Question

Obtain a relation, `k_2/k_1 = ((t_(1/2))_2)/((t_(1/2))_1)`, where k₁ and k₂ are rate constants while (t_1/2)₁  and  (t_1/2)₂  are half-life periods of the first order reaction at temperatures T₁ and T₂ respectively. Write the relation for activation energy.

Answer 1

The rate constant k and half-life period t_1/2 are related as:

`(t_(1/2))_1 = 0.693/k_1   "and"   (t_(1/2))_2 = 0.693/k_2`

`∴ ((t_(1/2))_2)/((t_(1/2))_1) = (0.693/k_2)/(0.693/k_1)`

`∴ k_2/k_1 = ((t_(1/2))_1)/((t_(1/2))_2)`

If E_a is the energy of activation, then

`"log"_10k_2/k_1 = (E_a(T_2 - T_1))/(2.303  R xx T_1 xx T_2)`

`∴ "log"_10((t_(1/2))_1)/((t_(1/2))_2) = (E_a(T_2 - T_1))/(2.303  R xx T_1 xx T_2)`