CO$2$: Relate the understanding on rate laws, stoichiometry tables, reaction kinetics, reaction mechanism and dependence of rate constant on temperature to the conversion and reactor sizing under isothermal and non$-$isothermal operations.

Question

Consider the thermal decomposition of ozone into oxygen as described by the balanced stoichiometric equation;

$2O_3$longrightarrow$3O_2$

Assuming that the reaction follows the given mechanism of $3$ elementary reactions;

$O_{3}longrightarro{w}^{k_1}{O}_2+O.$

$O_2+O*longrightarro{w}^{k_2}{O}_3$

$O*+O_{3}longrightarro{w}^{k_3}2O_2$

with $O-$ represents the active intermediate within the mechanism.

The constants $k_1,k_2$ and $k_3$ are the rate constants for each of the elementary reaction and the second reaction is believed to be the slowest amongst the three. Derive the rate expression of the decomposition of ozone.