Question

A first-order reaction is 25% complete in 30 minutes. How much time will it take for the reaction to be 75% complete? 
 

• A. 90 min
• B. 60 min
• C. 145 min
• D. 120 min

Hint:

\textbf{Tip:} For first-order reactions, use the integrated rate law and logarithmic relationships; log(4) ≈ 0.602, log(1.33) ≈ 0.1249.

Answer 1

The Correct Option is C

The solution employs first-order reaction kinetics and the concept of half-life.

1. First-order reaction formula:
$ \ln \frac{[A]_0}{[A]} = kt $

2. Provided data:
- 25% completion in 30 minutes signifies 75% remaining:
$ \frac{[A]}{[A]_0} = 0.75 $
- Objective: Determine time $t$ for 75% completion, meaning 25% remains:
$ \frac{[A]}{[A]_0} = 0.25 $

3. Rate constant $k$ calculation:
$ k = \frac{1}{t} \ln \frac{[A]_0}{[A]} = \frac{1}{30} \ln \frac{1}{0.75} = \frac{1}{30} \ln \frac{4}{3} $

4. Time for 75% completion calculation:
$ t = \frac{1}{k} \ln \frac{1}{0.25} = \frac{1}{k} \ln 4 $

5. Substitution of $k$:
$ t = 30 \times \frac{\ln 4}{\ln \frac{4}{3}} $

6. Approximate values:
$ \ln 4 \approx 1.386 $
$ \ln \frac{4}{3} \approx 0.2877 $
$ t \approx 30 \times \frac{1.386}{0.2877} = 30 \times 4.82 = 144.6 \, \text{minutes} $

7. Interpretation:
Approximately 145 minutes are required for 75% completion.

Final Answer:
The time required for 75% completion is approximately $ {145\, \text{minutes}} $.