`2A + B to D +E`, for the reaction proposed mechanism `A + B to C +D` (slow), `A + C to E` (fast). The rate law expression for the reaction is

Options

A. `r=K[A]^(2)[B]`
B. r=K[A][B]
C. `r=K[A]^(2)`
D. r=K[A][C]

Explanation

To derive the rate law expression for the given reaction and its proposed mechanism, we can follow these steps:

### Step 1: Identify the overall reaction and the proposed mechanism
The overall reaction is:
\[ 2A + B \rightarrow D + E \]

The proposed mechanism consists of two steps:
1. \( A + B \rightarrow C + D \) (slow step)
2. \( A + C \rightarrow E \) (fast step)

### Step 2: Determine the rate-determining step
In a reaction mechanism, the rate-determining step (RDS) is the slowest step. This step controls the rate of the overall reaction. Here, the slow step is:
\[ A + B \rightarrow C + D \]

### Step 3: Write the rate law based on the rate-determining step
The rate law for a reaction is expressed in terms of the concentrations of the reactants involved in the rate-determining step. Since the slow step involves reactants \( A \) and \( B \), the rate law can be written as:
\[ \text{Rate} = k[A]^m[B]^n \]
where \( k \) is the rate constant, and \( m \) and \( n \) are the orders of the reaction with respect to \( A \) and \( B \), respectively.

### Step 4: Determine the orders of the reaction
For the slow step \( A + B \rightarrow C + D \), the stoichiometry suggests that the order with respect to \( A \) is 1 and the order with respect to \( B \) is also 1. Therefore:
\[ m = 1 \quad \text{and} \quad n = 1 \]

### Step 5: Write the final rate law expression
Combining all the information, the rate law expression for the overall reaction is:
\[ \text{Rate} = k[A]^1[B]^1 \]
or simply:
\[ \text{Rate} = k[A][B] \]

### Conclusion
The rate law expression for the reaction \( 2A + B \rightarrow D + E \) based on the proposed mechanism is:
\[ \text{Rate} = k[A][B] \]