The chemical reaction `2O_(3)rarr3O_(2)` proceeds as follows :
`O_(3)rarrO_(2)O" "` (fast)
`O+O_(3)rarr2O_(2)" "` (slow)
The rate law expression should be :

Options

A. `r=k[O_(3)]^(2)`
B. `r=k[O_(3)]^(2)[O_(2)]^(-1)`
C. `r=k[O_(3)][O_(2)]`
D. `r=k[O_(3)][O_(2)]^(2)`

Explanation

To determine the rate law expression for the reaction \(2O_3 \rightarrow 3O_2\) given the two-step mechanism, we will follow these steps:

### Step 1: Identify the Steps of the Reaction
The reaction proceeds in two steps:
1. \(O_3 \rightarrow O_2 + O\) (fast step)
2. \(O + O_3 \rightarrow 2O_2\) (slow step)

### Step 2: Determine the Rate-Determining Step
In a multi-step reaction, the slow step is the rate-determining step. Here, the second step is slow, so it will dictate the rate of the overall reaction.

### Step 3: Write the Rate Law Expression
For the slow step \(O + O_3 \rightarrow 2O_2\), the rate law expression can be written as:
\[ \text{Rate} = k[\text{O}][\text{O}_3] \]
where \(k\) is the rate constant for the slow step.

### Step 4: Express Concentration of Intermediate Species
The species \(O\) (nascent oxygen) is an intermediate and is not present in the overall balanced equation. To express its concentration in terms of the reactants, we can use the fast step. The fast step can reach an equilibrium, and we can express the concentration of \(O\) in terms of \(O_3\) and \(O_2\).

From the fast step:
\[ K_{eq} = \frac{[O_2][O]}{[O_3]} \]
Rearranging gives:
\[ [O] = \frac{K_{eq}[O_3]}{[O_2]} \]

### Step 5: Substitute the Expression for \(O\) into the Rate Law
Substituting this expression for \(O\) back into the rate law gives:
\[ \text{Rate} = k \left(\frac{K_{eq}[O_3]}{[O_2]}\right)[O_3] \]
This simplifies to:
\[ \text{Rate} = k K_{eq} \frac{[O_3]^2}{[O_2]} \]

### Step 6: Define a New Constant
Let \(K' = k K_{eq}\), then we can write:
\[ \text{Rate} = K'[O_3]^2[O_2]^{-1} \]

### Final Rate Law Expression
Thus, the final rate law expression for the reaction \(2O_3 \rightarrow 3O_2\) is:
\[ \text{Rate} = K'[O_3]^2[O_2]^{-1} \]