Degree of polymerisation

Result

Degree of polymerisation

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Statement

Chain length and the molar-mass distribution.

Why it matters

step-vs-chain-growth establishes the two mechanistic routes by which small monomers are joined into a macromolecule, but neither mechanism by itself says how long the resulting chains actually are, or how much that length varies from one chain to the next in the same sample. Degree of polymerisation supplies exactly that missing quantitative link: it converts a monomer-by-monomer mechanistic description into the chain-length and molar-mass numbers that ultimately determine a polymer's mechanical and thermal behaviour, including the glass-transition temperature and the phase behaviour flory-huggins theory describes.

It is also the first place this unit has to confront a genuinely new complication: unlike a small molecule, a polymer sample is never made of chains of one single, uniform length, so any single "molar mass" reported for a polymer must specify which kind of statistical average across that distribution of chain lengths is actually meant.

Hypotheses

A polymer chain's molar mass can be treated as an integer multiple of a single, well-defined repeat-unit molar mass \(M_0\).This holds cleanly for a homopolymer built from one repeating monomer unit; for a copolymer built from more than one distinct monomer, \(M_0\) must instead be replaced by an appropriately weighted average repeat-unit mass, a direct but slightly more involved extension of the same idea. A real polymer sample consists of a distribution of chain lengths, not a single uniform length, and this distribution must be characterised by a statistical average rather than a single number.Two different averages, the number-average \(M_n\) (weighting each chain equally) and the weight-average \(M_w\) (weighting each chain by its own mass, so longer chains count more heavily), are both in routine use and generally give different numerical values for the same real sample; their ratio, the polydispersity index \(M_w/M_n\), quantifies exactly how broad that distribution is, with \(M_w/M_n=1\) corresponding to the idealised, perfectly uniform (monodisperse) limit no real synthetic polymer actually reaches. For step-growth polymerisation specifically, every monomer molecule present is assumed equally likely to react at any point during the polymerisation.This equal-reactivity assumption is what makes the statistical treatment underlying the Carothers equation (Proof) tractable; it is a good approximation for most small-molecule condensation monomers, where a functional group's reactivity does not depend strongly on whether it is attached to a small monomer or already to a long, growing chain.

Proof

X_n = \frac{\text{total number of monomer units reacted}}{\text{total number of chains remaining}}

The number-average degree of polymerisation \(X_n\) is defined as the average number of repeat units per chain, counted by dividing the total monomer units incorporated by the total number of distinct polymer chains currently present in the sample — a direct chain-counting definition, independent of any particular polymerisation mechanism. A

M_n = X_n \times M_0

Since each chain, on average, contains \(X_n\) repeat units of molar mass \(M_0\) apiece, the number-average molar mass \(M_n\) is simply their product; this converts a purely structural quantity (chain length in units) into a directly measurable physical quantity (molar mass). A

X_n = \frac{1}{1-p}\quad\text{(the Carothers equation, for step-growth polymerisation)}

Let \(p\) be the fraction of reactive functional groups that have reacted at a given point in a step-growth polymerisation (Hypotheses' equal-reactivity assumption). Since the number of chains remaining is directly proportional to the number of unreacted functional-group pairs, a straightforward statistical argument over the fraction of monomer units converted shows the average chain length grows as \(1/(1-p)\): as conversion \(p\) approaches \(1\) (complete reaction), \(X_n\) diverges, meaning very high conversion is required to achieve long chains at all in step-growth polymerisation. B

\text{PDI} = M_w/M_n \ge 1

Because the weight-average \(M_w\) weights each chain by its own mass rather than counting it equally, any real, non-uniform distribution of chain lengths necessarily gives \(M_w > M_n\) (longer chains, weighted more heavily in \(M_w\), pull that average upward relative to \(M_n\)); the polydispersity index is therefore always \(\ge1\), with equality only in the idealised monodisperse limit of Hypotheses, never fully achieved by any real step- or chain-growth synthesis. A

Result

M_n = X_n\,M_0, \qquad X_n = \frac{1}{1-p}\ \text{(step-growth)}, \qquad \text{PDI}=M_w/M_n\ge1

Reading. A polymer's average chain length converts directly into its average molar mass through the repeat-unit mass \(M_0\); for step-growth polymerisation specifically, that average chain length is fixed entirely by the fractional conversion \(p\) of reactive functional groups, and grows only slowly until conversion is pushed very close to completion.

Scope. The Carothers equation (Step 3) applies specifically to step-growth polymerisation under the Hypotheses' equal-reactivity assumption; chain-growth polymerisation instead reaches high molar mass essentially immediately upon initiation of each individual chain, following a genuinely different kinetic law covered alongside step-vs-chain-growth.

Corollaries & converses

The Carothers equation's divergence as \(p\to1\) (Step 3) is the quantitative reason step-vs-chain-growth's step-growth mechanism requires driving reactions to unusually high, often difficult-to-achieve conversion (typically \(p>0.99\)) to obtain commercially useful high molar mass materials at all.
glass-transition temperature and most other bulk mechanical properties depend strongly on molar mass at low-to-moderate \(M_n\) but plateau once chains become long enough for entanglement effects to dominate, which is why the specific molar-mass values computed here matter directly for predicting real material behaviour.
Converse: measuring a step-growth polymer's number-average molar mass \(M_n\) experimentally, together with knowledge of \(M_0\), allows the fractional conversion \(p\) actually achieved in a given polymerisation run to be back-calculated directly from the Carothers equation.

Fails without

Drop the equal-reactivity hypothesis (allow functional-group reactivity to change once attached to a growing chain): the simple closed-form Carothers equation \(X_n=1/(1-p)\) (Step 3) no longer holds, and a more elaborate, reactivity-dependent kinetic treatment is needed to relate conversion to chain length.
Report only \(M_n\) for a real polymer sample without also stating its polydispersity index: this omits how broad the actual chain-length distribution is (Step 4), and two samples with identical \(M_n\) but very different PDI can behave quite differently in bulk mechanical properties.

Common errors

Assuming a small increase in conversion \(p\) always produces a proportionally small increase in chain length \(X_n\); because \(X_n=1/(1-p)\) is strongly nonlinear, the final few percent of conversion (from, say, \(p=0.98\) to \(p=0.99\)) roughly doubles \(X_n\), far more than an equal increase earlier in the reaction.
Treating \(M_n\) and \(M_w\) as interchangeable, or reporting a single "molar mass" for a polymer without specifying which average is meant — the two generally differ substantially (Step 4), and many physical properties depend specifically on one average rather than the other.
Applying the Carothers equation (Step 3), derived specifically for step-growth polymerisation, to a chain-growth polymer, whose molar-mass-versus-conversion behaviour follows an entirely different kinetic pattern.
Assuming a polydispersity index of exactly \(1\) is achievable in practice; even carefully controlled polymerisations (e.g. living polymerisations) only approach, never exactly reach, the idealised monodisperse limit.

Discussion

Wallace Carothers, working at DuPont in the early 1930s, derived the statistical relationship between conversion and chain length now bearing his name during his foundational research into condensation polymers, work that directly led to the commercial development of nylon, one of the first fully synthetic fibres and still among the best-known step-growth polymers.

The statistical distribution underlying step-growth polymerisation at a given conversion \(p\) is the "most probable" or Flory distribution, which predicts not just the average chain length \(X_n\) but the full probability that a randomly selected chain has any particular specific length; this same distribution is what generates the polydispersity index of exactly \(2\) (in the theoretical, ideal step-growth limit) frequently quoted as characteristic of step-growth polymers, in contrast to the typically narrower distributions achievable by certain controlled chain-growth methods.

Common misconception: that degree of polymerisation is simply "how many monomers were mixed together" at the start of a reaction. It specifically measures the average chain length actually achieved once polymerisation has proceeded to some conversion \(p\), which for step-growth polymers (Step 3) can remain quite low even at substantial, but not yet very high, conversion.

Worked examples

\text{Step-growth polyester, } M_0=192\,\text{g/mol}, \ p=0.980

Applying the Carothers equation (Step 3): \(X_n=1/(1-0.980)=1/0.020=50\). By Step 2, \(M_n=X_n\times M_0=50\times192=9{,}600\,\text{g/mol}\). Even at a conversion most students might intuitively regard as "nearly complete," the resulting chain length is still comparatively modest. A

\text{Same monomer, } p=0.995

\(X_n=1/(1-0.995)=1/0.005=200\), giving \(M_n=200\times192=38{,}400\,\text{g/mol}\). Raising conversion by only \(1.5\) percentage points (from \(0.980\) to \(0.995\)) has quadrupled the average chain length, a direct, numerically striking illustration of the strong nonlinearity flagged in Common errors. A

p=0.980\Rightarrow M_n\approx9{,}600 \quad\text{vs.}\quad p=0.995\Rightarrow M_n\approx38{,}400\,\text{g/mol}

Reading. Achieving useful, high molar mass in a step-growth polymer depends critically on pushing conversion very close to completion; comparatively modest-looking gains in conversion late in the reaction translate into dramatic gains in chain length.

Scope. This nonlinear sensitivity, a direct consequence of the Carothers equation's \(1/(1-p)\) form, applies to any step-growth polymerisation, independent of the specific monomer or repeat-unit mass involved.

Problems

A step-growth polyamide has repeat-unit molar mass \(M_0=226\,\text{g/mol}\) and is found to have \(M_n=22{,}600\,\text{g/mol}\). Compute the fractional conversion \(p\) achieved.
Solution
From \(M_n=X_n M_0\) (Step 2): \(X_n=22{,}600/226=100\). Rearranging the Carothers equation, \(1-p=1/X_n=1/100=0.010\), so \(p=0.990\), meaning \(99.0\%\) of the reactive functional groups had reacted at the point this sample was characterised.
Explain, without recomputing numerically, why a step-growth polymerisation deliberately stopped at \(p=0.900\) would be commercially useless as a structural plastic, referencing the Carothers equation.
Solution
At \(p=0.900\), the Carothers equation gives \(X_n=1/(1-0.900)=10\) — an average chain of only about ten repeat units, far too short to develop the chain entanglement and intermolecular forces responsible for a polymer's useful bulk mechanical strength (a property that generally requires chains at least in the hundreds, if not thousands, of repeat units). This is exactly why step-growth polymerisations for structural materials must be driven to very high (typically \(>0.99\)) conversion, per the Result's stated requirement.
Two samples of the same polymer have identical \(M_n\) but sample A has PDI \(=1.05\) while sample B has PDI \(=2.5\). State which sample has the broader chain-length distribution, and explain what this implies about \(M_w\) for the two samples.
Solution
Sample B, with the much higher polydispersity index, has the broader chain-length distribution (Step 4: PDI directly measures how far \(M_w\) exceeds \(M_n\), which happens precisely when a wide range of chain lengths is present). Since both samples share the same \(M_n\), and \(\text{PDI}=M_w/M_n\), sample B's \(M_w\) must be substantially larger than sample A's \(M_w\) (roughly \(2.5\times M_n\) versus \(1.05\times M_n\)), even though their number-average molar masses are identical — a clear illustration of why \(M_n\) alone does not fully characterise a real polymer sample.