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 $$C_{P,\mathrm{m}}-C_{V,\mathrm{m}}=R,$$ where Template:Math is the molar heat at constant pressure, Template:Math is the molar heat at constant volume and Template:Math is the gas constant.

For more general homogeneous substances, not just ideal gases, the difference takes the form, $$C_{P,\mathrm{m}}-C_{V,\mathrm{m}}=V_{\mathrm{m}}T\frac{{\alpha }_V^2}{{\beta }_T}$$ (see relations between heat capacities), where $V_{\mathrm{m}}$ is the molar volume, $T$ is the temperature, ${\alpha }_V$ is the thermal expansion coefficient and $\beta$ is the isothermal compressibility.

From this latter relation, several inferences can be made:

• Since the isothermal compressibility ${\beta }_T$ is positive for nearly all phases, and the square of thermal expansion coefficient $\alpha$ 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: $$C_{P,\mathrm{m}}\ge C_{V,\mathrm{m}}.$$ 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 (unusual) cases where $C_{P,\mathrm{m}}<C_{V,\mathrm{m}}$.
• For incompressible substances, Template:Math and Template:Math 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 Template:Math and Template:Math also approaches zero. Exceptions to this rule might be found in systems exhibiting residual entropy due to disorder within the crystal.