The reaction `2NO + Br_(2) to 2NOBr`, obeys the following mechanism:
`NO + Br_(2) overset("fast")(hArr) NOBr_(2), NOBr_(2) + NO overset("Slow")(to) 2NOBr` The rate expression of the above reaction can be written as

Options

A. `r = k [NO]^(2) [Br_(2)]_(2)`
B. `r = k [NO][Br_(2)]`
C. `r = k [NO][Br_(2)]`
D. `r = k [NOBr_(2)]`

Explanation

To derive the rate expression for the reaction \(2NO + Br_2 \rightarrow 2NOBr\) based on the provided mechanism, we can follow these steps:

### Step 1: Identify the Rate-Determining Step
The mechanism provided states:
1. \(NO + Br_2 \overset{\text{fast}}{\rightleftharpoons} NOBr_2\)
2. \(NOBr_2 + NO \overset{\text{slow}}{\rightarrow} 2NOBr\)

The slow step is the rate-determining step (RDS), which means the rate of the overall reaction is determined by this step.

### Step 2: Write the Rate Expression for the Slow Step
For the slow step, the rate expression can be written as:
\[ \text{Rate} = k \cdot [NOBr_2] \cdot [NO] \]
where \(k\) is the rate constant for the slow step.

### Step 3: Express the Intermediate in Terms of Reactants
Since \(NOBr_2\) is an intermediate, we need to express its concentration in terms of the reactants. From the fast equilibrium step, we can write:
\[ K_{eq} = \frac{[NOBr_2]}{[NO][Br_2]} \]
Thus, we can rearrange this to find:
\[ [NOBr_2] = K_{eq} \cdot [NO] \cdot [Br_2] \]
where \(K_{eq}\) is the equilibrium constant for the first step.

### Step 4: Substitute the Expression for \(NOBr_2\) into the Rate Equation
Now, we substitute the expression for \(NOBr_2\) back into the rate expression:
\[ \text{Rate} = k \cdot (K_{eq} \cdot [NO] \cdot [Br_2]) \cdot [NO] \]
This simplifies to:
\[ \text{Rate} = k \cdot K_{eq} \cdot [NO]^2 \cdot [Br_2] \]

### Step 5: Final Rate Expression
We can denote \(k' = k \cdot K_{eq}\) as a new rate constant for the overall reaction. Thus, the final rate expression is:
\[ \text{Rate} = k' \cdot [NO]^2 \cdot [Br_2] \]

### Conclusion
The rate expression for the reaction \(2NO + Br_2 \rightarrow 2NOBr\) is:
\[ \text{Rate} = k' \cdot [NO]^2 \cdot [Br_2] \]