## Abstract

The statistical theory of branching or cascade processes leads to simple methods for evaluating statistical parameters of polymerization systems. As a good first approximation, these methods are here extended to cover equilibrium f-functional polycondensation systems which depart from randomness, not only by a (first-shell) substitution effect, but also through intramolecular (cyclization) reactions which compete with intermolecular (crosslinking) reactions. The results are tested against the inorganic equilibrium polymer system POCl$_3$/P$_2$O$_5$ for which excellent data were made available by Van Wazer and his co-workers. The three-parameter theory agrees within experimental error with their n.m.r. measurements of the concentrations of the four types of P atoms (bearing zero, one, two or three Cl atoms) in six equilibrium mixtures. The concentration of pyrophosphoryl dichloride calculated for one mixture agrees almost exactly with two concordant experimental methods; but too much confirmatory value must not be placed on this particular result. The maximum amount of cyclization calculated (i.e. at the gel point) is 0.64 $\pm$ 0.10 ring per molecule present. This is in good agreement with forecasts made by adapting the classical random-flight model, and using known bond lengths and angles. Contrary to previous estimates, arrived at by neglecting cylization, of a substantial non-linear substitution effect in POCl$_3$/P$_2$O$_5$, a barely significant (linear) effect, in the opposite direction, is here deduced. The theory is based on the process of imagining all rings split open and classifying the resulting functionalities as a special type ('ring-closing' functionalities) on the resultant tree-molecules. A zero approximation would consist in distributing these special functionalities at random on tree-molecules which obey the usual distributions calculable in absence of cyclization; but this has little value in practice. The next (first) approximation is amply sufficient: it rests on the recognition that the chance of a given functionality being ring-closing is almost exactly proportional to the number of crosslinks borne by the repeat unit to which it belongs. The theory, which should apply even in presence of fair degrees of fused-ring formation, reduces for difunctional systems exactly to a generalization of the Jacobson-Stockmayer treatment. The theory is also asymptotically correct for systems of any functionality as the degree of cyclization goes to zero. It is well adapted for calculating statistical averages, but not the equilibrium concentrations of individual species, except probably the simplest ones, such as pyrophosphoryl dichloride mentioned above.