The depletion of ozone involves the following steps :
Step 1 : `O_3underset(K_1)overset(K_2)hArrO_2+O` (fast)
Step 2 : `O_3+Ounderset(K)rarr2O_2` (slow)
The predicted order of the reaction will be

Options

A. First
B. 1.5
C. `-1.5`
D. Zero

Answer

To determine the order of the reaction involving the depletion of ozone, we will analyze the given steps and apply the rate-determining step (RDS) concept.

### Step-by-Step Solution:

1. **Identify the Steps**:
- Step 1: \( O_3 \underset{K_1}{\overset{K_2}{\rightleftharpoons}} O_2 + O \) (fast)
- Step 2: \( O_3 + O \underset{K}{\rightarrow} 2 O_2 \) (slow)

2. **Determine the Rate-Determining Step**:
- The slow step (Step 2) is the rate-determining step (RDS). The overall rate of the reaction will depend on this step.

3. **Write the Rate Law for the Slow Step**:
- The rate law for the slow step can be expressed as:
\[ \text{Rate} = k [O_3][O] \]
- Here, \( k \) is the rate constant for the slow step.

4. **Consider the Fast Step**:
- In the fast step (Step 1), \( O \) is produced and consumed. We need to express the concentration of \( O \) in terms of \( O_3 \).
- At equilibrium for Step 1, we can use the equilibrium constant \( K \):
\[ K = \frac{[O_2][O]}{[O_3]} \]
- Rearranging gives us:
\[ [O] = \frac{K [O_3]}{[O_2]} \]

5. **Substitute \( [O] \) into the Rate Law**:
- Substitute the expression for \( [O] \) into the rate law:
\[ \text{Rate} = k [O_3] \left(\frac{K [O_3]}{[O_2]}\right) \]
- This simplifies to:
\[ \text{Rate} = kK \frac{[O_3]^2}{[O_2]} \]

6. **Determine the Overall Order of the Reaction**:
- The rate expression \( \text{Rate} = kK \frac{[O_3]^2}{[O_2]} \) indicates that the reaction is second order with respect to \( O_3 \) and first order with respect to \( O_2 \).
- Therefore, the overall order of the reaction is:
\[ \text{Order} = 2 + (-1) = 1 \]

### Conclusion:
The predicted order of the reaction is **1** (first order).

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