Chapter 17, Problem 17E.2ST

Interpretation Introduction

Interpretation:

The rate law for the decomposition of the ozone in the reaction has to be derived.

Concept introduction:

Rate law: It is generally the rate equation that consists of the reaction rate with the concentration or the pressures of the reactants and constant parameters.

Explanation of Solution

The given reaction involves multiple mechanism steps, by adding the entire individual steps gives rise to an overall reaction equation. Hence, the reaction equation is as follows,

  $\underset{\_}{\begin{array}{l}\underset{\_}{\begin{array}{l}{O}_3\to O_2+O{k}_a\\ O_2+O\stackrel{}{\to }{O}_3{k}_a^{\text{'}}\\ O+O_3\stackrel{}{\to }{O}_2+O_2{k}_b\end{array}}\\ 2O_3+2O+O_2\stackrel{}{\to }{O}_3+3O_2+O\end{array}}$.

The overall equation is given as $2O_3\stackrel{}{\to }3O_2$.

The intermediate is $\text{O}$, the net rates of change of its concentration is,

  $\frac{\text{d}\left[\text{O}\right]}{\text{dt}}\text{=}{\text{k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]\text{-}{\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]\left[\text{O}\right]\text{-}{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]\left[\text{O}\right]=0$.

From the above, the concentration of $\left[\text{O}\right]$ intermediate becomes,

  $\begin{array}{l}{\text{k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]\text{-}{\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]\left[\text{O}\right]\text{-}{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]\left[\text{O}\right]=0\\ \\ {\text{k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]{\text{= k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]\left[\text{O}\right]\text{+}{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]\left[\text{O}\right]\\ \\ \left[\text{O}\right]\text{=}\frac{{\text{k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]}{\left({\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]\text{+}{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]\right)}\end{array}$.

The net rate of change of concentration of $\left[{\text{O}}_{\text{3}}\right]$ is obtained as,

  $\frac{\text{d}\left[{\text{O}}_3\right]}{\text{dt}}\text{=}-{\text{k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]\text{+}{\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]\left[\text{O}\right]\text{-}{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]\left[\text{O}\right]$.

By substituting the expression of $\left[\text{O}\right]\text{=}\frac{{\text{k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]}{\left({\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]\text{+}{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]\right)}$ into the above expression gives,

  $\begin{array}{l}\frac{\text{d}\left[{\text{O}}_3\right]}{\text{dt}}\text{=}-{\text{k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]+\left[\text{O}\right]\left\{{\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]-{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]\right\}\\ \\ =-{\text{k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]+\frac{{\text{k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]}{\left({\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]\text{+}{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]\right)}\left\{{\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]-{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]\right\}\\ \\ =-{\text{k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]{\text{+k}}_{\text{a}}\left[{\text{O}}_{\text{3}}\right]\frac{{\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]-{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]}{{\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]\text{+}{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]}\\ \\ =\frac{\cancel{-{\text{k}}_{\text{a}}{\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{3}}\right]\left[{\text{O}}_{\text{2}}\right]}-{\text{k}}_{\text{a}}{\text{k}}_{\text{b}}{\left[{\text{O}}_{\text{3}}\right]}^2\cancel{+{\text{k}}_{\text{a}}{\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{3}}\right]\left[{\text{O}}_{\text{2}}\right]}-{\text{k}}_{\text{a}}{\text{k}}_{\text{b}}{\left[{\text{O}}_{\text{3}}\right]}^2}{{\text{k}}_{\text{a}}^{\text{'}}\left[{\text{O}}_{\text{2}}\right]\text{+}{\text{k}}_{\text{b}}\left[{\text{O}}_{\text{3}}\right]}\\ \\ =\frac{-2k_a{k}_b{\left[O_3\right]}^2}{k_a^{\text{'}}\left[O_2\right]+k_b\left[O_3\right]}.\end{array}$.

Therefore, the rate law for the decomposition of the ozone in the reaction is $\frac{-2k_a{k}_b{\left[O_3\right]}^2}{k_a^{\text{'}}\left[O_2\right]+k_b\left[O_3\right]}.$