Le Chatelier's principle

The gas-phase dimerization 2 A ⇌ B is perturbed four times: the volume is changed, A is added or magically removed, B is added or magically removed, and the volume is changed back. Each perturbation shifts the reaction quotient Q away from K, and then as the reaction continues, Q approaches K until equilibrium is reached when Q = K. At equilibrium the forward and back reactions keep going at equal rates. Hover or drag across the plots to read off values.

2 A kfkr B

Partial pressures of A and B

P(A)P(B)Total P

Rates of the forward and back reactions converge

Rate of forming B, kfP(A)²Rate of forming A, krP(B)

Net forward rate

forward − back (positive: net formation of B)

The reaction quotient Q approaches K

Q = P(B)/P(A)²K = kf/kr
How this is calculated

For the elementary reaction 2 A ⇌ B, written with partial pressures (ideal gases):

rate of forming B = kf · P(A)² (second order) rate of forming A = kr · P(B) (first order) d P(B)/dt = kf P(A)² − kr P(B) d P(A)/dt = −2 ( kf P(A)² − kr P(B) ) two A are used per B made

At equilibrium the two rates are equal, which gives K = kf/kr = P(B)/P(A)². The reaction quotient Q = P(B)/P(A)² has the same form but uses whatever pressures are present at the moment. When Q < K the net reaction forms B; when Q > K it forms A.

The perturbations. Compressing the gas by a factor n multiplies both partial pressures by n. Because Q has P(A) squared in the denominator, Q falls by a factor n, so the reaction shifts toward B, the side with fewer molecules. Adding A or B changes one partial pressure suddenly; the amounts added are given as partial pressures at the volume in effect at the time. If a removal would make a pressure negative, it is set to zero.

How fast equilibrium returns. Near equilibrium, a small displacement decays with relaxation time

1/τ = kr + 4 kf P(A)eq = √( kr² + 8 kf kr [P(A) + 2 P(B)] )

Because of the P(A) term, the reaction re-equilibrates faster at higher pressure. The page uses this formula, with the larger of the starting and post-addition pressures, to choose the time scale, as the MATLAB version does.

Numerical integration. Because the forward rate depends on P(A)², the rate equations are nonlinear and the page solves them numerically, stepping forward in time with the fourth-order Runge–Kutta method. The step size is kept small compared with the local relaxation time, so the result matches MATLAB's ode45.

Web version of the MATLAB function Sim_twoAtoBr3 (J. Kahn, UMD). Doing the real experiment would need magical tanks of A and B that do not react while being added.