Question

61.The decomposition of ozone by light can be described by the mechanism.

$$\begin{gathered} {\mathrm{O}}_3+\mathrm{Light}\stackrel{{\mathrm{k}}_1}{\to }{\mathrm{O}}_2+\mathrm{O} \\ \mathrm{O}+{\mathrm{O}}_2+\mathrm{M}\stackrel{{\mathrm{k}}_2}{\to }{\mathrm{O}}_3+\mathrm{M} \\ \mathrm{O}+{\mathrm{O}}_3\stackrel{{\mathrm{k}}_1}{\to }2{\mathrm{O}}_2 \end{gathered}$$

with the overall reaction being

$2{\mathrm{O}}_3+\mathrm{light}\stackrel{}{\to }3{\mathrm{O}}_2$

The rate constant k₁ depends on the light intensity and the type of light source used. By making a steady-state approximation for the concentration of oxygen atoms, express the rate of formation of O₂ in terms of the O₂, O₃, and M concentrations and the elementary rate constants. Show that only the ratio k₃/k₂, and not the individual values of k₂ and k₃, affects the rate.

Short answer

$\mathrm{Rate}={\mathrm{k}}_1\left[{\mathrm{O}}_3\right]\frac{{\mathrm{k}}_1\left[{\mathrm{O}}_3\right]\left[\mathrm{M}\right]}{\left({\mathrm{k}}_2\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[{\mathrm{O}}_3\right]\right)}\left[{\mathrm{O}}_2\right]+\frac{{\mathrm{k}}_3}{{\mathrm{k}}_2}\mathrm{x}\frac{{\mathrm{k}}_1\left[{\mathrm{O}}_3\right]}{\left({\mathrm{k}}_2\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[{\mathrm{O}}_3\right]\right)}\left[{\mathrm{O}}_3\right]........\left(6\right)$

For constant light intensity k₁remains constant and k₂ and k₃ differs and hence finally$\frac{{\mathrm{k}}_3}{{\mathrm{k}}_2}$ affects the rate.

Step-by-step solution

Rate equation:

For the reaction

$$\begin{gathered} {\mathrm{O}}_3+\mathrm{Light}\stackrel{{\mathrm{k}}_1}{\to }{\mathrm{O}}_2+\mathrm{O} \\ \mathrm{O}+{\mathrm{O}}_2+\mathrm{M}\stackrel{{\mathrm{k}}_2}{\to }{\mathrm{O}}_3+\mathrm{M} \\ \mathrm{O}+{\mathrm{O}}_3\stackrel{{\mathrm{k}}_1}{\to }2{\mathrm{O}}_2 \end{gathered}$$

Rate of generation of oxygen$\left({\mathrm{O}}_2\right)$is calculated as following

$\mathrm{Rate}={\mathrm{k}}_1\left[{\mathrm{O}}_3\right]-{\mathrm{k}}_2\left[\mathrm{O}\right]\left[{\mathrm{O}}_2\right]+{\mathrm{k}}_3\left[\mathrm{O}\right]\left[{\mathrm{O}}_3\right].........\left(1\right)$

Change in concentration of O ;

In the above-given equation O is the nascent oxygen , and is considered as reactive intermediate,it is formed by reaction of ozone with light, its formation is given as ${\mathrm{k}}_1\left[{\mathrm{O}}_3\right]$ and its decomposition is given as $-{\mathrm{k}}_2\left[\mathrm{O}\right]\left[{\mathrm{O}}_2\right],{\mathrm{k}}_3\left[\mathrm{O}\right]\left[{\mathrm{O}}_3\right]$.

The over all change in concendration of is given as

$\frac{\mathrm{d}\left[\mathrm{O}\right]}{\mathrm{dt}}={\mathrm{k}}_1\left[{\mathrm{O}}_3\right]-{\mathrm{k}}_2\left[\mathrm{O}\right]\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[\mathrm{O}\right]\left[{\mathrm{O}}_3\right].......\left(2\right)$

k₂ and k₃ are having – sign as O is decomposed in 2^nd and third step.

If a reaction contains more then one step, along with the slowest rate determining step, then there comes the use of steady stae approxinmation theory which states that concendration of reactive intermediate reamains constant threough the reaction,which means that rate of change of intermediate remains constant through out the reaction.

$\frac{\mathrm{d}\left[\mathrm{O}\right]}{\mathrm{dt}}=0.....\left(3\right)$

From the equation 2 and 3

$$\begin{gathered} {\mathrm{k}}_1\left[{\mathrm{O}}_3\right]-{\mathrm{k}}_2\left[\mathrm{O}\right]\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[\mathrm{O}\right]\left[{\mathrm{O}}_3\right]=0......\left(3\right) \\ {\mathrm{k}}_1\left[{\mathrm{O}}_3\right]={\mathrm{k}}_2\left[\mathrm{O}\right]\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]-{\mathrm{k}}_3\left[\mathrm{O}\right]\left[{\mathrm{O}}_3\right] \\ \mathrm{Taking}\mathrm{common}\mathrm{terms}\mathrm{out} \\ {\mathrm{k}}_1\left[{\mathrm{O}}_3\right]=\left[\mathrm{O}\right]\left({\mathrm{k}}_2\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[{\mathrm{O}}_3\right]\right) \\ \left[\mathrm{O}\right]=\frac{{\mathrm{k}}_1\left[{\mathrm{O}}_3\right]}{\left({\mathrm{k}}_2\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[{\mathrm{O}}_3\right]\right)}...........\left(4\right) \end{gathered}$$

Final rate equation;

$$\begin{gathered} \mathrm{Rate}={\mathrm{k}}_1\left[{\mathrm{O}}_3\right]-{\mathrm{k}}_2\left[\mathrm{O}\right]\left[{\mathrm{O}}_2\right]+{\mathrm{k}}_3\left[\mathrm{O}\right]\left[{\mathrm{O}}_3\right]......\left(1\right) \\ \mathrm{Substituting}\left[\mathrm{O}\right]\mathrm{valueinequation}-1 \\ \left[\mathrm{O}\right]=\frac{{\mathrm{k}}_1\left[{\mathrm{O}}_3\right]}{\left({\mathrm{k}}_2\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[{\mathrm{O}}_3\right]\right)} \\ \mathrm{Rate}={\mathrm{k}}_1\left[{\mathrm{O}}_3\right]-{\mathrm{k}}_2\frac{{\mathrm{k}}_1\left[{\mathrm{O}}_3\right]}{\left({\mathrm{k}}_2\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[{\mathrm{O}}_3\right]\right)}\left[{\mathrm{O}}_2\right]+{\mathrm{k}}_3\frac{{\mathrm{k}}_1\left[{\mathrm{O}}_3\right]}{\left({\mathrm{k}}_2\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[{\mathrm{O}}_3\right]\right)}\left[{\mathrm{O}}_3\right]........\left(5\right) \end{gathered}$$

Rate constant affectiong the reaction rate:

Divide the rate expression by 5 by k_{2 then} following expression is obtained.

$\mathrm{Rate}={\mathrm{k}}_1\left[{\mathrm{O}}_3\right]\frac{{\mathrm{k}}_1\left[{\mathrm{O}}_3\right]\left[\mathrm{M}\right]}{\left({\mathrm{k}}_2\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[{\mathrm{O}}_3\right]\right)}\left[{\mathrm{O}}_2\right]+\frac{{\mathrm{k}}_3}{{\mathrm{k}}_2}\mathrm{x}\frac{{\mathrm{k}}_1\left[{\mathrm{O}}_3\right]}{\left({\mathrm{k}}_2\left[{\mathrm{O}}_2\right]\left[\mathrm{M}\right]+{\mathrm{k}}_3\left[{\mathrm{O}}_3\right]\right)}\left[{\mathrm{O}}_3\right]........\left(6\right)$

It is said in the equation that k₁ depend upon the light intensity and particular light source, so for constant light intensity k₁remains constant and k₂ and k₃ differs and hence finally$\frac{{\mathrm{k}}_3}{{\mathrm{k}}_2}$ affects the rate.