Reasons for Nonlinearity in Third-Order Plot

The failure to fit the data over the complete conversion range from 0 to 100% to a third-order plot has sometimes been ascribed to failure of the assumption of equal functional group reactivity, but this is an invalid conclusion. The nonlinearities are not inherent characteristics of the polymerization reaction. Similar nonhnearities have been observed for nonpolymerization esterification reactions such as esterifications of lauryl alcohol with lauric or adipic acid and diethylene glycol with caproic acid [Flory, 1939 Fradet and Marechal, 1982b]. 

The low conversion region fits an overall 2 reaction order better than it does 3-order. A change from 2 to 3-order as the reaction medium becomes less polar is compatible with a 

Another problem is the very high concentrations of reactants present in the low-conversion region. The correct derivation of any rate expression such as Eqs. 2-20 and 2-22 requires the use of activities instead of concentrations. The use of concentrations instead of activities assumes a direct proportionality between concentration and activity. This assumption is usually valid at the dilute and moderate concentrations where kinetic studies on small molecules are t5ipically performed. However, the assumption often fails at high concentrations and those are the reaction conditions for the typical step polymerization that proceeds with neat reactants. 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 ho is often the more appropriate measure of acidity for nonaqueous systems or systems containing high acid concentrations [Ritchie, 1990]. Unfortunately, the appropriate ho 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 esterification). This presents no problem if concentration is plotted against time as in Eq. 2-20. However, a plot of 1/(1 — pf 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]. EUas [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 esterification and polyesterification. 

2-2a-2-b HIgh-Conversion Region. The nonlinearity observed in the third-order plot in the final stages of the polyesterification (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 he difficult to conjecture that the factors responsible for nonlinearity at low conversions are present at high conversion hut absent in between. It is more likely that other factors are responsible for the nonlinear region above 93% conversion. 

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