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

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

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

• Since the isothermal compressibility ${\displaystyle \beta _{T}}$ is positive for nearly all phases, and the square of thermal expansion coefficient ${\displaystyle \alpha }$ 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: $${\displaystyle C_{P,\mathrm {m} }\geq C_{V,\mathrm {m} }.}$$ There are no known exceptions to this principle for gases or liquids, but certain solids are known to exhibit negative compressibilities ^[2] and presumably these would be (unusual) cases where ${\displaystyle C_{P,\mathrm {m} }<C_{V,\mathrm {m} }}$.
• For incompressible substances, C_P,m and C_V,m 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 C_P,m and C_V,m also approaches zero. Exceptions to this rule might be found in systems exhibiting residual entropy due to disorder within the crystal.