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A related problem is that neither concentration nor activity may be the appropriate measure of the ability of the reaction system to donate a proton to the carboxyl group The acidity function h0 is often the more appropriate measure of acidity for nonaqueous systems or systems containing high acid concentrations [Ritchie, 1990] Unfortunately, the appropriate h0 values are not available for polymerization systems Yet another possiblity for the nonlinearity in the low conversion region is the decrease in the volume of the reaction mixture with conversion due to loss of one of the products of reaction (water in the case of esteri cation) This presents no problem if concentration is plotted against time as in Eq 2-20 However, a plot of 1= 1 p 2 against time (Eq 2-22) has an inherent error since the formulation of Eq 2-21 assumes a constant reaction volume (and mass) [Szabo-Rethy, 1971].



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(6.20) (6.21)

Elias [1985] derived the kinetics of step polymerization with correction for loss of water, but the results have not been tested It is unclear whether this effect alone can account for the nonlinearity in the low conversion region of esteri cation and polyesteri cation 2-2a-2-b High-Conversion Region The nonlinearity observed in the third-order plot in the nal stages of the polyesteri cation (Fig 2-1) is probably not due to any of the above mentioned reasons, since the reaction system is fairly dilute and of relatively low polarity Further, it would be dif cult to conjecture that the factors responsible for nonlinearity at low conversions are present at high conversion but absent in between It is more likely that other factors are responsible for the nonlinear region above 93% conversion.





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From these two equations, we can write the first-order expansion in the fine structure constant of equation (6.18) as (6.22) where As,a is given by

Polyesteri cations, like many step polymerizations, are carried out at moderate to high temperatures not only to achieve fast reaction rates but also to aid in removal of the small molecule by-product (often H2O) The polymerization is an equilibrium reaction and the.

In the last step in (6.23), we have used equation (6.20) and neglected the contribution to mass renormalization in (6.23). Substituting (6.23) in (6.22) and taking the expectation value for the case where initially the atom is in an arbitrary atomic energy levella} and the field in the vacuum state, we obtain (6.24) In the case of the vacuum contribution (6.19), Avac is of zero order in the coupling constant, so the atomic operators have to be computed up to first order in the coupling constant. It is convenient to write the perturbation expansion of (6.19), up to first order in the fine structure constant, as

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equilibrium must be displaced to the right (toward the polymer) to achieve high conversions (synonymous with high molecular weights) Partial vacuum (often coupled with purging of the reaction system by nitrogen gas) is usually also employed to drive the system toward high molecular weight Under these conditions small amounts of one or the other or both reactants may be lost by degradation or volatilization In the case of polyesteri cation, small degradative losses might arise from dehydration of the diol, decarboxylation of the diacid, or other side reactions (Sec 2-8d) Although such losses may not be important initially, they can become very signi cant during the later stages of reaction Thus a loss of only 03% of one reactant can lead to an error of almost 5% in the concentration of that reactant at 93% conversion.

H atom ) vac = -

Kinetic studies on esteri cation and polyesteri cation have been performed under conditions which minimized the loss of reactants by volatilization or side reactions [Hamann et al, 1968] An ester or polyester was synthesized in a rst stage, puri ed, and then used as the solvent for a second-stage esteri cation or polyesteri cation The initial reactant concentrations of the second-stage reactions corresponded to 80% conversion relative to the situation if the rst-stage reaction had not been stopped but with an important difference the reactant concentrations are accurately known The second-stage reaction showed third-order behavior up to past 98 99% conversion (higher conversions were not studied) This compares with the loss of linearity in the third-order plot at about 93% conversion if the rst-stage reactions were carried out without interruption Another possible reason for the observed nonlinearity is an increase in the rate of the reverse reaction.

2~qc1i {[Vdoul' Pal' Avac(O) + Avac(O) . [V6oul> Pal} ,

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