Kinetics
Chain Reactions
Initiation → propagation (radical/ion regenerated each turn) → branching → termination — combustion, polymerization, atmospheric ozone
A chain reaction is a multi-step mechanism in which a chain carrier — typically a radical or ion — is regenerated each propagation cycle, so a single initiation event triggers many turnovers. The four canonical step types are initiation (forms the carrier from stable molecules), propagation (consumes substrate, regenerates carrier), branching (one carrier produces several), and termination (destroys carriers). Mean chain length ranges from ~105 in H2-Br2 to thousands in styrene polymerization. Nikolay Semenov and Cyril Hinshelwood shared the 1956 Nobel Prize in Chemistry for explaining how branching produces the H2-O2 explosion limits — the same kinetics that controls every internal-combustion engine and every flame on Earth.
- StepsInitiation · Propagation · Branching · Termination
- H2-Br2 chain length~105
- NobelSemenov & Hinshelwood, Chemistry 1956
- Branching exampleH· + O2 → OH· + O·
- Bodenstein rate lawd[HBr]/dt ∝ [H2][Br2]1/2/(1+k[HBr]/[Br2])
- Polymerization scale>100 Mt/yr — LDPE, PVC, PS, acrylics
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Why chain reactions matter
- Combustion runs on chains. Every hydrocarbon flame, from a candle to a jet engine, propagates by a radical chain in which OH·, H·, O·, and HO2· cycle through propagation and branching. The branching step H· + O2 → OH· + O· is the "fuse" — its rate constant is known to ~3% from shock-tube experiments because every combustion model on Earth depends on it.
- Polymerization scale. Free-radical polymerization makes >100 Mt/year of LDPE, polystyrene, PVC, and polyacrylates. Each initiator radical (typically AIBN, t1/2 ≈ 10 h at 60 °C, or benzoyl peroxide) launches a chain that adds 103–104 monomer units before terminating, setting the molecular weight distribution.
- Branching causes explosions. If branching rate exceeds termination, the carrier population grows exponentially. Semenov's 1934 theory of thermal explosions and Hinshelwood's H2-O2 explosion-limit map (a P–T plot with three boundaries) earned the joint 1956 Nobel — the canonical kinetics Nobel.
- Stratospheric ozone destruction. A single Cl· atom from CFC photolysis catalyzes ~105 ozone destructions before being deactivated by reaction with NO2 or HOCl. The Antarctic ozone hole is a chain reaction with chain length on the order of 100,000 — Molina, Rowland, and Crutzen's 1995 Nobel work.
- Cracking of petroleum. Steam cracking of ethane to ethylene at 850 °C runs on a free-radical chain through ·CH3 and ·C2H5 intermediates. Ethylene production is >200 Mt/yr globally — the largest-volume organic chemical, all chain-driven.
- Atmospheric NOx and ozone smog. Tropospheric ozone forms by chain initiated by OH·, propagating through HO2·, RO2·, NO2· cycles. Models that drop chain kinetics underpredict tropospheric ozone by orders of magnitude.
- Inhibitor design. A long chain length amplifies trace inhibitors. 1 ppm of butylated hydroxytoluene (BHT) can suppress autoxidation of edible oils ten-fold by intercepting peroxyl radicals; the same logic underlies all antioxidant additive packages.
Common misconceptions
- "Chain reactions are nuclear." Nuclear chain reactions (uranium fission) and chemical chain reactions are conceptually similar — a self-propagating carrier — but the chemistry articles refer specifically to chemical chains: radicals, atoms, or ions. The kinetics framework is the same; the carrier is just different (neutron vs. radical).
- "Branching is the same as propagation." Propagation conserves the carrier count (one in, one out). Branching multiplies it (one in, two or more out). Only branching can lead to explosion; pure propagation chains decay if initiation stops.
- "Long chain length means fast reaction." Chain length describes amplification per initiation, not absolute rate. A reaction with ν = 106 but a slow initiation step is slow overall. The overall rate is rate of initiation × ν.
- "All radical reactions are chains." Many radical reactions are non-chain — e.g. coupling of two radicals to a stable product. A chain requires regeneration of the carrier in propagation. Persistent-radical reactions (TEMPO trapping) deliberately break the chain.
- "Termination at walls is irrelevant in solution." In gas-phase chains termination on container walls (Cl· + wall → ½ Cl2) is often the dominant chain-ending step at low pressure — surface-to-volume ratio matters, which is why Hinshelwood used different vessels to map the first explosion limit. Surface chemistry shifts explosion boundaries.
- "Chain reactions follow simple rate laws." The Bodenstein-Lind rate law d[HBr]/dt = (ka[H2][Br2]1/2)/(1 + kb[HBr]/[Br2]) has half-order in Br2 and product inhibition — both signatures of chain kinetics that conventional power-law fits cannot capture.
Mechanism
A canonical chain has four step types. Initiation: a stable molecule splits homolytically or photolytically to form one or more chain carriers, e.g. Br2 + M → 2Br· + M (thermal at >500 K), or AIBN → 2R· + N2 at 60 °C. Initiation is usually slow — it has the highest activation energy in the mechanism — and a single initiation event has to be amplified for the reaction to proceed at observable rates. Propagation: the carrier reacts with a substrate to form product plus a (different) carrier, which then reacts again to regenerate the original. The two-step propagation cycle for H2-Br2 is Br· + H2 → HBr + H· (slow) and H· + Br2 → HBr + Br· (fast), each turn producing two HBr molecules.
Branching multiplies carriers. In H2-O2: H· + O2 → OH· + O·, then O· + H2 → OH· + H·, then OH· + H2 → H2O + H·. Net: one H· in, three carriers out (one H·, two OH·, one O· that becomes another OH·). If the branching step rate exceeds the termination rate, [carrier] grows exponentially with time, leading to thermal runaway and explosion. Termination: two carriers combine, or one carrier reacts with a wall (a third body M is usually needed to dissipate the bond energy), e.g. 2Cl· + M → Cl2 + M or Br· + wall → ½ Br2. Termination removes carriers without consuming substrate.
Apply the steady-state approximation to each carrier under steady conditions: d[X·]/dt ≈ 0. For H2-Br2, this gives [Br·] = (ki/kt)1/2[Br2]1/2 from initiation-termination balance, [H·] from propagation steady-state, and the overall rate d[HBr]/dt = (ka[H2][Br2]1/2)/(1 + kb[HBr]/[Br2]). The half-order in [Br2] arises because two Br· are produced per initiation; the product inhibition by HBr arises from H· + HBr → H2 + Br· competing with H· + Br2. Bodenstein measured this rate law in 1907; Christiansen and Herzfeld derived it from a chain mechanism in 1919–1920 — one of the first kinetic successes of the chain framework.
Comparison: H2-Br2 · CH4 chlorination · polymerization · branching
| System | Carrier | Propagation | Branching? | Chain length |
|---|---|---|---|---|
| H2-Br2 → 2HBr (gas, ~500 K) | Br·, H· | Br·+H2→HBr+H·, then H·+Br2→HBr+Br· | No (linear) | ~105 |
| CH4 + Cl2 chlorination (photo, ambient) | Cl·, CH3· | Cl·+CH4→HCl+CH3·, CH3·+Cl2→CH3Cl+Cl· | No | ~104 |
| H2 + O2 combustion (gas, >800 K) | H·, OH·, O·, HO2· | OH·+H2→H2O+H·, etc. | Yes — H·+O2→OH·+O· | Diverges (explosion) |
| Free-radical polymerization (e.g. styrene, 60 °C) | Polymer end-group radical R· | R·+monomer→RM·, repeat | No (degenerate transfer occasional) | 103–104 (kinetic chain) |
| Anionic polymerization (e.g. styrene + n-BuLi) | Carbanion end-group | R−+monomer→RM− | No | Living: chains do not terminate |
| Cl/O3 stratospheric destruction | Cl·, ClO· | Cl·+O3→ClO·+O2, ClO·+O→Cl·+O2 | No | ~105 per Cl atom |
| Nuclear fission (U-235) | Neutron | n + U-235 → 2 fragments + 2.5 n | Yes (keff>1) | Diverges (criticality) |
Initiation, propagation, branching, termination
| Step type | Carrier balance | Typical rate constant k | Effect on chain | Example |
|---|---|---|---|---|
| Initiation | 0 → 1 (or 2) | 10-5 – 10-1 s-1 (slow) | Starts chain | AIBN → 2R· + N2 |
| Propagation | 1 → 1 (conserved) | 103 – 109 M-1s-1 | Sustains chain | R·+M → RM· |
| Branching | 1 → 2 (or more) | varies | Multiplies carriers; can cause explosion | H·+O2→OH·+O· |
| Termination by combination | 2 → 0 | 108 – 1010 M-1s-1 (diffusion-limited) | Ends chain | 2R· → R-R |
| Termination by disproportionation | 2 → 0 | 107 – 109 M-1s-1 | Ends chain; gives mixed products | 2 RM· → RM(H) + RM(=) |
| Termination at wall | 1 → 0 | surface-area dependent | Ends chain; controls 1st explosion limit | H· + wall → ½ H2 |
| Chain transfer | 1 → 1 (different carrier) | 10-2 – 102 M-1s-1 | Caps chain length without ending it | RM·+R'SH → RM-H + R'S· |
Applications
- Thermal cracking of ethane. C2H6 → C2H4 + H2 at 850 °C runs through CH3·, ·C2H5, and H· chains. Industrial steam crackers operate at residence times of 0.1–0.3 s and produce >200 Mt/yr ethylene — the largest-volume organic chemical. Mechanism kinetic models with hundreds of elementary steps (Curran, Konnov mechanisms) predict yield to within a few percent.
- H2-O2 explosion limits. Hinshelwood's classic explosion-limit map for stoichiometric H2-O2 at 500–700 °C shows three pressure boundaries forming a "peninsula": below the 1st limit (~1 torr) wall termination wins, between 1st and 2nd (~100 torr) branching wins and explosion happens, between 2nd and 3rd thermal balance suppresses branching, above the 3rd (~10 atm) thermal explosion takes over. The map is a 1928 measurement still in textbooks.
- Free-radical polymerization of styrene. AIBN initiation at 60 °C, propagation rate constant kp ≈ 145 M-1 s-1, termination kt ≈ 7.2×107 M-1 s-1. The molecular weight is Mn ≈ kp[M]/(2 f kd[I])1/2 · M0, giving polystyrene of Mn = 105–106. Adding a chain transfer agent like dodecanethiol caps the chain length at the desired molecular weight.
- Stratospheric ozone destruction. Cl· + O3 → ClO· + O2, ClO· + O → Cl· + O2. One Cl· catalyzes ~105 O3 destructions before reaction with NO2, HOCl, or polar stratospheric clouds removes it. The Montreal Protocol bans CFCs precisely because of this huge chain length amplification.
- Antioxidant inhibition in food and lubricants. Lipid autoxidation propagates through ROO· peroxyl radicals; chain length 50–500 in butter at 25 °C. Adding 200 ppm BHT or vitamin E intercepts ROO· at near-diffusion rates (k ≈ 105–106 M-1 s-1) and extends shelf life by 5–10×. Engine oils use ZDDP and hindered phenols on the same principle.
Frequently asked questions
What are the four steps of a chain reaction?
Initiation: a stable molecule splits to form chain carriers, often by thermal homolysis (Cl2 → 2Cl·) or photolysis. Propagation: the carrier reacts with substrate to form product and regenerate a carrier of the same kind, e.g. Cl· + CH4 → HCl + CH3·, then CH3· + Cl2 → CH3Cl + Cl·. The cycle repeats. Branching (only in some chains): one carrier produces multiple carriers, e.g. H· + O2 → OH· + O· in H2-O2 combustion. Termination: two carriers combine or are destroyed at a wall, e.g. 2Cl· + M → Cl2 + M, removing them from the cycle. The mean chain length, the ratio of propagation rate to initiation rate, sets the overall rate's sensitivity to inhibitors.
What is chain length and why does it matter?
Chain length ν is the average number of propagation cycles (and therefore product molecules) per initiation event. Computed as ν = rate of propagation / rate of initiation. For H2-Br2 at ~500 K, ν ≈ 105 — one Br atom from thermal Br2 dissociation triggers 100,000 cycles before terminating. For free-radical polymerization of styrene at 60 °C, ν is the kinetic chain length (number of monomer additions per initiator radical) and ranges from 103 to 104, setting the molecular weight. Long chains amplify trace impurities into observable kinetics — 1 ppm of an inhibitor can reduce the rate ten-fold by intercepting the carrier before propagation completes.
What are branching chain reactions?
Branching steps produce more than one carrier from one. The classic example is H· + O2 → OH· + O· in H2-O2 combustion, where one H atom becomes one OH and one O — net gain of one carrier. If the branching rate exceeds termination, the carrier population grows exponentially with time and the system explodes. Semenov and Hinshelwood mapped the H2-O2 explosion limits: a pressure-temperature plot showing three boundaries — first limit (low pressure), second limit (intermediate), and third limit (high pressure thermal). Inside the first-second 'peninsula' the mixture explodes; outside it reacts slowly. The branching rate constant for H· + O2 has been measured to ~3% in shock-tube experiments because it controls every flame on Earth.
How is the steady-state approximation applied?
Apply d[carrier]/dt = 0 to each chain carrier separately; in a long-chain limit propagation rates are equal, so all carrier concentrations are linked. For the H2-Br2 chain, set d[Br·]/dt = 0 and d[H·]/dt = 0. Combining gives [H·] in terms of [Br·], [Br·] in terms of initiation and termination rates, and ultimately the rate of HBr formation as a complicated expression involving ki, kp, kterm, and concentrations of H2 and Br2. The result is the famous Bodenstein rate law d[HBr]/dt = (ka[H2][Br2]1/2)/(1 + kb[HBr]/[Br2]), which is non-integer order in Br2 and inhibited by HBr — both signatures of a chain mechanism, observed by Bodenstein and Lind in 1907 and explained by Christiansen and Herzfeld a decade later.
What is free-radical polymerization?
A linear (non-branching) chain reaction where the propagating carrier is a polymer end-group radical. Initiation: an azo or peroxide initiator decomposes (e.g. AIBN at 60 °C, half-life 10 h) to give R·. Propagation: R· + M → RM·, then RM· + M → RMM·, repeating to add monomer one at a time. Termination: combination (RM·n + RM·m → RM(n+m)R) or disproportionation. Free-radical polymerization makes polystyrene, polyethylene (LDPE), PVC, and polyacrylates — over 100 megatons per year. The kinetic chain length determines molecular weight; chain transfer agents (e.g. dodecanethiol) deliberately limit it to control polymer Mn.
Who got the Nobel Prize for chain reactions?
Nikolay Semenov (USSR) and Cyril Hinshelwood (UK) shared the 1956 Nobel Prize in Chemistry "for their researches into the mechanism of chemical reactions." Semenov's 1934 monograph "Chemical Kinetics and Chain Reactions" formalized branching theory and the theory of thermal explosions. Hinshelwood independently mapped the H2-O2 explosion limits and connected them to branching kinetics. Earlier, Frederick Lindemann (1922) provided the unimolecular framework, Bodenstein (1894) proposed atomic chain carriers in H2-Cl2, and Christiansen and Herzfeld (1919) explained the H2-Br2 rate law. The 1956 prize is the canonical Nobel for kinetics.