Mayer's relation explained

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 thatC_ - C_ = R,where is the molar specific heat at constant pressure, is the molar specific heat at constant volume and is the gas constant.

For more general homogeneous substances, not just ideal gases, the difference takes the form,C_ - C_ = V_ T \frac(see relations between heat capacities), where

Vm

is the molar volume,

T

is the temperature,

\alphaV

is the thermal expansion coefficient and

\beta

is the isothermal compressibility.

From this latter relation, several inferences can be made:[1]

\betaT

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_ \geq C_. 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

CP,m<CV,m

.

Notes and References

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