Pre-exponential function

Stress relaxation for polydisperse linear chains Assume that the distribution of relaxation rates P s) in Eq. (8.184) has a sharp cutoff at some characteristic lowest rate e in the form of a stretched exponential with some exponent. x. /1(e) is a weaker pre-exponential function (such as a power law) ... 

The Ea is the activation energy of diffusion (kcal/mole). Do is the pre-exponential function (an empirical constant its units are cmVs). 

Where k is the rate constant, A is the pre-exponential function a function of the reaction (units of k and A depend upon the stoichiometry of the reaction), is the activation energy (J mol ), R is the gas constant (J moP K ) and T is temperature (K). The dependence on temperature is clear. This equation is directly applicable if the reaction mixture is well-mixed and/or the phases are fully miscible. 

The classical experiment tracks the off-gas composition as a function of temperature at fixed residence time and oxidant level. Treating feed disappearance as first order, the pre-exponential factor and activation energy, E, in the Arrhenius expression (eq. 35) can be obtained. These studies tend to confirm large activation energies typical of the bond mpture mechanism assumed earlier. However, an accelerating effect of the oxidant is also evident in some results, so that the thermal mpture mechanism probably overestimates the time requirement by as much as several orders of magnitude (39). Measurements at several levels of oxidant concentration are useful for determining how important it is to maintain spatial uniformity of oxidant concentration in the incinerator. 

The solution of the simultaneous differential equations implied by the mechanism can be expressed to give the time-varying concentrations of reactants, products, and intermediates in terms of increasing and decreasing exponential functions (8). Expressions for each component become comphcated very rapidly and thus approximations are built in at the level of the differential equations so that these may be treated at various limiting cases. In equations 2222 and 2323, the first reaction may reach equiUbrium for [i] much more rapidly than I is converted to P. This is described as a case of pre-equihbrium. At equihbrium, / y[A][S] = k [I]. Hence,... 

Cordes discusses the magnitudes of pre-exponential terms with reference to the partition function for the activated complex in which the following cases are recognized. 

We again assume that the pre-exponential factor and the entropy contributions do not depend on temperature. This assumption is not strictly correct but, as we shall see in Chapter 3, the latter dependence is much weaker than that of the energy in the exponential terms. The normalized activation energy is also shown in Fig. 2.11 as a function of mole fraction. Notice that the activation energy is not just that of the rate-limiting step. It also depends on the adsorption enthalpies of the steps prior to the rate-limiting step and the coverages. 

A more spread-out dependence of OHads on - (H20)/R-0Hads occur when interaction energies between chemisorbed species become significant however, it is always the potential difference E — (HjO/Pt-OHads determines the availability of active metal sites, reflected by the 1 OH term in the pre-exponential factor. The pre-exponential factor can consequently be expressed directly as a function of — -E (H20)/Pt-OH d by replacing the 1 — 0qh term in (1.10) by a term derived from (1.11), to yield... 

As a consequence of these various defined quantities, care must be taken in assigning values of rate constants and corresponding pre-exponential factors in the analysis and modeling of experimental data. This also applies to the interpretation of values given in the literature. On the other hand, the function [ [ c and the activation energy EA are characteristics only of the reaction, and are not specific to any one species. 

An increase in the number of ways to store energy increases the entropy of a system. Thus, an estimate of the pre-exponential factor A in TST requires an estimate of the ratio g /gr. A common approximation in evaluating a partition function is to separate it into contributions from the various modes of energy storage, translational (tr), rotational (rot), and vibrational (vib) ... 

In the simplest case, where (+)-AH and (-)AD are isotopically pure, a = [a]H[AH]0 and a2 = [a]D[AD]0 where a is the specific rotation of the AH and AD isotopomers, respectively, and [AH]0 and [AD]0 are the concentrations of the substrates in g ml-1 at time t = 0. When the substrate is neither isotopically nor enantiomerically pure, corrections must be made in calculating fli and a2 (Bergson et al., 1977). It is important to note that the pre-exponential factors, a and a2, which contain the information about the starting conditions, can be determined with high accuracy. The extreme, ae (the maximum or minimum value of the optical rotation in the optical rotation versus time plot) and the corresponding reaction time, te, are functions of the rate constant ratio (5 = kHlkD) (65) and the difference between the rate constants (66), respectively. 

Another approach is known as the Distributed Activation Energy Model (DAEM). This model recognizes that devolatilization occurs through many simultaneous reactions. To express this process in a mathematically tractable manner, these reactions are all presumed to be first order and to be describable by a continuous distribution of kinetic rates with a common pre-exponential and a defined distribution function of activation energy [43],... 

The inversion operator i acts on the electronic coordinates (fr = —r). It is employed to generate gerade and ungerade states. The pre-exponential factor, y is the Cartesian component of the i-th electron position vector (mf. — 1 or 2). Its presence enables obtaining U symmetry of the wave function. The nonlinear parameters, collected in positive definite symmetric 2X2 matrices and 2-element vectors s, were determined variationally. The unperturbed wave function was optimized with respect to the second eigenvalue of the Hamiltonian using Powell s conjugate directions method [26]. The parameters of were... 

Fig. 12 Migration activation enthalpy H (a) and mobility pre-exponential factor Ao (b) for 38.2° (C>)and 40.5° ( tilt grain boundaries as a function of Ga concentration.

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... 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

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