Q4(c). Equilibrium constant of A + B ⇌ C from two relaxation times

Reading: K₁ and K₋₁ over the arrows are the rate constants k₁, k₋₁; [A] and [B] are the equilibrium concentrations at which τ was measured.

1. After a small jump (temperature or pressure), let x be the deviation of [C] from its new equilibrium value. Then [A] and [B] each deviate by -x: 3pt] (dx)/(dt)=k₁([A]-x)([B]-x)-k₋₁([C]+x)≈-k₁([A]+[B])+k₋₁ x 3pt] (the equilibrium terms cancel and x² is neglected). So x decays as e^(-t/τ) with (1)/(τ)=k₁([A]+[B])+k₋₁
2. Run 1: [A]+[B]=2.0 M, 1/τ₁=1/(2.0×10⁻⁶)=5.00×10⁵ s⁻¹=2.0 k₁+k₋₁
3. Run 2: [A]+[B]=1.0 M, 1/τ₂=1/(3.0×10⁻⁶)=3.33×10⁵ s⁻¹=1.0 k₁+k₋₁
4. Subtracting: k₁=(5.00-3.33)×10⁵/1.0, k₁=1.67×10⁵dm³ mol⁻¹ s⁻¹
5. Back in run 2: k₋₁=3.33×10⁵-1.67×10⁵, k₋₁=1.67×10⁵s⁻¹
6. K=k₁k₋₁=1.67×10⁵1.67×10⁵=1.0dm³ mol⁻¹

Check: with K=1.0, an equilibrium mixture with [A]=[B]=1.0 M holds [C]=K[A][B]=1.0 M. In the plot of 1/τ against [A]+[B] (panel) the slope is k₁ and the intercept k₋₁.

Application. relaxation methods (temperature jump, pressure jump, ultrasonic absorption) measure reactions with half-lives down to microseconds, too fast for mixing: Eigen's study of H⁺ + OH⁻ → H₂O (k near the diffusion limit) used the same relation.