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A rate law is an equation that describes how the rate of a chemical reaction depends on the concentration of its reactants. For the decomposition of ozone, we are interested in how the concentrations of the substances involved affect the speed at which ozone breaks down. In our given problem, the rate law is determined by the proposed mechanism of the reaction. The derived rate law is \(\text{Rate} = k' \frac{[\mathrm{O}_{3}]^{2}}{[\mathrm{O}_{2}]}\), which tells us:

• The rate of reaction is directly proportional to the square of the concentration of ozone \(\mathrm{O}_{3}\).
• It is inversely proportional to the concentration of oxygen \(\mathrm{O}_{2}\).

This means that the breakdown of ozone happens more rapidly when there is a high concentration of \(\mathrm{O}_{3}\) and a low concentration of \(\mathrm{O}_{2}\). Understanding the factors in this rate law helps us predict how the reaction rate will change under different conditions.

Intermediate species are molecules or atoms that are formed in one step of a reaction mechanism and consumed in another, without appearing in the overall chemical equation. In the decomposition of ozone, the intermediate entity is the oxygen atom \(\mathrm{O}\). It is produced in the first step when \(\mathrm{O}_{3}\) decomposes into \(\mathrm{O}_{2}\) and \(\mathrm{O}\), and it is consumed in the second step where \(\mathrm{O}_{3}\) and \(\mathrm{O}\) form \(\mathrm{O}_{2}\).Identifying intermediates is crucial because they help us understand the entire mechanism of a reaction. This understanding is essential for deriving the correct rate law, as the intermediate often figures into the equation, even if it doesn't appear in the net reaction.

Equilibrium expressions arise when one or more steps in a reaction mechanism are reversible. These expressions help us calculate the concentrations of reactants and products at equilibrium. In the ozone decomposition, the first step is an equilibrium reaction:\[\mathrm{O}_{3} \leftrightarrows \mathrm{O}_{2} + \mathrm{O}\]For this reaction, the equilibrium constant \(K\) is defined as:\[K = \frac{[\mathrm{O}_{2}][\mathrm{O}]}{[\mathrm{O}_{3}]}\]This expression is valuable because it allows us to express intermediates (like \(\mathrm{O}\)) in terms of the original reactants \(\mathrm{O}_{3}\) and products \(\mathrm{O}_{2}\). It provides a connection between the equilibrium concentrations and the rate law, completing the picture of how the reaction proceeds.

The rate-determining step is the slowest step in a reaction mechanism, much like the busy checkout line that determines how fast you can leave a grocery store. It effectively controls the rate of the entire reaction, as it forms a bottleneck that slows everything down. In this decomposition process, the second step:\(\mathrm{O}_{3} + \mathrm{O} \rightarrow 2 \mathrm{O}_{2}\)is the rate-determining step. Knowing which step it is helps us figure out the initial form of the rate law. The species involved in this step, \(\mathrm{O}_{3}\) and the intermediate \(\mathrm{O}\), set the rate equation. Hence, we began by expressing the rate as \(\text{Rate} = k_{2} [\mathrm{O}_{3}][\mathrm{O}]\), later substituting the intermediate \([\mathrm{O}]\) with its equilibrium expression.