Isotope effect theory

It is important to point out here, in an early chapter, that the Born-Oppenheimer approximation leads to several of the major applications of isotope effect theory. For example the measurement of isotope effects on vapor pressures of isotopomers leads to an understanding of the differences in the isotope independent force fields of liquids (or solids) and the corresponding vapor molecules with which they are in equilibrium through use of statistical mechanical theories which involve vibrational motions on isotope independent potential functions. Similarly, when one goes on to the consideration of isotope effects on rate constants, one can obtain information about the isotope independent force constants which characterize the transition state, and how they compare with those of the reactants. 

The theory of isotope effects is well established and has been presented in detail in the books by Collins and Bowman (1970), Melander (1960), Melander and Saunders (1980) and Willi (1983). Only some general principles of isotope effects on chemical equilibria are presented here mainly to introduce the formulations and parlance of isotope effect theory. A frame of reference is given which is intended to allow the interpretations of the isotope effect studies to be followed with regard to the specific problems described. 

A. Application of Isotope Effect Theory to Secondary Isotope Effects 123... 

In the particular reaction used as an illustration, there are one primary and two secondary isotope effects and a good deal of ancillary kinetic evidence to fall back on. Generally, however, the burden of uncertainty is much too great for a minute and not always a very precisely determined secondary isotope effect to bear. Small wonder then that many workers in the field have preferred to lash themselves firmly to the formal framework of isotope effect theory After a model for the transition state is assumed, bond lengths and angles can be altered, force constants increased and decreased ad libitum, until the experimental result is more or less comfortably accommodated in the Procrustean bed of theory. How much useful mechanistic information is derived by this procedure is open to question. 

All of these properties are affected by isotopic substitution, and any practitioner of what LeflEler (16) calls molecular psychology is intuitively certain that isotope effects on them must somehow be related to chemical isotope effects— whether the formal relation to isotope effect theory is immediately obvious or not. 

The chemical implications of this conclusion are self-evident and far-reaching. The next step must be to determine to what extent the idea of electronic isotope effects can be reconciled with the formal framework of isotope effect theory. Then we must consider the experimental evidence on equilibria and kinetics, which is much more extensive than that on physical properties, to see to what degree and in what direction our provisional ideas must be modified or extended. 

Isotope effect theory has been worked out in detail, and kinetic isotope effects have been reviewed authoritatively, from two some-... 

Eyring (56) is of the opinion that transition state theory in its present form cannot—nor was it intended to— confidently handle rate differences of the magnitude encountered in secondary isotope effects. Still, if these are to be treated formally at all, it must be done in the framework of kinetic isotope effect theory—which is all we have. Fortunately, to use the words of Bigeleisen and Wolfsberg (49a) the effect of isotopes on the various quantities can often be predicted with more reliability than the quantities themselves, so that one might try to predict quantitative isotope effects for reactions so complex that quantitative predictions of fc would not be attempted. ... 

There is no reason why the relative effects of CDa and CH3 cannot be discussed in the terms just employed for those of methyl vs. tert-butyl. Now of course, the equivalent of Figure 6 would not be a real potential energy curve, but would show how the cu>erage potential energy varies with the reaction coordinate. For the formalist s peace of mind, it is sufficient to recall that the considerations of Sec. IIIB allow translation into the language of isotope effect theory, in principle if not always in practice. 

Logically, the experimental results which should be presented first are secondary isotope effects in the vapor phase, or at least in nonpolar solvents. Instead we shall defer these to Sec. IVB, and begin with acid-base equilibria in that most polar of solvents—water. The principal reason for adopting this sequence is that the secondary isotope effect on the ionization of formic acid is the only one to date that has been rigorously evaluated in terms of formal isotope effect theory. As such it provides a natural transition between the theoretical considerations that have concerned us up to now and the empirical approach that seems to be the most profitable way of dealing with the more complicated systems that will occupy us for the rest of the chapter. 

We now carry the argument over to transition state theory. Suppose that in the transition state the bond has been completely broken then the foregoing argument applies. No real transition state will exist with the bond completely broken—this does not occur until the product state—so we are considering a limiting case. With this realization of the very approximate nature of the argument, we make estimates of the maximum kinetic isotope effect. We write the Arrhenius equation for the R-H and R-D reactions... 

A more rigorous theory of kinetic isotope effects begins with the transition state equation k = (kTlh)K. Writing this for and ito leads to... 

Values of kH olki3. o tend to fall in the range 0.5 to 6. The direction of the effect, whether normal or inverse, can often be accounted for by combining a model of the transition state with vibrational frequencies, although quantitative calculation is not reliable. Because of the difficulty in applying rigorous theory to the solvent isotope effect, a phenomenological approach has been developed. We define <[), to be the ratio of D to H in site 1 of a reactant relative to the ratio of D to H in a solvent site. That is. 

Collisions at low ion energies (where Equation 1 can be applied) lead to a short-lived complex between the ion and the molecule—i.e., both collision partners move with the same linear velocity in the direction of the incident ion. The decay of the complex may be described by the theory of unimolecular rate processes if its excess energy can fluctuate between the various internal degrees of freedom. For example, the isotope effect in the reaction of Ar+ with HD may be explained by the properties of... 

Intramolecular Isotope Effects. The data in Figure 2 clearly illustrate the failure of the experimental results in following the predicted velocity dependence of the Langevin cross-section. The remark has been frequently made that in the reactions of complex ions with molecules, hydrocarbon systems etc., experimental cross-sections correlate better with an E l than E 112 dependence on reactant ion kinetic energy (14, 24). This energy dependence of reaction presents a fundamental problem with respect to the nature of the ion-molecule interaction potential. So far no theory has been proposed which quantitatively predicts the E l dependence, and under these circumstances interpreting the experiment in these terms is questionable. 

In the following, a detailed exposition of Bartell s (1961a) theory of steric isotope effects will be given (Section II, A), and an alternative model will be developed, based on somewhat different assumptions about the timing in the transition state, which leads to predictions at variance with the experimental results (Section II, B). In both of these sub-sections, special reference will be made to the work of Melander and Carter (1964). Finally, a selective non-comprehensive review of other experimental work in this field will be presented (Section III). 

Solvolytic experiments specifically designed to test Bartell s theory were carried out by Karabatsos et al. (1967), who were primarily interested in an assessment of the relative contributions of hyperconjugation and non-bonded interactions to secondary kinetic isotope effects. Model calculations of the (steric) isotope effect in the reaction 2- 3 were performed, as well as that in the solvolyses of acetyl chloride... 

In apparent contradiction with Bartell s (1960, 1961a, b) theory, Heitner and Leffek (1966) reported the absence of an isotope effect in... 

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