= Cyclization in step-growth polymerization
In <cyclization in step-growth polymerization>, compatible <functional groups> on the same chain react and close it into a ring. They consume groups without merging two <molecules>. If $R$ ring-closing events and $J$ intermolecular joins occur in an otherwise balanced bifunctional feed, the group conversion is $p=(J+R)/N_0$, while $N=N_0-J=N_0(1-p)+R$. Thus
$$
X_n=\frac{N_0}{N_0(1-p)+R}.
$$
For example, an $AB$ dimer can close its remaining $A$ and $B$ ends: two group pairings have occurred among two starting <monomer molecules>, giving $p=1$, but the resulting single cyclic <molecule> has only two <monomeric units>. This counterexample shows exactly where the linear <Carothers equation> fails. Rings have no chain ends, so counting every chain-bearing <molecule> as two ends also fails.
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