Symmetry numbers factor

From Equation 4.79, it is then recognized that the isotope effect is given by a symmetry number factor and terms which depend only on the normal mode vibrational frequencies. There are no terms in the equality that depend explicitly on atomic and molecular masses or on moments of inertia. 

Isotope effects on equilibria have been formulated earlier in this chapter in terms of ratios of (s2/si)f values, referred to as reduced isotopic partition function ratios. From Equation 4.80, we recognize that the true value of the isotope effect is found by multiplying the ratio of reduced isotopic partition function ratios by ratios of s2/si values. Using Equation 4.116 one now knows how to calculate s2/si from ratios of factorials. Note well that symmetry numbers only enter when a molecule contains two or more identical atoms. Also note that at high temperature (s2/si)f approaches unity so that the high temperature equilibrium constant is the symmetry number factor. 

In the following an explicit mathematical proof is presented to show that symmetry numbers factors do not lead to isotope enrichment. That result should come as no surprise since the factor on the right hand side of Equation 4.118 can be identified as... 

The symmetry number factors are derived from the reduced isotopic partition function ratio of the RHt species. 

It should be emphasized that the symmetry number ratio (s2/si) is entered as a multiplier of (s2/si)f so that the symmetry number factors which lead to no isotopic enrichment in themselves are left out. To obtain the complete isotope effect one has to multiply the above expression by symmetry number ratios so that the symmetry number ratios in front of the f expressions are removed. So the symmetry number factor in Equations 4.143 and 4.144 is given by... 

The vibrational sum rule (Equation 4.99) applies to transition states even when one of the frequencies is imaginary (and if is negative for that frequency). In that case one finds for ki /k2, with omission of the symmetry number factor, the analogue of Equation 4.105 for the exchange equilibrium constant... 

Ka/b so defined is not a true equilibrium constant but one in which the symmetry-number factor has been omitted (the true Ka/b = /a//b), since the purely classical symmetry numbers cannot lead to isotopic fractionation (7). Ka/b, as defined by Equation 7, might more properly be called a reduced equilibrium constant. 

Estimates of maximum regular and inverse isotope effects may be made quite simply. For simplicity of notation the classical symmetry number factors will be omitted. In order to arrive at maximal values of regular isotope effects, we assume no isotope effect in the activated complex, i.e., /+ = 1. This assumption is equivalent to no isotopic binding in the activated complex. The maximum value of jv2L is (m2/m1) /l, where the m s are the masses of the isotopic atoms. Values of / for various isotopic species have been tabulated in the literature.49 Using the values of / for strongly bonded species, one may then estimate maximum values of isotope effects. These are tabulated in Table I which is... 

Greiff approximation to the partition function ratio, Qj /Q2 Eq. (8), and eliminate the symmetry number factor, then... 

The factor of 2 in the denominator of the H2 molecule s rotational partition function is the "symmetry number" that must be inserted because of the identity of the two H nuclei. 

Thus the factor 1/2 in Equation 4.109, above, arises naturally in the high temperature (classical) limit and is just the reciprocal of the symmetry number of the homonu-clear diatomic molecule. 

From the quantum mechanical standpoint the appearance of the factor 1/2 = 1/s for the diatomic case means the configurations generated by a rotation of 180° are identical, so the number of distinguishable states is only one-half the classical total. Thus the classical value of the partition function must be divided by the symmetry number which is 1 for a heteronuclear diatomic and 2 for a homonuclear diatomic molecule. 

Equation 4.117 makes complete sense. One of the first things one learns in dealing with phase space integrals is to be careful and not over-count the phase space volume as has already been repeatedly pointed out. In quantum mechanics equivalent particles are indistinguishable. The factor n ni is exactly the number of indistinguishable permutations, while A accounts for multiple minima in the BO surface. It is proper that this factor be included in the symmetry number. Since the BO potential energy surface is independent of isotopic substitution it follows that A is also independent of isotope substitution and cannot affect the isotopic partition function ratio. From Equation 4.116 it follows... 

Electronic absorption spectra have been recorded for a large number of oxomolybdenum(IV) species but, in addition to d-d transitions, there is the possibility of two-electron excitations, charge transfer bands and splitting of degenerate energy levels due to low symmetry. These factors make it difficult to interpret or even to compare spectra.5... 

We now refer to Appendix 1 to write the partition functions in terms of their translational, rotational, and vibrational components. Of the quantities appearing in the expressions for these components, only the molecular mass M, the moments of inertia /, the vibrational frequencies uu and the symmetry numbers a are different for the isotopic molecules all other factors cancel, leaving Equation A2.10. 

In connection with transition-state theory, one will also occasionally meet the concept of a statistical factor [13]. This factor is defined as the number of different activated complexes that can be formed if all identical atoms in the reactants are labeled. The statistical factor is used instead of the symmetry numbers that are associated with each rotational partition function (see Appendix A.l) and, properly applied, the... 

The values of Kn for the systems involving monodentate ligands, summarized in Table IV, can be compared with one another after an appropriate correction for the statistical factor in each reaction. This correction is made by considering the symmetry numbers of reactant (crR) and product species ([Pg.143]

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