RPFRs reduced partition function ratios

In Section 4.8, Equations 4.78,4.79 and Table 4.1 develop the connections between the harmonic oscillator rigid rotor partition function and isotope chemistry as expressed by the reduced partition function ratio, RPFR = (s/s ) f. RPFR is defined in Equation 4.79 as the product over oscillators of ratios of the function [u exp(—u/2)/ (1 - exp(u))]... 

It has been previously noted that the first quantum correction to the classical high temperature limit for an isotope effect on an equilibrium constant is interesting. Each vibrational frequency makes a contribution c[>(u) to RPFR and this contribution can be expanded in powers of u with the first non-vanishing term proportional to u2/24, the so called first quantum correction. Similarly, for rates one introduces the first quantum correction for the reduced partition function ratios, includes the Wigner correction for k /k2 and makes use of relations like Equation 4.103 for small x and small y, to find a value for the rate constant isotope effect (omitting the noninteresting symmetry number term)... 

In a system conlaining condensed and gases phases, Bigeleisen derived Ihe relationship between the reduced partition function ratio (RPFR), /, and the measured... 

Equations (1) and (2) refer to measurements on separated Isotopes. It is also Important to develop a relation between the reduced partition function ratio (RPFR) and measured separation factors, a, referring to single stage Isotopic enrichments (on distillation) for dilute nonideal solutions. In the case of infinite dilution the molar volumes of the two Isomers are Identical (they are determined by environment) and one obtains (19)... 

In the calculation of equilibrium constants and in the theory of isotope effects, the so-called reduced partition function ratio (RPFR) introduced by Bigeleisen and Mayer (1947) plays an important role. The RPFR (/) is the quantum mechanical partition function divided by the classical partition function, and in the case of an ideal gas is given by the expression... 

Equation 4.79a points out the Reduced Isotopic Partition Function Ratio (RPFR) may be considered as the product of three factors the product factor (PF), the excitation factor (EXC), and the zero-point energy factor (ZPE). Note that in terms of RPFR s, the isotope effects corresponding to Equations 4.65, 4.66, and 4.68 can be written... 

In earlier sections of this chapter we learned that the calculation of isotope effects on equilibrium constants of isotope exchange reactions as well as isotope effects on rate constants using transition state theory, TST, requires the evaluation of reduced isotopic partition function ratios, RPFR s, for ordinary molecular species, and for transition states. Since the procedure for transition states is basically the same as that for normal molecular species, it is the former which will be discussed first. 

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