Question

Find the rate expression for the reaction showing mechanism:
$2{O_3}(g) \to 3{O_2}(g)$
$(i){O_3} \rightleftarrows {O_2} + O\_\_\_(Fast)$
$(ii){O_3} + O \to 2{O_2}\_\_\_\_\_(Slow)$

Answer 1

Hint: Rate laws or rate expressions or rate equations are the mathematical expressions that define the relationship between the concentration of the reactants and the rate of a chemical reaction. For writing the rate expression for a chemical reaction, we need to analyse it and identify the slow step which helps in determining the rate of a reaction.

Complete answer:
Rate laws or rate expressions or rate equations are the mathematical expressions that defines the relationship between the concentration of the reactants and the rate of a chemical reaction
In general, a rate law (or differential rate law, as it is sometimes called) takes this form:
$$rate = k{\left[ A \right]^m}{\left[ B \right]^n}{\left[ C \right]^p}$$
Where, $$\left[ A \right],{\text{ }}\left[ B \right],{\text{ }}and{\text{ }}\left[ C \right]$$ represent the molar concentrations of reactants.
k is the rate constant, which is specific for a particular reaction at certain pressure and temperature.
The given chemical reaction is:
$2{O_3}(g) \to 3{O_2}(g)$
$(i){O_3} \rightleftarrows {O_2} + O\_\_\_(Fast)$
$$(ii){O_3} + O \to 2{O_2}\_\_\_\_\_(Slow)$$
Now, we need to analyse it and identify the slow step which helps in determining the rate of a reaction because slowest step is always the rate determining step. Therefore, rate for the above reaction is:
$r = k[{O_3}][O]\_\_\_\_(1)$
Where k is rate constant.
Now, we now from the reaction that ${O_3}$ is a reactant but O is not a reactant. So, we will now find out the expression for [O] and replace it.
Now, from equation (i), we will find out the equilibrium constant ${K_{eq}}$ :
${K_{eq}} = \dfrac{{[{O_2}][O]}}{{[{O_3}]}}$
Now, from this equation, we will find the expression of [O]
$[O] = \dfrac{{{K_{eq}}[{O_3}]}}{{[{O_2}]}}\_\_\_\_\_(2)$
Now, we will put the value of [O] from equation $(2)$ in equation $(1)$, we get
$r = k{K_{eq}}\dfrac{{{{[{O_3}]}^2}}}{{[{O_2}]}}$
Therefore, on simplifying, we get the rate expression as:
$r = k'{[{O_3}]^2}{[{O_2}]^{ - 1}}$
Where, $k' = k{K_{eq}}$

Note:
 Always remember that the slowest step is the rate determining step and rate is written in terms of it. Then we should also keep in mind that rate expression is always written in terms of reactants of a chemical reaction and not the products or intermediate. If they appear in rate expression, then we should replace them by determining their value expression.