= Finite chain-growth bookkeeping
{title2=$X_n=S/C$}
Stipulate a feed of exactly $M_0=40$ <monomer molecules>, four successfully initiated linear chains, one <polymerization-active site> per chain, no <chain transfer in chain polymerization>, no loss of incorporated units, and no further <chain-growth initiation>. Count only initiated chains in $C$; exclude unreacted <monomers>. Assume each consumed <monomer molecule> contributes exactly one <monomeric unit>. With incorporated feed fraction $q=S/M_0$, the number-average <degree of polymerization> is the counting identity
$$
X_n=\frac{\sum_{i=1}^{C}j_i}{C}=\frac{S}{C}=\frac{qM_0}{C}.
$$
This is conditional conservation bookkeeping, not a universal empirical law. Immediately after <chain-growth initiation>, the four lengths are $(1,1,1,1)$, so $(C,A,S)=(4,4,4)$ and $q=0.1$. Stipulate twelve subsequent <chain-growth propagation> events, three on each chain: the lengths become $(4,4,4,4)$, giving $(C,A,S)=(4,4,16)$, $q=0.4$ and $X_n=4$. Sixteen units have been incorporated, while twenty-four free <monomer molecules> remain. The chain-length assignment is deliberately chosen, not a prediction of equal growth in an experiment.
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