Certainly! Let's analyze the given reaction mechanism and the provided information step-by-step to identify the rate-determining step.
### Overall Reaction
The overall reaction is:
[tex]\[ 2 \, O_3 \longrightarrow 3 \, O_2 \][/tex]
### Given Rate Law
The rate law is given as:
[tex]\[ \text{Rate} = k [O_3][O] \][/tex]
This rate law indicates that the rate of the reaction depends on the concentration of both [tex]\(O_3\)[/tex] and [tex]\(O\)[/tex]. This implies that the rate-determining step (the slowest step in the mechanism) involves one molecule of [tex]\(O_3\)[/tex] and one molecule of [tex]\(O\)[/tex].
### Analyzing the Options
1. Option 1: [tex]\(O_3 + 2 \, O \rightarrow 3 \, O_2\)[/tex]
- This step involves one [tex]\(O_3\)[/tex] molecule and two [tex]\(O\)[/tex] atoms.
- The rate law derived from this step would not match [tex]\(k[O_3][O]\)[/tex] because it involves a second-order dependency on [tex]\(O\)[/tex].
2. Option 2: [tex]\(O_2 + O \rightarrow O_3\)[/tex]
- This step involves one [tex]\(O_2\)[/tex] molecule and one [tex]\(O\)[/tex] atom.
- This does not directly involve [tex]\(O_3\)[/tex] as a reactant, which contradicts the given rate law that depends on the concentration of [tex]\(O_3\)[/tex].
3. Option 3: [tex]\(O_3 \Leftrightarrow O_2 + O\)[/tex]
- This is a reversible step involving [tex]\(O_3\)[/tex] breaking down into [tex]\(O_2\)[/tex] and [tex]\(O\)[/tex].
- Although it involves [tex]\(O_3\)[/tex], it is a reversible step and does not fit the typical rate-determining step criteria where the rate law directly represents the slowest irreversible step.
4. Option 4: [tex]\(O_3 + O \rightarrow 2 \, O_2\)[/tex]
- This step involves one [tex]\(O_3\)[/tex] molecule and one [tex]\(O\)[/tex] atom.
- The rate law derived from this step would be [tex]\(k[O_3][O]\)[/tex], which matches the given rate law perfectly.
### Conclusion
Based on the analysis above, the correct rate-determining step that fits the given rate law of [tex]\(k[O_3][O]\)[/tex] is:
[tex]\[ O_3 + O \rightarrow 2 \, O_2 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
