Coverage θ 0 (mean field 0)TOF 0 /s/site (mean field 0)Ea 0 eVVolcano top E_b* 0 eVModel time 0loadingFPS 0

Drag to orbit, wheel or pinch to zoom. Gold atoms hold an adsorbed CO₂; H₂O and CO leave when a reaction event fires. The dotted amber curve is the energy profile (binding well, barrier, exit). Bottom-right: the same profile as a plot.

Catalyst Surface: Sabatier Volcano, BEP Barrier and Turnover Rate

A metal surface must hold the reactant strongly enough to catch it, but weakly enough to let it react. Tune the binding energy and watch a kinetic Monte Carlo lattice trace out the Sabatier volcano.

Catalyst

Conditions

Animation

Live readouts

Sabatier volcano: log10 TOF against binding energy

Coverage history (lattice, windowed)

Missions

Pick a mission; its status updates live from the mean-field rates.

Independent numerical checks

Controls

Drag / one finger: orbit; wheel / pinch: zoom. The Camera button cycles hero, surface close-up and side views.
Binding energy E_b is how strongly the surface holds the adsorbed molecule; α is the BEP slope (how much of the extra binding raises the reaction barrier); E₀ is the barrier at zero binding.
Events per frame sets how many lattice events are simulated for each picture. Statistics use all of them; the molecules you see flying are a sample.
Pressure is a dimensionless ratio on a log scale; it sets the arrival rate on an empty site.

The science

Model. A molecule A (CO₂ in the picture) arrives on an empty site at rate k_a = r_a p, desorbs at k_d = ν exp(−E_b/k_BT) and reacts (H₂ is assumed to be in excess, products H₂O and CO leave at once) at k_r = ν exp(−E_a/k_BT), with ν = 10¹³ s⁻¹ and r_a = 10⁶ s⁻¹ (model values, not measurements).

BEP relation. E_a = E₀ + α E_b: a more strongly bound adsorbate is harder to convert. This is the Brønsted-Evans-Polanyi linear free-energy relation, here with illustrative constants.

Steady state. Mean field gives θ = k_a/(k_a + k_d + k_r) and turnover frequency TOF = k_r θ. Setting d ln TOF/dE_b = 0 gives exactly k_d = k_a α/(1 − α), i.e. E_b* = k_BT ln[ν (1 − α)/(α k_a)] - the top of the Sabatier volcano. Weak binding: bare surface, rate limited by arrival. Strong binding: surface blocked and barrier too high.

Kinetic Monte Carlo. The lattice (10 × 8 sites) uses the Gillespie (BKL) algorithm: the waiting time is −ln(u)/R_tot and the event is chosen in proportion to its rate. Sites do not interact, so KMC must agree with the mean-field result; the checks below confirm it.

Simplifications. One adsorbed species on a perfect surface; no adsorbate-adsorbate interactions, no H₂ coverage, no steps or defects, no reverse reaction, no entropy terms. The pictured intermediates (CO₂*, formate-like species) are illustrative. Not a calculation for any real metal [перевірити за джерелом before comparing with a real catalyst]. Educational virtual model. Models built with Blender, scene by Three.js.