Calculation of the Activation Energy

Within the framework of the transition state theory [112,113], the observed activation energy, Eobs, for a monomolecular catalytic process in the heterogeneous case is Eobs = E0 + A//ads [act. complex], where E0 is the energy of the reaction without a catalyst and A//.lds act. complex] is the adsorption enthalpy of the activated complex [114], In the monomolecular cracking of n-alkanes catalyzed by 

FIGURE 9.5 Fitting of Equation 9.21 with the data of the observed activation energy for the cracking of C4-C10 alkanes in H-ZSM-5 vs. the number of carbon atoms (m) reported in Ref. [100], 

The Physical Chemistry of Materials Energy and Environmental Applications 

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From the most humble, qualitative resonance depiction given by 5a and 5b to the advanced ab initio calculations of the activation energies of [1,3]-sigmatropic rearrangements of allylboranes21 (Eq. 5),... 

Ab initio Hartree-Fock and density functional theory (DFT) calculations were performed to study transition geometries in the intramolecular hetero-Diels-Alder cycloaddition reactions of azoalkenes 20 (LJ = CH2, NFI, O) (Equation 1). The order of the reactivities was predicted from frontier orbital energies. DFT calculations of the activation energies at the B3LYP level were in full agreement with the experimental results described in the literature <2001JST(535)165>. 

The IPM as a semiempirical model of an elementary bimolecular reaction appeared to be very useful and efficient in the analysis and calculation of the activation energies for a wide variety of radical abstraction and addition reactions [108-113]. As a result, it became possible to classify diverse radical abstraction reactions and to differentiate in each class the groups of isotypical reactions. Later this conception was applied to the calculations of activation energies and rate constants of bimolecular reactions of chain generation [114]. In the IPM, the radical abstraction reaction, for example,... 

The results of the calculation of the activation energies and the rate constants of peroxyl, alkoxyl, and alkyl radicals with alkanes and cycloalkanes are presented in Table 6.3-Table 6.5. 

The IPM parameters for hydrogen transfer atom in alkoxyl radicals are presented in Table 6.12. Isomerization proceeds via the formation of a six-membered activated complex, and the activation energy for the thermally neutral isomerization of alkoxyl radicals is equal to 53.4 kJ mol-1. These parameters were used for the calculation of the activation energies for isomerization of several alkoxyl radicals via Eqns. (6.7, 6.8, 6.12) (see Table 6.14). The activation energies for the bimolecular reaction of hydrogen atom (H-atom) abstraction by the alkoxyl radical and intramolecular isomerization are virtually the same. 

The quantum-chemical calculation of the activation energies of reactions EtOO + EtH and EtOO + EtOH by the DFT method supports this result [54]. For the reaction of EtOO with ethane, the calculated E value is 82 kJ mol-1, AH = 76 kJ mol-1, and Ee0 = 56 kJ mol-1. For the reaction of EtOO with ethanol is = 52.7. The structure of TS is presented in Figure 7.1. 

All these reactions are exothermic, and the AH values are negative. All these reactions should seemingly occur equally rapidly. The question to how easily the aminyl radicals react with the H—O and H—C bonds of the peroxyl radicals can be answered by analyzing these reactions in terms of the IPM model of free radical reaction (see Chapter 6). This model gives a tool to perform the calculation of the activation energy for a thermally neutral reaction of each class. Analysis of experimental data has shown (see Chapter 15) that, when aminyl... 

The estimation of TMRld from isothermal DSC experiments is illustrated by an example taken from Figures 11.4 and 11.5. For calculation of the activation energy, two reference points, 500 Wkg1 at 200°C and 90Wkg 1 at 170°C, are taken ... 

Calculations of the activation energies for the 1,3-shifts of substituents in a-imidoylketenes, which interconvert with o -oxoketenimines via a four-membered cyclic transition state, show that barriers are lowest for substituents with accessible lone pairs (halogens, OR, NR2) (Scheme 31).54... 

The structure of the TS for the studied SG SC mutual transformation reactions is shown in Figure 7.3b. The calculations of the activation energies for the forward and reverse reactions yield the values of 20 and 23 kcal/mol, respectively, in a satisfactory agreement with the experimentally measured activation barriers. 

The calculation of the activation energy is based on the approximation of the following integral equation, resulting from the combination of the Arrhenius equation and the formula for a kinetic desorption process ... 

Measurements of the rate constants for at least three different temperatures allows for the calculation of the activation energy and prediction of the temperature dependencies of the drug degradation based on the Arrhenius equation. The relationship between the rate constant and the temperature is given by the Arrhenius equation ... 

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