Expressions for the Rate Constant

With the help of U, an expression for the rate constant for the reaction... 

Since the decay is associated with passing through the barrier, the quantity a t) is nothing but the step function a = Q[x — x). Differentiating (3.93) and finally setting r = 0 one obtains [Chandler 1987] the expression for the rate constant. 

At last, the formally exact quantal expression for the rate constant is... 

Although the correlation function formalism provides formally exact expressions for the rate constant, only the parabolic barrier has proven to be analytically tractable in this way. It is difficult to consistently follow up the relationship between the flux-flux correlation function expression and the semiclassical Im F formulae atoo. So far, the correlation function approach has mostly been used for fairly high temperatures in order to accurately study the quantum corrections to CLST, while the behavior of the functions Cf, Cf, and C, far below has not been studied. A number of papers have appeared (see, e.g., Tromp and Miller [1986], Makri [1991]) implementing the correlation function formalism for two-dimensional PES. 

Working in the same weak-coupling approximation, it takes little effort to produce the expression for the rate constant in the asymmetric case, by simply replacing J in (2.42)-(2.44) by the energy bias . 

The derivation of the transition state theory expression for the rate constant requires some ideas from statistical mechanics, so we will develop these in a digression. Consider an assembly of molecules of a given substance at constant temperature T and volume V. The total number N of molecules is distributed among the allowed quantum states of the system, which are determined by T, V, and the molecular structure. Let , be the number of molecules in state i having energy e,- per molecule. Then , is related to e, by Eq. (5-17), which is known as theBoltzmann distribution. 

With these assumptions, it is possible to develop an expression for the rate constant k. Although the expression is too complex to discuss here, we show some of the results in Table 11.3. 

You will recall from Section 11.4 that the collision model yields the following expression for the rate constant ... 

With these equations, the expression for the rate constant becomes... 

Both Marcus27 and Hush28 have addressed electron transfer rates, and have given detailed mathematical developments. Marcus s approach has resulted in an important equation that bears his name. It is an expression for the rate constant of a net electron transfer (ET) expressed in terms of the electron exchange (EE) rate constants of the two partners. The k for ET is designated kAS, and the two k s for EE are kAA and bb- We write the three reactions as follows ... 

The rate law of an elementary reaction that depends on collisions of A with B is Rate = fc[AJ[B], where k is the rate constant. We can therefore identify the expression for the rate constant as... 

As in collision theory, the rate of the reaction depends on the rate at which reactants can climb to the top of the barrier and form the activated complex. The resulting expression for the rate constant is very similar to the one given in Eq. 15, and so this more general theory also accounts for the form of the Arrhenius equation and the observed dependence of the reaction rate on temperature. 

In order to obtain an expression for the rate constant, we introduce the normalized particle densities /o (z) and p (z), which obey the conditions p (—oo) = p (oo) = 1. They are related to the potential of mean force through ... 

From the Instantaneous values of the properties reported In Table I, It is possible to determine a maximum of four kinetic parameters. Explicit expressions for the rate constants can be obtained directly from equations 12 to 16 In terms of the parameters t and 8, and from these the values of the rate constants can be obtained for a variety of reaction schemes. 

A number of papers are devoted to the effect of dissipation on tunneling.81"83,103,104 Wolynes81 was one of the first to consider this problem using the Feynman path integral approach to calculate the correlation function of the reactive flux involved in the expression for the rate constant,... 

As an example, consider an early calculation of isotope effects on enzyme kinetics by Hwang and Warshel [31]. This study examines isotope effects on the catalytic reaction of carbonic anhydrase. The expected rate-limiting step is a proton transfer reaction from a zinc-bound water molecule to a neighboring water. The TST expression for the rate constant k is... 

Influenced by the form of the van t Hoff equation, Arrhenius (1889) proposed a similar expression for the rate constant kAin equation 3.1-2, to represent the dependence of (-rA) on T through the second factor on the right in equation 3.1-1 ... 

Energies of reorganization are typically of the order of 0.5 - 1.5 eV applied overpotentials are often not higher than 0.1 - 0.2 V. For small overpotentials, when A 2> eo, the quadratic term in the energy of activation may be expanded to first order in eo this gives the following expression for the rate constant of the oxidation reaction ... 

Derive an expression for the rate constant of a reaction having order n. Also calculate the half-life period. 

It is important to emphasize here that the model in such a form allows one to simulate the influence of the reaction conditions. The temperature dependence of all the relative probabilities appears in the exponential expressions for the rate constants and the equilibrium constants. The olefin pressure influences the isomerization-insertion relative probabilities. As a result, both, temperature and olefin pressure influence the values of the absolute probabilities for all the reactive events considered. In the following use has been made of calculated reaction rates [13f] to evaluate all stochastic probabilities, unless otherwise stated. 

The result looks familiar and well it should. Equation 14.28 is just the expression for the rate constant in conventional canonical transition state theory with the... 

Elimination ofK t gives the transition state expression for the rate constant k as... 

It is natural to conclude that the high rate constants for electron attachment reactions in nonpolar liquids are associated with the high mobility of electrons. Early studies [96,104,105] of attachment to biphenyl and SFg emphasized the dependence of on mobility. This relationship is apparent if the expression for the rate constant for a diffusion-controlled reaction ... 

Many addition reactions such as the OH-SOz reaction are in the falloff region between second and third order in the range of total pressures encountered from the troposphere through the stratosphere. Troe and co-workers have carried out extensive theoretical studies of addition reactions and their reverse unimolecular decompositions as a function of pressure (e.g., see Troe, 1979, 1983). In this work they have developed expressions for the rate constants in the falloff region these are now most commonly used to derive the... 

The transition state for the dissociative chemisorption and for desorption is strongly constrained. This assumption allows us to set the partition function in the transition state equal to 1, and the prefactors in the Arrhenius expressions for the rate constants of desorption of AB and re-desorption of A2 thus become... 

The final form of the expression for the rate constant A Et, for energy transfer in solution, expressed in the units of litre mol-1 s-1 is... 

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