Computed activation energy

Table 10.4 lists the values of trap density and binding energy obtained in the quasi-ballistic model for different hydrocarbon liquids by matching the calculated mobility with experimental determination at one temperature. The experimental data have been taken from Allen (1976) and Tabata et ah, (1991). In all cases, the computed activation energy slightly exceeds the experimental value, and typically for n-hexane, 0/Eac = 0.89. Some other details of calculation will be found in Mozumder (1995a). It is noteworthy that in low-mobility liquids ballistic motion predominates. Its effect on the mobility in n-hexane is 1.74 times greater than that of diffusive trap-controlled motion. As yet, there has been no calculation of the field dependence of electron mobility in the quasi-ballistic model. 

Fig. 19. Relationship between computed activation energies (B3LYP/6-311+G ) and p-orbital contribution in the NLMO hybridization of the lone pairs on the coordinating atoms (156).

Computed Activation Energies and Their Components, in kcal/mol, for Reactions 21.9,a 21.8,b and 21.11c... 

CaviPro and CaviMax processors, 28-29 cavitational heating, 31 computed activation energy, 30 degree of in situ calcination, 28 reaction rate calculation, 30 sample from lowest temperature oven calcination, 29-30 shock wave, 30... 

T3rpically, surface sites constituted of atoms with a low degree of coordinative saturation are the most reactive (25). The lowest computed activation energy for CO dissociation has been found for the Ru(1121) surface, with a reported value of 60 kj/mol (7). The activation energy for CO dissociation on a comparable site on a cobalt surface is 40 kJ/mol higher. 

Figure 12 shows computed activation energies and relative energies of reaction intermediates for "Ci hydrogenation. Two different ruthenium surfaces are considered. It is important to realize that the relative stabilities of the various CH species on the two surfaces differ considerably. These differences in stability are crucial because they determine the relative concentrations of the CH species, which in turn control to a significant extent the relative rates of the CH —CHj, recombination reactions. 

We observed a very good agreement of the computed activation energies with experiment. 

TABLE 9.2 Computed Activation Energy (in kcal mol for Reaction 9.3 within the Citrate Synthase Enzyme... 

The major advance of the past decade is that, using quantum-chemical computations, activation energies (Eact) as well as activation entropies (AS ) can be predicted a priori for systems of catalytic interest. This implies much more reliable use of the transition-state reaction rate expression than before, since no assumption of the transition state-structure is necessary. This transition-state structure can now be predicted. However, the estimated absolute accuracy of computed transition states is approximately of the order of 20-30 kJ/mol. Here, we do not provide an extensive introduction to modern quantum-chemical theory that has led to this state of affairs excellent introductions can be found elsewhere [38,39]. Instead, we use the results of these techniques to provide structural and energetic information on catalytic intermediates and transition states. 

The elementary rate constant for proton activation is weakly dependent on the micropore size as long as steric constraints do not affect the transition state. Because of the zwitterionic nature of the transition state, dielectric screening by the oxygen atoms of the micropore tends to decrease the cluster-calculated transition state energies to 10 to 30% of the activation energies. Steric constraints on the transition state may substantially increase the cluster-computed activation energies by similar amounts. These steric constraints can be computed from periodical DFT calculations or from transition-state model structures using Monte Carlo adsorbate-zeolite pore interaction calculations. 

The computed activation energy of 19.6 kcal/mol in vacuum at 298 K is in excellent agreement with the experimental value of 19.7 kcal/mol [70]. In toluene, the experimental activation enthalpy is 15.8 1.4 kcal/mol, with an activation entropy of —38 4 cal/mol K [71]. Therefore the computed 19.1 kcal/mol gas phase activation enthalpy of acrolein and butadiene should be lowered by 3.3 kcal/mol in toluene. In a related Diels-Alder reaction of cyclopentadiene and methyl acrylate, the experimental activation enthalpy is 15.1 kcal/mol in toluene, which is ca. 0.7 kcal/mol lower than that reported for acrolein and butadiene [81],... 

The computed activation energies for the combination reaction suggest the following order in which the metals facilitate the reaction,... 

Let us now evaluate exo-endo selectivity in cyclopropene addition to benzo[c]heterocycles. On the basis of bond order analysis, the exo addition of cyclopropene to benzo[c]furan was selected over the endo addition while in the case of benzo[c]pyrrole and benzo[c]thiophene, it was suggested that SOI might be responsible for formation of an exo cycloadduct product. The computed activation energies for cyclopropene addition to the benzo[c]-fused heterocycle clearly favors formation of the exo cycloadduct. The activation barrier with benzo[c]furan was... 

The AM 1 computed activation barrier for the elimination of nitrogen from the cycloadduct between cyclooctyne and 2,5-bis(trifluoromethyl)-l,3,4-oxadiazole was near 7.2 kcal/mol. The fact that the AMI computational method overestimated activation barriers, the activation barrier for nitrogen elimination must be even smaller than 7.2 kcal/mol. In any case, computed activation energy for decomposition of this cycloadduct was substantially lower than it was for its formation. Therefore, we can safely state that trapping a bicyclic product would be very difficult and that the product of the reaction is definitively a derivative of furan as outlined in Scheme 11. 

At this point, it is relevant to note that theoretical wodc using ab initio MO-CI methods indicates that the reactions of twisted singlet excited ethylene, to give directly either the lowest singlet state of vinylidene+H2 or ethylidene, are easy pathways that have relatively low computed activation energies, in partial contradiction with Laufer s observation reported above. The liagmentadon of... 

Choo and Benson (1981) estimated the pieexponential factors for the rate coefficients of alkoxyl radical decay by comparison with similar reactions. With these values and measured rate coefficients, they computed activation energies. Their results are listed in Table 12. The preexponential factors are all given by log A, sec = 13.6-14.1, considerably lower than the measured values. Drew and Kerr (1983) pointed out that the values of Choo and Benson are about 10 times lower than predicted from equilibrium constants at 423 K. 

A three-stage mechanism of extraction has been proposed and mathematical equations derived to fit this concept. In a study of the extraction of bituminous coals at 200°C-400°C (390°F-750°F) with polycyclic compounds, the kinetics of extraction up to SSO C (660 F) were computed activation energies between 14 and 40 kcal/mol were derived and were dependent on the rank of the coal and on the particular solvent employed. 

The computed activation energy for this was 32 kcal/mol [64], which is less than for any of the other processes investigated for either PETN or Si-PETN. In addition, this rearrangement is overall quite exothermic, with a predicted AE of about —45 kcal/mol, and therefore can provide energy for further decomposition. In the case of PETN, on the other hand, the corresponding process has a much higher activation barrier of about 80 kcal/mol. 

CO2 diffusivity can be calculated from the estimated computational activation energy, using the following equation [45] ... 

The success of silica potential evolution is demonstrated by Litton and Garofalini,29 who investigated the kinetics of Si and O self-diffusion in the bulk of vitreous silica. These authors report computed activation energies for self-diffusion in bulk vitreous silica in good agreement with experimental values obtained from diffusion-coupled tracer studies. 

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