# 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:

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

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,

$C_{P}-C_{V}=VT{\frac {\alpha _{V}^{2}}{\beta _{T}}}\,$ (see relations between heat capacities), where $C_{P}$ is the heat capacity of a body at constant pressure, $C_{V}$ is the heat capacity at constant volume, $V$ is the volume, $T$ is the temperature, $\alpha _{V}$ is the thermal expansion coefficient and $\beta$ is the isothermal compressibility.

From this relation, several inferences can be made:

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