Elementary step reversible

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. 

Atoms and free radicals are highly reactive intermediates in the reaction mechanism and therefore play active roles. They are highly reactive because of their incomplete electron shells and are often able to react with stable molecules at ordinary temperatures. They produce new atoms and radicals that result in other reactions. As a consequence of their high reactivity, atoms and free radicals are present in reaction systems only at very low concentrations. They are often involved in reactions known as chain reactions. The reaction mechanisms involving the conversion of reactants to products can be a sequence of elementary steps. The intermediate steps disappear and only stable product molecules remain once these sequences are completed. These types of reactions are refeiTcd to as open sequence reactions because an active center is not reproduced in any other step of the sequence. There are no closed reaction cycles where a product of one elementary reaction is fed back to react with another species. Reversible reactions of the type A -i- B C -i- D are known as open sequence mechanisms. The chain reactions are classified as a closed sequence in which an active center is reproduced so that a cyclic reaction pattern is set up. In chain reaction mechanisms, one of the reaction intermediates is regenerated during one step of the reaction. This is then fed back to an earlier stage to react with other species so that a closed loop or... 

Usually, one of the elementary steps is rate eontrolling (that is, it is very slow relative to all the other steps). Suppose that A -1- Xj —> X2 is the rate-eontrolling step and the reverse reaetion is ignored, then... 

The reverse of the single elementary step in the one-step mechanism is... 

Pj release occurs at a relatively apparent slow rate (kobs = 0.005 s" ), so that the transient intermediate F-ADP-Pj in which P is non-covalently bound, has a life time of 2-3 minutes (Carlier and Pantaloni, 1986 Carlier, 1987). While the y-phosphate cleavage step is irreversible as assessed by 0 exchange studies (Carlier et al., 1987), the release of Pi is reversible. Binding of H2PO4 (Kp 10 M) causes the stabilization of actin filaments and the rate of filament growth varies linearly with the concentration of actin monomer in the presence of Pi (Carlier and Pantaloni, 1988). Therefore, Pi release appears as the elementary step responsible for the destabilization of actin-actin interactions in the filament. 

Fig. 3.1 (Kapteijn et al., 1999) shows the model commonly u.sed to pre.sent a reversible reaction (A B) taking place on the surface of a solid catalyst. Three elementary steps are distinguished, i.e. adsorption of A on an active site, reaction of this adsorbed complex to adsorbed complex B, and desorption of B from the active site.

This bi-exponential behavior confirms the presence of reversible isomerization steps coupled with irreversible degradation steps and accounts for the role of the di-cis isomers as reaction intermediates, according to the general reaction scheme presented in Figure 12.1. The dependence of the rate constant of each elementary step on temperature allowed the calculation of the respective activation... 

Let us consider the basic enzyme catalysis mechanism described by the Michaelis-Menten equation (Eq. 2). It includes three elementary steps, namely, the reversible formation and breakdown of the ES complex (which does not mean that it is at equilibrium) and the decomposition of the ES complex into the product and the regenerated enzyme ... 

The transfer reaction utilizes a sacrificial alkene to remove the dihydrogen from the pincer or anthraphos complex first, before the oxidative addition of the target alkane. The elementary reaction steps are slightly different from the thermal reaction, which is discussed in the next section, both in their order and their direction. For simplicity, we describe the symmetric reaction where the sacrificial alkene is ethylene and the reactant is ethane (21b). The elementary reaction steps for the mechanism of this transfer reaction involve IVR, IIIR, VIR, VI, III and IV, where the superscript R stands for the reverse of the elementary steps listed in Section III. These reverse steps (IVR, IIIR, and VIR) involve the sacrificial alkene extracting dihydride from the metal to create the Ir(I) species 8, while steps VI, III and IV involve oxidative addition of target alkane, p-H transfer and olefin loss. 

The ion 28 loses H2 by CID with argon to form [(PHOX)Ir(styrene)]+ (29). Compound 29 then undergoes H-D exchange with D2 gas to form the mixture of iso-topomers 29, 29-dh and 29-d2 (Scheme 13.3). When combined, these observations show that the oxidative addition of H2 to 29 is followed by alkene hydride insertion, and that both these steps occur rapidly and reversibly in the gas phase. These results thereby provide gas-phase analogues for catalytic elementary steps that are proposed to occur in solution. Support for this proposed sequence of steps was obtained from a solution-phase catalytic deuteration of styrene. Analysis showed no deuterium incorporation in the unreacted styrene at various conversions, and clean formation of dideuterio ethylbenzene as sole product. 

Because of its relevance to the chemistry of air at elevated temperatures the homogeneous decomposition of nitric oxide has received considerable attention from gas kineticists. References to early studies are given in the more recent work discussed below. The mechanisms for the decomposition and for the reverse reaction, the formation of NO from air, are well established and good quantitative data (Table 12) are available for the rate coefficients of the elementary steps. 

There is ample evidence that the reductive elimination of alkanes (and the reverse) is a not single-step process, but involves a o-alkane complex as the intermediate. Thus, looking at the kinetics, reductive elimination and oxidative addition do not correspond to the elementary steps. These terms were introduced at a point in time when o-alkane complexes were unknown, and therefore new terms have been introduced by Jones to describe the mechanism and the kinetics of the reaction [5], The reaction of the o-alkane complex to the hydride-alkyl metal complex is called reductive cleavage and its reverse is called oxidative coupling. The second part of the scheme involves the association of alkane and metal and the dissociation of the o-alkane complex to unsaturated metal and free alkane. The intermediacy of o-alkane complexes can be seen for instance from the intramolecular exchange of isotopes in D-M-CH3 to the more stable H-M-CH2D prior to loss of CH3D. 

From other experiments involving the reaction between nitrogen tetroxide and nitrogen dioxide, chemists know that both the forward and reverse reactions involve elementary steps. Thus, you can write rate equations for the reactions. 

Since all of the above-mentioned interconversion reactions are reversible, any kinetic analysis is difficult. In particular, this holds for the reaction Sg - Sy since the backward reaction Sy -+ Sg is much faster and, therefore, cannot be neglected even in the early stages of the forward reaction. The observation that the equilibrium is reached by first order kinetics (the half-life is independent of the initial Sg concentration) does not necessarily indicate that the single steps Sg Sy and Sg Sg are first order reactions. In fact, no definite conclusions about the reaction order of these elementary steps are possible at the present time. The reaction order of 1.5 of the Sy decomposition supports this view. Furthermore, the measured overall activation energy of 95 kJ/mol, obtained with the assumption of first order kinetics, must be a function of the true activation energies of the forward and backward reactions. The value found should therefore be interpreted with caution. 

Rate equations for simple reversible reactions are often developed from mechanistic models on the assumption that the kinetics of elementary steps can be described in terms of rate constants and surface concentrations of intermediates. An application of the Langmuir adsorption theory for such development was described in the classic text by Hougen and Watson (/ ), and was used for constructing rate equations for a number of heterogeneous catalytic reactions. In their treatment it was assumed that one step would be rate-controlling for a unique mechanism with the other steps at equilibrium. 

The reverse reaction, steam cracking of methane, involves the same elementary steps as the methanation reaction. The kinetics for that reaction have been developed for a single direct mechanism by Snagovskii and Ostrovskii (39). 

In the process of carbonyl insertion the 1,1 migratory insertion of the coordinated CO ligand into the metal-carbon bond results in the formation of a metal-acyl complex (Figure 1-7). This process, as nearly all elementary steps discussed so far, is reversible, but even when using atmospheric CO pressure the equilibrium is mostly shifted towards insertion. In the process of insertion a vacant coordination site is also produced on the metal, where further reagents might be attached. Of the metals covered in this book palladium is by far the most frequently utilized in such transformations. 

So far all the reaction steps have been considered as being totally irreversible. This choice has been made in the interests of keeping the model at its simplest possible level. The fact that the model shows oscillations under such conditions is revealing, as it clearly demonstrates that oscillatory behaviour does not correspond to particular elementary steps sometimes proceeding forwards, at other times running backwards. We should also show, on the other hand, that oscillations are not a consequence of our simplification. All the qualitative results derived above should be seen in the model with reverse reactions included, and the quantitative relationships for these more general forms should clearly reduce to those already obtained in the limit of high values for the equilibrium constants for the various steps. 

A multistep pathway analogous to the mechanism of alkene hydrogenation has been shown to be operative in the rhodium-catalyzed hydroboration of alkenes.363 Deuterium labeling studies furnished evidence that the reversibility of the elementary steps is strongly substrate-dependent. The key step is hydride rather than boron migration to the rhodium-bound alkene. 

Many of the elementary steps of enantioselective reactions are reversible, and the first irreversible step that involves diastereomeric tran-... 

The Langmuir-Hinshelwood treatment of the kinetics of surface catalyzed reactions affords a useful representation of some of the characteristics of catalytic hydrogenation. It is a limiting form of more exact equations which recognize that, even though the elementary steps are reversible, few if any will be at equilibrium (ref. 15). Not surprisingly, alternative assumptions regarding the relative rates of the forward and reverse elementary reactions can lead to approximate equations of the same form. 

Kinetic treatment is more difficult for mechanisms with more than one elementary step. Here we shall restrict the discussion to two commonly encountered special cases. Let us look first at a simple two-step process (Equations 2.37 and 2.38) in which we are justified by the chemistry in ignoring reverse reactions. 

The most difficult problem is the adequate determination of the "abnormally (i.e. unexpected) slow transition process. For this purpose, we must imagine the simplest system of expectations. It is based on the hypotheses about reaction mechanisms. A catalytic reaction is represented as a combination of elementary steps (see Chap. 3). We admit some hypotheses concerning values of the corresponding rate coefficients. Typical concentrations of gas-phase substances and of surface compounds are also assumed to be known. One can also introduce a concept of the characteristic time of a step. For example, the characteristic time for the step A B can be determined as 1 l(k+ + k ), where k and k are the rate constants for the direct and reverse reactions. For the reaction A + B - C one can introduce two characteristic times 1 [kCA and 1 /kCB, where CA and CB are the characteristic concentrations of A and B. 

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