Conisder the reaction mechanism:
`A_(2) overset(k_(eq))hArr 2A ("fast")` (where `A` is the intermediate.)
`A+B underset(k_(1))rarr P (slow)`
The rate law for the reaction is

Options

A. `k_(1)[A][B]`
B. `k_(1)k^(1//2)[A_(2)]^(1//2)[B]`
C. `k_(1)k^(1//2)[A][B]`
D. `k_(1)k^(1//2)[A]^(2)[B]`

Explanation

To determine the rate law for the given reaction mechanism, we can follow these steps:

### Step 1: Identify the Fast and Slow Steps
The reaction mechanism consists of two steps:
1. **Fast Step:** \( A_2 \overset{k_{eq}}{\rightleftharpoons} 2A \)
2. **Slow Step:** \( A + B \overset{k_1}{\rightarrow} P \)

### Step 2: Write the Equilibrium Expression for the Fast Step
For the fast step, we can write the equilibrium constant expression:
\[ K_{eq} = \frac{[A]^2}{[A_2]} \]

### Step 3: Solve for the Concentration of the Intermediate (A)
From the equilibrium expression, we can rearrange it to find the concentration of \( A \):
\[ [A]^2 = K_{eq} \cdot [A_2] \]
Taking the square root of both sides gives:
\[ [A] = \sqrt{K_{eq} \cdot [A_2]} \]

### Step 4: Write the Rate Law for the Slow Step
The slow step is the rate-determining step, and the rate law for this step can be expressed as:
\[ \text{Rate} = k_1 [A][B] \]

### Step 5: Substitute the Expression for [A] into the Rate Law
Now, substitute the expression we found for \( [A] \) into the rate law:
\[ \text{Rate} = k_1 \left(\sqrt{K_{eq} \cdot [A_2]}\right) [B] \]

### Step 6: Finalize the Rate Law Expression
Thus, the rate law for the overall reaction can be expressed as:
\[ \text{Rate} = k_1 \cdot K_{eq}^{1/2} \cdot [A_2]^{1/2} \cdot [B] \]

### Conclusion
The final rate law for the reaction is:
\[ \text{Rate} = k \cdot [A_2]^{1/2} \cdot [B] \]
where \( k = k_1 \cdot K_{eq}^{1/2} \).