Question

Ozone, \(\mathrm{O}_{3,}\) in the Earth's upper atmosphere decomposes according to the equation $$ 2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g}) $$ The mechanism of the reaction is thought to proceed through an initial fast, reversible step followed by a slow, second step. Step 1: Fast, reversible $$ \mathrm{O}_{3}(\mathrm{g}) \rightleftarrows \mathrm{O}_{2}(\mathrm{g})+\mathrm{O}(\mathrm{g}) $$ Step 2: Slow $$ \mathrm{O}_{3}(\mathrm{g})+\mathrm{O}(\mathrm{g}) \rightarrow 2 \mathrm{O}_{2}(\mathrm{g}) $$ (a) Which of the steps is rate-determining? (b) Write the rate equation for the rate-determining step.

Short answer

Step 2 is the rate-determining step; rate equation: \(\text{Rate} = k[\mathrm{O}_{3}][\mathrm{O}]\).

Step-by-step solution

Identify the Rate-Determining Step

The rate-determining step is usually the slowest step in a reaction mechanism. Here, Step 2 is labeled as 'Slow.' Therefore, Step 2 is the rate-determining step.

Write the Rate Equation for the Slow Step

The rate equation can be derived from the rate-determining step. In Step 2, the reactants are \(\mathrm{O}_{3}(\mathrm{g})\) and \(\mathrm{O}(\mathrm{g})\). The rate law is based on these reactants, thus the rate equation is: \(\text{Rate} = k[\mathrm{O}_{3}][\mathrm{O}]\), where \(k\) is the rate constant.

Key concepts

Reaction Mechanism

A reaction mechanism is a sequence of elementary steps that leads to the overall chemical reaction. Each step represents a single transformation where bonds are broken and formed, involving a few molecules at a time. Understanding the mechanism helps us predict the rate of reaction and describe how the reactants transform into products.
In the exercise about ozone decomposition, the reaction mechanism consists of two steps:

• Step 1: Fast and reversible; \(\mathrm{O}_{3}(g) \rightleftharpoons \mathrm{O}_{2}(g) + \mathrm{O}(g)\)
• Step 2: Slow and irreversible; \(\mathrm{O}_{3}(g) + \mathrm{O}(g) \rightarrow 2\mathrm{O}_{2}(g)\)

The first step involves the formation of \(\mathrm{O}(g)\) and is quickly reversible, reaching a temporary equilibrium. In contrast, the second step consumes \(\mathrm{O}(g)\), producing oxygen molecules. Together, these steps describe how ozone decomposes in the upper atmosphere.

Rate-determining Step

In a multistep reaction, one step typically controls the reaction speed, called the rate-determining step. It's often the slowest step because it forms a bottleneck that dictates how quickly the overall reaction can proceed.
From the given reaction mechanism of ozone decomposition, the second step (\(\mathrm{O}_{3}(g) + \mathrm{O}(g) \rightarrow 2\mathrm{O}_{2}(g)\)) is labeled as 'Slow.' This indicates that it is the rate-determining step. Since it's the slowest step, it limits the speed at which the ozone decomposes into oxygen molecules.
Understanding which step is rate-determining allows chemists to focus on that part to control or optimize the reaction rate. If this step can be sped up, the entire reaction could potentially proceed faster.

Rate Equation

The rate equation quantifies the speed of a reaction, expressed usually in terms of the concentration of reactants for the rate-determining step. This mathematical representation allows us to calculate the reaction rate and understand how changes in concentration affect it.
For the ozone decomposition mechanism, the rate-determining step involves \(\mathrm{O}_{3}(g)\) and \(\mathrm{O}(g)\). Therefore, its rate equation is: \[\text{Rate} = k[\mathrm{O}_{3}][\mathrm{O}]\] Here:

• \(k\) is the rate constant, a factor that includes the reaction's temperature sensitivity and other conditions.
• The concentration terms \([\mathrm{O}_{3}]\) and \([\mathrm{O}]\) indicate that the rate depends on both ozone and atomic oxygen concentrations.

This equation is crucial for calculating the reaction rate and performing experiments that can validate the proposed mechanism or help discover unknown components affecting the reaction.