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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.

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 $N=N_0-J=N_0(1-p)+R$. ThusFor 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.

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.

In (i), any pair of compatible functional groups can react: monomers, oligomers and long chains all participate. For an equimolar, strictly bifunctional mixture without cyclization in step-growth polymerization, the Carothers equation givesHere $p$ is functional-group conversion and ${\overline{X}}_n$ counts the original monomer molecules. Large degree of polymerization therefore requires nearly complete conversion.

In (ii), growth occurs only at the metal-bound chain end. If initiation is rapid and chain transfer in chain polymerization is negligible, the number of chains is fixed andThis is the linear conversion dependence of living polymerization. Actual zirconocene systems can undergo chain transfer, so linear growth is a limiting model, not a consequence of coordination polymerization alone.

In (iv), a newly initiated chain rapidly attains a large size and then undergoes radical termination. Further conversion mostly creates additional chains. Thus high degree of polymerization can occur at low conversion, without the systematic linear increase of an ideal living system. temperature, initiator depletion and monomer depletion can change that size during the run.

An experimental comparison joined a two-ended diol chain, HOPCLOH, with a bifunctional isocyanate, at reported isocyanate-to-diol ratio $1.14:1$. The researchers used oven-dried $25\mathrm{mL}$ round-bottom flasks and an IKA RET magnetic hotplate with a metal carrier. They varied solvent, concentration, reaction temperature ($50$ or $61{}^{\circ }\mathrm{C}$), and duration ($1$ or $8\mathrm{h}$).^[ref]

A Bruker $500\mathrm{MHz}$ NMR spectroscopy instrument measured residual hydroxy group-adjacent signal relative to an ester signal. Independently, size-exclusion chromatography used ResiPore columns at $70{}^{\circ }\mathrm{C}$, a refractive-index detector, and polystyrene calibration. Figure 8 reports sharply increasing apparent number-average molar mass beyond the reported conversion $0.95$. A deliberately water-exposed isocyanate comparison reproduced a $1625\mathrm{c}{\mathrm{m}}^{-1}$ infrared spectroscopy band assigned to urea-containing products. The authors warn that tiny residual NMR spectroscopy signals reduce integration precision.^[ref]

The inference is increasing chain size alongside depletion of reactive groups, consistent with step-growth polymerization. It is not an exact measurement of $X_n$ in the toy model. The feed contains already extended diol chains: counting their constituent repeat units differs from counting each feed molecule as one unit. Their varying molar masses, calibration against a different polymer, residual starting material, and competing chemical reactions further obstruct a direct conversion-to-$X_n$ substitution. Report which functional groups the conversion readout measures before comparing it with stoichiometric imbalance in step growth.

Useful proposed controls, distinct from the reported water comparison, are pre-reaction spectra and matched mixtures omitting each joining partner in turn. Keep solvent, heating, and signal normalization matched. Disappearance of an isocyanate signal in the omitted-diol control would reveal competing consumption rather than establish chain joining. Agreement between residual-group and independently calibrated chain-size readouts is stronger evidence than calculating both quantities from the same Carothers equation.

For a strictly bifunctional $AA+BB$ feed, label the limiting functional groups $A$, set $0<r=A_0/B_0\le 1$, and define $p=J/A_0$ as in functional-group conversion in step growth. The initial molecule count is $N_0=(A_0+B_0)/2$, and each intermolecular join consumes one $A$ and one $B$. ThusThis version of the Carothers equation uses limiting-group conversion, not the fraction of all groups reacted.^[ref]

As an original finite example, mix $95$ $AA$ monomer molecules with $100$ $BB$ monomer molecules. Then $(N_0,A_0,B_0)=(195,190,200)$ and $r=0.95$. At $p=0.90$, $171$ joins leave $24$ molecules, giving $X_n=195/24=8.125$. At $p=1$, $190$ joins leave five chains and ten unreacted $B$ groups; every chain has two $B$ ends, and $X_n=39$. Continuing to wait cannot remove the missing-$A$ constraint within this model.

For $r<1$, complete limiting-group conversion gives the ceiling $X_{n,\max }=(1+r)/(1-r)$. Reaching a target $X_n=K>1$ therefore requires $r\ge (K-1)/(K+1)$, even before considering incomplete conversion. For $K=200$, $r\ge 199/201\approx 0.99005$. Equality requires $p=1$; any lower conversion demands closer balance. The imbalance $1-r$ is measured relative to the initial number of $B$ groups, not relative to the total number of groups. A monofunctional impurity changes the initial groups-per-molecule count, so it cannot be inserted into this strictly bifunctional formula by retaining an unexamined $r$.