For the reaction,
`2NO+Cl_(2)rarr2NOCl`
the following mechanism has been proposed.
`NO+Cl_(2)hArr NOCl_(2)" "` (fast)
`NOCl_(2)+NOrarr2NOCl" " ` (slow)
The rate for the reaction is :

Options

A. rate = `k[NO]^(2)[Cl_(2)]`
B. rate = `k[NO][Cl_(2)]^(2)`
C. rate = `k[NOCl_(2)]`
D. rate = `k[NOCl]^(2)`

Explanation

To determine the rate of the reaction \(2NO + Cl_2 \rightarrow 2NOCl\) based on the proposed mechanism, we will analyze the steps involved in the mechanism and apply the appropriate principles of chemical kinetics.

### Step-by-Step Solution:

1. **Identify the Steps of the Mechanism:**
- The first step is a fast equilibrium:
\[ NO + Cl_2 \rightleftharpoons NOCl_2 \]
- The second step is the slow step:
\[ NOCl_2 + NO \rightarrow 2NOCl \]

2. **Determine the Rate-Determining Step:**
- The rate of the overall reaction is determined by the slow step of the mechanism. Therefore, we will focus on the slow step to derive the rate expression.

3. **Write the Rate Expression for the Slow Step:**
- For the slow step, the rate can be expressed as:
\[ \text{Rate} = k_2 [NOCl_2][NO] \]
where \(k_2\) is the rate constant for the slow step.

4. **Express \(NOCl_2\) in Terms of the Equilibrium Constant:**
- Since the first step is fast and at equilibrium, we can express the concentration of \(NOCl_2\) using the equilibrium constant \(K\):
\[ K = \frac{[NOCl_2]}{[NO][Cl_2]} \]
Rearranging gives:
\[ [NOCl_2] = K[NO][Cl_2] \]

5. **Substitute \(NOCl_2\) into the Rate Expression:**
- Now, substitute \(NOCl_2\) into the rate expression:
\[ \text{Rate} = k_2 (K[NO][Cl_2])[NO] \]
This simplifies to:
\[ \text{Rate} = k' [NO]^2[Cl_2] \]
where \(k' = k_2 K\) is the new rate constant.

6. **Final Rate Expression:**
- The final rate expression for the overall reaction is:
\[ \text{Rate} = k' [NO]^2[Cl_2] \]

### Conclusion:
The rate of the reaction \(2NO + Cl_2 \rightarrow 2NOCl\) is given by:
\[ \text{Rate} = k [NO]^2[Cl_2] \]