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

$$C\_\{P,\backslash mathrm\{m\}\}\; -\; C\_\{V,\backslash mathrm\{m\}\}\; =\; R,$$

where {{Math|C{{sub|P,m}}}} is the molar heat at constant pressure, {{Math|C{{sub|V,m}}}} is the molar heat at constant volume and {{Math|R}} is the gas constant.

For more general homogeneous substances, not just ideal gases, the difference takes the form,

$$C\_\{P,\backslash mathrm\{m\}\}\; -\; C\_\{V,\backslash mathrm\{m\}\}\; =\; V\_\{\backslash mathrm\{m\}\}\; T\; \backslash frac\{\backslash alpha\_V^2\}\{\backslash beta\_\{T\}\}$$

(see relations between heat capacities), where $V\_\{\backslash mathrm\{m\}\}$ is the molar volume, $T$ is the temperature, $\backslash alpha\_\{V\}$ is the thermal expansion coefficient and $\backslash beta$ is the isothermal compressibility.

From this latter relation, several inferences can be made:{{cite book | last2=Boles|first1=Yunus A. | last1 = Çengel | first2 = Michael A.|title=Thermodynamics: an engineering approach| publisher=McGraw-Hill | location=New York | isbn=0-07-736674-3 |edition=7th}}

• Since the isothermal compressibility $\backslash beta\_\{T\}$ is positive for nearly all phases, and the square of thermal expansion coefficient $\backslash alpha$ 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: $$C\_\{P,\backslash mathrm\{m\}\}\; \backslash geq\; C\_\{V,\backslash mathrm\{m\}\}.$$ There are no known exceptions to this principle for gases or liquids, but certain solids are known to exhibit negative compressibilities {{cite journal |last1=Anagnostopoulos |first1=Argyrios |last2=Knauer |first2=Sandra |last3=Ding |first3=Yulong |last4=Grosu |first4=Yaroslav |title=Giant Effect of Negative Compressibility in a Water–Porous Metal–CO2 System for Sensing Applications |journal=ACS Applied Materials and Interfaces |date=2020 |volume=12 |issue=35 |page=35 |doi=10.1021/acsami.0c08752 |s2cid=221200797 |url=https://pubs.acs.org/doi/10.1021/acsami.0c08752 |access-date=26 March 2022}} and presumably these would be (unusual) cases where .
• For incompressible substances, {{Math|C{{sub|P,m}}}} and {{Math|C{{sub|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 {{Math|C{{sub|P,m}}}} and {{Math|C{{sub|V,m}}}} also approaches zero. Exceptions to this rule might be found in systems exhibiting residual entropy due to disorder within the crystal.