Problem 70 - General Chemistry: Principles and Modern Applications

Question

A simplified rate law for the reaction \(2 \mathrm{O}_{3}(\mathrm{g}) \longrightarrow\) \(3 \mathrm{O}_{2}(\mathrm{g})\) is $$\text { rate }=k=\frac{\left[\mathrm{O}_{3}\right]^{2}}{\left[\mathrm{O}_{2}\right]}$$ For this reaction, propose a two-step mechanism that consists of a fast, reversible first step, followed by a slow second step.

Step-by-Step Solution

Answer

The proposed two-step mechanism consists of a fast, reversible first step: \(O_{3} \longleftrightarrow O_{2} + O\), and a slow second step \(O_{3} + O -> 2 O_{2}\)

Step 1: Identify Rate Law Expression and Given Reaction

The given rate law expression is rate = \(k \left[\mathrm{O}_{3}\right]^{2}/\left[\mathrm{O}_{2}\right]\). And the overall reaction is \(2 O_{3}(g) → 3 O_{2}(g)\).

Step 2: Propose the First Step

Propose a fast reversible step that matches with the rate law expression. Thus, the first step could be \(O_{3} \longleftrightarrow O_{2} + O\). This step is fast and reversible, meeting the requirements in the question.

Step 3: Propose the Second Step

After the first step, the O radical should combine with another O3 to form O2; this must be the second step and is slow because the radical needs to collide with another O3 molecule. Thus, the second step could be \(O_{3} + O -> 2 O_{2}\). This step is slow, so it acts as the rate-determining step in this reaction mechanism.

Key Concepts

Rate Laws

Rate laws are essential to understanding how the concentration of reactants affects the speed of a chemical reaction. In the context of this problem, the rate law is given as \( \text{rate} = k \frac{[\text{O}_3]^2}{[\text{O}_2]} \). This expression tells us how the concentration of ozone \([\text{O}_3]\) and oxygen \([\text{O}_2]\) influences the overall rate of the reaction. The rate constant \(k\) is a proportionality factor that varies with conditions like temperature.

In general, rate laws can be broken down into the form \( \text{rate} = k[A]^m[B]^n... \), where \(m\) and \(n\) represent the order of the reaction with respect to each reactant. The sum of these exponents gives the overall order of the reaction, which can help predict how changes in concentration affect reaction rate.

• The numerator \([\text{O}_3]^2\) shows that the reaction is second-order with respect to ozone.
• The denominator \([\text{O}_2]\) indicates a negative first-order relationship with oxygen.

This specific mechanism involves a complex interplay between reactants, conjugates, and the balancing of fast and slow reaction steps.

Chemical Kinetics

Chemical kinetics studies the rates of chemical processes and the factors affecting them. It involves analyzing reaction mechanisms and determining which steps are fast or slow, and identifies the rate-determining step. In this exercise, we have a two-step reaction mechanism involving ozone decomposition.

The first step is fast and reversible, \( \text{O}_3 \longleftrightarrow \text{O}_2 + \text{O} \), meaning it quickly reaches an equilibrium state where forward and backward reactions occur at equal rates. This step does not significantly impact the overall reaction rate. Instead, it establishes a reactive intermediate that participates in the second step.

• This intermediate, \(\text{O}\), is crucial because it is consumed in the slow, rate-determining step \( \text{O}_3 + \text{O} \rightarrow 2 \text{O}_2 \).
• The term 'rate-determining step' refers to the slowest part of the reaction, limiting how fast the overall reaction proceeds.

Understanding the roles of each step and intermediate helps chemists manipulate conditions to optimize reaction speed and efficiency.

Ozone Decomposition

Ozone decomposition is a complex chemical process crucial in atmospheric studies and environmental science. Ozone, \( \text{O}_3 \), slowly decomposes into oxygen, \( \text{O}_2 \), a process studied to understand its kinetics and impact on the environment.

The proposed mechanism—fast reversible formation of radicals followed by a slow step—models how ozone breaks down naturally and how its concentration is affected by environmental factors. This knowledge is crucial for developing strategies to manage harmful ground-level ozone and understand ozone's role in the upper atmosphere, where it filters ultraviolet radiation.

• Step 1: This fast equilibrium generates oxides capable of participating in further reactions, establishing available reactants.
• Step 2: The slow degradation of \( \text{O}_3 \) in the presence of radical \( \text{O} \) is what predominantly influences reactant levels and reaction speed.

This decomposition process also serves as an example of how kinetic studies can be applied to larger-scale environmental issues such as ozone layer depletion and air quality control.