The rate-determining step approximation

When k2 Pt k, in the mechanism of Equation 4.7, i.e. the intermediate B proceeds to the product C much faster than it reverts to the reactant A, the predicted rate law of Equation 4.9 

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. 

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The next problem of the Langmuir-Hinshelwood kinetics, the validity of the rate-determining step approximation, has not been rigourously examined. However, as has been shown (e.g. refs. 57 and 63), the mathematical forms of the rate equations for the Langmuir-Hinshelwood model and for the steady-state models are very similar and sometimes indistinguishable. This makes the meaning of the constants in the denominators of the rate equations somewhat doubtful in the Langmuir—Hinshelwood model, they stand for adsorption equilibrium constants and in the steady-state models, for rate coefficients or products and quotients of several rate coefficients. 

The rate-determining step approximation is made to determine a rate law for a mechanism in which one step occurs at a rate substantially slower than any others. The slow step is a bottleneck, and the overall reaction rate cannot be larger than the slow step. The rate law for the rate-determining step is written first. If a reaction intennediate appears as a reactant in this step, its concentration term must be eliminated from the rate law. The final rate law only has concentration terms for reactants and products. 

The rate-determining step approximation is made to determine a rate law for a mechanism in which one step occurs at a rate substantially slower than any others. 

Finally, reaction of hydroxide with 29 furnishes 31 approximately 20 times faster than methoxide promoted rearrangement of 29 to 23 (R = Ph), suggesting ring closure is the rate-determining step. 

In this approximation we assume that one elementary step determines the rate while all other steps are sufficiently fast that they can be considered as being in quasi-equilibrium. If we take the surface reaction to AB (step 3, Eq. 134) as the rate-determining step (RDS), we may write the rate equations for steps (1), (2) and (4) as ... 

It is important to realize that the assumption of a rate-determining step limits the scope of our description. As with the steady state approximation, it is not possible to describe transients in the quasi-equilibrium model. In addition, the rate-determining step in the mechanism might shift to a different step if the reaction conditions change, e.g. if the partial pressure of a gas changes markedly. For a surface science study of the reaction A -i- B in an ultrahigh vacuum chamber with a single crystal as the catalyst, the partial pressures of A and B may be so small that the rates of adsorption become smaller than the rate of the surface reaction. 

In order to derive approximate laws for the growth of a two-dimensional layer, we consider a simplified model in which all isolated clusters, i.e. clusters that do not touch another cluster, axe circular. For the moment, consider a single such cluster of radius r(t). New particles can only be incorporated at its boundary. Assuming that this incorporation is the rate-determining step, the number N(t) of particles belonging to the cluster obeys the equation ... 

Since reaction B —> C is the rate-determining step, the concentration of C obeys the steady-state approximation leading to... 

Since the bond to the isotopic atom is not formed or broken in the transition state of the rate-determining step of the reaction, the difference between the rate constant for the reaction of the undeuterated and deuterated substrates is usually small. As a result, secondary deuterium KIEs are usually close to unity, i.e. the maximum secondary deuterium KIE is 1.25 per deuterium (Shiner, 1970a) and most of these KIEs are less than 1.10 (Westaway, 1987a). Therefore, careful kinetic measurements with an error of approximately 1 % in each rate constant or specially designed competitive methods are required to determine them with an acceptable degree of accuracy. 

Rhee and Shine39 used an impressive combination of nitrogen and carbon kinetic isotope effects to demonstrate that a quinonoidal-type intermediate is formed in the rate-determining step of the acid-catalyzed disproportionation reaction of 4,4 -dichlorohydrazobenzene (equation 26). When the reaction was carried out at 0°C in 60% aqueous dioxane that was 0.5 M in perchloric acid and 0.5 M in lithium perchlorate, extensive product analyses indicated that the major pathway was the disproportionation reaction. In fact, the disproportionation reaction accounted for approximately 72% of the product (compounds 6 and 7) while approximately 13% went to the ortho-semidine (8) and approximately 15% was consumed in the para-semidine (9) rearrangement. 

This equation shows that the rate will be reduced with pressure, but according to Eq. (80) this reduction will be absorbed into kt, which is really constant. The rate constants kt and k-t have been removed in the initial approximation, and nothing can be said about the pressure dependences of the steps 1 and —1. The interpretation will be that the rate-determining step 2 becomes slower with pressure, while in fact the rate determination has been displaced to step 1. It is immediately clear that such an interpretation would be disastrous for the clarification of the high-pressure mechanism. The condition for a relatively simple rate equation of the random ternary-complex two-substrate mechanism was a small ka. This constant k3 may not be as small at high pressures, and the whole rate equation breaks down. 

With a single rate-determining step, the affinity of elementary steps other than the rate-determining step is negligible, and the overall reaction affinity — AG approximately equals the affinity - 4gr multiplied by the stoichiometric number of the rate-determining step in Eqn. 7-44 as has been shown in Eqn. 

Reaction rates have first-order dependence on both metal and iodide concentrations. The rates increase linearly with increased iodide concentrations up to approximately an I/Pd ratio of 6 where they slope off. The reaction rate is also fractionally dependent on CO and hydrogen partial pressures. The oxidative addition of the alkyl iodide to the reduced metal complex is still likely to be the rate determining step (equation 8). Oxidative addition was also indicated as rate determining by studies of the similar reactions, methyl acetate carbonylation (13) and methanol carbonylation (14). The greater ease of oxidative addition for iodides contributes to the preference of their use rather than other halides. Also, a ratio of phosphorous promoter to palladium of 10 1 was found to provide maximal rates. No doubt, a complex equilibrium occurs with formation of the appropriate catalytic complex with possible coordination of phosphine, CO, iodide, and hydrogen. Such a pre-equilibrium would explain fractional rate dependencies. 

The proposed mechanism will give the experimental kinetic equation (25) by assuming that the rate-determining step is the formation or decomposition of the complex and that the metal (III)-metal (II) ratio is approximately constant during reaction. 

Rogne253 and Arcoria and his co-workers254-256 (Fig. 7) shows only 3-furyl deviating markedly certainly using a values for the five-membered rings would predict reaction acceleration compared with phenyl. This approximate correlation indicates that the rate-determining step is addition of the nucleophile (Scheme 4) rather than synchronous attack of amine and displacement of chloride. 

If the adsorption of A is the rate determining step in the sequence of adsorption, surface reaction and desorption processes, then equation 3.71 will be the appropriate equation to use for expressing the overall chemical rate. To be of use, however, it is first necessary to express CA, Cv and Cs in terms of the partial pressures of reactants and products. To do this an approximation is made it is assumed that all processes except the adsorption of A are at equilibrium. Thus the processes involving B and P are in a state of pseudo-equilibrium. The surface concentration of B can therefore be expressed in terms of an equilibrium constant KB for the adsorption-desorption equilibrium of B ... 

This represents the case where there is a fast preequilibrium preceding the rate-determining step. It is again not necessary to work through the steady-state approximation if this situation is known to exist. If the intermediate I is in equilibrium with the reactant A, then... 

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