Pre-Exponential Factor in the Expression

Consider first of all the experimental values of the pre-exponential factor A. An idea about its order of magnitude can be obtained from data on the decomposition and isomerization of various compounds [501]. The abundance of log A values is illustrated in Fig. 26. 

It follows from these data that 75% of the log A o values are of the order 12—15, i.e. 13.5 on the average. A similar value would be expected from the theoretical expression exp (—Ea/RT) obtained for the oscillator model 

Values of the same order can be obtained also from Eq. (17.7). Thus, the log A values of the order 13 or close to 13 (often referred to as normal) obtain theoretical support so that this result can be treated as experimental evidence of the theor  

Large pre-exponential factors, by three and more orders higher than the normal ones, are usually explained by the high ratio of partition functions ab/ ab in the expression (11.1) for the rate constant [256]. When rewritten 

A similar explanation would obviously be invalid for low pre-exponential factors (log A 11 and lower). Indeed, the formally possible allowance for negative activation entropy would have no sense because the activated complex structure would have to be more rigid than that of the initial molecule which is quite improbable. The occurence of reactions with low pre-exponential factors seems to be due to non-adiabaticity of these reactions (see Section III.9) The transmission coefficient omitted in Eq. (18.1) for non-adiabatic reactions can be much lower than unity. Some non-adiabatic decomposition reactions, mainly of three-atom molecules, have been studied recently over a wide temperature range [489]. 

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In this connection we should mention the enormous magnitude of the pre-exponential factor in the expression of the probability of evaporation in the case of multi-atomic molecules (see Langmuir [6]). 

In these formulas a is a numerical constant which is to be found by integration (see 9), B and D are the pre-exponential factors in the expressions for the equilibrium constant and the rate coefficient. 

The theoretical description of the kinetics of electron transfer reactions starts fi om the pioneering work of Marcus [1] in his work the convenient expression for the free energy of activation was defined. However, the pre-exponential factor in the expression for the reaction rate constant was left undetermined in the framework of that classical (activate-complex formalism) and macroscopic theory. The more sophisticated, semiclassical or quantum-mechanical, approaches [37-41] avoid this inadequacy. Typically, they are based on the Franck-Condon principle, i.e., assuming the separation of the electronic and nuclear motions. The Franck-Condon principle... 

For both these exothermic reactions, the potential barriers are early , that is, reactant-like , in accordance with Polanyi s rules . One consequence is that the transitional modes in the transition states have low frequencies and the partition function for the internal modes in the transition state (see earlier, eqn (1.16)) will be strongly temperature-dependent, providing at least a partial reason for the positive temperature-dependence of the pre-exponential factor in the expressions for the rate constants. 

This mechanism for the process results in a modification of the pre-exponential factor in the expression for the exchange current as compared to that given in the absolute reaction rate theory. The symmetry factor a depends, in general, on the overpotential, this dependence being determined by properties of a regular part of the potential curve near the bottom of the potential well and near the barrier top. 

However, it should be considered that for the low pressure region, due to the negative temperature dependence of the pre-exponential factor in the expression for kg of a unimolecular second-order reaction, the experimental activation energy in the Arrhenius equation... 

The idea that the cathode potential with respect to ]lt(H20)/Pt-0Hads determines the value of the pre-exponential factor in the ORR rate expression was inspired by a comment by Andy Gewirth (Urbana) in his talk in Leiden, pointing to the value of Pourbaix diagrams for understanding ORR electrocatalysis. Indeed, the information on these ORR-mediating and facilitating M/M-OH surface redox systems is to be found in Pourbaix s Atlas. 

The rate constant for decomposition of the activated complex is simply the pre-exponential factor in the high-pressure Arrhenius fit to kun. This constant may be available from experimental measurement. Alternately, Eq. 10.98 provides an estimate for A,, . We can use this result and Eq. 10.150 to obtain an expression for kd. ... 

Using transition state theory, find the temperature dependence of the pre-exponential factor in the Arrhenius expression for the reaction NO + NO3 —> NO2 + NO2. In other words for A (X Tn, find n. Assume that both NO3 and the activated complex are nonlinear. Furthermore assume that hv << kgT and that the electronic degeneracies are all one. 

Exponential expression in eqns. (8) and (10) coincides with that in eqn. (7), obtained on the basis of the simple qualitative reasoning. The pre-exponential factor in the equation for W depends on both the electron characteristics of the atom and the electric field. 

The partial rate factor is equal to a ratio of rate constants. It may be used as a measure of the substituent effect on the activation energy for the reaction, provided it is assumed that the pre-exponential factors in the kinetic equation for the reaction with benzene and its derivatives are the same. On the basis of the Arrhenius equation and the theory of absolute reaction rates the rate constant can be expressed as... 

From this expression originates the statement that entropy effects are determining the pre-exponential factor in the Arrhenius expression. Thus we have the following. 

An exception to this approach is to be found in the work of Cohen [44] who has used the transition state methods developed by Benson and his colleagues [38] as a framework for evaluating experimental data for bimolecular metathetical reactions such as those of H, O and OH with alkanes. Using a model of the transition state, a theoretical value of the pre-exponential factor in the rate expression is derived which may be combined with experimental data at one temperature to give the exponential term. The rate expression so derived may be used to calculate values at other temperatures. By treating families of reactions adjustments can be made to the transition states to make them compatible with the experimental data on the whole range of reactions considered. 

In the estimation methods discussed so far the quantities estimated have either been the rate constant or the pre-exponential factor in the Arrhenius expression. Methods for estimating the activation energy of bimolecular reactions are much less developed. Theoretical prediction, at the level required, is beyond current computational techniques except in some exceptional, simple cases. However, there have been empirical attempts to relate the activation energy for a series of related reactions e.g., H abstraction by methyl radicals from hydrocarbons, to the thermodynamics of the process. 

To use this equation it is necessary to know how the energies associated with the states of several vibrational modes vary along the minimum energy path. This model explains why the pre-exponential factors in the rate constant expressions, as well as the activation energies, depend on the vibrational state of the reagents. 

Nuclear tunneling has also been included as a pre-exponential factor, in the rate constant expression. It is given by = 3xl0 V , where v is the nuclear vibrational frequency of the reactants and... 

Activation Energy and Pre-Exponential Factors in the Reaction Rate Constant Expression... 

Table 3.5 NMR parameters obtained from the Li NMR experiments in polymer electrolytes. T ax is the temperature at the spin-relaxation rate maximum, to is a pre-exponential factor in the Arrhenius expression, Eq. [3.1], and Ea is the activation energy for the motion causing the relaxation...

In the same manner, the activation energy and the pre-exponential factor of the combustion are determined Irom an Arrhenius plot. As can be seen the kinetic equation of the combustion can be expressed as ... 

Another problem which can appear in the search for the minimum is intercorrelation of some model parameters. For example, such a correlation usually exists between the frequency factor (pre-exponential factor) and the activation energy (argument in the exponent) in the Arrhenius equation or between rate constant (appears in the numerator) and adsorption equilibrium constants (appear in the denominator) in Langmuir-Hinshelwood kinetic expressions. 

The effect of the surface oxide species on the rate of the ORR is explained by metal site blocking, and can be described mathematically by including a 1 — 6ox term in the pre-exponential factor of the rate expression. 

The E° difference is a necessary but not a sufficient condition. The rate constant for either ET (in general, / et) may be described in a simple way by equation (4). The activation free energy AG is usually expressed as a quadratic function of AG°, no matter whether we deal with an outer-sphere ET or a dissociative ET. However, even if the condition (AG")c < (AG°)sj holds (hereafter, subscripts C and ST will be used to denote the parameters for the concerted and stepwise ETs, respectively), the kinetic requirements (intrinsic barriers and pre-exponential factors) of the two ETs have to be taken into account. While AGq depends only slightly on the ET mechanism, is dependent on it to a large extent. For a concerted dissociative ET, the Saveant model leads to AG j % BDE/4. Thus, (AGy )c is significantly larger than (AG )sj no matter how significant AGy, is in (AG( )gj (see, in particular. Section 4). In fact, within typical dissociative-type systems such as... 

In some cases experimental data indicate that the Arrhenius expression for the rate constant (Eq. 11.107) is modified by the coverage (concentration) of some surface species. Many functional forms for such coverage-dependence are possible. We describe one such choice that allows both the pre-exponential factor and the activation energy to be functions of the surface coverage of any surface species. The general modification of the Arrhenius rate expression is... 

Finally, these expressions can be related to the Arrhenius equation, where the activation energy Ea is identified from RT2d n k/dT. Although the rate constant is independent of volume (or pressure), we note that the exponential factor, as well as the pre-exponential factor, in Eq. (6.61) depends on volume (or pressure). Thus, we choose to perform the differentiation under constant pressure. Using that l/ce = V/N = ksTIp, we get... 

According a simplified theory (Alexandrov, 1976), a concerted reaction occurs as a result of the simultaneous transition (taking approximately 10 13 s) of a system of independent oscillators, with the mean displacement of nuclei ground state, to the activated state in which this displacement exceeds for each nuclei a certain critical value (q>cr). If activation energy of the concerted process Esyn > nRT, the theory gives the following expression for the synchronization factor which is the ratio of the pre-exponential factors of the synchronous and simple processes ... 

The same considerations made before are valid for this case and it is very important to have an available validated reaction mechanism. It can be obtained from three main sources (Blelski et al., 1985 Buxton et al., 1988 Stefan and Bolton, 1998) and it is shown in Table 5. With the available information about the constant k2, k, k, fcg, and k-j, it could be possible to solve a system of four differential equations and extract from the experimental data, the missing constants 4> and k (that in real terms is k /Co2)-This method would provide good information about the kinetic constants, but it is not the best result for studying temperature effects if the same information is not available for the pre-exponential factors and the activation energies. Then, it is better to look for an analytical expression even if it is necessary to make some approximations. This is particularly true in this case, where the direct application of the micro steady-state approximation (MSSA) is more difficult due to the existence of a recombination step that includes the two free radicals formed in the reaction. From the available information, it is possible to know that to calculate the pseudo-steady-state... 

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