Bigeleisen-Mayer formulation

Bell and Crooks (85) specifically taking all the vibrational frequencies of the four molecules, shown in Table IV, into account, calculate a theoretical value of AAP° = 51 cal., in excellent agreement with experiment. Their calculation was carried out once directly with eq. (III-5), into which translational, rotational, and vibrational partition functions were explicitly introduced, and once with eq. (III-6). The results of both modes of calculation were virtually identical, a fact that offers strong support for the validity of the assumptions underlying the Bigeleisen-Mayer (3) formulation, even when dealing with effects as small as this. 

Introducing the masses of exchanging isotopes (m and m°, respectively), the ratio of partition functions for crystalline components can be related to that of the primitive unit cell (Kieffer, 1982), thus defining reduced partition function ratio / (whose formulation is equivalent to that obtained by Bigeleisen and Mayer, 1947, and Urey, 1947, for gaseous molecules) ... 

The theory of IEs was formulated by Bigeleisen and Mayer.9 The IE on the acid-base reaction of Equation (1) is defined as the ratio of its acidity constant KA to the acidity constant of the isotopic reaction, Equation (2). The ratio KJ KA is then the equilibrium constant XEIE for the exchange reaction of Equation (3). That equilibrium constant may be expressed in terms of the partition function Q of each of the species, as given in Equation (4), which ignores symmetry numbers. 

This phenomenon forms the basis for the formulations of Urey (1947) and Bigeleisen and Mayer (1947) for the temperature dependence of isotopic exchange between two molecules. With the nearly simultaneous development of the isotope-ratio mass spectrometer by Nier et al. (1947), the potential for application of stable isotopes was created. Other isotopic fractionation processes are observed in kinetics, diffusion, evaporation-condensation, crystallization, and biology (e.g., photosynthesis, respiration, nitrogen fixation, sulfate reduction, and transpiration). The concomitant isotopic fractionations can also be used to provide details of the relevant process. 

The theory of calculation of ratios of isotopic partition functions has been developed in detail for exchange equilibria in gases. The formulation of Bigeleisen and Goeppert-Mayer,10 which writes everything in terms of the vibrational frequencies of the molecules, will be reviewed. Classically QAJQAx is where the tn/s are... 

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