The time required for x₀ to reach the value (1/ e) x₀ is called the relaxation time . After a few relaxation times, x becomes very small and the new equilibrium is established.

For other equilibria too, expressions for relaxation times can be derived. For example, for the reaction

A + B

(27.12)

the relaxation time is given by

= 1 / k₁ ([A] eq , new + [B] eq, new) + k_-1

(27.13)

Given in the following table are the values of k₁ and k _-1 for a few reversible acid base reactions.

Table 27.1 values of k₁ and k _-1 for some reversible acid base reactions at 298 K

Reaction	k₁ (in dm³ mol ^–1 s ^–1)	k₁ (in s^–1)

H⁺ (aq) + OH ^– (aq) H₂O(l)	1.4 * 10 ¹¹	10^–3

H⁺ (aq) + CH₃COO^–(aq) CH₃COOH(aq)	4.5 * 10¹⁰	7.8 * 10^–5

H⁺ (a q) + HCO₃^– (aq) H₂CO₃ (aq)	4.7 * 10¹⁰	8 * 10⁶

H⁺(aq) + C₆H ₅COO^– (aq) C₆H₅COOH(aq)	3.5 *10¹⁰	2.2 * 10⁶

For the reaction A

is calculated from k_f and k_r. For the reactions in the table, in addition to k₁ and k _-1, we need the equilibrium constants of the reactions to get

. For the reaction H⁺ + OH ^–

H₂O, the conductivity of the solution increases with a temperature jump. Measuring the time dependent conductivity, the value of

has been found to be 3.7 * 10 ^-5 s. The value of [H⁺] [OH^–] = K_w for water is 10 ^–14and the value of K_c = [H₂O] / [H⁺] [ OH^– ] = [H₂O] /K_w = 5.5 * 10 ^{+ 15} mol ^-1 dm ^–3. Using these values, show that k₁ = 1.4 * 10 ¹¹ dm³ mol ^–1 s ^–1.

Relaxations methods work best when equilibria are disturbed by small amounts to get new equilibria. These methods combine the methods of thermodynamics and kinetics.

The time required for x0 to reach the value (1/ e) x0 is called the relaxation time . After a few relaxation times, x becomes very small and the new equilibrium is established.

The time required for x₀ to reach the value (1/ e) x₀ is called the relaxation time . After a few relaxation times, x becomes very small and the new equilibrium is established.