Question

Consider the following mechanism for ozone thermal decomposition: \\[ \begin{array}{l} \mathrm{O}_{3}(g) \stackrel{k_{1}}{\rightleftharpoons} \mathrm{O}_{2}(g)+\mathrm{O}(g) \\ \mathrm{O}_{3}(g)+\mathrm{O}(g) \stackrel{k_{2}}{\longrightarrow} 2 \mathrm{O}_{2}(g) \end{array} \\] a. Derive the rate law expression for the loss of \(\mathrm{O}_{3}(g)\) b. Under what conditions will the rate law expression for \(\mathrm{O}_{3}(g)\) decomposition be first order with respect to \(\mathrm{O}_{3}(g) ?\)

Short answer

a. The rate law expression for the loss of \(\mathrm{O}_{3}(g)\) is: Rate of loss of O3 = \(\ -[\mathrm{O}_{3}](k_{1} + k_{2}[\mathrm{O}] - k_{-1}[\mathrm{O}_{2}][\mathrm{O}])\) b. The rate law expression for \(\mathrm{O}_{3}(g)\) decomposition will be first order with respect to \(\mathrm{O}_{3}(g)\) when either the concentration of atomic oxygen (\([\mathrm{O}]\)) or molecular oxygen (\([\mathrm{O}_{2}]\)) is very small.

Step-by-step solution

Write down the rate expressions for the two reactions

Write the rate expressions for the given reactions using their rate constants: Reaction 1: \(r_{1} = k_{1}[\mathrm{O}_{3}] - k_{-1}[\mathrm{O}_{2}][\mathrm{O}]\) Reaction 2: \(r_{2} = k_{2}[\mathrm{O}_{3}][\mathrm{O}]\) where: - \(r_{1}\) is the rate of the first reaction - \(r_{2}\) is the rate of the second reaction - \(k_{1}\) is the forward rate constant of the first reaction - \(k_{-1}\) is the reverse rate constant of the first reaction - \(k_{2}\) is the rate constant of the second reaction - \([\mathrm{O}_{3}]\) is the concentration of ozone - \([\mathrm{O}_{2}]\) is the concentration of molecular oxygen - and \([\mathrm{O}]\) is the concentration of atomic oxygen

Find the rate law expression for the loss of O3

The rate of loss of \(\mathrm{O}_{3}(g)\) is given by the sum of the rates of both reactions: Rate of loss of O3 = \(\ \ -(r_{1} + r_{2})\) Using the rate expressions from Step 1, we have: Rate of loss of O3 = \(\ -(k_{1}[\mathrm{O}_{3}] - k_{-1}[\mathrm{O}_{2}][\mathrm{O}] + k_{2}[\mathrm{O}_{3}][\mathrm{O}])\)

Simplify the rate expression

Rearrange the rate expression to combine like terms: Rate of loss of O3 = \(\ -(k_{1}[\mathrm{O}_{3}] + k_{2}[\mathrm{O}_{3}][\mathrm{O}] - k_{-1}[\mathrm{O}_{2}][\mathrm{O}])\) Factor out the O3 concentration: Rate of loss of O3 = \(\ -[\mathrm{O}_{3}](k_{1} + k_{2}[\mathrm{O}] - k_{-1}[\mathrm{O}_{2}][\mathrm{O}])\) This is the rate law expression for the loss of \(\mathrm{O}_{3}(g)\).

Determine when the rate law expression will be first order with respect to O3

For the rate law expression to be first order with respect to \(\mathrm{O}_{3}(g)\), the rate should only depend on the concentration of \(\mathrm{O}_{3}(g)\). The term containing both O2 and O must be negligible. This can happen when the concentration of atomic oxygen (\([\mathrm{O}]\)) is very small or when molecular oxygen concentration (\([\mathrm{O}_{2}]\)) is very small. In such cases, the rate law expression becomes: Rate of loss of O3 = \(\ -[\mathrm{O}_{3}]k_{1}\) The rate law expression is now first order with respect to \(\mathrm{O}_{3}(g)\). So, the required condition is that either the concentration of atomic oxygen or molecular oxygen should be very small.

Key concepts

Rate Law

The rate law is a mathematical expression that describes the rate of a chemical reaction in terms of the concentration of its reactants. It is crucial in understanding reaction dynamics and gives insight into how different concentrations affect the speed of the reaction. For ozone decomposition, the rate law involves various constants and concentrations:

• Forward and reverse rate constants (\(k_1\) and \(k_{-1}\))
• Concentrations of ozone \([\mathrm{O}_3]\), oxygen \([\mathrm{O}_2]\), and atomic oxygen \([\mathrm{O}]\)

In the reaction, different terms combine to give a comprehensive rate expression, allowing us to predict the reaction behavior under various conditions.

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical processes occur. It helps us understand how different variables affect a reaction's speed. In the case of ozone decomposition, two elementary steps contribute to the overall kinetics. Reaction kinetics provides tools to analyze how factors like temperature and concentration influence the reaction rate. By writing rate expressions for each step, we can track the progression of the reaction and use this information to understand complex reaction systems.

First-Order Reaction

A first-order reaction is characterized by the reaction rate being directly proportional to the concentration of a single reactant. For ozone decomposition to be first-order concerning \(\mathrm{O}_3\), the rate should depend only on its concentration. This simplification occurs when either the concentration of atomic or molecular oxygen becomes negligible. In such cases, the rate expression simplifies to\(-[\mathrm{O}_3]k_1\), allowing a clearer understanding of the relationship between concentration and rate.

Chemical Mechanism

The chemical mechanism involves a step-by-step breakdown of a reaction into elementary processes. For ozone decomposition, this mechanism comprises reactions forming and consuming atomic oxygen. By analyzing these steps, we gain a deeper understanding of the reaction pathway. The mechanism helps explain how intermediates, like atomic oxygen, play a role in the overall process, bridging the gap between molecular interactions and observable reaction rates.