Problem 45 - Chemistry and Chemical Reactivity

Question

Ozone, \(\mathbf{O}_{3},\) 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: \(\quad\) Fast, reversible \(\mathbf{O}_{3}(\mathrm{g}) \rightleftarrows \mathrm{O}_{2}(\mathrm{g})+\mathrm{O}(\mathrm{g})\) Step 2: \(\quad\) Slow \(\quad \mathbf{O}_{3}(\mathrm{g})+\mathbf{O}(\mathrm{g}) \rightarrow 2 \mathrm{O}_{2}(\mathrm{g})\) (a) Which of the steps is rate-determining? (b) Write the rate equation for the rate-determining step.

Step-by-Step Solution

Answer

(a) Step 2 is rate-determining. (b) Rate = k[O₃][O].

Step 1: Identifying the rate-determining step

In reaction mechanisms, the slowest step is known as the rate-determining step because it dictates the speed of the overall reaction. Here, Step 2 is described as the slow step, which means it is the rate-determining step.

Step 2: Writing the rate equation for the rate-determining step

The rate of a reaction is determined by the slowest step. For Step 2, the equation is: \( \text{O}_3(g) + \text{O}(g) \rightarrow 2 \text{O}_2(g) \). Thus, the rate equation is based on the reactants in this step, which are \( \text{O}_3 \) and \( \text{O} \). The rate equation is: \( \text{Rate} = k[\text{O}_3][\text{O}] \), where \( k \) is the rate constant.

Key Concepts

Rate-Determining Step

The rate-determining step is a crucial concept in the study of chemical reactions and their mechanisms. Think of it as the slowest step in a sequence of reactions, similar to how a bottleneck slows down traffic. This step determines the overall speed of the reaction because no matter how fast other steps occur, the overall progress cannot exceed the speed of the slowest one.
In the decomposition of ozone, the chemical reaction is divided into multiple steps. Here, Step 2 is the rate-determining step. It is specified as slow, which means all other steps occur faster and hence, play a supporting role in the overall reaction rate.

Rate Equation

Understanding the rate equation helps in quantifying the speed of a chemical reaction. The rate equation is a mathematical expression that relates the reaction rate to the concentratons of the reactants. In the context of the rate-determining step, we derive the rate equation based on the slow step in the reaction.
For ozone decomposition, Step 2 is our focus. The equation for this step is:

• \( \text{O}_3(g) + \text{O}(g) \rightarrow 2 \text{O}_2(g) \)

This means the rate equation can be written as:

• \( \text{Rate} = k[\text{O}_3][\text{O}] \)

where \( k \) is the rate constant. This equation shows that the reaction rate depends on the concentrations of both ozone (\( \text{O}_3 \)) and atomic oxygen (\( \text{O} \)).

Ozone Decomposition

Ozone decomposition is a process of breaking down ozone molecules into simpler oxygen molecules. This chemical reaction is significant in the context of the Earth's upper atmosphere, playing a role in the protective ozone layer.
The given reaction for ozone decomposition is:

• \( 2 \text{O}_3(\text{g}) \rightarrow 3 \text{O}_2(\text{g}) \)

This process involves multiple steps, starting with a fast, reversible reaction that creates atomic oxygen, followed by a slower step which arranges the molecules into two stable \( \text{O}_2 \) molecules. Understanding the steps in this mechanism helps scientists to predict how changes in conditions affect ozone concentration in the atmosphere.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and mechanisms. It explores how reactions proceed over time and what factors influence their speed. By studying kinetics, chemists can understand why certain substances react quickly while others take longer.
In the case of ozone decomposition, chemical kinetics involves analyzing the reaction pathway. By breaking down the reaction into individual steps, scientists can identify the rate-determining step and write corresponding rate equations. This knowledge allows for the prediction of reaction behavior under different conditions. Understanding chemical kinetics helps us to control or enhance processes, such as slowing down ozone depletion or optimizing industrial chemical production.