Mayer's relation

In the 19th century, German chemist and physicist Julius von Mayer derived a relation between the molar heat capacity at constant pressure and the molar heat capacity at constant volume for an ideal gas. Mayer's relation states that
where is the molar heat capacity at constant pressure, is the molar heat capacity at constant volume and is the gas constant.

General form

For more general homogeneous substances, not just ideal gases, the difference takes the form,
, where is the molar volume, is the temperature, is the thermal expansion coefficient and is the isothermal compressibility.
From this latter relation, several inferences can be made:

Since the isothermal compressibility is positive for nearly all phases, and the square of thermal expansion coefficient is always either a positive quantity or zero, the specific heat at constant pressure is nearly always greater than or equal to specific heat at constant volume: There are no known exceptions to this principle for gases or liquids, but certain solids are known to exhibit negative compressibilities and presumably these would be cases where.
For incompressible substances, and are identical. Also for substances that are nearly incompressible, such as solids and liquids, the difference between the two specific heats is negligible.
As the absolute temperature of the system approaches zero, since both heat capacities must generally approach zero in accordance with the Third Law of Thermodynamics, the difference between and also approaches zero. Exceptions to this rule might be found in systems exhibiting residual entropy due to disorder within the crystal.