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 $C_{P, m} - C_{V, m} = R,$ where $C P,m$ is the molar heat capacity at constant pressure, $C V,m$ is the molar heat capacity at constant volume and $R$ is the gas constant.

General form

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

From this latter relation, several inferences can be made:

Since the isothermal compressibility $β_{T}$ is positive for nearly all phases, and the square of thermal expansion coefficient $α$ 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: $C_{P, m} \geq C_{V, m} .$ 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 $C_{P, m} < C_{V, m}$ .
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

↑ Çengel, Yunus A.; Boles, Michael A. Thermodynamics: an engineering approach (7th ed.). New York: McGraw-Hill. ISBN 0-07-736674-3.
↑ 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.

[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.