= Carothers equation
{c}
{title2=$X_n=1/(1-p)$}
The balanced <Carothers equation> follows immediately from <linear step-growth molecule count> and <functional-group conversion in step growth>:
$$
X_n=\frac{1}{1-p}.
$$
The same <Carothers equation> extends to unequal group counts once the limiting-group convention is specified.<Generalization of the Carothers equation for linear step growth polymers by Emilio Meaurio>[Introduction, equations (1)–(6): Carothers equation, end-group count and limiting-$A$ conversion]{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> $dX_n/dp=(1-p)^{-2}$ 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.
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