# 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:

${\displaystyle C_{P,m}-C_{V,m}=R}$,

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,

${\displaystyle C_{P}-C_{V}=VT{\frac {\alpha _{V}^{2}}{\beta _{T}}}\,}$

(see relations between heat capacities), where ${\displaystyle C_{P}}$ is the heat capacity of a body at constant pressure, ${\displaystyle C_{V}}$ is the heat capacity at constant volume, ${\displaystyle V}$ is the volume, ${\displaystyle T}$ is the temperature, ${\displaystyle \alpha _{V}}$ is the thermal expansion coefficient and ${\displaystyle \beta }$ is the isothermal compressibility.

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

• Since isothermal compressibility ${\displaystyle \beta _{T}}$ is positive for all phases and the square of thermal expansion coefficient ${\displaystyle {\alpha }}$ is a positive quantity or zero, the specific heat at constant-pressure is always greater than or equal to specific heat at constant-volume.
${\displaystyle C_{P,m}}$${\displaystyle C_{V,m}}$
• 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.

## References

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