of state can be arranged to give: The following partial derivatives are obtained from the above equation of state: The following simple expressions are obtained for thermal expansion coefficient
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