Theory of ring formation in irreversible system: AB model

Abstract

An irreversible AB-type reaction with ring formation is analyzed. The distributions of chains and cyclics are deduced as functions of statei, using a method in which every particle is enumerated, where statei is related to the extent of the reaction,D=i/N ₀. The numbers of the chain and the cyclic x-mers in statek are given, respectively, by

$$\begin{gathered} N_{1,k} = N_0 - \sum\limits_{i = 1}^k {[2 \cdot P_{i - 1} \{ 1,L\} + P_{i - 1} \{ 1,R\} ]} , \hfill \\ \hfill \\ N_{x,k} = \sum\limits_{i = 1}^k {\left[ {\sum\limits_{j = 1}^{x - 1} {N_{x - j,i - 1} \cdot P_{i - 1} \{ j,L\} /(N_0 - i + 1)} } \right.} \hfill \\ \hfill \\ {\text{ }}\left. \begin{gathered} \hfill \\ - 2 \cdot P_{i - 1} \{ x,L\} - P_{i - 1} \{ x,R\} \hfill \\ \hfill \\ \end{gathered} \right]{\text{ for }}x\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2, \hfill \\ \hfill \\ N_{Rx,k} = \sum\limits_{i = 1}^k {[P_{i - 1} \{ x,R\} ]} \hfill \\ \end{gathered} $$

whereP _i-1 {x, L} andP _i-1 {x, R} are probabilities of chain-propagation and ring formation, respectively, in statei-1.

The chain distribution from the above equations agrees with the most probable one in the case where the chain propagation occurs exclusively (e.g., a concentrated solution), while the ring distribution obeys the exponential lawN _Rx ∞x ^−5/2, which is identical with the equilibrium case. The theory was examined using cycloalkane formation by Knipe and Stirling. Agreement between the theory and experimental observations is found to be considerably favorable for five- and six-membered ring formation.