Transition state theory, rate determining steps

Electrode kinetics lend themselves to treatment usiag the absolute reaction rate theory or the transition state theory (36,37). In these treatments, the path followed by the reaction proceeds by a route involving an activated complex where the element determining the reaction rate, ie, the rate limiting step, is the dissociation of the activated complex. The general electrode reaction may be described as ... 

Contrary to experimental evidence, the CO insertion step is predicted as the rate-determining step of the catalytic cycle at all reported levels of theory. The difference between of the computed results and the experiment has been attributed [17] to effects of solvation. Oxidative addition is the only step that involves an unsaturated reactant. The solvent is supposed to stabilize all transition states (TS) in the same extent, but further stabilize the unsaturated complex, which would increase the activation barrier. When a single ethene molecule was used to model the solvent, the activation barrier of H2 oxidative addition increased [17], to almost the same size as the CO insertion barrier. At this point, it seems that theory has not yet managed to distinguish which is the faster step. 

Finally, yet another issue enters into the interpretation of nonlinear Arrhenius plots of enzyme-catalyzed reactions. As is seen in the examples above, one typically plots In y ax (or. In kcat) versus the reciprocal absolute temperature. This protocol is certainly valid for rapid equilibrium enzymes whose rate-determining step does not change throughout the temperature range studied (and, in addition, remains rapid equilibrium throughout this range). However, for steady-state enzymes, other factors can influence the interpretation of the nonlinear data. For example, for an ordered two-substrate, two-product reaction, kcat is equal to kskjl ks + k ) in which ks and k are the off-rate constants for the two products. If these two rate constants have a different temperature dependency (e.g., ks > ky at one temperature but not at another temperature), then a nonlinear Arrhenius plot may result. See Arrhenius Equation Owl Transition-State Theory van t Hoff Relationship... 

The theory of radiationless transition considers the transition to occur in two steps (i) horizontal transition from one energy state to the other at the isocnergetic point, for the two combining states and (ii) vibrational relaxation of the lower energy state. The step (i) is the rate determining step. The rate constant is given by the theory of Robinson and Frosch as,... 

The rate constant, k, for most elementary chemical reactions follows the Arrhenius equation, k = A exp(— EJRT), where A is a reaction-specific quantity and Ea the activation energy. Because EA is always positive, the rate constant increases with temperature and gives linear plots of In k versus 1 IT. Kinks or curvature are often found in Arrhenius plots for enzymatic reactions and are usually interpreted as resulting from complex kinetics in which there is a change in rate-determining step with temperature or a change in the structure of the protein. The Arrhenius equation is recast by transition state theory (Chapter 3, section A) to... 

Complicated rate-law focus on rate determining step. The intermediate formed at this step can be modeled using transition-state-theory. The steady-state approximation works for reactions with unstable intermediates. 

Figure 5.39 shows the schematic diagram of the transition state for an exothermic reaction. The transition-state theory assumes that the rate of formation of a transition-state intermediate is very fast and the decomposition of the unstable intermediate is slow and is the rate-determining step. On the other hand, the collision theory states that the rate of the reaction is controlled by collisions among the reactants. The rate of formation of the intermediate is very slow and is followed by the rapid decomposition of the intermediates into products. Based on these two theories, the following expression can be derived to account for the temperature dependence of the rate constant ... 

According to the transition state theory, the reaction rate of the rate determining step can be computed from expression (3.30). 

The difference between the QM/MM-calculated energy barriers for the rate-determining steps of the two enzymatic reaction systems is consistent with the experimental observation that the fccat value (1.6 x 10" s ) [124] for AChE-catalyzed hydrolysis of ACh was about 150-fold larger than that (fccat = 1.07 X 10 s ) [97] for BChE-catalyzed hydrolysis of (+)-cocaine. Based on the widely used classical transition-state theory (CTST) [125], the experimental fccat difference of 150-fold suggests an energy barrier difference of 3.0 kcal/mol when T = 298.15 K, which is in good agreement with the calculated barrier difference of 3.7 kcal/mol [113]. 

For sufficiently fast consumption of to form product (b c), the rate-determining step is the activation process governed by = a. On the other hand, in the opposite limit b c), the ratio of R and R is maintained at the equilibrium value and transition state theory (TST) may be employed. Now the R P process is the rate-determining step ... 

Here is the forward rate constant, aj is the activity of species j in the rate-determining reaction, mj and are constants, and R and T are the gas constant and absolute temperature, respectively. The sign of the rate indicates whether the reaction goes forward or backward. The relationship of this equation to transition state theory and irreversible kinetics has been discussed in the literature (Lasaga, 1995 Alekseyev et al., 1997 Lichtner, 1998 Oelkers, 2001b). The use of this equation with = 1 is generally associated with a composite reaction in which all the elementary reactions are near equilibrium except for one step which is ratedetermining. This step must be shared by both dissolution and precipitation. 

The transition state theory gives us a framework to relate the kinetics of a reaction with the thermodynamic properties of the activated complex (Brezonik, 1990). In kinetics, one attempts to interpret the stoichiometric reaction in terms of elementary reaction steps and their free energies, to assess breaking and formation of new bonds, and to evaluate the characteristics of activated complexes. If, in a series of related reactions, we know the rate-determining ele-mentaiy reaction steps, a relationship between the rate constant of the reaction, k (or of the free energies of activation, AG ), and the equilibrium constant of the reaction step, K (or the free energy, AG°), can often be obtained. For two related reactions. 

In this case neither original reactant appears in the rate-determining step, but both appear in the rate-law expression. Each step is a reaction in itself. Transition state theory tells us that each step has its own activation energy. Because step 3 is the slowest, we know that its activation energy is the highest, as shown in Figure 16-12. 

Transition-state theory may be useful in testing the dissolution mechanisms presented above. According to TST, for any elementary chemical reaction the reactants should pass through a free-energy maximum, labeled the activated complex , before they are converted to products. It is assumed that the reaction rate-determining step is related to the decomposition of this activated complex ... 

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