A first order reaction completes 60% in 20 minutes. The time required for the completion of 90% of the reaction is approx………..

Options

A. 30 minutes
B. 40 minutes
C. 50 minutes
D. 60 minutes

Answer

To solve the problem of determining the time required for the completion of 90% of a first-order reaction, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Reaction Order**:
The reaction is a first-order reaction. For a first-order reaction, the rate constant \( k \) can be determined using the formula:
\[ t = \frac{2.303}{k} \log \left( \frac{[A_0]}{[A]} \right) \]
where \( [A_0] \) is the initial concentration and \( [A] \) is the concentration at time \( t \).

2. **Calculate the Rate Constant \( k \)**:
Given that 60% of the reaction is completed in 20 minutes, we can express this as:
- Initial concentration, \( [A_0] = A \)
- Concentration after 60% completion, \( [A] = 0.4A \) (since 60% has reacted, 40% remains).

Plugging these values into the first-order rate equation:
\[ 20 = \frac{2.303}{k} \log \left( \frac{A}{0.4A} \right) \]
This simplifies to:
\[ 20 = \frac{2.303}{k} \log(2.5) \]

3. **Rearranging to Find \( k \)**:
Rearranging gives:
\[ k = \frac{2.303 \log(2.5)}{20} \]

4. **Calculate \( k \)**:
We know that \( \log(2.5) \approx 0.3979 \):
\[ k \approx \frac{2.303 \times 0.3979}{20} \approx 0.0457 \, \text{min}^{-1} \]

5. **Calculate Time for 90% Completion**:
Now, we need to find the time required for 90% completion:
- After 90% completion, \( [A] = 0.1A \) (10% remains).
Using the same formula:
\[ t_{90\%} = \frac{2.303}{k} \log \left( \frac{A}{0.1A} \right) \]
This simplifies to:
\[ t_{90\%} = \frac{2.303}{k} \log(10) \]

6. **Substituting \( k \)**:
Since \( \log(10) = 1 \):
\[ t_{90\%} = \frac{2.303}{k} \]
Substituting the value of \( k \):
\[ t_{90\%} = \frac{2.303}{0.0457} \approx 50.4 \, \text{minutes} \]

7. **Final Answer**:
Therefore, the time required for the completion of 90% of the reaction is approximately **50 minutes**.