Preexponential factor equation

Some relationships describing the effect of the above factors on the drying constant are presented in Table 4.11. Equations Tll.l and T11.2 are Arrhenius-type equations, which take into account the temperature effect only. The effect of water activity can be considered by modifying the activation energy (Equation Tll.l) on the preexponential factor (Equation T11.2). Equations Tll.l and T11.2 consider the same factors in a different form. Equation T11.4 takes into account only the air velocity effect, whereas Equation T11.5 considers all the factors affecting the drying constant. Table 4.12 lists parameter values for typical equations of Table 4.11. 

The Arrhenius equation relates the rate constant k of an elementary reaction to the absolute temperature T R is the gas constant. The parameter is the activation energy, with dimensions of energy per mole, and A is the preexponential factor, which has the units of k. If A is a first-order rate constant, A has the units seconds, so it is sometimes called the frequency factor. 

The preexponential factor of the Arrhenius equation is approximately given by... 

Collision theory leads to this equation for the rate constant k = A exp (-EIRT) = A T exp (,—EIRT). Show how the energy E is related to the Arrhenius activation energy E (presuming the Arrhenius preexponential factor is temperature independent). 

A more interesting possibility, one that has attracted much attention, is that the activation parameters may be temperature dependent. In Chapter 5 we saw that theoiy predicts that the preexponential factor contains the quantity T", where n = 5 according to collision theory, and n = 1 according to the transition state theory. In view of the uncertainty associated with estimation of the preexponential factor, it is not possible to distinguish between these theories on the basis of the observed temperature dependence, yet we have the possibility of a source of curvature. Nevertheless, the exponential term in the Arrhenius equation dominates the temperature behavior. From Eq. (6-4), we may examine this in terms either of or A//. By analogy with equilibrium thermodynamics, we write... 

Some workers in this field have used Eyring s equation, relating first-order reaction rates to the activation energy d(7, whereas others have used the Arrhenius parameter E. The re.sults obtained are quite consistent with each other (ef. ref. 33) in all the substituted compounds listed above, AG is about 14 keal/mole (for the 4,7-dibromo compound an E value of 6 + 2 keal/mole has been reported, but this appears to be erroneous ). A correlation of E values with size of substituents in the 4- and 7-positions has been suggested. A/S values (derived from the Arrhenius preexponential factor) are... 

Several points are worth noting about these formulae. Firstly, the concentrations follow an Arrhenius law except for the constitutional def t, however in no case is the activation energy a single point defect formation energy. Secondly, in a quantitative calculation the activation energy should include a temperature dependence of the formation energies and their formation entropies. The latter will appear as a preexponential factor, for example, the first equation becomes... 

In the following, the traditional treatment of the rate equation (3) will be adopted, taking the preexponential factor as a constant. Evidently, no other procedure is available at present. Even if a quantitative theory of the outlined problems were available, mathematical difficulties would render it possible to present only selected computerized data. 

By using the kinetic equations developed in Sect. 5.2, the degradation yield as a function of strain rate and temperature can be calculated. The results, with different values of the temperature and preexponential factor, are reported in Fig. 51 where it can be seen that increasing the reaction temperature from 280 K to 413 K merely shifts the critical strain rate for chain scission by <6%. 

The observed linear variation in activation energy, E, and in the preexponential factor r0/ r° (Figs. 4.35 to 4.37), which conform to the equations ... 

For Q = Q , this density function describes electronic motions for given nuclear positions, while for Q = Q it describes the quantal correlation of nuclear positions at time f, which should be small for classical-like variables. The equation of motion for the density function could be derived from the original LvN equation. Instead, it is more convenient to construct it from the wavefunctions. The phase factor and the preexponential factor are trial functions to be determined from the TDVP. The procedure followed here parallels that in ref. (23). 

Arrhenius proposed his equation in 1889 on empirical grounds, justifying it with the hydrolysis of sucrose to fructose and glucose. Note that the temperature dependence is in the exponential term and that the preexponential factor is a constant. Reaction rate theories (see Chapter 3) show that the Arrhenius equation is to a very good approximation correct however, the assumption of a prefactor that does not depend on temperature cannot strictly be maintained as transition state theory shows that it may be proportional to 7. Nevertheless, this dependence is usually much weaker than the exponential term and is therefore often neglected. 

Hence, to write the rate in the form of the Arrhenius equation, we replace the energy barrier AE by the activation energy AE + Vi k T, which means that the preexponential factor contains the additional factor eJ ... 

This expression corresponds to the Arrhenius equation (14.1) and basically provides the possibihty of calculating the preexponential factor (a calculation of is, in fact, not easy). It also shows that in the Arrhenius equation it will be more correct to use the parameter AG rather than A//. However, since AGt = Aff TASt, it follows that the preexponential factor of Eq. (14.4) will contain an additional factor exp(ASi/R) reflecting the entropy of formation of the transition state when the enthalpy is used in this equation. 

Preexponential factor, molecules/sec cm2, in the equation r0 = r0 exp( —E/RT). c Order with respect to ethane. d Order with respect to hydrogen. 

Holroyd (1977) finds that generally the attachment reactions are very fast (fej - 1012-1013 M 1s 1), are relatively insensitive to temperature, and increase with electron mobility. The detachment reactions are sensitive to temperature and the nature of the liquid. Fitted to the Arrhenius equation, these reactions show very large preexponential factors, which allow the endothermic detachment reactions to occur despite high activation energy. Interpreted in terms of the transition state theory and taking the collision frequency as 1013 s 1- these preexponential factors give activation entropies 100 to 200 J/(mole.K), depending on the solute and the solvent. 

The estimation based on the equations of the parabolic model indicates that a reaction of the type (ArO + H02 —> ArOH + 02) involving phenoxyl radicals also requires no activation energy (in this case, AH> A emin = 57kJ mol-1). However, the addition of the peroxyl radical to the aromatic ring of the phenoxyl radical occurs very rapidly. Hence, the rate constant for this reaction is determined by diffusion processes. The data on the Ee0 values are also consistent with this. For the ArO + HOOR reactions with the O H O reaction center and for Am + HOOR reactions with the N H O reaction center, these values are 45.3 and 39.8 kJ mol-1, respectively [23]. At the same time, the calculation of the preexponential factor in terms of the parabolic model indicates that the rate constant k 7 for the reaction of ROOH with the participation of the aminyl radical is several times higher than that for the reaction involving the phenoxyl radical, where the enthalpies of these reactions... 

To set up the calculation, we take a quartz sand of the same porosity as in the calculations in Section 26.1 and assume that the quartz reacts according to the same rate law (Eqn. 26.1). We let the rate constant vary with temperature according to the Arrhenius equation (Eqn. 26.7), using the values for the preexponential factor and activation energy given in Section 26.2. As in the previous section, we need only be concerned with the time available for water to react as it flows through the aquifer. We need not specify, therefore, either the aquifer length or the flow velocity. 

The simultaneous desorption peaks observed at 560-580 K in TPR are of reaction-limited desorption. The peak temperatures of these peaks do not depend on the coverage of methoxy species, indicating that the desorption rate (reaction rate) on both surfaces has a first-order relation to the coverage of methoxy species. Activation energy (Ea) and the preexponential factor (v) for a first-order process are given by the following Redhead equation [12] ... 

Equation (10.23) describes the relations of the preexponential factors to the analyte concentration in the same way as relative concentration of free and bound forms given by Eqs. (10.12) and (10.13). The preexponential factor analyte response function may be shifted toward lower or higher analyte concentrations compared to those obtained from the absorbance or/and intensity measurements (Figure 10.6) because of the apparent dissociation constant (Kd) given by Eq. (10.24). 

The Horiuti group treats the temperature coefficient of the rate differently from the way it is usually treated in TST. They clearly identify E as the experimentally observed activation energy, but according to TST [cf. Eq. (5)] the (E — RT) quantity of Eq. (52) is the enthalpy of activation. The RT term in Eq. (5) arises because the assumption that the Arrhenius plot is linear is equivalent to the assumption that the preexponential factor A of the Arrhenius equation is constant, whereas, according to TST, A always contains the factor (kT/h). In addition, the partition function factors of Table I are also part of A, and most of them are functions of T. Since the Horiuti group takes this temperature dependency of the preexponential factor into account, the factor exp[(5/2)(vi -I- V2)] (where 5/2 is replaced by 3 for nonlinear molecules) arises. 

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