Collision-Theory Reactor: Dimerisation Equilibrium, K_c and Le Chatelier Shifts

About 160 hard spheres collide in a glass reactor. Two A molecules that hit hard enough fuse into a dimer D; a dimer splits again by thermal activation. Push the mixture around and watch it settle at the equilibrium constant that collision theory predicts.

Reduced units. Molecule diameter = 1, mass = 1, k_B = 1, so energies are in units of k_B at T = 1. This is a model reaction (A + A → D), not a specific real chemical.

Scenario

Conditions

Add or remove molecules while it runs: the mixture re-balances (Le Chatelier).

Live readout

Reaction quotient Q against K_c

Experiment: temperature sweep

Runs five fresh reactors (T = 0.8 to 2.2, 50 + 100 time units each) and compares the measured K_c with collision theory. The slope of ln K against 1/T is the bond energy (van't Hoff).

Independent numerical checks

Also run in Node: node src/check.js adds the particle simulation (K_c at three temperatures, wall pressure vs Carnahan–Starling, compression shift).

Controls

Drag / one finger: orbit; wheel / pinch: zoom; Front / Close-up / Above buttons.
Temperature, volume (the piston), barrier and bond energy change the reaction while it runs; scenarios apply a standard disturbance.
Speed sets how many simulated time units pass per second; Pause freezes the reactor.

The science

Collision theory. Two A molecules react when the relative kinetic energy along the line of centres exceeds the barrier E_f. The forward rate is dn_D/dt = k_f n_A² with k_f = ½ πd² √(8k_BT/πμ) e^−E_f/k_BT and reduced mass μ = 1/2.

Reverse step. A dimer splits by thermal activation over the barrier E_r = E_f + ΔE: k_r = ν e^{−(E_f+ΔE)/k_BT}, ν = 2 per time unit [model value].

Equilibrium. Forward and reverse rates are equal when K_c = n_D/n_A² = k_f/k_r ∝ √T e^ΔE/k_BT. E_f cancels: a higher barrier (or a catalyst that lowers it) changes how fast equilibrium is reached, not where it lies. With the total number N = n_AV + 2n_DV fixed, n_A follows from a quadratic.

Le Chatelier. Compare Q = n_D/n_A² with K_c: Q < K_c runs the forward reaction, Q > K_c the reverse. Compression multiplies every concentration by the same factor f, so n_D grows by f but n_A² by f²: Q drops below K_c and the forward reaction runs until the side with fewer molecules (D) has grown. Heating favours the reverse (dissociation) because the forward direction releases energy ΔE.

Van't Hoff. d ln K_c/d(1/T) = ΔE + ½ d ln T/d(1/T) (the second term is the small √T prefactor).

Simplifications. Hard spheres without internal degrees of freedom; the reaction heat goes into a heat bath (Andersen thermostat, rate 1 per molecule per time unit) so T stays fixed; a real A + A → D step needs a third body to carry the energy away; the rate formula is the dilute-gas limit without the Enskog contact-value enhancement (agreement is within a few per cent here); concentrations use the volume reachable by molecule centres (the walls exclude about 0.55). Pressure is the wall momentum transfer, compared with the Carnahan–Starling equation of state.