Mayer's relation

In the 19th century, German chemist and physicist Julius von Mayer derived a relation between specific heat at constant pressure and the specific heat at constant volume for an ideal gas. Mayer's relation states that $$C_{P,\mathrm{m}} - C_{V,\mathrm{m}} = R,$$ where $CP,m$ is the molar specific heat at constant pressure, $CV,m$ is the molar specific heat at constant volume and $R$ 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: $$C_{P,\mathrm{m}} \geq C_{V,\mathrm{m}}.$$ There are no known exceptions to this principle for gases or liquids, but certain solids are known to exhibit negative compressibilities and presumably these would be (unusual) cases where $$C_{P,\mathrm{m}} < C_{V,\mathrm{m}}$$.
 * 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.
 * 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 $CP,m$ and $CV,m$ also approaches zero. Exceptions to this rule might be found in systems exhibiting residual entropy due to disorder within the crystal.