= Stoichiometric imbalance in step growth
{title2=$r=A_0/B_0$}
For a strictly bifunctional $AA+BB$ feed, label the limiting <functional groups> $A$, set $0<r=A_0/B_0\leq1$, 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 $B$. Thus
$$
N=\frac{A_0+B_0}{2}-pA_0,\qquad
X_n=\frac{1+r}{1+r-2rp}.
$$
This version of the <Carothers equation> uses limiting-group conversion, not the fraction of all groups reacted.<Generalization of the Carothers equation for linear step growth polymers by Emilio Meaurio>[Introduction, equations (2)–(6), especially the paragraph before equation (3): A is the limiting reactant used to define p]{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 $r\geq(K-1)/(K+1)$, even before considering incomplete conversion. For $K=200$, $r\geq199/201\approx0.99005$. 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 $r$.
\Image[/step-growth-conversion-stoichiometry-sensitivity.png]
{title=<Carothers equation> sensitivity to limiting-group conversion and initial <stoichiometric imbalance in step growth>}
{id=image-step-growth-conversion-stoichiometry-sensitivity}
{description=Original analytic model curves with logarithmic vertical axes. Left: $X_n=(1+r)/(1+r-2rp)$ and the balanced example points. Right: $X_{n,\max}=(1+r)/(1-r)$ versus $100(1-r)$, the initial excess-$B$ fraction relative to $B_0$. These are ideal linear no-ring calculations, not fits to the <polyurethane> study.}
Back to article page
