Single event rate coefficients

The rate constants for isomerization steps are similar in the forward and the reverse directions. For convenience, we choose to parameterize the kinetic model in terms of the forward rate constants. In this respect, we use the concept of single-event rate coefficients developed by Froment (127). According to Froment, the rate constant for a particular step is obtained by multiplying a single-event rate coefficient by the number of single events, ne, possible for the reactant. The expression for ne is... 

Written according to Arrhenius the temperature dependency of the single event rate coefficient becomes... 

The Number of Single Event Rate Coefficients, Assuming that the rates of hydride shift and methyl shift do not depend upon the number of C-atoms in a chain, but only on the type of carbenium ions involved (secondary or tertiary), the following single event rate coefficients are retained for the isomerizations,... 

Assuming a planar structure for the activated complex, very similar to that of the olefin, leads to single event rate coefficients for protonation independent of the olefin, so that only two values have to be retained kp,(s) and kp,(t). 

Deriving the single event rate coefficients from experimental data requires the rates to be expressed in terms of observables. Baltanas et al (1989) and Froment (1991) derived the following equation for the rate of formation of a paraffin ... 

It should be added here that these huge reaction networks have to be generated by computer [Baltanas Froment, 1985]. Booleans relation matrices were used to describe the molecules and carbenium ions. Since a component analysis of a vacuum gas oil is not entirely feasible some lumping is inevitable but the rate coefficients for the reactions between the lumps can be constructed from those of the single events entering in the reactions of the components of the lump [Vynckier Froment, 1991]. 

Note that assumptions (2) and (3) are about timescales. Denoting by x, and tlz the characteristic times (inverse rates) of the electron transfer reaction, the solvent relaxation, and the Landau-Zener transition, respectively, (the latter is the duration of a single curve-crossing event) we are assuming that the inequalities Tr A Ts tlz hold. The validity of this assumption has to be addressed, but for now let us consider its consequences. When assumptions (1)—(3) are satisfied we can invoke the extended transition-state theory of Section 14.3.5 that leads to an expression for the electron transfer rate coefficient of the form (cf. Eq. 14.32)... 

The proportionality constant fe(T) is a function of the temperature of the system and is called the (binary) reaction rate coefficient. Thus, rate coefficient is a quantity defined for an ensemble of a sufficient number of particles, in contrast with cross-section which is a quantity characterizing a single event of collision. 

Polymerization rate and average polymer chain length are controlled by the relative rates of initiation, propagation, termination and chain transfer events in the system. Starting from these basic mechanisms, this section derives simple kinetic expressions for a single monomer system. Complicating (but industrially important) secondary reactions are then discussed, followed by extension of the kinetics to multi-monomer systems. Dispersed throughout are values for important FRP rate coefficients, and descriptions of how they are experimentally obtained. More extensive overviews of FRP kinetics and mechanisms are presented in references 1-3. 

This relation permits the calculation of the activation energy. Eg, for any elementary step or single event pertaining to a certain type, provided a, the transfer coefficient and E°, the intrinsic activation barrier of a reference step of that type are available. They are the only 2 independent rate parameters for this type of step. Use of modern quantum chemical packages, such as GAUSSIAN, is essential for the calculation of AHr, which is the difference between the heats of formation of reactant and product. 

The single event concept and the Evans-Polanyi relationship drastically reduce the number of independent rate coefficients and thus enable addressing the complex problems encountered in industrial processes. 

Experimental data lead to parameter estimates revealing that the (de) protonation steps are in quasi-equilibrium and also that the concentration of on the active sites of the zeolite is extremely low compared to the total concentration of sites, so that (2.4.4-10) reduces to Q= C. The consequence is that in this type of data collection the protonation equilibrium constant cannot be determined independently from the single event isomerization and cracking rate coefficients, leading to composite parameters for the latter. 

Reduction of the Number of Parameters. An initiation of the type represented by reaction (1) can occur twice, that represented by reaction (2) six times. In other words an elementary step like (1) consists of what may be called two single events, one like (2) of six single events. When the rate coefficients are expressed in terms of these single events, values are derived from the experimental data which are determined only by the nature of the C-atoms on the bond ruptured in the initiation and this is reflected in the type of radicals produced. Consequently, the following set of rate coefficients suffices to account for initiations in paraffins, regardless of their chain length ... 

From the technology of combustion we move to the molecular mechanism of flame propagation. We shall give a molecular-kinetic expression for the heat release rate by calculating the frequency v of collisions of fuel molecules with other molecules (v is proportional to the molecular velocity and inversely proportional to the mean free path), further taking into account that only a small (1/j/) part of all collisions are effective. The quantity 1/v—the probability of reaction taken with respect to a single collision— depends on the activation heat of an elementary reaction event, as well as on the fraction of all molecules comprised of those radicals or atoms by means of which the reaction occurs. The molecular-kinetic expression for the coefficient of thermal conductivity follows from formulas (1.2.4) and (1.2.3). 

If X6 is not a lacs, but X is zero, the last matrix row contributes one single term bi i2 23 34 45 caiCHcNCtPN with the same concentration co-factors as that in the fifth row, so that eqn 8.53 remains valid with only a different significance of fcd.J The first two steps of the cycle are likely to be at quasi-equilibrium. If so, the first denominator term in eqn 8.53 is negligible. The rate equation in one-plus form then has only three phenomenological coefficients. In any event, the reaction orders are plus one for nickel, between zero and plus one for HCN and 4-pentenenitrile, and between zero and minus two for the organic phosphine. 

The stoichiometric number uj, introduced by Horiuti (1962) for reactions with a single rate-controlling step, is the number of times that step occurs in an overall reaction j as written. This number is computable with reasonable precision from simultaneous observations of the forward and reverse reactions, and has been investigated for many reaction.systems. Note, however, that (jj is necessarily unity if one constructs the overall reaction j as a composite event in the manner just described. Thus, a reported stoichiometric number is simply the ratio of the investigator s chosen stoichiometric coefficients to those of the actual composite event, and is not needed further once the composite event has been constructed. 

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