The reaction of `O_(3)` with chlorine atom is given as :
`O_(3)(g)+Cl(g)rarrO_(2)(g)+ClO(g),k_(1)=5.2xx10^(9)L"mol"^(-1)sec^(-1)`
`ClO(g)+O(g)rarrCl(g)+O_(2)(g),k_(2)=2.6xx10^(10)L"mol"^(-1)sec^(-1)`
Which of theses values is closest to the rate constant of the overall reaction ?
`O_(3)(g)+O(g)rarr2O_(2)(g)`

Options

A. `5.2xx10^(9)" L mol"^(-1) s^(-1)`
B. `2.6xx10^(10)" L mol"^(-1) s^(-1)`
C. `3.1xx10^(10)" L mol"^(-1) s^(-1)`
D. `1.4xx10^(20)" L mol"^(-1) s^(-1)`

Explanation

To determine the rate constant of the overall reaction \( O_3(g) + O(g) \rightarrow 2O_2(g) \) based on the provided elementary steps, we can follow these steps:

### Step 1: Identify the Elementary Steps
The reaction occurs in two steps:
1. \( O_3(g) + Cl(g) \rightarrow O_2(g) + ClO(g) \) with rate constant \( k_1 = 5.2 \times 10^9 \, L \, mol^{-1} \, sec^{-1} \)
2. \( ClO(g) + O(g) \rightarrow Cl(g) + O_2(g) \) with rate constant \( k_2 = 2.6 \times 10^{10} \, L \, mol^{-1} \, sec^{-1} \)

### Step 2: Determine the Rate-Determining Step
In a reaction mechanism, the slowest step is the rate-determining step. The rate constant of the overall reaction is determined by this step.

- Compare the values of \( k_1 \) and \( k_2 \):
- \( k_1 = 5.2 \times 10^9 \)
- \( k_2 = 2.6 \times 10^{10} \)

Since \( k_1 < k_2 \), the first step is the rate-determining step.

### Step 3: Conclude the Rate Constant for the Overall Reaction
The rate constant for the overall reaction is equal to the rate constant of the rate-determining step. Therefore, the rate constant of the overall reaction \( O_3(g) + O(g) \rightarrow 2O_2(g) \) is:

\[ k_{\text{overall}} = k_1 = 5.2 \times 10^9 \, L \, mol^{-1} \, sec^{-1} \]

### Final Answer
The closest value to the rate constant of the overall reaction is \( 5.2 \times 10^9 \, L \, mol^{-1} \, sec^{-1} \).

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