Systems with different types of branch units

Systems with different types of branch units Let us consider a more general case where both A and B types of branch units occur and where there are also monofunctional and bifunctional A and B reactants, for example, polymerization of A + A-A + A/ + B + B-B + B/. 

To derive a general expression for the conversion of functional groups at the gel point, let symbols /, 6, and p denote, respectively, functionality ( 2), fraction of functional groups belonging to bifunctional reactant, and fraction of functional groups belonging to multifunctional (/ 2) reactant, with subscripts A and B 

The probability a is equal to the sum of the 4 individual probabilities of each of these 4 cases occurring in any selection taken at random, each case being equally valid for the purpose of evaluating a. If the extents of reaction, pA and PB, are the fractions of A and B groups, respectively, that have reacted, then the 4 probabilities are given by equations written below  

The sum of the above four probabihties gives the total probability a  

NApjOBpBPA + NaPaPbPa + NbPaPbPb + Nb9aPbPaPb (NapA + 77bPb)(1 - Oa bPaPb) 

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