Kinetics fractional kinetic orders

Table 2. Association Numbers and Fractional Kinetic Orders for Alkyllithium Initiators...

The kinetics of initiation reactions of alkyllithium compounds often exhibit fractional kinetic order dependence on the total concentration of initiator as shown in Table 2. For example, the kinetics of the initiation reaction of //-butyUithium with styrene monomer in benzene exhibit a first-order dependence on styrene concentration and a one-sixth order dependence on //-butyUithium concentration as shown in equation 13, where is the rate constant for... 

Steady-state approximation. Fractional reaction orders may be obtained from kinetic data for complex reactions consisting of elementary steps, although none of these steps are of fractional order. The same applies to reactions taking place on a solid catalyst. The steady-state approximation is very useful for the analysis of the kinetics of such reactions and is illustrated by Example 5.4.2.2a for a solid-catalysed reaction. 

There might be various reasons that lead to finding an apparent instead of the true activation energy. The use of power-law kinetic expressions can be one of the reasons. An apparent fractional reaction order can vary with the concentration, i.e. with conversion, in one experimental run. Depending upon the range of concentrations or, equivalently, conversions, different reaction orders may be observed. As an example, consider the a simple reaction ... 

It is unfortunate that many workers have not appreciated how essential a clue to the kinetics can be provided by the kinetic order of the whole reaction curve. The use of initial rates was carried over from the practice of radical polymerisation, and it can be very misleading. This was in fact shown by Gwyn Williams in the first kinetic study of a cationic polymerization, in which he found the reaction orders deduced from initial rates and from analysis of the whole reaction curves to be signfficantly different [111]. Since then several other instances have been recorded. The reason for such discrepancies may be that the initiation is neither much faster, nor much slower than the propagation, but of such a rate that it is virtually complete by the time that a small, but appreciable fraction of the monomer, say 5 to 20%, has been consumed. Under such conditions the overall order of the reaction will fall from the initial value determined by the consumption of monomer by simultaneous initiation and propagation, and of catalyst by initiation, to a lower value characteristic of the reaction when the initiation reaction has ceased. 

Fractional kinetic orders of homogenous reactions in solution may point to association of a particular reagent. The kinetics of the initiation step of styrene polymerization in the presence of n-BuLi (equation 33) is in accordance with the assumption that this organolithium compound in a nonbonding solvent forms aggregates of six molecules on the average" . 

Later, Smith and coworkers succeeded in measuring rate constants of the reaction of MeLi with a carbonyl compound at various reagent concentrations with a stopped-flow/rapid scan spectroscopic method, and demonstrated that the reaction also exhibited a fractional kinetic order . Thus, the reaction of 2,4-dimethyl-4 -methylmercaptobenzophenone with MeLi in diethyl ether at 25 °C showed one-fourth order in MeLi in the concentration range of MeLi between 3.9 mM and 480 mM (Figure 1). The rate constant was 200 7 M s . Under these conditions, the monomer was considered the reactive species that exists in equilibrium with the tetramer. Addition of LiBr or Lil depressed the reaction rate but did not change the kinetic order. The same... 

Thus kp for lithium counterion is 1/300 of kp for potassium counterion. The low reactivity and association of lithium alkoxide was reported in the anionic polymerization of epoxides.We have found that two fold increase of the lithium initiator concentration has led to a decrease of the kp nearly to one half. This indicates that the kinetic order with respect to the initiator would be near to zero, suggesting a very high degree of association of the active species. Thus the propagation reaction appears to proceed in practice through a very minor fraction of monomeric active species in case of lithium catalyst. 

The kinetics of initiation reactions of alkyllithium compounds often exhibit fractional kinetic order dependence on the total concentration of initiator, consistent with initiation by the unassociaied form or the alkyllithium. 

The kinetics of catalytic reactions in a number of cases also agrees only qualitatively with the equations obtained on the basis of the model of an ideal adsorption layer experimental data often lead to fractional reaction orders (such as cannot be accounted for by dissociative adsorption). 

The independence of a on IS pressure confirms that IS does not capture nor release electrons, whereas the fractional kinetic order 0.35 shows that IS reacts in an adsorbed phase, since this value is very close to the apparent order of adsorption 0.3 found for the surface coverage in IS according to a Langmuir model in the pressure range investigated (13-60 kPa). The o- Pq relationship corroborates that OJ species control the adsorption equilibrium for the pressures chosen, while 0T sites are saturated. Since A is unaffected by oxygen pressure, it is deduced that the active oxygen species are associated with 0T ion-radicals. 

The kinetic orders in eqn. (62) are also unusual, being entirely predictable from the stoichiometric coefficients of the reaction. The resulting orders are mostly fractional. One could easily misunderstand this outcome as indicating Freundlich adsorption of the reactants, a warning against too facile an... 

As seen in Table 2.1, the overall order of an elementary step and the order or orders with respect to its reactant or reactants are given by the molecularity and stoichiometry and are always integers and constant. For a multistep reaction, in contrast, the reaction order as the exponent of a concentration, or the sum of the exponents of all concentrations, in an empirical power-law rate equation may well be fractional and vary with composition. Such apparent reaction orders are useful for characterization of reactions and as a first step in the search for a mechanism (see Chapter 7). However, no mechanism produces as its rate equation a power law with fractional exponents (except orders of one half or integer multiples of one half in some specific instances, see Sections 5.6, 9.3, 10.3, and 10.4). Within a limited range of conditions in which it was fitted to available experimental results, an empirical rate equation with fractional exponents may provide a good approximation to actual kinetics, but it cannot be relied upon for any extrapolation or in scale-up. In essence, fractional reaction orders are an admission of ignorance. 

The enolate has a direct impact on the polymerization kinetics. Indeed, the kinetics order with respect to [P ] is 1 when /sTda[P ] < 1 (equation 17) and fractional order with respect to [P ] is the rule when 7sTda[P ] 1 (equation 18). A fractional order in initiator is thus the signature of aggregation. 

This is a simple second order equation which usually applies to description of dissociative chemisorption from the gas phase on the homogeneous surface. However, Eq. (43a) describes well only an initial part of kinetic chemisorption isotherm for interaction of trimethylchlorosilane on dehydrated pyrogenic silica [113] and mixed alumina-silica and titania-silica [114] surface. At the same time, whole kinetic chemisorption isotherms are described using equations derived for the heterogeneous surface. Thus, high fractional reaction orders in respect to the silica surface OH groups obtained by Hertl and Hair for chemisorption of silanes and siloxanes [106,107] and in other studies [113,114] may be explained by heterogeneity of the oxides surface. 

By analogy with the mechanism proposed for the solvated anions, these fractional kinetic orders can be ascribed to association-dissociation phenomena involving the growing chains. Because ionic dissociation is not a viable assumption for such low dielectric media, ion-pair association is assumed. Thus,... 

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