A reaction is represented by :
`A to B` slow and `A+B to C` (fast) where `k_1 and k_2` are the rate constants of the mechanistic steps. The rate of production of B will be given by :

Options

A. `K_1[A][B]`
B. `K_1[A]`
C. `K_1K_2[A]`
D. `K_2[A][B]`

Answer

To determine the rate of production of B in the given reaction mechanism, we will analyze the two steps of the reaction:

1. **Step 1**: A → B (slow)
2. **Step 2**: A + B → C (fast)

### Step-by-Step Solution:

**Step 1: Identify the rate-determining step.**
- The first step (A → B) is the slowest step and thus is the rate-determining step. The rate of production of any intermediate (in this case, B) is primarily determined by the slowest step.

**Step 2: Write the rate expression for the rate-determining step.**
- The rate of the reaction for the slow step can be expressed as:
\[ \text{Rate} = k_1 [A] \]
where \( k_1 \) is the rate constant for the slow step and \([A]\) is the concentration of reactant A.

**Step 3: Relate the rate to the production of B.**
- Since B is produced from A in the first step, the rate of production of B (\( \frac{d[B]}{dt} \)) is equal to the rate of the slow step:
\[ \frac{d[B]}{dt} = k_1 [A] \]

**Step 4: Conclusion.**
- Therefore, the rate of production of B is given by:
\[ \frac{d[B]}{dt} = k_1 [A] \]
- This matches with option 2 from the provided choices.

### Final Answer:
The rate of production of B is \( k_1 [A] \).

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