If 75% of a first order reaction was completed in 90 minutes, 60% of the same reaction would be completed in approximately (in minutes ) ..............
(Take : log 2 = 0.30 , log 2.5 = 0.40)

Answer

To solve the problem, we need to determine the time it takes for 60% of a first-order reaction to complete, given that 75% of the reaction is completed in 90 minutes.

### Step-by-Step Solution:

1. **Understanding the Reaction**:
- In a first-order reaction, the relationship between the time taken and the concentration of reactants can be expressed using the formula:
\[ T = \frac{2.303}{k} \log \left( \frac{A_0}{A} \right) \]
- Here, \( T \) is the time, \( k \) is the rate constant, \( A_0 \) is the initial concentration, and \( A \) is the concentration at time \( T \).

2. **Setting Up the Initial Conditions**:
- Let’s denote the initial concentration \( A_0 = 100 \) (as a percentage).
- After 75% completion, the remaining concentration \( A \) is:
\[ A = 100 - 75 = 25 \]
- Therefore, for 75% completion in 90 minutes:
\[ T_{75} = 90 \text{ minutes} \]

3. **Using the Formula for 75% Completion**:
- Plugging into the formula:
\[ T_{75} = \frac{2.303}{k} \log \left( \frac{100}{25} \right) \]
- We know that \( \log(100) = 2 \) and \( \log(25) = 1.3979 \) (since \( 25 = 10^{1.3979} \)).
- Thus, \( \log \left( \frac{100}{25} \right) = \log(100) - \log(25) = 2 - 1.3979 = 0.6021 \).

4. **Setting Up for 60% Completion**:
- For 60% completion, the remaining concentration \( A \) is:
\[ A = 100 - 60 = 40 \]
- Therefore, for 60% completion, we have:
\[ T_{60} = \frac{2.303}{k} \log \left( \frac{100}{40} \right) \]
- We can calculate \( \log \left( \frac{100}{40} \right) \):
\[ \log(100) = 2 \quad \text{and} \quad \log(40) = 1.6021 \quad (\text{since } 40 = 10^{1.6021}) \]
- Thus, \( \log \left( \frac{100}{40} \right) = 2 - 1.6021 = 0.3979 \).

5. **Relating \( T_{60} \) and \( T_{75} \)**:
- Now we can relate the two times:
\[ \frac{T_{60}}{T_{75}} = \frac{\log \left( \frac{100}{40} \right)}{\log \left( \frac{100}{25} \right)} \]
- Substituting the values:
\[ \frac{T_{60}}{90} = \frac{0.3979}{0.6021} \]
- Therefore,
\[ T_{60} = 90 \times \frac{0.3979}{0.6021} \]

6. **Calculating \( T_{60} \)**:
- Calculating the fraction:
\[ T_{60} \approx 90 \times 0.660 = 59.48 \text{ minutes} \]

7. **Final Answer**:
- Rounding to the nearest whole number, we find that approximately 60 minutes are required for 60% completion of the reaction.

### Final Result:
The time required for 60% completion of the reaction is approximately **60 minutes**.