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Concept
Relations between heat capacities
Summary
In thermodynamics, the heat capacity at constant volume, , and the heat capacity at constant pressure, , 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 thermal expansion coefficient:
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.
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.
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.
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