Mayer's relation

Julius von Mayer derived a relation between specific heat at constant pressure and the specific heat at constant volume for an ideal gas. The relation is:


where 'CP,m' is the molar specific heat at constant pressure, 'CV,m' is the specific heat at constant volume and R is the Universal Gas Constant.

For more general homogeneous substances, not just ideal gases, the difference takes the form,

(see relations between heat capacities), where is the heat capacity of a body at constant pressure, is the heat capacity at constant volume, is the volume, is the temperature, is the thermal expansion coefficient and is the isothermal compressibility.

From this relation, several inferences can be made:[1]

  • Since isothermal compressibility is positive for all phases and the square of thermal expansion coefficient is a positive quantity or zero, the specific heat at constant-pressure is always greater than or equal to specific heat at constant-volume.
  • As the absolute temperature of the system approaches zero, the difference between CP,m and CV,m also approaches zero.
  • For incompressible substances, CP,m and CV,m are identical. Also for substances that are nearly incompressible, such as solids and liquids, the difference between the two specific heats is negligible.


  1. ^ Boles, Yunus A. Çengel, Michael A. Thermodynamics : an engineering approach (7th ed.). New York: McGraw-Hill. ISBN 0-07-736674-3.