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Question: The chemical reaction, \(2{{\text{O}}_3} \to 3{{\text{O}}_2}\)proceeds as \({{\text{O}}_3} \right...

The chemical reaction, 2O3→3O22{{\text{O}}_3} \to 3{{\text{O}}_2}proceeds as
O3⇌O2+[O]{{\text{O}}_3} \rightleftharpoons {{\text{O}}_2} + \left[ {\text{O}} \right] (fast)
[O]+O3→2O2\left[ {\text{O}} \right] + {{\text{O}}_3} \to 2{{\text{O}}_2} (slow)
The rate law expression will be
A.Rate=k[O][O3]{\text{Rate}} = k\left[ O \right]\left[ {{O_3}} \right]
B.Rate=k[O2]−1[O3]2{\text{Rate}} = k{\left[ {{O_2}} \right]^{ - 1}}{\left[ {{O_3}} \right]^2}
C.Rate=k[O3]2{\text{Rate}} = k{\left[ {{O_3}} \right]^2}
D.Rate=k[O][O2]{\text{Rate}} = k\left[ O \right]\left[ {{O_2}} \right]

Explanation

Solution

To answer this question, you must recall the rate law. For a multi- step reaction, the rate of the overall reaction is given by the slowest reaction which is also known as the Rate determining step or RDS.

Complete step by step answer:
We know that the rate of a reaction is determined by the rate determining step. The slower of the two reactions is the rate determining step. The rate law equation will contain the concentrations of the reactants of this step.
[O]+O3→2O2\left[ {\text{O}} \right] + {{\text{O}}_3} \to 2{{\text{O}}_2}
Oxygen atom is the intermediate form and thus its concentration is unknown. So we must write its concentration in known parameters.
Form the first reaction, O3⇌O2+[O]{{\text{O}}_3} \rightleftharpoons {{\text{O}}_2} + \left[ {\text{O}} \right]
We can write the equilibrium constant for this reaction as, Keq=[O2][O][O3]{K_{eq}} = \dfrac{{\left[ {{O_2}} \right]\left[ O \right]}}{{\left[ {{O_3}} \right]}}.
From this, we can determine the concentration of O as [O]=Keq[O3][O2]\left[ O \right] = \dfrac{{{K_{eq}}\left[ {{O_3}} \right]}}{{\left[ {{O_2}} \right]}}.
Now, using the rate law, we can write the rate of this reaction as K[O][O3]K\left[ O \right]\left[ {{O_3}} \right]
Substituting the value, we get,
Rate=KKeq[O3][O2][O3]{\text{Rate}} = K\dfrac{{{K_{eq}}\left[ {{O_3}} \right]}}{{\left[ {{O_2}} \right]}}\left[ {{O_3}} \right]
∴Rate=k[O3]2[O2]\therefore {\text{Rate}} = k{\left[ {{O_3}} \right]^2}\left[ {{O_2}} \right]

Hence, the correct answer is B.
Additional information: While studying a chemical reaction, it is important to consider other factors in the reaction other than the chemical properties of the reactants, like, the conditions under which the reaction takes place, the mechanism by which the reaction proceeds, the equilibrium toward which it is moving, and the rate at which it is occurring.

Note:
The rate law is an experimentally determined expression that is used to predict the relationship between the rate of the reaction and the concentrations of reactants. For elementary reactions, the rate equation is generally derived using the first principles given by the collision theory. The rate equation of a reaction with a multi-step mechanism cannot be calculated simply using the stoichiometric coefficients of the overall reaction.