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The balanced Carothers equation follows immediately from linear step-growth molecule count and functional-group conversion in step growth:The same Carothers equation extends to unequal group counts once the limiting-group convention is specified.^[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 $d{X}_n/dp=(1-p{)}^{-2}$ 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.

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 cyclization in step-growth polymerization, compatible functional groups on the same chain react and close it into a ring. They consume groups without merging two molecules. If $R$ ring-closing events and $J$ intermolecular joins occur in an otherwise balanced bifunctional feed, the group conversion is $p=(J+R)/N_0$, while $N=N_0-J=N_0(1-p)+R$. ThusFor example, an $AB$ dimer can close its remaining $A$ and $B$ ends: two group pairings have occurred among two starting monomer molecules, giving $p=1$, but the resulting single cyclic molecule has only two monomeric units. This counterexample shows exactly where the linear Carothers equation fails. Rings have no chain ends, so counting every chain-bearing molecule as two ends also fails.

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.

In a balanced bifunctional feed there are initially $N_0$ $A$ groups and $N_0$ $B$ groups. Define the functional-group conversion in step growth by $p=J/N_0$: the reacted fraction of either group type. Each join consumes one of each. Therefore $N=N_0(1-p)$. In the eight-$AB$ example, three joins give $p=3/8$ and four give $p=1/2$. In the latter state only three of eight starting monomer molecules remain isolated: their disappearance fraction is $5/8$, different from $p=1/2$. Neither this group-based $p$ nor disappearance of isolated monomers is the incorporated-feed fraction in the ideal living polymerization count.

With unequal initial counts $A_0\le B_0$, this article always uses $p=J/A_0$, the conversion of the limiting $A$ groups. The reacted fraction of $B$ groups is $J/B_0=rp$, where $r=A_0/B_0$. The fraction of all initial functional groups consumed is $2J/(A_0+B_0)=2rp/(1+r)$. At nonzero conversion these three fractions coincide only at $r=1$. Naming the measured group type prevents an apparently correct equation from using the wrong conversion.

Use an ideal feed in which every starting molecule has exactly two suitable functional groups: either one $A$ and one $B$, or two $A$ groups mixed with two $B$ groups. Only an $A$–$B$ pairing can form a joining chemical bond. Assume no loss of feed-derived material, no splitting of chains, no competing consumption of the counted functional groups, no branching, and no cyclization in step-growth polymerization. Linear here describes connectivity; a chain can bend in space. Small byproducts of polycondensation do not enter the chain-bearing molecule count.

Let $N_0$ count all initial feed molecules, $J$ the number of intermolecular joins, and $N$ all remaining chain-bearing molecules, including unreacted monomers. Each join turns two molecules into one, so $N=N_0-J$ exactly. If chain $i$ contains $j_i$ feed-derived monomeric units, conservation gives ${\sum }_{i=1}^N{j}_i=N_0$. Consequently the number-average degree of polymerization isThis is a count of both feed species' monomeric units, rather than a count of interior constitutional repeating units. For $AA+BB$ chemistry an interior repeat often contains two feed-derived units, as explained in monomer. Equal functional group reactivities are not needed for this bookkeeping at a specified $J$; predicting a time course or a chain-length probability distribution requires additional assumptions.

For the figure's eight $AB$ feed molecules, three joins can give lengths $(2,3,1,1,1)$, so $N=5$ and $X_n=8/5$. Joining the first two chains gives $(5,1,1,1)$: the incorporated total stays eight, $N$ becomes four, and $X_n=2$. A graph makes the bookkeeping visible: each starting monomer molecule is a vertex of a graph, and each intermolecular joining chemical bond connects two previously distinct connected components of a graph. The connected pieces are the counted molecules.

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 monomer is a chemical substance whose molecules can undergo polymerization and contribute units to the essential structure of a polymer. A monomer molecule is the individual reactant; its incorporated monomeric unit is the portion identifiable as derived from that reactant. Incorporation can change chemical bonds and release small byproducts, so the unit in the chain need not have the complete molecular formula of the feed molecule. See IUPAC: monomer molecule and IUPAC, A brief guide to polymer terminology: monomeric units.

Three different counts answer three different questions. Monomer molecules describe the feed; monomeric units trace that feed's contributions to the product; a constitutional repeating unit describes the smallest chemical pattern repeating along a regular chain. In Nylon 6,10, two different monomers supply one interior constitutional repeating unit,Its molecular formula is ${\mathrm{C}}_{16}{\mathrm{H}}_{30}{\mathrm{N}}_2{\mathrm{O}}_2$. The diamine feed is ${\mathrm{C}}_6{\mathrm{H}}_{16}{\mathrm{N}}_2$, and the diacid-chloride feed is ${\mathrm{C}}_{10}{\mathrm{H}}_{16}\mathrm{C}{\mathrm{l}}_2{\mathrm{O}}_2$; subtracting two $\mathrm{HCl}$ gives the interior-repeat budget. For a finite linear chain made from $n$ of each reactant, there are $2n-1$ joining connections, hence $2n-1$ formal $\mathrm{HCl}$ eliminations, rather than $2n$. One amine end and one acid-chloride end remain in that ideal unmodified chain. This is a stoichiometry check on the depicted reaction, not a claim that the washed experimental thread retains those ends. The reacting end-group chemistry is documented in Hammond, lecture 5, p. 4; the specified Nylon 6,10 reactant pair is reported in Polymer Day, nylon demonstration.

Ethene shows the reverse counting mismatch. One $\mathrm{C}{\mathrm{H}}_2=\mathrm{C}{\mathrm{H}}_2$ monomer molecule contributes a two-carbon $-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}{\mathrm{H}}_2-$ monomeric unit to ideal linear polyethylene, without a small byproduct in that growth step. But the smallest constitutional repeating unit of its uniform backbone is just $-\mathrm{C}{\mathrm{H}}_2-$: a chain containing $m$ incorporated ethene monomeric units contains $2m$ such interior one-carbon repeats. End groups are additional bookkeeping. The frequently drawn $[-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}{\mathrm{H}}_2-{]}_m$ is useful for tracing the feed, although it is not the smallest constitutional repeat. The minimum-pattern definition is given in IUPAC: constitutional repeating unit; the ethene growth chemistry is described by ^[ref].

A segment of a freely jointed chain or Gaussian chain instead belongs to a statistical model. It can represent many chemical bonds and several constitutional repeating units; its length and the segment count are chosen to describe chain conformation at the intended scale. The three-repeats-per-segment drawing in Figure "From two liquid layers to a Nylon 6,10 thread: monomers, a constitutional repeating unit, and statistical segments" is a deliberately arbitrary coarse-graining example, not a measured Kuhn length. Always state the counting convention before substituting a segment count for a degree of polymerization. This separation lets chemistry track which chemical atoms entered the material while statistical models track how the material bends and extends.

An incorporated monomeric unit is the largest constitutional unit identifiable as derived from one monomer molecule. It need not be a whole unreacted molecule or the smallest constitutional repeating unit. See IUPAC: monomeric unit.

Polymerization converts a monomer, or a mixture of monomers, into a polymer: repeated chemical reactions establish a connected molecular chain or network. The central question is how connections form, not merely whether a material becomes solid. Joining small molecules can produce a thread that can be handled, while the individual molecules remain invisible. See IUPAC, Basic Definitions of Terms Relating to Polymers, definition 2.2.

In step-growth polymerization, complementary reactive groups on monomers and on shorter or longer chains can react with one another. A two-unit chain can join a three-unit chain to make a five-unit chain; growth does not require a specially active end to add only free monomers. An example is the Nylon 6,10 reaction of ${\mathrm{H}}_2\mathrm{N}(\mathrm{C}{\mathrm{H}}_2{)}_6\mathrm{N}{\mathrm{H}}_2$ with $\mathrm{ClC}(=\mathrm{O})(\mathrm{C}{\mathrm{H}}_2{)}_8\mathrm{C}(=\mathrm{O})\mathrm{Cl}$: each amine end can form an amide connection with an acid-chloride end, with formal elimination of $\mathrm{HCl}$. Both reactants have two suitable ends, allowing repeated joining. See Paula Hammond, MIT 10.569 lecture 2: step-growth joining and Hammond, lecture 5, p. 4: acid chloride–amine reaction.

In chain-growth polymerization, a monomer reacts with an active site on a growing chain, and the growth step regenerates an active site. Schematically, $P_j^{\ast }+M⟶P_{j+1}^{\ast }$, where $j$ counts incorporated monomeric units and the star denotes reactivity rather than a particular electric charge. Ethene forming polyethylene is an example: the double chemical bond is converted into single chemical bonds as the carbon backbone grows. Initiation creates the active chains; subsequent reactions can deactivate them or transfer reactivity to another molecule. See IUPAC: chain polymerization and John McMurry, Organic Chemistry, 31.1: chain growth. Whether growth is stepwise or carried by an active site is a different question from whether small byproducts are released: chain-growth polymerization can also release them.

A motivating experiment: drawing nylon from a liquid interface. Feldwinn's UCSB demonstration places $25\mathrm{mL}$ of an aqueous chemical solution of hexamethylenediamine, ${\mathrm{H}}_2\mathrm{N}(\mathrm{C}{\mathrm{H}}_2{)}_6\mathrm{N}{\mathrm{H}}_2$, in a $100\mathrm{mL}$ beaker, then gently adds $25\mathrm{mL}$ of sebacoyl chloride, $\mathrm{ClC}(=\mathrm{O})(\mathrm{C}{\mathrm{H}}_2{)}_8\mathrm{C}(=\mathrm{O})\mathrm{Cl}$, in an organic solvent. The procedure specifies hexane; other parts of that handout list cyclohexane. The layers remain separate. The instructor catches material at their interface with tweezers and winds it slowly onto a glass rod; the handout reports continued spooling of Nylon 6,10. It specifies freshly prepared acid-chloride chemical solution and washing the thread before touching it. These are observations and conditions of a supervised laboratory demonstration, not measurements of chain lengths. See Feldwinn, UCSB Demonstration Library, Demo 044, pp. 1–3.

The mechanistic interpretation is that the monomers encounter one another near the interface, where amide connections produce the film; withdrawing it exposes a region where more material can form. An authored research demonstration independently reports a rapidly formed interfacial film collapsing into a thread on withdrawal, using cyclohexane and an aqueous chemical solution containing potassium hydroxide. See Polymer Day: Outreach Experiments for High School Students, Journal of Chemical Education (2017), nylon demonstration. This is the striking observation: a thread much longer than the beaker can emerge because material is continually supplied to the formation region, rather than because a pre-existing thread was hidden in the liquid.

Useful proposed controls, not results reported in those demonstrations, omit one monomer at a time while retaining the same solvents, liquid amounts, and pulling procedure. The prediction is loss of a renewable drawable Nylon 6,10 film; merely finding cloudiness would not establish polymerization, since other chemical reactions, including acid-chloride hydrolysis, can occur. See McMurry, 21.4: competing acid-chloride hydrolysis. Stopping and restarting withdrawal would probe renewal at the interface. Even successful drawing does not measure the degree of polymerization, establish a chain-length probability distribution, or prove the entire growth mechanism. Chemical identification and a molecular-size measurement would be needed for those stronger conclusions. These controls and limits are inferences from the setup, not invented experimental data.

Why almost complete reaction matters in step growth. For ideal linear step-growth polymerization, assume exactly two reactive groups per starting molecule, equal numbers of complementary groups, only joining of different chains, no rings, no branching, and no loss of chain material. Let $N_0$ count starting monomer molecules, including both reactant species, and let $p$ be the fraction of either type of reactive group consumed. There are initially $N_0$ groups of each type, so $p{N}_0$ connections reduce the number of chain-bearing molecules to $N=N_0(1-p)$; the count excludes small byproducts but includes unreacted monomers. The number-average degree of polymerization, measured in monomeric units, is thereforeThus $p=0.90$, $0.99$, and $0.999$ give $X_n=10$, $100$, and $1000$. Losing the last fraction of unreacted groups matters enormously. See Hammond, lecture 3: balanced step-growth conversion. This counts residues of both feed species; a Nylon 6,10 interior constitutional repeating unit contains two such monomeric units.

For an $AA+BB$ feed with $A_0\le B_0$ initial reactive-group counts, put $r=A_0/B_0$ and now let $p$ denote the fraction of the limiting $A$ groups consumed. Under the same linear, no-ring assumptions, $N_0=(A_0+B_0)/2$ and $N=N_0-p{A}_0$, givingFor $r=0.98$, even complete consumption of $A$ gives $X_n=(1.98)/(0.02)=99$. Stoichiometry controls chain length as well as yield. See Hammond, lecture 2, pp. 2–3: stoichiometric imbalance. The liquid-interface demonstration is not a homogeneous, balanced reactor, so its thread length cannot be inserted into this formula as a molecular chain length.

A contrasting chain count. In ideal living polymerization, assume no chain termination or irreversible chain transfer, all chains initiated before the counted growth, one chain per effective initiator, a fixed chain count $C$, and every consumed monomer molecule incorporated into those chains. If $M_0$ is the initial number of feed monomer molecules and $q$ their incorporated fraction, conservation gives $X_n=q{M}_0/C=qR$ in monomeric units, excluding initiator fragments. For $R=200$, incorporating half the feed gives $X_n=100$. If initiation consumes one monomer molecule per chain, the fixed-chain interval starts at $q=1/R$, rather than $q=0$. See Hammond, lecture 21, p. 1: ideal living-chain count. The mean does not imply identical chain lengths. Living polymerization does not require reversible loss of monomers; the defining absence of chain termination and irreversible chain transfer is explained in Devon Shipp, IUPAC Polymer Education workshop, Radical Polymerizations II, slide 50. Ordinary chain-growth polymerization with new chains, termination, or transfer need not follow the straight line.

The useful connection is between local chemistry and a macroscopic material: selecting reactive ends and their proportions controls connectivity, while the resulting polymer's conformation is a separate problem described by models such as the freely jointed chain or Gaussian chain. Formation of chemical bonds, the count of monomeric units, and the count of statistical segments must be kept distinct.

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.

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 $r$.