Which of the following relative value of `k_(f)` (rate constant of forward reaction) and `k_(b)` (rate constant of backward reaction) results in an equilibrium mixture that contain large amount of reactants and small amounts of products?

Options

A. `k_(f) gt k_(b)`
B. `k_(f) = k_(b)`
C. `k_(f) lt k_(b)`
D. None of these

Answer

To solve the question regarding the relative values of \( k_f \) (rate constant of the forward reaction) and \( k_b \) (rate constant of the backward reaction) that lead to an equilibrium mixture containing a large amount of reactants and a small amount of products, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Reaction**:
Consider the equilibrium reaction:
\[ A + B \rightleftharpoons C + D \]
Here, \( A \) and \( B \) are the reactants, and \( C \) and \( D \) are the products.

2. **Equilibrium Condition**:
At equilibrium, the concentrations of the reactants and products remain constant. The equilibrium constant \( K \) for this reaction can be expressed as:
\[ K = \frac{[C][D]}{[A][B]} \]
Where \( [C] \), \( [D] \), \( [A] \), and \( [B] \) are the molar concentrations of the respective species.

3. **Given Condition**:
The problem states that the equilibrium mixture contains a large amount of reactants and a small amount of products. This implies:
\[ [A] + [B] > [C] + [D] \]
Therefore, the ratio \( \frac{[C][D]}{[A][B]} \) must be less than 1, indicating that \( K < 1 \).

4. **Relating Rate Constants to Equilibrium Constant**:
The relationship between the rate constants and the equilibrium constant is given by:
\[ K = \frac{k_f}{k_b} \]
Where \( k_f \) is the rate constant for the forward reaction and \( k_b \) is the rate constant for the backward reaction.

5. **Analyzing the Condition**:
Since we established that \( K < 1 \) (because the concentration of reactants is greater than that of products), we can infer:
\[ \frac{k_f}{k_b} < 1 \]
This implies that:
\[ k_f < k_b \]

6. **Conclusion**:
Thus, for the equilibrium mixture to contain a large amount of reactants and a small amount of products, the condition required is:
\[ k_f < k_b \]

### Final Answer:
The relative value of \( k_f \) and \( k_b \) that results in an equilibrium mixture containing a large amount of reactants and a small amount of products is:
\[ k_f < k_b \]