For a first order reaction 75% reaction complete in 90 min. for the same reaction 60% complete in what time?

Options

A. 45 min
B. 37 min
C. 60 min
D. 40 min

Answer

To solve the problem, we need to determine the time taken for 60% completion of a first-order reaction, given that 75% completion takes 90 minutes. We will use the first-order reaction kinetics formula.

### Step-by-Step Solution:

1. **Understanding the Reaction**:
- For a first-order reaction, the relationship between time and concentration is given by the equation:
\[ t = \frac{2.303}{k} \log \frac{[A_0]}{[A]} \]
- Here, \( [A_0] \) is the initial concentration, and \( [A] \) is the concentration at time \( t \).

2. **Setting Up the Known Values**:
- Given that 75% of the reaction is complete in 90 minutes, we can find the remaining concentration:
- Initial concentration, \( [A_0] = 100\% \)
- Remaining concentration after 75% completion, \( [A] = 100\% - 75\% = 25\% \)
- Therefore, we can write:
\[ t_{75\%} = 90 \text{ minutes} \]

3. **Applying the Formula for 75% Completion**:
- Using the formula for \( t_{75\%} \):
\[ 90 = \frac{2.303}{k} \log \frac{100}{25} \]
- Simplifying the logarithm:
\[ \log \frac{100}{25} = \log 4 = 2 \log 2 \approx 2 \times 0.301 = 0.602 \]
- Thus, we can rewrite the equation:
\[ 90 = \frac{2.303}{k} \times 0.602 \]

4. **Setting Up the Equation for 60% Completion**:
- Now, we need to find the time for 60% completion:
- Remaining concentration after 60% completion, \( [A] = 100\% - 60\% = 40\% \)
- Using the formula for \( t_{60\%} \):
\[ t_{60\%} = \frac{2.303}{k} \log \frac{100}{40} \]
- Simplifying the logarithm:
\[ \log \frac{100}{40} = \log 2.5 \approx 0.398 \]

5. **Dividing the Two Equations**:
- To eliminate \( k \), we can divide the two equations:
\[ \frac{t_{60\%}}{t_{75\%}} = \frac{\log \frac{100}{40}}{\log \frac{100}{25}} \]
- Substituting the values:
\[ \frac{t_{60\%}}{90} = \frac{0.398}{0.602} \]
- Cross-multiplying gives:
\[ t_{60\%} = 90 \times \frac{0.398}{0.602} \approx 59.48 \text{ minutes} \]

6. **Final Answer**:
- The time taken for 60% completion is approximately 59.48 minutes. Rounding this, we find the closest option is:
\[ \text{Answer: } 60 \text{ minutes} \]