Matrix of rate constant

Direct calculation of the whole matrix of rate constants is a rather difficult problem, even if the intermolecular potential is well known. Actually, it was done only once for a N2-Ar mixture in the semiclassical centrifugal... 

Figure 12b illustrates an example of estimating relative rate constants for HDS of dibenzothiophene catalyzed by a NiMoSjr/Al203 using wellspaced conversion experiments. The data point at very high conversion is essential in obtaining a unique solution to the overall matrix of rate constants as the data obtained below 90% conversion could be simulated with... 

Example 2.17 Matrix of rate constants for consecutive reactions... 

Let us first consider that only the species A is excited, and that the fluorescence decays of A, B and C can be measured in independent experiments (the presence of emission overlap will be discussed latter). Then, each of the three rows of the experimental pre-exponentials matrix A is affected by a constant depending on (1) the number of counts accumulated, (2) the fraction of the total emission collected at the measurement wavelength and (3) the instrumental response at that wavelength, i.e. Ajj = 5,fly. The relation between matrixes A and a is given by Eq. (15.52), and its substitution in Eq. (15.51) yields the explicit relation of the rate constants matrix to the experimental matrixes of rate constants X and preexponential coefficients, A (Eq. 15.53). 

Let us now analyse the information that can be extracted from steady-state fluorescence data. First we note that the matrix of rate constants k (Eq. 15.57) contains two pseudo-unimolecular rate constants, k.iEH" ] and (Eq. 15.58,... 

The data analysis can be carried out by calculating the matrix of rate constants k with an initial guess of Si, S2 and kpx and optimising these values by minimisation of the differences between the experimental and calculated values of k, kA, which result of intercepts 1 and 2 and slopes 1 and 2 in Eqs. (15.59) and (15.60). 

The overall reaction network results in ten-simultaneous equations which can be solved in a step-wise fashion due to the irreversible reactions. The matrix of rate constants is lower triangular involving 20 kinetic constants (plus deactivation function parameters). 

The fundamental rate constants will (almost) always be denoted by the symbol k, with subscripts to indicate different rate constants for example it is common to use and to indicate the forward and backward steps of an individual transition, or to use kg for the rate of the transition from the ith state to the yth state. As the reader will have noticed, the latter form is used in this volume. An equilibrium constant, the ratio of two rate constants, will normally be denoted by an upper case K, with suitable subscripts. The measured time constants, t, will be quoted with a subscript where necessary. The relation between k, t and k, the eigenvalues of the matrix of rate constants, was discussed in section 2.1 and will be illustrated in sections 4.2 and 5.1. 

Since Cj t) and Cg(r) are not zero, the determinant of the matrix of rate constants must be zero and the characteristic equation can be written as ... 

Comparison of Rate Constants and Activation Energies of Bimolecular Reactions in Solution and Polymer Matrix... 

Comparison of Rate Constants and Steric Factors of Bimolecular Reactions with Frequency of Reactant Rotation and Orientation in Polymer Matrix (T = 300 K)... 

FIGURE 19.2 The correlation of rate constants of various free radical reactions with molecular mobility of nitroxyl radical in the polymer matrix of different polymers with addition of plastificator I in IPP, II in preliminary oxidized IPP, III in PE, and IV in PS. Line 1 for the reaction of 2,6-bis(l,l-dimethy-lethyl)phenoxyl radical with hydroperoxide groups at T — 295 K line 2 for the reaction of 2,2,6, 6-tetramethyl-4-bcnzoyloxypiperidinc-/V-oxyl with 1-naphthol at T = 333 K line 3 for the reaction of 2,2,6,6-tetramethyl-4-benzoyloxypiperidine-iV-oxyl with 2,6-bis(l,l-dimethylethyl)phenol at T = 333 K line 4 for the same reaction at 7 — 303 K line 5 for the same reaction at T = 313 K and line 6 for the same reaction at T — 323 K [18]. 

Dependence of Rate Constant on the Volume of Reactants for Bimolecular Reactions of Nitroxyl Radicals with Phenols in Polymer Matrix [8,10,11]... 

The mechanism of antioxidant action on the oxidation of carbon-chain polymers is practically the same as that of hydrocarbon oxidation (see Chapters 14 and 15 and monographs [29 10]). The peculiarities lie in the specificity of diffusion and the cage effect in polymers. As described earlier, the reaction of peroxyl radicals with phenol occurs more slowly in the polymer matrix than in the liquid phase. This is due to the influence of the polymeric rigid cage on a bimolecular reaction (see earlier). The values of rate constants of macromolecular peroxyl radicals with phenols are collected in Table 19.7. 

It should be taken into account that the reaction of chain propagation occurs in polymer more slowly than in the liquid phase also. The ratios of rate constants kjlkq, which are so important for inhibition (see Chapter 14), are close for polymers and model hydrocarbon compounds (see Table 19.7). The effectiveness of the inhibiting action of phenols depends not only on their reactivity, but also on the reactivity of the formed phenoxyls (see Chapter 15). Reaction 8 (In + R02 ) leads to chain termination and occurs rapidly in hydrocarbons (see Chapter 15). Since this reaction is limited by the diffusion of reactants it occurs in polymers much more slowly (see earlier). Quinolide peroxides produced in this reaction in the case of sterically hindered phenoxyls are unstable at elevated temperatures. The rate constants of their decay are described in Chapter 15. The reaction of sterically hindered phenoxyls with hydroperoxide groups occurs more slowly in the polymer matrix in comparison with hydrocarbon (see Table 19.8). 

In the chemical reaction networks that we study, there is no small parameter with a given distribution of the orders of the matrix nodes. Instead of these powers of we have orderings of rate constants. Furthermore, the matrices of kinetic equations have some specific properties. The possibility to operate with the graph of reactions (cycles surgery) significantly helps in our constructions. Nevertheless, there exists some similarity between these problems and, even for... 

The Jacobian matrix (81) should have the lower triangle form in coordinates a, (75) for all nonnegative values of rate constants and concentrations. This is equivalent to the lower triangle form of all matrices M,y in these coordinates. Because usually there are many zero matrices among Mrj, it is convenient to describe the set of nonzero matrices. 

During the selectivity kinetic parameter estimation, the relationship for x in terms of C5 - is determined from Eq. (12). For an assumed set of rate constants K, x is calculated for each composition data point such that the experimentally measured C5- equals that estimated from Eq. (12). Selectivity composition profiles as a function of C5- are generated in this manner. The proper selectivity matrix K will be that which minimizes the deviation between experimental and predicted profiles for the hydrocarbons other than C5-, as illustrated in Fig. 10. 

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