 Stoichiometric imbalance in step growth

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 $r$.