If `75%` of a first order reaction completed in `15 min`. Then `90%` of the same reaction completed in

Options

A. `20 min`
B. `25 min`
C. `30 min`
D. `150 min`

Explanation

To solve the problem, we need to determine how long it takes for 90% of a first-order reaction to complete, given that 75% completion takes 15 minutes.

### Step-by-Step Solution:

1. **Understanding the Completion Percentages**:
- We know that 75% of the reaction is complete in 15 minutes. This means that 25% of the reactant remains.

2. **Using the First-Order Reaction Formula**:
- For a first-order reaction, the time taken to reach a certain percentage completion can be related to the half-life (T_half) of the reaction. The relationship is given by:
\[ T_{75\%} = 2 \times T_{half} \]
- From the question, we have \( T_{75\%} = 15 \text{ minutes} \).

3. **Calculating T_half**:
- We can rearrange the equation to find \( T_{half} \):
\[ T_{half} = \frac{T_{75\%}}{2} = \frac{15 \text{ min}}{2} = 7.5 \text{ min} \]

4. **Finding Time for 90% Completion**:
- For a first-order reaction, the time taken to reach 90% completion can be calculated using the formula:
\[ T_{90\%} = \frac{10}{3} \times T_{half} \]
- Substituting the value of \( T_{half} \):
\[ T_{90\%} = \frac{10}{3} \times 7.5 \text{ min} \]

5. **Calculating T_{90\%}**:
- Now, calculate \( T_{90\%} \):
\[ T_{90\%} = \frac{10 \times 7.5}{3} = \frac{75}{3} = 25 \text{ min} \]

### Final Answer:
The time taken for 90% of the reaction to complete is **25 minutes**.