Limiting behavior of partition functions of models with quadratically dependent energy spectra and consequences regarding theoretical study of chemical reactivity

Abstract

The conventional partition functions of models with quadratically dependent energy spectra ( e . g . , translation in a box or free internal rotation), which are widely used in literature, exhibit an incorrect limit behavior in the region of low temperatures and/or small values of the characterizing molecular parameter. In particular, this leads to the absurd value -2 for the contribution of these motions to the entropy term at these limit conditions. This incorrect behavior appears to be due to the replacement of rigorous summation by integration in conventional derivation of these partition functions. The derived improved practical formulas with the correct limit behavior indicate that it will be necessary to carry out quite substantial nunierical corrections in the case of many reactivity characteristics reported in the literature. Particularly, the improved formulas should be applied with (small) molecular complexes at the typical temperatures of their observation, e.g., van der Waals molecules with free internal rotation. Simultaneously discussed is the possibility of application of the functional relation valid for a rigorous partition function of the considered type of energy spectrum. The reasoning in the case of the translational partition function indicates that, in principle, it can depend on the vessel shape. Possible consequences for thermodynamics of the ideal gas are analyzed.