The Kinetic Scheme

If the data follow the kinetic scheme presented here, the values of k, calculated for the different runs should be constant ... 

A postulated reaction mechanism is a description of all contributing elementary reactions (we will call this the kinetic scheme), as well as a description of structures (electronic and chemical) and stereochemistry of the transition state for each elementary reaction. (Note that it is common to mean by the term transition state both the region at the maximum in the energy path and the actual chemical species that exists at this point in the reaction.)... 

The overall reaction stoichiometry having been established by conventional methods, the first task of chemical kinetics is essentially the qualitative one of establishing the kinetic scheme in other words, the overall reaction is to be decomposed into its elementary reactions. This is not a trivial problem, nor is there a general solution to it. Much of Chapter 3 deals with this issue. At this point it is sufficient to note that evidence of the presence of an intermediate is often critical to an efficient solution. Modem analytical techniques have greatly assisted in the detection of reactive intermediates. A nice example is provided by a study of the pyridine-catalyzed hydrolysis of acetic anhydride. Other kinetic evidence supported the existence of an intermediate, presumably the acetylpyridinium ion, in this reaction, but it had not been detected directly. Fersht and Jencks observed (on a time scale of tenths of a second) the rise and then fall in absorbance of a solution of acetic anhydride upon treatment with pyridine. This requires that the overall reaction be composed of at least two steps, and the accepted kinetic scheme is as follows. 

In the process of establishing the kinetic scheme, the rate studies determine the effects of several possible variables, which may include the temperature, pressure, reactant concentrations, ionic strength, solvent, and surface effects. This part of the kinetic investigation constitutes the phenomenological description of the system. 

Consecutive reactions involving one first-order reaction and one second-order reaction, or two second-order reactions, are very difficult problems. Chien has obtained closed-form integral solutions for many of the possible kinetic schemes, but the results are too complex for straightforward application of the equations. Chien recommends that the kineticist follow the concentration of the initial reactant A, and from this information rate constant k, can be estimated. Then families of curves plotted for the various kinetic schemes, making use of an abscissa scale that is a function of c kit, are compared with concentration-time data for an intermediate or product, seeking a match that will identify the kinetic scheme and possibly lead to additional rate constant estimates. 

Scheme XVI is more difficult to analyze than are any of the kinetic schemes treated earlier in this chapter, because it includes a second-order reaction.

When reactions other than first-order processes are included in the kinetic scheme, reactant concentrations may appear in the denominator of the rate equation. Scheme XIX is an example. 

The quantitative description of enzyme kinetics has been developed in great detail by applying the steady-state approximation to all intermediate forms of the enzyme. Some of the kinetic schemes are extremely complex, and even with the aid of the steady-state treatment the algebraic manipulations are formidable. Kineticists have, therefore, developed ingenious schemes for writing down the steady-state rate equations directly from the kinetic scheme without carrying out the intermediate algebra." -" ... 

Sections 3.1 and 3.2 considered this problem Given a complex kinetic scheme, write the differential rate equations find the integrated rate equations or the concentration-time dependence of reactants, intermediates, and products and obtain estimates of the rate constants from experimental data. Little was said, however, about how the kinetic scheme is to be selected. This subject might be dismissed by stating that one makes use of experimental observations combined with chemical intuition to postulate a reasonable kinetic scheme but this is not veiy helpful, so some amplification is provided here. 

This manner of implicitly including the rate equations in the kinetic scheme is veiy convenient. It is amplified with the statement that when AX is acetyl chloride, ki/k i is very large and the reaction occurs essentially only via the I route. When... 

In these circumstances a decision must be made which of two (or more) kinet-ically equivalent rate terms should be included in the rate equation and the kinetic scheme (It will seldom be justified to include both terms, certainly not on kinetic grounds.) A useful procedure is to evaluate the rate constant using both of the kinetically equivalent forms. Now if one of these constants (for a second-order reaction) is greater than about 10 ° M s-, the corresponding rate term can be rejected. This criterion is based on the theoretical estimate of a diffusion-controlled reaction rate (this is described in Chapter 4). It is not physically reasonable that a chemical rate constant can be larger than the diffusion rate limit. 

Now suppose that only the monoanionic form of the dibasic acid H2S undergoes reaction and that neither the hydronium nor the hydroxide ion is directly involved. The kinetic scheme is, therefore. 

Sketch the kinetic scheme that is kinetically equivalent to Scheme VII. 

The kinetic scheme applicable to the Valinomycin carrier system is given in Fig. 18 where S is the carrier and MS+ is the carrier-cation complex. There are five unknown parameters, the four rate constants and Ns, the interfacial concentration of... 

In the case of catalysts activated with the help of reduction by CO, reactions (8) and (9) are excluded from the kinetic scheme. Such catalysts contain the centers Nr before their contact with the polymerization medium. 

According to some recent results (Sect. 5.5), the dependence of K on e and qs is more involved than suggested by Eq. (94). The dependence of K on ris is much weaker than a direct proportionality and the correct flow parameter to be used in Eq. (94) should be the local fluid kinetic energy ( v2) rather than the strain rate (e). For a constant flow geometry, however, the two variables v and e are interchangeable. At the present stage and in order not to complicate unduly the kinetic scheme, it will be assumed that the rate constant K varies with the MW and strain rate as ... 

For most real systems, particularly those in solution, we must settle for less. The kinetic analysis will reveal the number of transition states. That is, from the rate equation one can count the number of elementary reactions participating in the reaction, discounting any very fast ones that may be needed for mass balance but not for the kinetic data. Each step in the reaction has its own transition state. The kinetic scheme will show whether these transition states occur in succession or in parallel and whether kinetically significant reaction intermediates arise at any stage. For a multistep process one sometimes refers to the transition state. Here the allusion is to the transition state for the rate-controlling step. 

Relaxation kinetics with a reaction intermediate. Show that the kinetic scheme with a steady-state intermediate I corresponds to the single relaxation time shown ... 

Typical single-substrate enzymatic reactions can be described by the kinetic scheme (see Refs. 1 and 2 for more extensive discussions). 

The results of experiments obtained with phosphate analogues lead to another additional step in the kinetic scheme for the hydrolysis of ATP or GTP on F-actin or microtubules. This is described as follows ... 

The oxidation of polymers is most commonly depicted in terms of the kinetic scheme developed by BoUand [14]. The scheme is summarized in Figure 15.1. The key to the process is the initial formation of a free-radical species. At high temperatures and at large shear forces, it is likely that free-radical formation takes place by cleavage of C-C and C-H bonds. 

Figure 2. Kinetic schemes for Na channel gating (right) and the graphed time-course for single channels (solid lines, the higher position is open ) and for the population of many channels (broken line, the fraction open increases upwardly). Numbers at the arrows of the kinetic scheme are the rate constants, in 10 sec" The period of simulation is 5 msec. Computerized model courtesy of Dr. Daniel Chemoff.

In the proposed model, the penpenultimate effect of norbomene in a growing polymer chain is considered. The kinetic scheme fijr propagation ruction of ethylene (M,) and noibomene (Mj ) is shown in Eqs. (1) (6) ... 

To illustrate how a bifunctional catalyst operates, we discuss the kinetic scheme of the isomerization of pentane [R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York]. The first step is the dehydrogenation of the alkane on the metal ... 

Plots of k versus [Cr ] at fixed [CP] are linear and allow A , to be calculated from the slopes as 2.2x 10 l .mole . sec , at 25 °C and = 1.00 M. Product yields of Cr(H20)sCP and Cr(H20)g obtained experimentally were in excellent agreement with those calculated on the basis of the kinetic scheme. Dulz and Sutin conclude that two routes exist for the chloride-catalysed oxidation, Cr(H20)sCP being formed in both paths, viz. inner-sphere process... 

Figure 3.37. Computed UVRR difference spectra for successive intermediates in the kinetic scheme of Figure 3.36. The deoxyHb minus ffbCO spectrum is shown at the top for comparison. (Reprinted from reference [156]. Copyright (2004), with permission from Elsevier.)...

In the kinetic scheme of Figure 8.2C, we see that once the active species is formed, it can go on to inactivate the enzyme directly or be released into solution. If the active species formed is a good affinity label (i.e., is highly electrophilic), there is a chance that this species will rebind and inactivate the enzyme as an affinity label. To be classified as a mechanism-based inactivator, the active species must be demonstrated to directly inactivate the enzyme while still bound, without reliance on dissociation from the EA complex. 

Based on analogies we have cited, the kinetic scheme proposed for heavy-atom fluorescence quenching is reasonable and would predict the following relationship for fluorescence quenching ... 

The kinetic scheme used to describe micelle-facilitated dissolution is... 

The appropriate ratio between chiral Schiff bases and Ti(Oi-Pr)4 from catalytical point of view has been recommended. On the basis of the 1H NMR measurements at — 20 °C, the kinetic scheme for sulphide oxidation has been proposed. 

The small cluster threshold behavior, suggested by the kinetics scheme presented earlier, is apparent in this data set. Assuming (1) the RRK form for k n, (2) the addition reactions are not activated, and (3) the number of participating modes are independent of the element type, the M-CO bond strengths can be grouped by energy. For... 

To see how this approach works, consider the scheme above. If a steady state can be established, then we can analyse the various components of the kinetic scheme by assuming in each case that the rate of transport to or from the electrode must equal the nett sum of all the possible reactions. 

---