75% of a first order reaction was found to complete in 32 minutes. When will 50% of the same reaction complete ?

Question

Accepted Answer

To determine when 50% of a first-order reaction will complete, given that 75% of the reaction completes in 32 minutes, we can use the properties of first-order reactions and the concept of half-life.

### Explanation:

1. **First-Order Reaction Characteristics:**
   - In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant.
   - The half-life ($ t_{1/2} $) of a first-order reaction is constant and does not depend on the initial concentration of the reactant.

2. **Calculation of Half-Life:**
   - The formula for the half-life of a first-order reaction is:
     $$
     t_{1/2} = \frac{0.693}{k}
     $$
   - For a first-order reaction, the time taken to reach a certain percentage completion can be calculated using the integrated rate law:
     $$
     t = \frac{1}{k} \ln \left(\frac{[A]_0}{[A]}\right)
     $$
   - Given that 75% of the reaction completes in 32 minutes, this means 25% of the reactant remains. Therefore:
     $$
     32 = \frac{1}{k} \ln \left(\frac{100}{25}\right) = \frac{1}{k} \ln(4)
     $$
   - Simplifying, we find:
     $$
     32 = \frac{1}{k} \times 1.386
     $$
   - Solving for $ k $:
     $$
     k = \frac{1.386}{32}
     $$

3. **Time for 50% Completion:**
   - For 50% completion, 50% of the reactant remains, so:
     $$
     t_{1/2} = \frac{1}{k} \ln \left(\frac{100}{50}\right) = \frac{1}{k} \ln(2)
     $$
   - Using the value of $ k $ from above:
     $$
     t_{1/2} = \frac{1}{k} \times 0.693
     $$
   - Substituting $ k = \frac{1.386}{32} $:
     $$
     t_{1/2} = \frac{32}{1.386} \times 0.693 \approx 16 \text{ minutes}
     $$

### Conclusion:

- **Option 2 (16 minutes)** is correct because the half-life of the reaction, which is the time taken for 50% completion, is calculated to be approximately 16 minutes.

### Explanation of Incorrect Options:

1. **Option 1 (24 minutes):** This option is incorrect because the calculation shows that the half-life is 16 minutes, not 24 minutes.

2. **Option 3 (8 minutes):** This option is incorrect because it underestimates the half-life. The calculated half-life is 16 minutes, which is double this option.

3. **Option 4 (4 minutes):** This option is incorrect as it significantly underestimates the half-life. The correct half-life is 16 minutes, which is four times this option.

Thus, the correct answer is Option 2, which is 16 minutes.