In the 19th century, German chemist and physicistJulius von Mayer derived a relation between themolar heat capacity at constant pressure and the molar heat capacity at constant volume for anideal gas.Mayer's relation states that$${\displaystyle C_{P,\mathrm{m}}-C_{V,\mathrm{m}}=R,}$$whereC_P,m is themolar heat at constantpressure,C_V,m is the molar heat at constantvolume andR is thegas constant.

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

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:$${\displaystyle C_{P,\mathrm{m}}\ge C_{V,\mathrm{m}}.}$$ There are no known exceptions to this principle for gases or liquids, but certain solids are known to exhibit negative compressibilities^[2] and presumably these would be (unusual) cases where.
• Forincompressible substances,C_P,m andC_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 theabsolute temperature of the system approaches zero, since both heat capacities must generally approach zero in accordance with theThird Law of Thermodynamics, the difference betweenC_P,m andC_V,m also approaches zero. Exceptions to this rule might be found in systems exhibitingresidual entropy due to disorder within the crystal.

• Thermodynamic equations