The overall rate `(d[P])/(dt)`, for the reaction `2A overset(K)hArr B,B+Coverset(k_f)rarrP` is given by

Options

A. `Kk_f[A]^2[C]`
B. `K[A][B]`
C. `k_f[B][C]`
D. `Kk_f[A]^2[B][C]`

Answer

To solve the problem, we need to derive the overall rate of the reaction given by:

\[ 2A \overset{K}{\rightleftharpoons} B \]
\[ B + C \overset{k_f}{\rightarrow} P \]

### Step-by-Step Solution:

1. **Identify the Reaction Components**:
- The first part of the reaction is an equilibrium reaction where 2 moles of A produce 1 mole of B.
- The second part of the reaction involves B and C reacting to form product P.

2. **Write the Equilibrium Constant Expression**:
- For the equilibrium reaction \( 2A \rightleftharpoons B \), the equilibrium constant \( K \) is given by:
\[ K = \frac{[B]}{[A]^2} \]

3. **Express Concentration of B**:
- Rearranging the equilibrium constant expression gives us:
\[ [B] = K \cdot [A]^2 \]

4. **Write the Rate of Formation of P**:
- The rate of the reaction forming P from B and C can be expressed using the rate constant \( k_f \):
\[ \text{Rate} = k_f [B][C] \]

5. **Substitute [B] into the Rate Expression**:
- Now substitute the expression for [B] from step 3 into the rate expression:
\[ \text{Rate} = k_f (K \cdot [A]^2) [C] \]

6. **Final Expression for Overall Rate**:
- Therefore, the overall rate of formation of P can be expressed as:
\[ \frac{d[P]}{dt} = k_f K [A]^2 [C] \]

### Conclusion:
The overall rate of the reaction is given by:
\[ \frac{d[P]}{dt} = k_f K [A]^2 [C] \]