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Given a reaction mechanism that has an overall reaction of [tex]2 O_3 \longrightarrow 3 O_2[/tex] and a rate of [tex]k[O_3][O][/tex], which is the correct rate-determining step?

A. [tex]O_3 + 2 O \rightarrow 3 O_2[/tex]
B. [tex]O_2 + O \rightarrow O_3[/tex]
C. [tex]O_3 \Leftrightarrow O_2 + O[/tex]
D. [tex]O_3 + O \rightarrow 2 O_2[/tex]

Sagot :

GinnyAnswer
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]
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