Mechanistic rate constant

What is one to make of the 1975 experiments275 277 278 leading to the conclusions that kI2 (k2 + k2 ) = 4 1 and kx kn- 0 They and the 1984 work which reported experimental values for all four mechanistic rate constants and the ratios ka (k2 + /c23) = 0.23 and k, kl2= 1.8 cannot both be correct. One response is to remain objective and quite neutral, and to consider the experimental situation simply unresolved. Another is to try to make an independent assessment of the two sets of experiments by critically evaluating the experimental designs, the synthetic procedures and characterizations of labeled phenylcyclo-propanes, the analytical methodologies employed and the number of experimental rate... 

The kinetic situations posed by the l,2-d2-cyclopropanes is summarized in Scheme 6. The experimentally accessible rate constants for overall loss of optical activity, k and for approach to cis,trans equilibrium, kh are related to the mechanistic rate constants by the equalities k, = (2ka + 2k) and k, = 4k, with k = (k, + k + kn). Here kt stands for the one-center epimerization rate constant when C(l) C(2) breaks, and for one-center turnover at C(l) when C(l)—C(3) breaks. There are thus two observables and four mechanistic rate constants. If kt kt and kn kn ratios are assigned reasonable values based on assumed kinetic isotope effects, one is left with only kx and kn as unknown mechanistic rate constants to be deduced from the observable k, and kr kinetic parameters. 

We shall use the representative mechanism in Equation 4.7 comprising three unimolecular steps to illustrate their application each elementary step is assigned a mechanistic rate constant ... 

Thus, if the assumptions are sound, a first-order rate law will be observed and the experimentally observed first-order rate constant may be equated with the mechanistic rate constant of the first step, ka bs = k. In this event, the overall rate of reaction is effectively controlled by the first step, and this is known as the rate-determining or rate-limitingr) step of the reaction. 

As long as the SSA is valid for the mechanism in Equation 4.7, but regardless of whether it either involves a pre-equilibrium or proceeds via an initial rate-limiting step (or neither), the same prediction is obtained - a first-order rate law will be observed. However, the correspondence between the measured first-order rate constant, k0 iil and mechanistic rate constants is different, and additional evidence is required to distinguish between the alternatives. 

It should be stressed that whilst these two mechanistic rate equations have the same form, the molecular processes they are derived from are different, and the respective second-order rate constants (corresponding to the mechanistic rate constants k and k4 for bimolecular processes ri and r4 in Scheme 4.6) extracted from the experimental results differ by three orders of magnitude, as shown in Table 4.3. The more credible of the two alternatives, in... 

The mechanism shown in Scheme 4.9 has been proposed for the hydrogen atom transfer from phenols (ArOH) to radicals (Y ) in non-aqueous solvents, a kinetic effect ofthe solvent (S) being expected when ArOH is a hydrogen bond donor and the solvent a hydrogen bond acceptor. Steps with mechanistic rate constants k, k-1 and k>, involve proton transfer (the latter two near to the diffusion-controlled limit), and kj involves electron transfer. The step with rate constant fco involves a direct hydrogen atom transfer, and the other path around the cycle involves a stepwise alternative. 

In buffered solutions, the term k2KsKi [S]/[SH+] is constant, so the expected overall rate law is again second order (i.e. pseudo first order in [Y ]) but the correspondence of fcQbs with mechanistic rate constants is different. Of course, if the equilibrium constant Ki is appreciable, the phenolate concentration must be taken into account in the mass balance for the total phenol, i.e. [ArOH]T [ArOH]free + [ArOH- -S] + [ArO-], whereupon the mechanistic rate equation becomes more complicated. 

Herzfeld and Langmuir-Hinshelwood-Hougen-Watson cycles, could be formulated and solved in terms of analytical rate expressions (19,53). These rate expressions, which were derived from mechanistic cycles, are phrased, however, in terms of the formation and destruction of molecular species without the need for computing the composition of reactive intermediates. Thus, these expressions are the relevant kinetics required for molecular models and are rooted to the mechanistic cycles only implicitly by the mechanistic rate constants. The molecular model, in turn, transforms a vector of reactant molecules into a vector of product molecules, either of which is susceptible to thermodynamic analysis. This thermodynamic analysis helps to organize these components into relevant boiling point or solubility product classes. Thus the sequence of mechanistic to molecular to global models is intact. 

Evaluation of mechanistic rate constants k3 and k4 from kobs is rather complex the full derivation is described in Ref. 23. The full rate expression for formation of product (SO) is given by Eq. (3)18 23 ... 

Person and coworkers followed this study with a related analysis of the stereomutation of phenylcyclopropane-2,3-d2. In this system one can separate 2 from 23 cannot separate ki from ki2- However the two studies together would, in principle, allow complete determination of all of the mechanistic rate constants. In the event, experimental difficulties did not allow this complete dissection of the rate constants. The study did, however, provide an independent measure of /cj, which was found here to be 0.15 x 10" s" again showing correlated double rotation to be by far the dominant process for stereomutation at C(l). 

In fact the results are not quite as dramatically different as they appear when presented this way. The mechanistic rate constants are derived from phenomenological rate constants that can be described as linear combinations of the mechanistic ones. The discrepancy between the two results can be traced to a difference in just one of the phenomenological rate constants—the one corresponding to the first order loss of optical activity for a 1 1 mixture of (1R,25)- and (lR,2R)-phenylcyclopropane-2-d. The peculiar feature of this problem is that the Berson group obtained an internally consistent set of results from two independent experiments with one value for this rate constant and the Baldwin group also obtained an internally consistent set from two independent experiments but with a value that differed by a factor of 2.7 from that found by the Berson group. It is hard to know how to reconcile such results and so, to be objective, one must probably say that despite the immense amount of effort put into the problem the mechanism of stereomutation of phenylcyclopropane remains something of a mystery. 

The change in concentration of the four species in Figure 16 as a function of time could be fit by assigning the following values to the mechanistic rate constants (at 242. r C) ... 

The three phenomenological rate constants, having the indicated experimental values, can be equated with combinations of mechanistic rate constants as follows ... 

As usual, the singly subscripted rate constants refer to one-center epimerizations and the doubly subscripted rate constants refer to two-center epimerizations at the carbons indicated by the numbers. There should, in principle, be a secondary deuterium isotope effect included in the mechanistic rate constants comprising but this would have its largest effect on k 2 and 23 and the previous experience in cyclopropane stereomutations suggests that these are likely to be minor contributors at best. 

The experimental ilc computed from Eq. (21) can be related to the mechanistic rate constants. In general, the relationship is very complex, but in some limiting cases the correspondance of the experimental and mechanistic rate constants can be made by inspection. For example, in the dissociative mechanism when CrL is formed but is negligible, k computed from Eq. (21) is ka- In an associative mechanism when CrLJ is not reformed by geminate recombination, = k while in a concerted mechanism Kx = rp + rp- However when geminate recombination reforms CrLJ, the excited state decay is nonexponential. At long times is reduced by the recombination and k is a more complicated function of the several rate constants that appear in the mechanistic description. 

Rate coefficients referring to (or believed to refer to) elementary reactions are called rate constants or, more appropriately, microscopic (hypothetical, mechanistic) rate constants. 

As in all rate expressions, there are no known cases where a mechanistic experimental activation energy is negative. Although the observation of a negative temperature coefficient for the rate of the overall reaction is possible (see Chapter 11), the observation of a negative activation energy for a mechanistic rate constant is not. 

While the mechanistic rate constants remain indeterminant, limits on their range are provided by the experimental rate constants, and it was argued that the predominant stereomode for utilization of the allylic framework must be antarafacial, with both retention and inversion at C3. The reason for this unusual stereochemistry arises from the fact that is twice as large as and since could only arise by epimerization at C2 or C3, there could be little contribution of... 

---