The square-root law, and the trade it forces
Under the steady-state assumption — radicals formed as fast as they are destroyed — the whole of conventional free-radical polymerization follows from three rate constants. Initiator decomposes, radicals add monomer, and two radicals meet and stop:
The exponent of one half is the whole story. Doubling the initiator multiplies the rate by only 1.41, and it divides the chain length by the same factor — because the kinetic chain length is the ratio of propagation to initiation, and initiation went up faster than propagation did.
You cannot have both. That single constraint is the reason controlled radical methods were worth inventing: RAFT, ATRP and NMP break the link between rate and chain length by making chains grow together and reversibly, instead of a few at a time and irreversibly.
Kinetics calculator
Chain transfer
Transfer moves the radical to another molecule. The kinetic chain carries on, so the rate is barely touched, but the growing chain is cut short. Mayo's equation adds each transfer route as an independent way of ending a chain:
A transfer constant near 1 is what makes a good regulator: agent and monomer are consumed at the same rate, so the ratio [S]/[M] holds steady and every chain is cut to the same length. A CS far above 1 exhausts the agent early and leaves the last chains long; far below 1 and the agent survives to the end and cuts the late chains hardest.
Initiator half-life
A rule of thumb that saves a lot of wasted reactions: run for about ten half-lives to consume the initiator, and pick an initiator whose half-life at your temperature is roughly a tenth of the intended run. An initiator with a ten-hour half-life at 60 °C will barely have started after twenty minutes there.
A worked example: styrene with AIBN at 60 °C
Bulk styrene is 8.7 mol/L. Take kp = 341 L mol−1s−1, kt = 6 × 107 L mol−1s−1, AIBN's kd = 9.6 × 10−6 s−1 and f = 0.6, with 0.01 M initiator:
Rp = 341 × 8.7 × 3.1×10−8 = 9.2 × 10−5 mol L−1 s−1
The radical concentration is the number to sit with: 31 nanomolar. There are roughly three hundred million monomer molecules for every growing radical in the flask, and each of those radicals lives about a second before it meets another one and dies. Everything else about free-radical polymerisation follows from that ratio – it is why the chains are long, why the reaction takes hours despite each addition step being fast, and why a trace of oxygen or inhibitor at micromolar levels can stop the whole thing.
The kinetic chain length works out at ν ≈ 800. Styrene terminates mainly by combination, so two growing chains join and the degree of polymerisation is about 2ν ≈ 1,600 – roughly 166 kg/mol. Reaching 10 % conversion takes a little under three hours.
| [AIBN] | Rp (mol L−1s−1) | Mn | Time to 10 % |
|---|---|---|---|
| 0.01 M | 9.2 × 10−5 | 166 kg/mol | 2.8 h |
| 0.04 M | 1.8 × 10−4 | 83 kg/mol | 1.4 h |
That second row is the trade in the heading above, made concrete. Quadrupling the initiator exactly doubles the rate and exactly halves the molecular weight, because rate goes as [I]½ while chain length goes as [I]−½. You buy speed with molecular weight at a fixed exchange rate, and there is no setting of [I] that gives you both. If you need a fast reaction and a long chain, initiator concentration is the wrong lever – lower the temperature and use a faster initiator, raise [M] by removing solvent, or move to a controlled method where chain length is set by stoichiometry instead of by kinetics.
Rate constants here are representative literature figures for illustration, on the same footing as the page's defaults; use values sourced for your own monomer, solvent and temperature for quantitative work. Note also that all of this describes the reaction at low conversion, before kt begins to fall.
Where these numbers come from
kp is the best-known of the three. Pulsed-laser polymerization with SEC gives it directly, and IUPAC has benchmarked values for the common monomers; it is reliable to a few percent.
kt is not a constant at all. Termination is diffusion-controlled, so it falls as the medium thickens — by orders of magnitude across a bulk polymerization. That is the Trommsdorff effect: kt collapses, radicals accumulate, the rate accelerates and the reaction can run away. Any single kt is a value at one conversion, and the numbers usually quoted are for low conversion. Treat every result here as the state of the reaction at the beginning.
f is typically 0.5–0.8 and is rarely measured for the system at hand. It appears under a square root, so an error of a factor of two in f moves Rp by only 41% — the one forgiving parameter here.
Enter constants from a source for your monomer, solvent and temperature. The defaults loaded above are order-of-magnitude figures for styrene with AIBN near 60 °C, present so the page does something on arrival, not as reference data.
Related tools
For a controlled polymerization the chain length is set by stoichiometry rather than kinetics — see the recipe calculator — and the breadth of the distribution by the dispersity predictor. For step-growth, where conversion rather than rate sets the chain length, see Step-Growth & Gel Point.