preequilibria

Preequilibria refer to rapid, reversible steps in a chemical reaction mechanism that establish equilibrium between reactants and one or more intermediates before a slower, rate-determining step proceeds. In the preequilibrium approximation, these fast equilibria are assumed to be established much more quickly than the subsequent transformation, allowing the intermediate concentrations to be expressed algebraically in terms of the reactants. This separation of timescales enables the derivation of an overall rate law from the slow step and the equilibrium relationships.

Theory and criteria for use hinge on a clear separation of timescales and sufficiently large equilibrium constants for the preequilibria. If a fast equilibrium E + S ⇌ ES with constant K_eq is established before a slower step ES → products, then [ES] can be written as K_eq [E][S], and the total enzyme concentration E_tot equals [E] + [ES]. Solving gives [E] = E_tot /(1 + K_eq [S]) and [ES] = K_eq [S] E_tot /(1 + K_eq [S]). The rate v, governed by the slow step, becomes v = k2 [ES] = Vmax [S] /(Km + [S]), where Vmax = k2 E_tot and Km = 1/K_eq = k-1/k1 under the standard rapid-equilibrium assumption (k-1 ≫ k2).

A common context is enzyme kinetics, where substrate binding to enzyme is fast and reversible, followed by a slower catalytic step. The preequilibrium approximation contrasts with the quasi-steady-state (Briggs–Haldane) approach, which assumes the intermediate concentration remains small but its formation and consumption balance out. Limitations arise when the fast equilibria are not sufficiently rapid or when multiple steps share comparable rates, making simple rate laws invalid.

See also: Michaelis–Menten kinetics, Briggs–Haldane derivation, quasi-equilibrium approximation, steady-state approximation.