Chain Reaction Simulator: Radical Propagation & Explosion Limits

simulator advanced ~12 min
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φ = −0.5 — steady combustion, termination exceeds branching

With branching factor 1.5 and termination rate 2 s⁻¹, the net branching is −0.5 — termination dominates and the radical population reaches a steady state. Increasing branching or reducing termination pushes toward explosion.

Formula

φ = f_branching − k_termination (net branching)
d[R•]/dt = r_i + (f − 1)k_p[R•][M] − k_t[R•]²
ν = k_p[M] / (2 × k_t × r_i)^0.5

The Chain Mechanism

Chain reactions are nature's amplifiers. A single initiation event — a photon breaking a bond, a spark igniting fuel — produces a reactive radical that enters a propagation cycle. Each cycle consumes and regenerates the radical while converting reactant to product. In branching chains, each cycle produces more than one radical, creating exponential growth that can lead to explosion.

Branching vs Termination

The fate of a chain reaction depends on the competition between chain branching (radical multiplication) and chain termination (radical removal by wall collisions, three-body recombination, or inhibitor reactions). When branching exceeds termination, the radical population grows exponentially — this is the mathematical condition for explosion. The simulation tracks this balance in real time.

Explosion Limits

The hydrogen-oxygen system beautifully illustrates chain reaction dynamics with its three explosion limits. The 'explosion peninsula' in pressure-temperature space reveals how wall termination, gas-phase branching, and three-body termination compete. This simulation models the net branching factor and shows the transition from steady burning to explosive behavior as parameters change.

Visualizing Radical Growth

The canvas shows individual radicals as particles, with propagation events creating new radicals (branching) or removing them (termination). In the steady regime, you see a constant population maintained by the balance of initiation and termination. In the explosion regime, the radical count grows visibly, filling the simulation space — a dramatic visual representation of exponential amplification.

FAQ

What is a chain reaction?

A chain reaction is a sequence of reactions where a reactive intermediate (radical) is consumed in one step and regenerated in another, creating a self-sustaining cycle. Each propagation step consumes one radical and produces one (straight chain) or more (branching chain). Chain reactions drive combustion, polymerization, and nuclear fission.

What determines whether a reaction explodes?

Explosion occurs when chain branching (radical multiplication) exceeds chain termination (radical removal). The net branching factor φ = branching rate − termination rate determines the regime: φ > 0 gives exponential radical growth (explosion), φ < 0 gives steady-state combustion.

What are the hydrogen-oxygen explosion limits?

The H₂/O₂ system has three explosion limits: below the first limit (~1 torr), radicals diffuse to walls and terminate. Between the first and second limits (~1–100 torr), gas-phase branching dominates — explosion occurs. Above the second limit, three-body termination quenches branching. Above the third limit (~several atm), thermal explosion occurs.

What is kinetic chain length?

Kinetic chain length ν is the average number of propagation cycles per initiation event. For free-radical polymerization, ν directly determines the degree of polymerization and thus molecular weight. Long chains (ν > 1000) produce high-molecular-weight polymers.

Sources

Embed

<iframe src="https://homo-deus.com/lab/chemical-kinetics/chain-reaction/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub