This is a 2D cross-section of the electrode surface itself: every dot is one radical, spawned right where a carboxylate ion loses an electron and CO2 at the anode, then diffusing in the thin reactive layer above it.
RCOO⁻ − e⁻ → RCOO• → R• + CO2 (anodic oxidation + fast decarboxylation, spawns a radical here)
2 R• → R–R (2nd-order: rate ∝ [R•]², needs a collision)
R• → byproduct (timeout) (1st-order: rate ∝ [R•], happens regardless of neighbors)
Each tick, every live radical does a random diffusion step. Two outcomes compete for each one:
- Coupling — if it collides with another radical (both still radicals) they combine into the dimer product. This needs two particles to find each other, so its rate scales with the square of the local radical concentration.
- Timeout — if it survives its characteristic lifetime without finding a partner, it is diverted into the non-Kolbe byproduct instead (over-oxidation / solvolysis). This is a single-particle process, so its rate scales linearly with concentration.
Current density sets how fast new radicals are generated at the anode. Raise it and the local radical population on the surface climbs — collisions (rate ∝ N²) become much more likely relative to solo timeouts (rate ∝ N), so a larger share of every radical generated ends up as dimer rather than byproduct. This is exactly the real selectivity trade-off in Kolbe electrolysis: dilute, slowly-generated radicals mostly self-destruct before they can find a partner; a crowded, fast-generated population couples efficiently.
Feed concentration raises the supply of carboxylate ions reaching the surface, which raises the radical generation rate the same way current density does. Anode material shifts the radical's characteristic lifetime and reactivity: platinum's clean one-electron chemistry gives radicals the longest lifetime (best Kolbe selectivity at a given current density); graphite shortens it (favors the timeout/byproduct channel); boron-doped diamond sits in between with a wide, evenly reactive surface.
Real-world relevance: this concentration-vs-kinetics trade-off is exactly why industrial Kolbe flow cells run at high current density with fast mass transport — it is the difference between a clean C–C-bond-forming synthesis and a reactor full of over-oxidized waste product.