Kinetic fractionation

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... 

When isotopes are fractionated kinetically during chemical reactions, the isotope ratio shift of the reaction products relative to the reactants often depends on reaction mechanisms and rates. This contrasts with isotopic fractionations between phases in isotopic equilibrium, where the isotopic differences are thermodynamic quantities and thus do not depend on reaction mechanisms or rates. In this section, we briefly review the well-developed theory for kinetic isotope effects that appears in the S isotope literature. This background serves as a guide for interpreting and predicting Se and Cr isotope systematics. 

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... 

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. 

Fractional dynamics is a made-to-measure approach to the description of temporally nonlocal systems, the kinetics of which is governed by a selfsimilar memory. Fractional kinetic equations are operator equations that are mathematically close to the well-studied, analogous Brownian evolution equations of the Klein-Kramers, Rayleigh, or Fokker-Planck types. Consequently, methods such as the separation of variables can be applied. More-... 

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. 

Before closing this chapter we would like to mention briefly a novel consideration of diffusion based on the recently developed concepts of fractional kinetics [29]. From our previous discussion it is apparent that if ds < 2, diffusion is recurrent. This means that diffusion follows an anomalous pattern described by (2.10) the mean squared displacement grows as (z2 (t)) oc t1 with the exponent 7 1. To deal with this, a consistent generalization of the diffusion equation (2.18) could have a fractional-order temporal derivative such as... 

Torget, R. W., Kim, J. S., and Lee, Y. Y., Fundamental aspects of dilute acid hydro lysis/fractionation kinetics of hardwood carbohydrates. 1. Cellulose hydrolysis. Industrial Eng Chem Res 2000, 39 (8), 2817-2825. 

G. M. Zaslavsky, Chaos, fractional kinetics and anomalous transport. Phys. Rep. 371,461 (2002). 

FIGURE 21.28 Effect of protein concentration on the evolution of protein fraction kinetic during chemical and electrochemical acidification. (Adapted from Bazinet, L., Ippersiel, D., and Mahdavi, B., Innovative Food Sci. Emerg. Technol., 5, 17, 2004.)... 

Krouse, H.R., McCready, R.G.L., Husain, S.A. and Campbell, J.N., 1968. Sulphur isotope fractionation kinetic studies of sulfite reduction in growing cells of Salmonella Heidelberg. Biophys. J., 8 109—124. 

I. Sokolov, J. Klafter, and A. Blumen, Fractional Kinetics. Phys. Today Nov. (2002). 

In like manner, combining the ideas embodied in the fractional diffusion Eq. (90) describing Cole-Cole relaxation and Eq. (96) describing Cole-Davidson relaxation, we may introduce the fractional kinetic equation... 

It thus appears, unlike the fractional kinetic equation of Section IV.A, namely Eq. (235), that the Barkai-Silbey [30] kinetic equation, Eq. (253), can provide a physically acceptable description of the high-frequency dielectric absorption behavior of an assembly of fixed axis rotators. The explanation of this appears to be the fact that in the equation proposed by Barkai and Silbey, the form of the Boltzmann equation, for the single-particle distribution function, is preserved that is, the memory function of which the fractional derivative is an example does not affect the Fiouville terms in the kinetic equation. Exactly the same conclusions apply to an assembly of rotators, which may rotate in space. 

Saichev, A.I., and G.M. Zaslavsky. 1997. Fractional kinetic equations Solutions and applications. Chaos 7 753-764. 

Zaslavski, G. M. and Edelman, M. (2000). Hierarchical structures in the phase space and fractional kinetics I classical systems. Chaos, 10 135-145. 

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,... 

The interfacial mechanism is characterized by (1) increased rates with increased electrostaticity of catalyst, (2) reaction rates are dependent on agitation rate, (3) fractional kinetic order with respect to the catalyst concentration, and (4) the value of substrate acidity is in the range 16-23. 

Since n-butyllithium is aggregated predominantly into hexamers in hydrocarbon solution [44], the fractional kinetic order dependency of the initiation process on the total concentration of initiator has been rationalized on the basis that unassociated n-butyllithium is the initiating species and that it is formed by the equilibrium dissociation of the hexamer as shown in Scheme 7.8. 

The kinetics of initiation for styrene and diene polymerization by alkyllithium compounds generally exhibit a fractional kinetic order dependence (e.g., 1/4 or 1/6) on the concentration of alkyllithium initiator. This can be rationalized in terms of the following steps ... 

A good example of this kinetic behavior was found in the study of the n-butyllithium-styrene system in benzene, in which a kinetic order dependency on n-butyllithium concentration was observed, consistent with the predominantly hexameric degree of association of n-butyllithium (Worsfold and By water, 1960). However, this expected correspondence between the degree of association of the alkyllithium compound and the fractional kinetic order dependence of the initiation reaction on alkyllithium concentration was not always observed (Young et al., 1984). One source of this discrepancy is the assumption that only the unassociated alkyllithium molecule can initiate polymerization. With certain reactive initiators, such as 5 c-butyllithium in hexane solution, the initial rate of initiation exhibits approximately a first-order dependence on alkyllithium concentration, suggesting that the aggregate can react directly with monomer to initiate polymerization (Bywater and Worsfold, 1967a). A further... 

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