The mechanism of the reaction : A+B `to`C is as under: A `to` D [slow], D `to` C [fast]. Choose the set showing rate equation and order with respect to A and B :

Options

A. `k_1[A],(1,0)`
B. `k_1[A]^2,(0,1)`
C. `k_1[A][B],(2,0)`
D. `k_1[A]^2,(2,0)`

Answer

To solve the problem regarding the mechanism of the reaction A + B → C, we need to analyze the given steps of the reaction and determine the rate equation and the order with respect to A and B.

### Step-by-Step Solution:

1. **Identify the Steps of the Reaction Mechanism:**
- The reaction mechanism is given as:
- Step 1: A → D (slow step)
- Step 2: D → C (fast step)

2. **Determine the Rate-Determining Step:**
- The slow step (A → D) is the rate-determining step (RDS). The rate of the overall reaction is determined by this step.

3. **Write the Rate Law:**
- The rate law is based on the rate-determining step. Since the slow step involves only reactant A, we can express the rate as:
\[ \text{Rate} = k[A]^n \]
- Here, \( n \) is the order with respect to A.

4. **Determine the Order with Respect to A:**
- From the stoichiometry of the slow step, the coefficient of A is 1. Therefore, the order with respect to A is:
\[ n = 1 \]
- Thus, the rate law becomes:
\[ \text{Rate} = k[A]^1 \]

5. **Determine the Order with Respect to B:**
- The concentration of B does not appear in the rate law because it is not involved in the rate-determining step. Therefore, the order with respect to B is:
\[ 0 \]

6. **Final Rate Equation and Orders:**
- The final rate equation is:
\[ \text{Rate} = k[A]^1 \]
- The order with respect to A is 1, and the order with respect to B is 0.

### Conclusion:
- The correct set showing the rate equation and the order with respect to A and B is:
- Rate equation: Rate = k[A]
- Order with respect to A: 1
- Order with respect to B: 0