Ideal kinetic model

The processes of the formation and transformation of polymer molecules proceed as a result of chemical reactions of their active centers, whose role can be played by functional groups, free valences in radicals, double bonds, and so on. Often, it may be suggested that the reactivity of the 

The Flory principle makes it possible in a simple way to relate the rate constants of the reactions of macromolecules (whose number is infinite) with the corresponding rate constants of elementary reactions. Since according to this principle all chemically identical active centers are kine-tically indistinguishable, the rate constant of the reaction between any two molecules is proportional to the rate constant of the reaction between their active centers and numbers of these centers in reacting molecules. Therefore, only a few rate constants of elementary reactions will enter in the material balance equations as kinetic parameters. 

The Flory principle is one of two main assumptions underlying the ideal kinetic model of any processes of synthesis and chemical modification of polymers. The second assumption is the neglect of the reaction between any active centers belonging to the same molecule. Clearly, in the absence of such intramolecular reactions, molecular graphs of all components of a reaction system will not contain cycles. The last affirmation applies just to sol molecules. As for the gel, in the framework of the ideal model, the cyclization reaction is admissible. 

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When there is a need to calculate only the statistical moments of the distribution of molecules for size and composition, rather than to find the very distribution, the task becomes essentially easier. The fact is that for the processes of polymer synthesis which may be described by the ideal kinetic model the set of equations for the statistical moments is always closed. 

The kernel (26) and the absorbing probability (27) are controlled by the rate constants of the elementary reactions of chain propagation kap and monomer concentrations Ma(x) at the moment r. These latter are obtainable by solving the set of kinetic equations describing in terms of the ideal kinetic model the alteration with time of concentrations of monomers Ma and reactive centers Ra. 

Monomers employed in a polycondensation process in respect to its kinetics can be subdivided into two types. To the first of them belong monomers in which the reactivity of any functional group does not depend on whether or not the remaining groups of the monomer have reacted. Most aliphatic monomers meet this condition with the accuracy needed for practical purposes. On the other hand, aromatic monomers more often have dependent functional groups and, thus, pertain to the second type. Obviously, when selecting a kinetic model for the description of polycondensation of such monomers, the necessity arises to take account of the substitution effects whereas the polycondensation of the majority of monomers of the first type can be fairly described by the ideal kinetic model. The latter, due to its simplicity and experimental verification for many systems, is currently the most commonly accepted in macromolecular chemistry of polycondensation processes. 

When the statistical moments of the distribution of macromolecules in size and composition (SC distribution) are supposed to be found rather than the distribution itself, the problem is substantially simplified. The fact is that for the processes of synthesis of polymers describable by the ideal kinetic model, the set of the statistical moments is always closed. The same closure property is peculiar to a set of differential equations for the probability of arbitrary sequences t//j in linear copolymers and analogous fragments in branched polymers. Therefore, the kinetic method permits finding any statistical characteristics of loopless polymers, provided the Flory principle works for all chemical reactions of their synthesis. This assertion rests on the fact that linear and branched polymers being formed under the applicability of the ideal kinetic model are Markovian and Gordonian polymers, respectively. 

The above-described "labeling-erasing" procedure is in common use in statistical chemistry of polymers (Kuchanov, 2000). It gives a chance to obtain a number of important theoretical results under kinetic modeling of polymerization and polycondensation processes, where the deviation from their description in terms of the ideal kinetic model is due to the short-range effects. 

The above reasoning allows a conclusion that once a researcher has decided upon the particular ideal kinetic model of polycondensation, he or she will be able to readily calculate any statistical characteristics of its products. The only thing he or she is supposed to do is to find the solution of a set of several ordinary differential equations for the concentrations of functional groups, using then the expressions known from literature. 

The active centers in this process are free radicals, whose reaction with double bonds of monomers leads to the growth of a polymer chain. In the framework of the ideal kinetic model, the reactivity of a macroradical is exclusively governed by the type of its terminal unit. According to this model, the sequence distribution in macromolecules formed at any moment is described by the Markov chain with elements controlled by the instantaneous composition of the monomer mixture in the reactor as... 

If the rate constant k of the elementary reaction of transformation A > B is supposed to be the same for all groups, the pattern of arrangement of units in macromolecules will be perfectly random. However, such an ideal kinetic model is not appropriate for a vast majority of real polymers because of the necessity to take into consideration under mathematical modeling of PARs proceeding in their macromolecules the short-range and long-range effects. The easiest way to take account... 

Figure 15.16. Broecker s (1974) idealized kinetic model for the marine cycles of biologically fixed elements. volume of river water entering the ocean per year expressed as volume per unit sea area (m m yr or m yr ) = 0.1 m yr Chv, concentration of an element in average river water (mol m ) x Cnv, input flux (mol yr ) Fmix. volume of water sinking into deep water box = volume of water rising to surface water box (volume m yr ) = 2(X) m yr Q...

Kinetics is the science which deals with the mechanisms and rates of chemical reactions, and ideally kinetic models should be incorporated into geochemical models, along with thermodynamics. This is being done increasingly, and is the subject of Chapter 11. The rest of this chapter outlines those aspects of thermodynamics needed to understand geochemical models. 

Similar results, however, were interpreted by others differently. For instance, butyl acrylate and butyl propionate polymerizations in benzene also fail to. meet ideal kinetic models. The results, however, were explained in terms of terminations of primary radicals by chain transferring. 

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