A hypothetical reaction `A_(2) + B_(2) rarr 2AB` follows the mechanism as given below:
`A_(2) hArr A+A ("fast")`
`A+B_(2) rarr AB+B` (slow)
`A+B rarr AB` (fast)
The order of the overall reaction is

Options

A. `2`
B. `1`
C. `1.5`
D. `0`

Explanation

To determine the order of the overall reaction for the given hypothetical reaction \( A_2 + B_2 \rightarrow 2AB \) that follows the specified mechanism, we will analyze each step of the mechanism and derive the rate law.

### Step 1: Identify the Mechanism
The mechanism consists of three steps:
1. \( A_2 \rightleftharpoons A + A \) (fast)
2. \( A + B_2 \rightarrow AB + B \) (slow)
3. \( A + B \rightarrow AB \) (fast)

### Step 2: Determine the Rate-Determining Step
The slow step in the mechanism is the rate-determining step. The rate law for the overall reaction will be based on this step. The rate law for the slow step is:
\[ R = k[A][B_2] \]
where \( k \) is the rate constant for the slow step.

### Step 3: Express Concentration of Intermediate
Since \( A \) is an intermediate, we need to express its concentration in terms of the reactants. The first step is an equilibrium step:
\[ A_2 \rightleftharpoons A + A \]
At equilibrium, we can express the equilibrium constant \( K_{eq} \) as:
\[ K_{eq} = \frac{[A]^2}{[A_2]} \]
From this, we can express the concentration of \( A \):
\[ [A] = \sqrt{K_{eq} [A_2]} \]

### Step 4: Substitute the Intermediate Concentration into the Rate Law
Now we substitute the expression for \( [A] \) into the rate law derived from the slow step:
\[ R = k[\sqrt{K_{eq} [A_2]}][B_2] \]
This simplifies to:
\[ R = k \cdot K_{eq}^{1/2} \cdot [A_2]^{1/2} \cdot [B_2] \]

### Step 5: Determine the Overall Order of the Reaction
From the rate expression, we can see that the order with respect to \( A_2 \) is \( \frac{1}{2} \) and with respect to \( B_2 \) is \( 1 \). Therefore, the overall order of the reaction is:
\[ \text{Overall Order} = \frac{1}{2} + 1 = \frac{3}{2} \]

### Final Answer
The order of the overall reaction is \( \frac{3}{2} \) or 1.5.

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