Problem 49 - Chemistry and Chemical Reactivity

Question

Ozone, \(\mathrm{O}_{3},\) in the earth's upper atmosphere decomposes according to the equation $$2 \mathrm{O}_{3}(\mathrm{g}) \longrightarrow 3 \mathrm{O}_{2}(\mathrm{g})$$ 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 Slow \(\quad \mathrm{O}_{3}(\mathrm{g})+\mathrm{O}(\mathrm{g}) \longrightarrow 2 \mathrm{O}_{2}(\mathrm{g})\) (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 equation: \( \text{rate} = k[\mathrm{O}_{3}][\mathrm{O}] \).

Step 1: Identify the Rate-Determining Step

The rate-determining step in a reaction mechanism is typically the slowest step. In the given mechanism, Step 2 is labeled as slow, indicating that it is the rate-determining step.

Step 2: Write the Rate Equation for Step 2

For the rate-determining step, the rate equation is based on the reactants involved in that step. In Step 2, the reactants are \( \mathrm{O}_{3}(\mathrm{g}) \) and \( \mathrm{O}(\mathrm{g}) \). Therefore, the rate equation is \( \text{rate} = k[\mathrm{O}_{3}][\mathrm{O}] \), where \( k \) is the rate constant for this step.

Key Concepts

Rate-Determining Step

In any multi-step chemical reaction, one step usually proceeds slower than the others. This slowest step is known as the **rate-determining step**. It acts like a bottleneck, controlling the overall rate of reaction. Think of it as the slowest cashier at a busy checkout line; no matter how fast the other cashiers are, the line will only move as fast as the slowest one.

In the exercise about ozone decomposition, Step 2 is labeled as the slow step, thus it's our rate-determining step. This means the overall reaction rate will depend primarily on the kinetics of this step. Identifying this step is crucial because it helps in understanding how changes in concentration of reactants affect the rate at which the product is formed.

• The slow step controls the speed of the entire reaction.
• The rate equation reflects the concentration of reactants in this step, determining its kinetics.

By understanding the rate-determining step, chemists can optimize reaction conditions or design catalysts that specifically target this bottleneck, effectively speeding up the entire reaction process.

Ozone Decomposition

Ozone, \({ \mathrm{O}_{3} }\), plays an essential role in the Earth's atmosphere by absorbing harmful ultraviolet radiation from the sun. However, ozone also undergoes natural decomposition processes, which can be studied through chemical mechanisms.

The decomposition of ozone can be quite complex, but let's simplify it into a two-step mechanism:

• **Step 1:** Fast and reversible, where ozone breaks apart to form oxygen gas and a reactive oxygen atom, \({ \mathrm{O}_{3} ( \mathrm{g} ) \rightleftarrows \mathrm{O}_{2} ( \mathrm{g} ) + \mathrm{O} ( \mathrm{g} ) }\).
• **Step 2:** Slow and rate-determining, where another ozone molecule reacts with the free oxygen atom forming more oxygen gas, \({ \mathrm{O}_{3} ( \mathrm{g} ) + \mathrm{O} ( \mathrm{g} ) \longrightarrow 2 \mathrm{O}_{2} ( \mathrm{g} ) }\).

This mechanism is significant because it provides insights into how ozone is naturally removed from the atmosphere. Understanding this process is key to explaining ozone balance and depletion, crucial for assessing environmental impacts.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the speed or rate of reactions and the mechanisms by which they occur. It helps understand the rate laws, the effects of various conditions on reaction speeds, and the details of molecular collisions.

For any reaction, the rate can depend on several variables:

• Concentration of reactants: Increasing concentration typically increases the rate.
• Temperature: Higher temperatures usually lead to faster reactions.
• Presence of catalysts: Catalysts lower the activation energy, speeding up reactions.

In the decomposition of ozone, kinetics tells us why the second step, although slow, determines the overall speed of the reaction. By focusing on this step, chemists can derive the rate equation, as seen with \({ \text{rate} = k[ \mathrm{O}_{3} ][ \mathrm{O} ] }\). This equation outlines how the concentrations of ozone and oxygen atoms influence the reaction rate.

Effective application of kinetics can lead to innovations in industries and research involving atmospheric and environmental chemistry.