Preequilibrium

The first situation corresponds to the first step being rate-determining. In the second case, it is the second step that is rate-determining, with the first step being a preequilibrium. 

The product ratio is therefore determined not by AG but by the relative energy of the two transition states A and B. Although the rate of the formation of the products is dependent upon the relative concentration of the two conformers, since AGJ is decreased relative to AG to the extent of the difference in the two conformational energies, the conformational preequilibrium is established rapidly, relative to the two competing... 

Solvent isotope effects are usually in the range / h20+ = 2-3. These values reflect the greater equilibrium acidity of deuterated acids (Section 4.5) and indicate that the initial protonation is a fast preequilibrium. 

The existence of the nitronium ion in sulfuric-nitric acid mixtures was demonstrated both by cryoscopic measurements and by spectroscopy. An increase in the strong acid concentration increases the rate of reaction by shifting the equilibrium of step 1 to the right. Addition of a nitrate salt has the opposite effect by suppressing the preequilibrium dissociation of nitric acid. It is possible to prepare crystalline salts of nitronium ions, such as nitronium tetrafluoroborate. Solutions of these salts in organic solvents rapidly nitrate aromatic compounds. ... 

A system of this type is commonly said to possess a fast preequilibrium step. Proton transfers constitute a veiy important class of fast preequilibria, as illustrated by Scheme XVII for acid-catalyzed ester hydrolysis. 

Figure 3-9. Companson of exact solutions and the preequilibrium assumption for Seheme XIV. For all of the calculations the values ki = 10 s", A i = 5 s were used. The values are given in the figure. The exact solutions were obtained with Eq. (3-135) and the approximate solution (corresponding to = 0) with Eq. (3-136).

Figure 3-9 shows plots of Eqs. (3-135) and (3-136) for some hypothetical systems. Obviously the equilibrium approximation is poor in the early stages of the reaction, but in the later stages the assumption can be quite good. The preequilibrium assumption, applied to Scheme XIV, amounts to the statement that 2 is negligible relative to ki and i. 

The time scale of a phenomenon whose characteristic quantity has the units seconds is given by the reciprocal of that quantity. In the present context, therefore, we can state that the preequilibrium assumption is valid if the A B reaction is. very fast on the time scale (1/ 2) of the B — C reaction. 

Further discussion of the preequilibrium assumption is given in the next subsection. 

In the preceding subsection we described the preequilibrium assumption. Let us now see how that assumption is related to the steady-state approximation. Scheme XIV will serve for the discussion. The equilibrium and steady-state expressions for the intermediate concentration are... 

It, therefore, appears that the equilibrium approximation is a special case of the steady-state approximation, namely, the case i > 2- This may be, but it is possible for the equilibrium approximation to be valid when the steady-state approximation is not. Consider the extreme but real example of an acid-base preequilibrium, which on the time scale of the following slow step is practically instantaneous. Suppose some kind of forcing function were to be applied to c, causing it to undergo large and sudden variations then Cb would follow Ca almost immediately, according to Eq. (3-153). The equilibrium description would be veiy accurate, but the wide variations in Cb would vitiate the steady-state description. There appear to be three classes of practical behavior, as defined by these conditions ... 

An important special case can be derived from this general result. This is the case of a fast preequilibrium, in which the A + B AB system rapidly equilibrates, the AB C step being much slower. Then the relaxation time for the first step is much shorter than that for the second, and some measure of uncoupling takes place. For such a system k,2, 21 23- 32, and we obtain Cn = k 12, a,2 = /c2, 021 = k, 2, 022 = 21- Equations (4-22) then give Tf = k, 2 + 21 and th 0. Because these are approximations, the result for ti is reasonable but that for Tn is not. To reach a reasonable result for Tn we use Eq. (4-24a),... 

In the first step the hydrated ion and ligand form a solvent-separated complex this step is believed to be relatively fast. The second, slow, step involves the readjustment of the hydration sphere about the complex. The measured rate constants can be approximately related to the constants in Scheme IX by applying the fast preequilibrium assumption the result is k = Koko and k = k Q. However, the situation can be more complicated than this. - °... 

The appearance of Cd in the denominator means that D is coupled to a reversible step prior to the rds. If k, and k- were so large that the fast preequilibrium assumption is valid, then the Cd term in the denominator would drop out, and we would have v = A 2 CaObCd, giving the composition of the rds transition state. If k2 is very much larger than it, and k i, Eq. (5-59) becomes v = A CaOb the first step is now the rds, and the rate equation gives the transition state composition. 

If the rate equation contains the concentration of a species involved in a preequilibrium step (often an acid-base species), then this concentration may be a function of ionic strength via the ionic strength dependence of the equilibrium constant controlling the concentration. Therefore, the rate constant may vary with ionic strength through this dependence this is called a secondary salt effect. This effect is an artifact in a sense, because its source is independent of the rate process, and it can be completely accounted for by evaluating the rate constant on the basis of the actual species concentration, calculated by means of the equilibrium constant appropriate to the ionic strength in the rate study. 

Al. A fast preequilibrium protonation of substrate followed by a slow ratedetermining reaction of the protonated substrate. Subsequent steps (such as attack by water) are fast. 

A2. A fast preequilibrium protonation followed by a slow rate-determining attack by nucleophile. 

Thermal isomerization and decomposition reaction of indole behind reflected shoek waves at 1050-1650 K have been explained by a preequilibrium between 17/-indole and 3//-indole (97JPC(A)7787). 

The reaction is less selective than the related benzoylation reaction (/pMe = 30.2, cf. 626), thereby indicating a greater charge on the electrophile this is in complete agreement with the greater ease of nuclophilic substitution of sulphonic acids and derivatives compared to carboxylic acids and derivatives and may be rationalized from a consideration of resonance structures. The effect of substituents on the reactivity of the sulphonyl chloride follows from the effect of stabilizing the aryl-sulphonium ion formed in the ionisation step (81) or from the effect on the preequilibrium step (79). 

Instead or in addition, the reactants A and B may associate in a fast preequilibrium, or one of them may bind to a third component. Such interactions will exert important rate effects and for that reason must be accounted for. The existence of equilibria in which the reactants participate may translate to an important effect on the chemical mechanism or to a trivial one. Either way, the issue must be addressed to arrive at a reliable mechanism. The matter can grow complicated, in that the concentration variables that affect the rate may do so because they really do enter the mechanism, or because they participate in extraneous equilibria. Of course, they may play both roles. Sorting out these matters sounds complicated, but it is not difficult if one proceeds systematically. 

The kinetics of enzyme reactions were first studied by the German chemists Leonor Michaelis and Maud Menten in the early part of the twentieth century. They found that, when the concentration of substrate is low, the rate of an enzyme-catalyzed reaction increases with the concentration of the substrate, as shown in the plot in Fig. 13.41. However, when the concentration of substrate is high, the reaction rate depends only on the concentration of the enzyme. In the Michaelis-Menten mechanism of enzyme reaction, the enzyme, E, and substrate, S, reach a rapid preequilibrium with the bound enzyme-substrate complex, ES ... 

A mechanism accommodating these data has been proposed the preequilibrium (41) is rapidly attained and is followed by... 

Comments on the thermal nitration of enol silyl ethers with TNM. The strikingly similar color changes that accompany the photochemical and thermal nitration of various enol silyl ethers in Table 2 indicates that the preequilibrium [D, A] complex in equation (15) is common to both processes. Moreover, the formation of the same a-nitroketones from the thermal and photochemical nitrations suggests that intermediates leading to thermal nitration are similar to those derived from photochemical nitration. Accordingly, the differences in the qualitative rates of thermal nitrations are best reconciled on the basis of the donor strengths of various ESEs toward TNM as a weak oxidant in the rate-limiting dissociative thermal electron transfer (kET), as described in Scheme 4.40... 

The electron-transfer paradigm in Scheme 1 (equation 8) is subject to direct experimental verification. Thus, the deliberate photoactivation of the preequilibrium EDA complex via irradiation of the charge-transfer absorption band (/ vCT) generates the ion-radical pair, in accord with Mulliken theory (equation 98). 

A more detailed examination shows that, in case of equilibrium approximation, the value of fCM corresponds to the inverse stability constant of the catalyst-substrate complex, whereas in the case of the steady-state approach the rate constant of the (irreversible) product formation is additionally included. As one cannot at first decide whether or not the equilibrium approximation is reasonable for a concrete system, care should be taken in interpreting KM-values as inverse stability constants. At best, the reciprocal of KM represents a lower limit of a stability constant In other words, the stability constant quantifying the preequilibrium can never be smaller than the reciprocal of the Michaelis constant, but can well be significantly higher. 

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