Statistical kinetics of polyesterification of adipic acid with pentaerythritol or trimethylol ethane

Abstract

On the basis of the statistical theory of branching processes, the kinetics of esterification of adipic acid (AA) with pentaerythritol are reinvestigated, and extended to the system in which trimethylol ethane (TME)(2,2-dihydroxymethylpropanol) replaces pentaerythritol (PE). The mathematical model is expressed in terms of eleven (TME) or fifteen (PE) simultaneous ordinary differential equations, or equivalently, in terms of two simultaneous partial differential equations. This model was shown previously to become asymptotically exact as the degree of cyclisation goes to zero, e.g., by extrapolation to zero volume. The substitution effect is allowed for, and it is found that each pre-existent ester link carried by a PE unit lowers the free energy of activation for forming a further such link by about 350 cal./mole. A small correction is also incorporated for the effect of the shrinkage on the reactant concentrations. This model allows extremely close fittings to be made to experimental rate curves on the assumption that the basic esterification reaction is first order in hydroxyls and second order in carboxyls (as postulated by Flory long ago). The model and this assumption are also supported by measurements of gel points as function of mole ratio [COOH/OH] and dilution by solvent. The model leads to the conclusion that in AA/PE near the gel point, about one in ten of the ester-links present was formed intramolecularly. (The proportion is higher if the system is diluted with solvent.) This high degree of cyclisation agrees quantitatively with that calculated from random-flight statistics. Together with the substitution effect, it seems to account for the lowering of the apparent reaction order to about 2·5 observed by some previous investigators of polyesterification reactions. The reactions have been followed continuously using measurements of pressure of the water evolved by means of a transducer and recorder. This method is superior to the classical procedure of intermittent titration of samples, with which it has been shown to give good agreement.The model based on the theory of branching processes has been briefly compared to the general stochastic theory of ring-chain competition processes by Whittle.