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 determine the relationship between the rate constants of the forward reaction (\(k_f\)) and the backward reaction (\(k_b\)) that results in an equilibrium mixture containing a large amount of reactants and a small amount of products, we can follow these steps:

### Step 1: Understand the Concept of Equilibrium
At equilibrium, the rate of the forward reaction equals the rate of the backward reaction. This can be expressed as:
\[ k_f [A] = k_b [B] \]
where \([A]\) is the concentration of reactants and \([B]\) is the concentration of products.

### Step 2: Define the Equilibrium Constant
The equilibrium constant \(K\) for the reaction can be defined as:
\[ K = \frac{[B]}{[A]} \]
A large value of \(K\) indicates that products are favored, while a small value of \(K\) indicates that reactants are favored.

### Step 3: Analyze the Given Condition
In the question, we need to find the conditions under which the equilibrium mixture contains a large amount of reactants and a small amount of products. This implies that:
\[ \frac{[B]}{[A]} \text{ is small} \]
Thus, \(K\) must be small.

### Step 4: Relate \(K\) to \(k_f\) and \(k_b\)
The equilibrium constant \(K\) can also be expressed in terms of the rate constants:
\[ K = \frac{k_f}{k_b} \]
From this, we can rearrange to find:
\[ \frac{k_f}{k_b} = K \]

### Step 5: Determine the Relationship Between \(k_f\) and \(k_b\)
Since we established that \(K\) is small (indicating that reactants are favored), it follows that:
\[ \frac{k_f}{k_b} < 1 \quad \Rightarrow \quad k_f < k_b \]
This means that the rate constant for the forward reaction (\(k_f\)) must be less than the rate constant for the backward reaction (\(k_b\)).

### Conclusion
Thus, the relative values of \(k_f\) and \(k_b\) that result in an equilibrium mixture containing a large amount of reactants and a small amount of products is:
\[ k_f < k_b \]