Statistical Mechanics of Isotopic Systems with Small Quantum Corrections. I. General Considerations and the Rule of the Geometric Mean

Abstract

It is shown that the differences in the thermodynamic properties of isotopic molecules subject to small quantum effects (u²/24 law) depend on the difference in the reciprocal masses of the atoms in the molecule and are, therefore, independent of all masses except of those atoms isotopically substituted. This theorem provides a rigorous proof of the rule of the geometric mean for gaseous molecules. It is shown that the partition function ratio for a pair of double‐labeled molecules, e.g., N¹⁵D₃/N¹⁵H₃ is equal to the ratio for the single‐labeled pair N¹⁴D₃/N¹⁴H₃. The application of the u²/24 law to isotopic isomer equilibria is pointed out.