Statistical kinetics

Finally, we shall consider certain combinations that give rise to linear plots of In (Y, - Yx) versus time, even though the scheme is biphasic. There are the trivial cases, k ki and k [Pg.76]

Spin trap, 102 Statistical kinetics, 76 Steady-state approximation, 77-82 Stiff differential equations, 114 Stoichiometric equations, 12 Stopped-flow method, 253-255 Substrate titration, 140 Success fraction approach, 79 Swain-Scott equation, 230-231... 

The first step consists in the attack of a proton on the W-H bond to yield a labile dihydrogen intermediate (Eq. (3)) that rapidly releases H2 to form a coordi-natively unsaturated complex (Eq. (4)). This complex adds water in the next step to form an aqua complex (Eq. (5)) that completes the reaction by substituting the coordinated water by the X anion (Eq. (6)). Steps (3)-(6) are repeated for each W-H bond and the factor of 3 in the rate constants appears as a consequence of the statistical kinetics at the three metal centers. The rate constants for both the initial attack by the acid (ki) and water attack to the coordinatively unsaturated intermediate (k2) are faster in the sulfur complex, whereas the substitution of coordinated water (k3) is faster for the selenium compound. 

Oxidation of 8-Fe(rr) to 8-Fe(oo) by a number of one-electron oxidants gives a single first-order process. After ruling out the more obvious reasons for this observation, it is concluded that condition (1.91) holds, one often referred to as statistical kinetics . For other examples, see Refs. 87 and 88. Even if (1.91) is not strictly satisfied, linear plots may still be obtained. 

STATISTICAL KINETIC MODELS 939 Reactant Transition state Product... 

The RRKM theory is the most widely used of the microcanonical, statistical kinetic models It seeks to predict the rate constant with which a microcanonical ensemble of molecules, of energy E (which is greater than Eq, the energy of the barrier to reaction) will be converted to products. The theory explicitly invokes both the transition state hypothesis and the statistical approximation described above. Its result is summarized in Eq. 2... 

The most sophisticated and computationally demanding of the variational models is microcanonical VTST. In this approach one allows the optimum location of the transition state to be energy dependent. So for each k(E) one finds the position of the transition state that makes dk(E)/dq = 0. Then one Boltzmann weights each of these microcanonical rate constants and sums the result to find fc ni- There is general agreement that this is the most reliable of the statistical kinetic models, but it is also the one that is most computationally intensive. It is most frequently necessary for calculations on reactions with small barriers occurring at very high temperatures, for example, in combustion reactions. 

Examples. There are by now several reactions for which the best available levels of ab initio electronic structure theory find a plateau on the PE hypersurface, in the vicinity of a singlet-state biradical. Several of these have been studied by MD simulation and/or experiment, and in each case the conclusion is that application of statistical kinetic models will give a misleading description of how the reaction really occurs. There is not room to describe each of these studies, and so just one is chosen as a representative, and described in Section 3.2.3.1. 

There is every reason to believe that the results found for the vinylcyclopropane rearrangement are typical of those to be expected for reactions involving biradicals (and presumably other reactive intermediates) on energetic plateaus. Such reactions simply cannot be understood within the context of any statistical kinetic model. 

As has long been known, every derivation of the bulk properties of matter from its atomic properties by statistical methods encounters essential difficulties of principle. Their effect is that in all but the simplest cases (i.e., equilibrium) the development does not take the form of a deductive science. This contrasts with the usual situation in physics e.g., Newtonian or relativistic mechanics, electromagnetism, quantum theory, etc. The present paper, after focusing on this difficulty, seeks a way out by exploring the properties of a special class of statistical kinetics to be called relaxed motion and to be defined by methods of generalized information theory. 

Statistical kinetics of polyesterification of adipic acid with pentaery thritol and trimethylolethane. J. Chem. Soc. 1967, 1. 

There thenfollowed reports by Katz [13] and Grubbs [14] and their co-workers on studies that aimed to simplify and confirm the analysis. The key remaining issue was whether a modified pairwise mechanism, in which another alkene can coordinate to the metal and equilibrate with the product prior to product displacement, would also explain the appearance of the anomalous cross-over products early in the reaction evolution. However, a statistical kinetic analysis showed that for a 1 1 mixture of equally reactive alkenes, the kinetic ratio of cross-metathesis should be 1 1.6 1 for the pairwise mechanism and 1 2 1 for the Chauvin mechanism. Any equilibration (substrate or product) would, of course, cause an approach towards a statistical distribution (1 2 1) and thus allow no distinction between the mechanisms. 

Kinetic regime. The first mechanism corresponds to the so-called kinetic regime (Popel 1994, Grigorenko et al. 1998) which is similar to the adsorption/desorption model (Blake 1993). Contrary to macroscopic hydrodynamic models, the adsorption/desorption model is based on the hypothesis that the motion of the triple line is ultimately determined by the statistical kinetics of atomic or molecular events occurring within the three-phase zone (Samsonov and Muravyev 1998). Such processes may be limiting at the very early stages of spreading of low... 

Standard representation of the TS in organic chemistry textbooks is the point of maximum energy on the reaction coordinate. More precise is the definition provided in Section 1.6 the TS is the col, a point where aU the gradients vanish, and all of the eigenvalues of the Hessian matrix are positive except one, which corresponds to the reaction coordinate. In statistical kinetic theories, a slightly different definition of the TS is required. 

Aoqing Tang, Statistic Kinetics of Macromolecule Reaction. Science Press. Beijing, China, 366 (1985). 

Therefore, also in this case only replacement of the last proton by a deuteron exhibits a non-statistical kinetic isotope effect. However, the HDD process will experience isotopic fractionation of the dissociation process. Moreover, the DDD process is additionally affected by isotopic fractionation. 

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