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Chain-growth initiation creates a chain capable of chain-growth propagation. WriteThis schematic includes the first incorporated monomeric unit; any preceding production of $I^{\ast }$ is a separate step. If $C$ counts polymer molecules, $A$ counts polymer-bound polymerization-active sites, and $S$ counts incorporated monomeric units, this event changes $(C,A,S)$ by $(1,1,1)$. The free $I^{\ast }$ was excluded from $A$, so this does not assert creation of reactivity from nothing. An initiator feed amount need not equal the number of successful chains: the efficiency of forming productive chains and the number of initiating species per initiator matter. ^[ref]

In chain-growth polymerization, each growth step joins a monomer to a reactive site on a polymer chain and regenerates that reactive site. This is the precise mechanistic restriction in IUPAC's definition: growth occurs exclusively through these monomer–chain-site reactions. The word chain refers to the succession of site-regenerating chemical reactions. It does not merely mean that the product looks like a long chain. Chain-growth initiation starts this succession; chain-growth propagation continues it; chain-growth termination and chain transfer in chain polymerization can change its fate.^[ref]

The contrast with step-growth polymerization concerns the permitted joining partners. There, suitable reactive groups can join two existing chains, for example a two-unit chain and a three-unit chain. In chain-growth propagation, a free monomer adds to a polymerization-active site. Release of a small byproduct is a separate classification question: IUPAC explicitly allows condensative chain polymerization.^[ref] The useful consequence is that a few growing chains can incorporate many monomer molecules while much of the feed remains unreacted. Their degree of polymerization depends on how those units are shared among chains, as finite chain-growth bookkeeping makes explicit.

In the one-site linear model, chain-growth propagation changes $(C,A,S)$ by $(0,0,1)$: one monomer molecule is consumed, one monomeric unit is incorporated, and the polymerization-active site is regenerated. Thus longer chains do not imply more active sites. Repeated chain-growth propagation can increase $S$ substantially with $A$ unchanged. This is a material count; it supplies no universal time dependence or empirical chemical kinetics formula.

In chain transfer in chain polymerization, a growing chain can lose its polymerization-active site while another molecule acquires reactivity. In a common atom-transfer picture, the old chain becomes inactive and the new carrier can initiate further chain-growth propagation.^[ref] The number of reactive carriers can survive even though the identity of the growing chain changes. If the new carrier successfully starts a new chain, $C$ increases; a fixed active-carrier count therefore does not imply a fixed polymer molecule count. Whether transfer is irreversible, reversible, or involves an already existing chain matters for the final bookkeeping.

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 identityThis 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.

The defining idealization of living polymerization excludes chain-growth termination and irreversible chain transfer in chain polymerization. Fast, effective chain-growth initiation is an additional condition behind a nearly fixed growing-chain population, not a guarantee contained in the name alone.^[ref] If all four chains in finite chain-growth bookkeeping remain active, completing the feed keeps $C=A=4$ and gives $X_n=10$. Stipulate six further additions to each chain, obtaining $(10,10,10,10)$. Other allocations with the same total also satisfy $X_n=10$: living does not mean identical lengths.

Alternatively, stipulate that the first two length-four chains undergo chain-growth termination without joining, while the other two each receive twelve more monomer molecules. The final lengths are $(4,4,16,16)$, with $(C,A,S)=(4,2,40)$. The mean degree of polymerization is still $10$, but two chains have stopped and two carry the remaining growth. The population variance of lengths is $36$, compared with zero in the deliberately equal living allocation. These are synthetic populations, not measured molecular distributions and not a claim that all terminating processes are broader than all living processes. Indeed, assigning all twenty-four remaining additions to one still-active chain gives the living allocation $(28,4,4,4)$, with the same mean but variance $108$. Comparable start times and opportunities for chain-growth propagation are extra assumptions behind an expectation of more uniform lengths.

If the two terminated length-four chains instead join, their inactive product has length eight; with the same remaining additions the final lengths are $(8,16,16)$. Now $C=3$, $A=2$, $S=40$ and $X_n=40/3$. Thus chain-growth termination can change both which chains grow and which chains enter the denominator. Continuing chain-growth initiation, inefficient initiation, chain transfer in chain polymerization, and reactivation require their own counts. Predicting an entire length distribution additionally needs a specified reaction mechanism and growth statistics. A measured mean alone cannot establish living polymerization.

A polymerization-active site is the reactive location that enables the next chain-growth propagation step. In the schematic$M$ is one free monomer molecule, $j$ is the chain's degree of polymerization in incorporated monomeric units, and the star marks the polymerization-active site. The site's reactive character survives the growth step, although the chemical atoms carrying it can change. The star is not an extra unit, a literal particle, or necessarily a positive or negative electric charge. Actual reaction mechanisms can involve an unpaired Electron, a cation, an anion, or a metal-associated site; the abstract notation does not identify which. ^[ref]

For bookkeeping below, each linear active chain has exactly one polymerization-active site. This is a stipulated model, not a property of every chain polymerization: a chain can have several reactive sites, and a reversibly inactive chain can remain capable of future growth. Count currently active sites separately from all chains capable of later reactivation.

In step-growth polymerization, suitable reactive groups on molecules of different chain lengths can join. The linear count and its assumptions are given in polymerization.

A one-unit monomer, a two-unit chain, and a twenty-unit chain can all be joining partners if their functional groups are compatible and accessible. Schematically, $P_j+P_k⟶P_{j+k}$: the subscripts count monomeric units, and end-group chemistry is suppressed. In chain-growth propagation, by contrast, $P_j^{\ast }+M⟶P_{j+1}^{\ast }$ requires a free monomer and a polymerization-active site. An arbitrary pair of existing chains is not a propagation pair in that mechanism. The practical consequence of step-growth polymerization is that most monomers can have joined something while the average chain is still short.

Whether a joining step releases a small molecule is a separate question. Polycondensation releases small products during the joining chemistry, as in the Nylon 6,10 demonstration in polymerization. Polyaddition joins suitable chains without that elimination: reacting a diol with a molecule bearing two isocyanate groups can build a polyurethane. Thus step-growth polymerization and polycondensation are not synonyms.^[ref]