Hong Life CalculatorsUnderstanding the step-growth polymerisation calculator (Carothers equation) — concept and principle
Polyesters, polyamides such as nylon, and polyurethanes are made by step-growth polymerisation, where functional groups react pair by pair. This calculator gives the number-average degree of polymerisation Xn, the resulting Mn, and the Flory dispersity for a given extent of reaction p and stoichiometric ratio r.
How it is calculated
- Xn = (1 + r) ÷ (1 + r − 2rp)
- With exact balance (r = 1): Xn = 1 ÷ (1 − p)
- Mn = Xn × M₀
- PDI = 1 + p (Flory's most-probable distribution, r = 1)
p is the fraction of functional groups that have reacted (0 to 1), r the ratio of the two kinds of groups with the scarcer one on top (r ≤ 1), and M₀ the mean molar mass of a monomer residue in g/mol.
The principle
Start with N₀ monomers. Each reaction between two groups joins two molecules into one, so after a fraction p of groups has reacted the number of molecules is N₀(1 − p). The average number of monomers per molecule is then N₀ divided by N₀(1 − p), which is 1/(1 − p). When one monomer is in excess, the scarce groups run out and chain ends of the excess type remain, which caps chain length; that is the role of r. Flory's statistical analysis shows that the chain-length distribution is then the most probable one, with Mw/Mn = 1 + p.
Worked example
- Defaults: p = 0.99, r = 1, M₀ = 113 g/mol, the average residue mass of nylon-6,6 (repeat unit 226 g/mol made of two monomers).
- Xn = (1 + 1) ÷ (1 + 1 − 2 × 1 × 0.99) = 2 ÷ 0.02 = 100.
- Mn = 100 × 113 = 11,300 g/mol.
- PDI = 1 + 0.99 = 1.99.
| Stoichiometric ratio r (≤ 1) | Degree of polymerisation Xn |
|---|---|
| 0.25 | 1.655629139 |
| 0.5 | 2.941176471 |
| 0.75 | 6.603773585 |
| 1 | 100 |
| 1.25 | -10 |
| 1.5 | -5.319148936 |
| 2 | -3.125 |
How to read the result
The curve is very steep at the end: p = 0.90 gives Xn = 10, p = 0.95 gives 20, p = 0.98 gives 50 and p = 0.995 gives 200. Useful engineering polymers usually need conversions above 99%, so purity, exact stoichiometry and removal of by-products such as water are critical. Even a small imbalance matters: with r = 0.99, Xn at p = 0.99 drops to about 67, and even at complete conversion it cannot exceed 199. A slight deliberate imbalance or an end-capper is used to control Mn.
Common mistakes
- Using p as percent instead of a fraction between 0 and 1.
- Entering r greater than 1; always divide the scarcer group by the more abundant one.
- Applying the equation to chain-growth (addition) polymerisation, where long chains form at low conversion.
Frequently asked questions
Why is Xn counted per monomer, not per repeat unit?
In an AA–BB system such as nylon-6,6, one repeat unit contains two monomer residues, so Xn counts residues; hence M₀ is half the repeat-unit mass.
What happens if a monomer has three functional groups?
Branching occurs and the system can form a cross-linked gel at a critical conversion, which this linear equation does not describe.
How is p measured in practice?
Often by titrating remaining end groups (for example acid number for polyesters) or by spectroscopy.
Sources: Carothers, W. H. (1936). Polymers and polyfunctionality. Transactions of the Faraday Society, 32, 39–49. · Flory, P. J. Principles of Polymer Chemistry. Cornell University Press, 1953.