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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:

• 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 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.

References

1. Çengel, Yunus A.; Boles, Michael A. Thermodynamics: an engineering approach (7th ed.). New York: McGraw-Hill. ISBN 0-07-736674-3.
2. Anagnostopoulos, Argyrios; Knauer, Sandra; Ding, Yulong; Grosu, Yaroslav (2020). "Giant Effect of Negative Compressibility in a Water–Porous Metal–CO2 System for Sensing Applications". ACS Applied Materials and Interfaces. 12 (35): 35. doi:10.1021/acsami.0c08752. S2CID 221200797. Retrieved 26 March 2022.

• Thermodynamic equations