Q4(b)(i). Rate constants of A ⇌ B + C from a temperature jump

Reading of the data: the data imply [A]_(eq)=[B][C]/K=2×10⁸ mol dm⁻³, which is unphysical; but [A] is not needed, since only K=k_f/kᵣ and [B]_(eq)+[C]_(eq) enter 1/τ.

Temperature jump: a capacitor discharge or laser pulse raises T by 5 to 10 K in about 1 µs. The rate constants take their new values at once, but the concentrations are still those of the old equilibrium. The system then relaxes to the new equilibrium, and a relaxation time of 3.0 µs is easily followed.

Derivation of the relaxation time

1. Net rate of change of A: (d[A])/(dt)=-k_f[A]+kᵣ[B][C]
2. Let x be the deviation from the new equilibrium: [A]=[A]_(eq)+x, [B]=[B]_(eq)-x, [C]=[C]_(eq)-x
3. Substituting, with d[A]/dt=dx/dt: 4pt] (dx)/(dt)=-k_f[A]_(eq)+kᵣ[B]_(eq)[C]_(eq)-k_f+kᵣ([B]_(eq)+[C]_(eq)) x+kᵣ x²
4. At the new equilibrium k_f[A]_(eq)=kᵣ[B]_(eq)[C]_(eq), so the first two terms cancel. The jump is small, so the term in x² is neglected (linearisation): (dx)/(dt)=-k_f+kᵣ([B]_(eq)+[C]_(eq)) x
5. This is first order in x, so x=x₀ e^(-t/τ) with (1)/(τ)=k_f+kᵣ([B]_(eq)+[C]_(eq))

Calculation

Given: τ=3.0×10⁻⁶ s; K=k_f/kᵣ=2.0×10⁻¹⁶ mol dm⁻³; [B]_(eq)=[C]_(eq)=2.0×10⁻⁴ mol dm⁻³.

1. (1)/(τ)=13.0×10⁻⁶ s=3.33×10⁵s⁻¹
2. With k_f=K kᵣ: (1)/(τ)=kᵣ(K+[B]_(eq)+[C]_(eq)), and K+[B]_(eq)+[C]_(eq)=2.0×10⁻¹⁶+4.0×10⁻⁴=4.0×10⁻⁴ mol dm⁻³
3. kᵣ=1τ (K+[B]_(eq)+[C]_(eq))=1(3.0×10⁻⁶)(4.0×10⁻⁴), kᵣ=8.3×10⁸ dm³ mol⁻¹ s⁻¹
4. k_f=K kᵣ=(2.0×10⁻¹⁶)(8.33×10⁸), k_f=1.7×10⁻⁷ s⁻¹

Reading the answer: the units agree: k_f in s⁻¹ divided by kᵣ in dm³ mol⁻¹ s⁻¹ gives K in mol dm⁻³, as printed. k_f supplies only 5×10⁻¹³ of 1/τ, so the relaxation is set almost wholly by the recombination of B and C, and k_f follows only through K. A value of kᵣ near 10⁹ dm³ mol⁻¹ s⁻¹ marks a reaction close to the diffusion-controlled limit. A, by contrast, dissociates slowly: t_(1/2)=2/k_f=4.2×10⁶ s, about 48 days.

Application. the same analysis for H₂O ⇌ H⁺ + OH⁻ (relaxation time 37 µs at 298 K) gives kᵣ=1.4×10¹¹ dm³ mol⁻¹ s⁻¹ for H⁺ + OH⁻, one of the fastest reactions in solution. Relaxation methods (temperature and pressure jump) are the standard way to measure such fast reactions.