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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 $r$.