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The balanced Carothers equation follows immediately from linear step-growth molecule count and functional-group conversion in step growth:The same Carothers equation extends to unequal group counts once the limiting-group convention is specified.^[ref]

For an original balanced feed of $2000$ monomer molecules, $1900$, $1960$, and $1990$ intermolecular joins give $p=0.95$, $0.98$, and $0.995$. The remaining counts are $100$, $40$, and $10$, hence number-average degree of polymerization values $20$, $50$, and $200$. The last thirty joins quadruple the average even though they consume only another $1.5\%$ of the initial groups of either type. These are exact synthetic counts, not experimental measurements.

The derivative $d{X}_n/dp=(1-p{)}^{-2}$ explains the sensitivity: near $p=0.99$, an absolute conversion uncertainty of $0.001$ changes the predicted degree of polymerization substantially. The exact balanced values at $p=0.989$ and $0.991$ are approximately $90.9$ and $111.1$, instead of $100$. An instrument reporting rounded “complete conversion” cannot distinguish those chain lengths. In a finite no-ring system, even one chain retains two ends; the maximum $p$ is $1-1/N_0$ and $X_n=N_0$. The formal divergence at $p=1$ is a large-system limit, not an infinite chain in a finite flask.

The functionality in polymerization is the number of reactive functional groups on a molecule for the specified joining chemistry. The linear step-growth molecule count assumes $f=2$ for every feed molecule. A diol supplies two hydroxy groups in the polyurethane example; an impurity supplying only one suitable group can cap a chain. A group count must reflect the selected chemical reaction, not simply every chemically recognizable group in the molecule.

Gelation is formation of an extended connected network. In step-growth polymerization, feed molecules with functionality in polymerization above two can introduce branching and permit gelation. The linear step-growth molecule count's two-ended-chain interpretation and bifunctional Carothers equation must not silently be applied to such networks. Each join between distinct connected pieces still reduces their count by one, but neither that identity nor average size alone specifies the onset of gelation.