Chain Reactions

Strike a match, and a self-sustaining cascade of radicals rips through the gas in milliseconds. This is a chain reaction. One initiation event spawns a repeating cycle. That cycle consumes reactant thousands of times before termination catches up. Chain reactions explain why some rate laws look nothing like their stoichiometry. The classic H₂ + Br₂ reaction has a rate law with a sum in the denominator. This is a fingerprint of chain chemistry. Understanding chains matters because they govern combustion, polymerization, atmospheric ozone chemistry, and industrial cracking. The same steady-state machinery you met for unimolecular reactions reappears here. Now it applies to a whole network of radical intermediates, instead of one excited molecule. This article builds the general chain mechanism from initiation through propagation to termination. You will derive the full H₂ + Br₂ rate law from first principles. Then you will see why chain length and inhibition emerge naturally from the same algebra. This sets up the leap to branching chains and explosions.

Key takeaways

Chain reactions cycle through initiation (radical birth), propagation (radical-to-radical steps that consume reactant), and termination (radical death).
The steady-state approximation applied to every radical intermediate collapses a multi-step mechanism into one closed-form rate law.
The H₂ + Br₂ reaction gives rate = k[H₂][Br₂]^(3/2) divided by ([Br₂] + k′[HBr]), explaining its famous inhibition by product.
Chain length, the number of propagation cycles per initiation event, can reach $10^3$ – $10^6$ and explains why trace initiators trigger huge conversions.
Inhibitors and radical scavengers add termination pathways, suppressing rate without changing thermodynamics.
Key idea: applying the steady-state approximation to each chain-carrying radical turns a tangled multi-step mechanism into a single algebraic rate law that often does not match the overall stoichiometry.

The core insight

A chain reaction is a closed loop of radical propagation steps, sandwiched between rare initiation and termination events, and applying the steady-state approximation to every radical intermediate — not just one — is what lets algebra tame this entire cycle into a single observable rate law.

The physical picture

A chain mechanism has three functional classes of elementary step. Initiation creates the first radicals, usually by thermal or photochemical bond homolysis, requiring a large activation energy since a strong bond is broken with no compensating bond formed. Propagation steps form a closed loop: a radical attacks a stable molecule, destroying it and regenerating a radical, so the cycle can repeat indefinitely without any net loss of radical population. Termination steps remove radicals, typically by recombination, which releases the enthalpy stored since initiation and requires a third body to carry away that energy. Because propagation steps repeat many times per initiation event, the reactant consumption is dominated by the propagation cycle, not by the rare initiation or termination acts. This is the source of the strange kinetic orders chains produce: the rate law reflects the algebra of the radical steady-state balance, not simple collision counting on the overall equation. The H₂ + Br₂ system is the textbook case, first rationalized by Bodenstein, Lind, and later Herzfeld and Polanyi with the mechanism below.

\text{Br}_2 \xrightarrow{k_1} 2\,\text{Br}\cdot \qquad \text{(initiation)}

\text{Br}\cdot + \text{H}_2 \xrightarrow{k_2} \text{HBr} + \text{H}\cdot \qquad \text{(propagation)}

\text{H}\cdot + \text{Br}_2 \xrightarrow{k_3} \text{HBr} + \text{Br}\cdot \qquad \text{(propagation)}

\text{H}\cdot + \text{HBr} \xrightarrow{k_4} \text{H}_2 + \text{Br}\cdot \qquad \text{(inhibition step)}

2\,\text{Br}\cdot \xrightarrow{k_5} \text{Br}_2 \qquad \text{(termination)}

The two propagation steps alternate: Br• abstracts H from H₂, and the resulting H• attacks Br₂ to regenerate Br•. Each pass through the cycle produces two molecules of HBr while consuming one H₂ and one Br₂, and the cycle can turn over many thousands of times before a termination event finally quenches a Br• radical.

Figure

The H₂ + Br₂ chain cycle: initiation feeds a self-sustaining propagation loop, closed by termination

The inhibition step, $k_4$ , is what makes this system famous: HBr, the product, feeds back and competes with Br₂ for the H• radical, slowing the reaction as product accumulates. This self-inhibition shows up directly in the denominator of the rate law derived below.

The core relations

\text{rate} = \frac{d[\text{HBr}]}{dt} = \frac{2k_2\left(\dfrac{k_1}{k_5}\right)^{1/2}[\text{H}_2][\text{Br}_2]^{3/2}}{[\text{Br}_2] + \dfrac{k_4}{k_3}[\text{HBr}]}

where: $k_1$ – $k_5$ are the rate constants of the five elementary steps; the $3/2$ order in $\text{Br}_2$ and the $[\text{HBr}]$ term in the denominator are the chain signature.

\text{chain length} \; \nu = \frac{\text{rate of propagation}}{\text{rate of initiation}}

where: $\nu$ counts how many times the propagation cycle turns over per radical generated in initiation.

\frac{d[\text{R}\cdot]}{dt} \approx 0 \quad \text{for every chain-carrying radical R}

where: this is the steady-state approximation, applied separately to each radical intermediate (here Br• and H•), the central algebraic tool of chain kinetics.

Derivation AThe H₂ + Br₂ rate law from the Herzfeld–Polanyi mechanism

Goal. Derive the rate law for $d[\text{HBr}]/dt$ from the five elementary steps, using the steady-state approximation on both radical intermediates, Br• and H•.

Step 1. Write the rate of HBr formation. HBr is produced in step 2 and step 3, and consumed in step 4:

\frac{d[\text{HBr}]}{dt} = k_2[\text{Br}\cdot][\text{H}_2] + k_3[\text{H}\cdot][\text{Br}_2] - k_4[\text{H}\cdot][\text{HBr}]

Step 2. Apply the steady-state approximation to H•. H• is produced in step 2 and consumed in steps 3 and 4:

\frac{d[\text{H}\cdot]}{dt} = k_2[\text{Br}\cdot][\text{H}_2] - k_3[\text{H}\cdot][\text{Br}_2] - k_4[\text{H}\cdot][\text{HBr}] = 0

Step 3. Apply the steady-state approximation to Br•. Br• is produced in initiation (2 per event) and in step 4, and consumed in steps 2 and by termination:

\frac{d[\text{Br}\cdot]}{dt} = 2k_1[\text{Br}_2] - k_2[\text{Br}\cdot][\text{H}_2] + k_3[\text{H}\cdot][\text{Br}_2] + k_4[\text{H}\cdot][\text{HBr}] - 2k_5[\text{Br}\cdot]^2 = 0

Step 4. Add the two steady-state equations. Adding the Step 2 and Step 3 expressions cancels every term that involves $k_2$ , $k_3$ , and $k_4$ pairwise (they appear with opposite signs across the two radicals), leaving only the initiation and termination terms:

2k_1[\text{Br}_2] - 2k_5[\text{Br}\cdot]^2 = 0

Step 5. Solve for the Br• steady-state concentration. Rearranging:

[\text{Br}\cdot] = \left(\frac{k_1}{k_5}\right)^{1/2}[\text{Br}_2]^{1/2}

This states that the Br• population tracks the square root of the Br₂ concentration, a direct consequence of first-order initiation balanced against second-order (radical–radical) termination.

Step 6. Solve the Step 2 equation for [H•]. Rearranging the H• steady-state balance:

[\text{H}\cdot]\bigl(k_3[\text{Br}_2] + k_4[\text{HBr}]\bigr) = k_2[\text{Br}\cdot][\text{H}_2]

[\text{H}\cdot] = \frac{k_2[\text{Br}\cdot][\text{H}_2]}{k_3[\text{Br}_2] + k_4[\text{HBr}]}

Step 7. Substitute back into the HBr rate expression. Insert $[\text{H}\cdot]$ from Step 6 into the Step 1 expression. The $k_3[\text{Br}_2]$ and $-k_4[\text{HBr}]$ terms combine cleanly because they share the same denominator, and after simplification:

\frac{d[\text{HBr}]}{dt} = 2k_2[\text{Br}\cdot][\text{H}_2]\cdot\frac{k_3[\text{Br}_2]}{k_3[\text{Br}_2] + k_4[\text{HBr}]}

Step 8. Insert the steady-state Br• concentration from Step 5. Substituting the Step 5 result for $[\text{Br}\cdot]$ and dividing numerator and denominator by $k_3$ :

\boxed{\dfrac{d[\text{HBr}]}{dt} = \dfrac{2k_2\left(\dfrac{k_1}{k_5}\right)^{1/2}[\text{H}_2][\text{Br}_2]^{3/2}}{[\text{Br}_2] + \dfrac{k_4}{k_3}[\text{HBr}]}}

This matches the experimental rate law found by Bodenstein and Lind in 1906, decades before the mechanism itself was proposed by Christiansen, Herzfeld, and Polanyi in 1919–1920.

Worked examples

Example 1 — Chain length in a simple radical chlorination. Consider the chain chlorination of methane, initiated photochemically:

\text{Cl}_2 \xrightarrow{h\nu} 2\,\text{Cl}\cdot \qquad r_i = 2.0\times10^{-8}\ \text{mol L}^{-1}\text{s}^{-1}

\text{Cl}\cdot + \text{CH}_4 \xrightarrow{k_p} \text{CH}_3\cdot + \text{HCl}

with $k_p = 1.0\times10^{4}\ \text{L mol}^{-1}\text{s}^{-1}$ and $[\text{CH}_4] = 0.50\ \text{mol L}^{-1}$ . Termination is second order in radical concentration with $k_t = 2.0\times10^{9}\ \text{L mol}^{-1}\text{s}^{-1}$ , and the steady-state radical concentration satisfies $r_i = k_t[\text{Cl}\cdot]^2$ (this is exactly the Br•-type balance from Step 4 above, applied to a single radical carrier). Solve for the steady-state radical concentration:

[\text{Cl}\cdot] = \left(\frac{r_i}{k_t}\right)^{1/2} = \left(\frac{2.0\times10^{-8}}{2.0\times10^{9}}\right)^{1/2}\ \text{mol L}^{-1}

[\text{Cl}\cdot] = (1.0\times10^{-17})^{1/2}\ \text{mol L}^{-1} = 3.16\times10^{-9}\ \text{mol L}^{-1}

The propagation rate is:

r_p = k_p[\text{Cl}\cdot][\text{CH}_4] = (1.0\times10^4)(3.16\times10^{-9})(0.50)\ \text{mol L}^{-1}\text{s}^{-1}

r_p = 1.58\times10^{-5}\ \text{mol L}^{-1}\text{s}^{-1}

The chain length is:

\nu = \frac{r_p}{r_i} = \frac{1.58\times10^{-5}}{2.0\times10^{-8}} = 790

Sanity check: each initiation event triggers roughly 790 propagation turnovers before a radical is destroyed, consistent with why trace Cl₂ photolysis drives essentially complete methane conversion, and with the general rule that chain lengths of $10^2$ – $10^6$ are typical for efficient radical chains.

Example 2 — Applying the full H₂ + Br₂ rate law under two regimes. Take $k_2 = 1.5\times10^{8}\ \text{L mol}^{-1}\text{s}^{-1}$ , $k_1/k_5 = 4.0\times10^{-14}\ \text{mol L}^{-1}$ (a combined constant absorbing the initiation/termination ratio at a given temperature), and $k_4/k_3 = 0.14$ (the experimentally known inhibition ratio near 550 K). Let $[\text{H}_2] = 0.20\ \text{mol L}^{-1}$ and $[\text{Br}_2] = 0.10\ \text{mol L}^{-1}$ .

Early in reaction ([HBr] ≈ 0): the denominator reduces to $[\text{Br}_2]$ alone:

\text{rate} = 2k_2\left(\frac{k_1}{k_5}\right)^{1/2}[\text{H}_2][\text{Br}_2]^{1/2}

\left(\frac{k_1}{k_5}\right)^{1/2} = (4.0\times10^{-14})^{1/2}\ \text{mol}^{1/2}\text{L}^{-1/2} = 2.0\times10^{-7}\ \text{mol}^{1/2}\text{L}^{-1/2}

\text{rate} = 2(1.5\times10^{8})(2.0\times10^{-7})(0.20)(0.10)^{1/2}\ \text{mol L}^{-1}\text{s}^{-1}

\text{rate} = (3.0\times10^{1})(0.20)(0.316) = 1.90\ \text{mol L}^{-1}\text{s}^{-1}

Later in reaction ([HBr] = 0.15 mol L⁻¹, [Br₂] dropped to 0.03 mol L⁻¹): now the denominator picks up the inhibition term:

\text{denominator} = [\text{Br}_2] + \frac{k_4}{k_3}[\text{HBr}] = 0.03 + (0.14)(0.15) = 0.03 + 0.021 = 0.051\ \text{mol L}^{-1}

\text{numerator} = 2k_2\left(\frac{k_1}{k_5}\right)^{1/2}[\text{H}_2][\text{Br}_2]^{3/2}

[\text{Br}_2]^{3/2} = (0.03)^{3/2} = 5.20\times10^{-3}\ \text{mol}^{3/2}\text{L}^{-3/2}

\text{numerator} = (3.0\times10^{1})(0.20)(5.20\times10^{-3}) = 3.12\times10^{-2}\ \text{mol}^{5/2}\text{L}^{-5/2}\text{s}^{-1}

\text{rate} = \frac{3.12\times10^{-2}}{0.051}\ \text{mol L}^{-1}\text{s}^{-1} = 0.612\ \text{mol L}^{-1}\text{s}^{-1}

Sanity check: even though this snapshot has less Br₂ consumed than a factor of three, the rate fell by roughly a factor of three, faster than the bare $[\text{Br}_2]^{3/2}$ dependence alone would predict, because accumulated HBr actively poisons the chain by intercepting H• radicals. This is the experimental signature that first alerted Bodenstein to a mechanism more complex than simple bimolecular collision.

Limiting cases & common traps

Situation	What happens	Why
$[\text{HBr}] \to 0$	Rate law reduces to $2k_2(k_1/k_5)^{1/2}[\text{H}_2][\text{Br}_2]^{1/2}$	Denominator collapses to $[\text{Br}_2]$ alone; no inhibition yet
$k_4/k_3 \to 0$	HBr no longer inhibits, rate law becomes simple $3/2$ order overall	Step 4 becomes negligible compared to step 3
Long chains, $\nu \gg 1$	Initiation/termination rates negligible for mass balance, but essential for setting $[\text{R}\cdot]$	Steady-state radical concentration still depends on $k_1$ and $k_5$ even though they barely affect total conversion
Radical–radical termination varies	Rate constant ratio $k_1/k_5$ appears only as a square root	Reflects the balance of first-order initiation against second-order termination

The chemist behind it

Max Bodenstein measured the peculiar H₂ + Br₂ rate law experimentally in 1906, decades before anyone could explain the fractional order or the product-inhibition term. The mechanism that finally rationalized his data — radical initiation, alternating propagation, and recombination termination — was worked out independently by Jens Anton Christiansen, Karl Herzfeld, and Michael Polanyi around 1919–1920. Their achievement showed that the newly developing steady-state method, soon formalized further, could convert a baffling empirical rate law into a fully mechanistic, testable prediction, cementing chain kinetics as central to reaction mechanism theory.

Origins · 1 minMax Bodenstein and how this was discovered

Max Bodenstein

1871–1942 · German physical chemist whose precise gas-phase kinetics on H2 + Br2 exposed the first fractional-order chain rate law, motivating the mechanistic search that followed

Ask the tutor: What changes in the steady-state algebra when a propagation step produces two radicals instead of one, and how does that lead to a chain-branching explosion?

Why does the H2 + Br2 rate law have a 3/2 order in Br2 instead of a whole number

The 3/2 order arises algebraically, not from any single elementary step being 3/2 order. Step 3 contributes one power of $[\text{Br}_2]$ directly, and the steady-state Br· concentration derived in Step 5 contributes another $[\text{Br}_2]^{1/2}$ , because Br· formation (first order in Br₂) is balanced against second-order radical–radical termination. Multiplying $[\text{Br}_2]^1$ by $[\text{Br}_2]^{1/2}$ gives the observed $[\text{Br}_2]^{3/2}$ , a signature that a single overall order can hide a two-radical chain mechanism.

Does every chain reaction show product inhibition like H2 + Br2 does

No. Product inhibition in the H₂ + Br₂ system comes specifically from step 4, where H• reacts with HBr to regenerate H₂ and Br·, competing with the productive step 3. Many chain reactions, such as simple radical halogenations of alkanes with no efficient back-reaction, show clean fractional or integer orders without any denominator term. Inhibition appears whenever a product molecule can intercept a chain-carrying radical and divert it from the main propagation cycle.

How is the chain-length concept related to catalysis

A chain reaction resembles catalysis in that a small number of radical carriers process a very large amount of reactant, but the radicals are themselves consumed and regenerated within the propagation loop rather than acting as an external catalyst. Chain length quantifies this amplification: a chain length of $10^4$ means each initiating event yields roughly ten thousand propagation turnovers. Unlike a catalyst, however, the chain carrier concentration is set by a dynamic initiation-termination balance, not fixed by a separate catalytic species added to the system.

What happens if termination becomes second order in a different way, say by reaction with a wall

If termination occurs at a vessel wall rather than by radical–radical recombination in the gas phase, the termination rate becomes first order in radical concentration and proportional to surface-to-volume ratio instead of second order. This changes the exponent on $[\text{Br}_2]$ that emerges from the steady-state balance in Step 5, illustrating that the observed reaction order is not a fixed property of the overall stoichiometry. It depends sensitively on which termination pathway dominates, which is exactly why chain kinetics measurements are so diagnostic of mechanism.

How does this connect to branching chains and explosions

Everything here assumes termination removes radicals at least as fast as initiation creates them, keeping the radical population in a low, steady concentration. If a propagation step instead produces more than one radical per radical consumed, a branching step, the radical population can grow exponentially instead of reaching steady state. That runaway growth is the mechanistic origin of chain-branching explosions, the subject that follows directly from the steady-state chain framework built here.