Steps with Similar Rates

In cases where more than one step has a slow rate, we vdll have to consider the rate for both of these steps. Suppose, for example, that steps (1) and (3) in the scheme of Eqs. (132-135) possess slow rates, whereas steps (2) and (4) may be considered at quasi-equilibrium, we would have the following set of equations  

The solution of the resultant set of differential equations is more complex than the situation involving one rate-determining step, but it is still simpler than the full solution. 

---

A catalytic cycle is a sequence of steps. When one step is much slower than the others, we say that this step is rate determining and we ignore, for kinetics purposes, the other (fast) steps. Nevertheless, sometimes there are two slow steps with similar rates, and sometimes the rate of a specific step changes in the course of the reaction or under different conditions. One common situation is that of two consecutive first-order reactions, as in Eq. (2.44). 

Whereas the effect of changing pH on a reaction involving only small molecules can be immediately informative, the interpretation of the effects of pH on enzyme reactions is more complex, since enzymes have many ionising groups and often react by a series of chemical steps with similar rates. 

In this latter case the Michaelis constant can be considerably smaller than the dissociation constant for the enzyme substrate complex, k2jkx2- The Michaelis constant defines the substrate concentration when half the enzyme, in the steady state, is in the form preceding the rate limiting step. Of course the situation can be less well defined when there are several steps with similar rates These comments are intended to underline the warning that Michaelis constants should not be used as measures for substrate affinities. A good example is the comparison of the values for the chymotrypsin catalysed hydrolysis of acetyl-L-tyrosine ethyl ester and the analogous amide. A hundredfold decrease in 23 results in an equivalent increase in... 

The rates are essentially independent of the distribution of metal in the MT with similar rates between Zn7—MTm (Cd, Zn)7-MT, and Cd7—MT. The values of the rate constants are ks = 6.9( 0.9) x 10 " s and kf=2.7( 1.2) x 10 s for the holo-protein. The slow rate constant is similar in magnitude to the first-order protein-dependent steps observed for reactions of DTNB (5,5 -dithiobis(2-nitrobenzoate)), EDTA (ethylenediamine tetraacetate), cisplatin, and other reagents, which has been attributed to a rearrangement of the protein. The fast step is more rapid by an order of magnitude, which suggests that other mechanisms are prevailing. 

Investigations of the kinetic isotope effect with NH3BD3, ND3BH3, ND3BD3 indicate that the rate-determining step involves either the simultaneous cleavage of the B-H and the N-H bonds or that both steps occur subsequently but with similar rate constants. The last case would be compatible with the mechanism outlined in Scheme 8.4. 

Up to now we have discussed only very simple electrode reactions with one well-defined electron transfer step. Real electrode reactions are usually much more complex and may involve several steps of similar rate, parallel reactions and other complications. Let us now look at a general electrode reaction where there are several reactants and products and more than one electron transfer step. However, we still assume that there is only one rate... 

More recently Minisci et al. (1986) compared the rate constants for phenylation of 4-cyanopyridine in the 2- and 3-positions by benzenediazonium ions, catalyzed by Cu+ and by Fe2+, with the rates of the same phenylations using benzoyl peroxide under similar conditions. The rate constants found for the phenylation steps were, within experimental error, the same. 

In neutral and alkaline media, the rate of exchange at the 3 and 6 position of 4-aminopyridazine is independent of acidity but decreases markedly when the media become more acidic. This was interpreted in terms of a rate-determining removal of the 6-proton by deuteroxide ion to give the ylid (XXIV), which reacts with deuterium oxide in a fast step. A similar result for the 3 and 6 positions of py-ridazin-4-one suggests the same mechanism. For reaction at the 5 position, the rate-acidity profile indicated reaction on the free base as did that for the 5 position of pyridazin-3-one, though the appearance of a maximum in the rate at — HQ = 0.8 was anomalous and suggested incursion of a further mechanism. 

The first step, as we have already seen (12-3), actually consists of two steps. The second step is very similar to the first step in electrophilic addition to double bonds (p. 970). There is a great deal of evidence for this mechanism (1) the rate is first order in substrate (2) bromine does not appear in the rate expression at all, ° a fact consistent with a rate-determining first step (3) the reaction rate is the same for bromination, chlorination, and iodination under the same conditions (4) the reaction shows an isotope effect and (5) the rate of the step 2-step 3 sequence has been independently measured (by starting with the enol) and found to be very fast. With basic catalysts the mechanism may be the same as that given above (since bases also catalyze formation of the enol), or the reaction may go directly through the enolate ion without formation of the enol ... 

CO oxidation reaction. The spectral changes in Cluster C are followed hy Cluster B reduction with a rate constant that is similar to the steady-state value. On the other hand, the rate of formation of the characteristic EPR signal for the CO adduct at Cluster A is much slower. Its rate constant matches that for acetyl-CoA synthesis, hut is several orders of magnitude slower than CO oxidation. Therefore, it was proposed that the following steps are involved in CO oxidation (1) CO hinds to Cluster C, (2) EPR spectral changes in Cluster C are accompanied hy oxidation of CO to CO2 hy Cluster C, (3) Cluster C reduces Cluster B, and (4) Cluster B couples to external electron acceptors (133). 

The NO + CO reaction is only partially described by the reactions (2)-(7), as there should also be steps to account for the formation of N2O, particularly at lower reaction temperatures. Figure 10.9 shows the rates of CO2, N2O and N2 formation on the (111) surface of rhodium in the form of Arrhenius plots. Comparison with similar measurements on the more open Rh(llO) surface confirms again that the reaction is strongly structure sensitive. As N2O is undesirable, it is important to know under what conditions its formation is minimized. First, the selectivity to N2O, expressed as the ratio given in Eq. (7), decreases drastically at the higher temperatures where the catalyst operates. Secondly, real three-way catalysts contain rhodium particles in the presence of CeO promoters, and these appear to suppress N2O formation [S.H. Oh, J. Catal. 124 (1990) 477]. Finally, N2O undergoes further reaction with CO to give N2 and CO2, which is also catalyzed by rhodium. 

If mechanism (A) applied the Cr(VI)+V(IV) system would be anomalous when compared with the Cr(VI) + Fe(II) and Ce(IV) + Cr(III) reactions which have similar rate laws and Cr(V) -> Cr(IV) transformations as rate-controlling steps. Apart from this there are other good reasons for rejecting mechanism (+). At 25 °C, K is 10 ° and k is 0.56 l.mole sec , allowing At2 to be calculated as... 

This great variety of pathways makes it difficult to decide which of the steps is the rate-determining step. It is most likely that at intermediate current densities the overall reaction rate is determined by the special kinetic features of step (15.24) producing the oxygen-containing species. The slopes of = 0.12 V observed experimentally are readily explained with the aid of this concept. Under different conditions, one of the steps in which these species react further may be the slow step, or several of the consecutive steps may occur with similar kinetic parameters. 

Without added acid or nucleophile the lability of the starting compound is very low (<1 x 10"8 s"1) most probably because of the efficient ring-closing process [18]. The second step depends linearly on the pH with a rate constant of 1.61 x 10"4 M 1 s 1. In the presence of added Cl-, the final product of the hydrolysis is m-DDP although the relevance of similar process in vivo was questioned [18]. It has been reported that carboplatin bound to DNA retains the dicarboxylate group, probably as a monodentate ligand [19]. 

---