The reaction of ozone with oxygen atoms in the presence of chlorine atoms can occur by a two step process shown below:
`O_(3)(g)+Cl^(.)rarrO_(2)(g)+ClO^(.)(g)…(i)`
`k_(i)=5.2xx10^(9)" L mol"^(-1)s^(-1)`
`ClO^(.)(g)+O^(.)(g)rarrO_(2)(g)+Cl^(.)(g).......(ii)`
`k_(ii)=2.6xx10^(10)"L mol"^(-1)s^(-1)`
The closest rate constant for the overall reaction
`O_(3)(g)+O^(.)(g)rarr2O_(2)(g)` is :

Options

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

Answer

To find the closest rate constant for the overall reaction \( O_3(g) + O(g) \rightarrow 2O_2(g) \), we will follow these steps:

### Step 1: Write down the two given reactions and their rate constants.
1. The first reaction is:
\[ O_3(g) + Cl^{\cdot} \rightarrow O_2(g) + ClO^{\cdot} \quad (i) \]
with a rate constant \( k_i = 5.2 \times 10^9 \, \text{L mol}^{-1} \text{s}^{-1} \).

2. The second reaction is:
\[ ClO^{\cdot} + O^{\cdot} \rightarrow O_2(g) + Cl^{\cdot} \quad (ii) \]
with a rate constant \( k_{ii} = 2.6 \times 10^{10} \, \text{L mol}^{-1} \text{s}^{-1} \).

### Step 2: Add the two reactions to find the overall reaction.
When we add the two reactions, we can cancel out the intermediates:
- \( Cl^{\cdot} \) appears on both sides of the first reaction.
- \( ClO^{\cdot} \) appears on both sides of the second reaction.

Thus, the overall reaction becomes:
\[ O_3(g) + O^{\cdot}(g) \rightarrow 2O_2(g) \]

### Step 3: Determine the rate constant for the overall reaction.
The rate constant for the overall reaction, \( k_{overall} \), can be calculated by multiplying the rate constants of the individual steps:
\[ k_{overall} = k_i \times k_{ii} \]
Substituting the values:
\[ k_{overall} = (5.2 \times 10^9) \times (2.6 \times 10^{10}) \]

### Step 4: Perform the multiplication.
Calculating:
\[ k_{overall} = 5.2 \times 2.6 \times 10^{9 + 10} = 13.52 \times 10^{19} \]

### Step 5: Express the result in scientific notation.
To express \( 13.52 \times 10^{19} \) in proper scientific notation:
\[ k_{overall} = 1.352 \times 10^{20} \, \text{L mol}^{-1} \text{s}^{-1} \]

### Step 6: Round to the appropriate significant figures.
Rounding \( 1.352 \) to two significant figures gives us:
\[ k_{overall} \approx 1.4 \times 10^{20} \, \text{L mol}^{-1} \text{s}^{-1} \]

### Final Answer:
The closest rate constant for the overall reaction \( O_3(g) + O(g) \rightarrow 2O_2(g) \) is:
\[ \boxed{1.4 \times 10^{20} \, \text{L mol}^{-1} \text{s}^{-1}} \]