Ideal Kinetics

A given polymerization conforming strictly to the reaction scheme (6.3) to (6.10) is commonly considered as an ideal polymerization. Any system deviating from this pattern of reaction is to be considered as a case of nonideal polymerization. The ideal behavior prescribes constancy of the term / [l] [M] ), according to Eq. (6.26) expressed as 

But studies of the kinetics of polymerization of different monomers, under different conditions and chemical environments, indicate that ideal behavior is probably more an exception than the rule. Most practical free-radical polymerizations will deviate to a greater or lesser extent from the standard conditions outlined in the reaction scheme (6.3) to (6.10) either because the actual reaction conditions are not entirely as postulated in the ideal kinetic scheme or because some of the assumptions that underlie the ideal scheme are not valid. 

According to the ideal kinetic scheme, which yields Eq. (6.161), the initiation rate and initiator efficiency / are independent of monomer con- 

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Photopolymerization of MMA was also carried out in the presence of visible light (440 nm) using /3-PCPY as the photoinitiator at 30°C [20]. The initiator and monomer exponent values were calculated as 0.5 and 1.0, respectively, showing ideal kinetics. An average value of kp /kt was 4.07 x 10 L-mol -s . Kinetic data and ESR studies indicated that the overall polymerization takes place by a radical mechanism via triplet carbene formation, which acts as the sources of the initiating radical. 

The Flory principle is one of two assumptions underlying an ideal kinetic model of any process of the synthesis or chemical modification of polymers. The second assumption is associated with ignoring any reactions between reactive centers belonging to one and the same molecule. Clearly, in the absence of such intramolecular reactions, molecular graphs of all the components of a reaction system will contain no cycles. The last affirmation concerns sol molecules only. As for the gel the cyclization reaction between reactive centers of a polymer network is quite admissible in the framework of an ideal model. 

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. 

Poly(acrylic acid) is not soluble in its monomer and in the course of the bulk polymerization of acrylic acid the polymer separates as a fine powder. The conversion curves exhibit an initial auto-acceleration followed by a long pseudo-stationary process ( 3). This behaviour is very similar to that observed earlier in the bulk polymerization of acrylonitrile. The non-ideal kinetic relationships determined experimentally in the polymerization of these two monomers are summarized in Table I. It clearly appears that the kinetic features observed in both systems are strikingly similar. In addition, the poly(acrylic acid) formed in bulk over a fairly broad range of temperatures (20 to 76°C) exhibits a high degree of syndiotacticity and can be crystallized readily (3). 

In an ideal kinetic resolution (common in enzyme-catalyzed processes), one enantiomer of a racemic substrate is converted tvhile the other is unreactive [70]. In such a kinetic resolution of 5-methyl-2-cyclohexenone, even with 1 equivalent of Me2Zn, the reaction should virtually stop after 50% conversion. This near perfect situation is found with ligand 18 (Fig. 7.10) [71]. Kinetic resolutions of 4-methyl-2-cyclohexenone proceed less selectively (s = 10-27), as might be expected from the lower trans selectivity in 1,4-additions to 4-substituted 2-cyclohexenones [69]. 

Bicyclohexyl groups act as an ideal kinetic protector of triplet carbene not only by quenching the intramolecular hydrogen-donating process but also by inhibiting dimerization of the carbene center. 

An ideal kinetic study would be made under conditions where the product is only graphite fluoride or polycarbon monofluoride with no byproducts formed. In terms of reaction kinetics, one method to follow the reaction is to measure the weight change as a function of the reaction time. Using this method the reaction rate of fluorine with carbon can be evaluated. Various carbon structures have been employed with sufficient fluorination contact time provided at a particular temperature for the carbon to reach fluorine saturation. The weight increase vs the temperature can be monitored at atmospheric pressure. Figure 515 shows the carbon structure and the temperature dependency of the fluorination reaction of various graphites. 

A similar method of analysis of transient state diffusion kinetics has been propos-ed 144,1491 based on the consideration that, in any experiment, the kinetic behaviour of the system represented by S(X), DT(X) will generally deviate from that of the corresponding ideal system represented by Se, De in either of two ways (i) ideal kinetics is obeyed, but with a different effective diffusion coefficient D , where n = 1,2,... denotes a particular kinetic regime (Dn is usually deduced from a suitable linear kinetic plot) or (ii) ideal kinetics is departed from, in which case one is reduced to comparison between the (non-linear) experimental plot and the corresponding calculated ideal line. 

In all late-time regimes notably those represented by Eqs. (60), (62), (64) and (66) ideal kinetics is obeyed 144.15°.15i.154.159.l6l) whereas this is not so in the short-time regimes presented by Eqs. (59), (63) and (65) 150 ,51 154,163,64) (which convey essentially the same information).151 The short time behaviour of lower permeation curves represented by Eq. (61) appears to occupy an intermediate position, in the sense that ideal kinetics appears to be followed only to a first approximation151. The relation between permeation and symmetrical sorption indicated by Eq. (70) is also notable. The respective kinetics become very similar at long times154 as indicated by the relevant relations151) D2M = Ds = D7 = D8 and... 

Some preliminary kinetic analyses of transient sorption and permeation data have been reported144,149. Examples from kinetic regimes where ideal kinetics is expected to be obeyed are shown in Fig. 17. The linearity of the relevant plots is satisfactory and the values of D3, D4 and D , Dd determined from these plots deviate from De in opposite directions, as expected l50-151 even though the detailed spatial dependence of S and DT in the membrane in question is two-dimensional145) (cf. previous subsection) and is, therefore, more complicated than envisaged in the theoretical... 

The Ideal World Ideal Kinetics and Ideal Reactors... 

The approach suggested provides the possibility of generalizing similar formulae for /i, and G for the case of variable specific partial heat capacities, more complex equations of state, non-ideal kinetics at V, T = const., etc. [16, 17]. Note that, at constant heat capacity, T can be treated as a "substance [18]. However, to extend this analogy to the general case is incorrect. 

Column performance is best compared under ideal kinetic conditions. Hence, test systems should be chosen which will produce the best column performance (e.g., low viscosity mobile phases, low molecular weight solutes, etc.). 

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... 

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