75% of a reaction of the first order was completed in 32 minutes, what is the time for its half completion?

Question

Accepted Answer

To determine the time for half completion of a first-order reaction, we can use the relationship between the rate constant and the time taken for a certain percentage of the reaction to complete. For a first-order reaction, the rate constant $ k $ can be calculated using the formula:

$$ k = \frac{2.303}{t} \log \left(\frac{[A]_0}{[A]}\right) $$

where:
- $ t $ is the time taken for the reaction to reach a certain completion percentage,
- $[A]_0$ is the initial concentration,
- $[A]$ is the concentration at time $ t $.

Given that 75% of the reaction is completed in 32 minutes, it means 25% of the reactant remains. Therefore, we can write:

$$ k = \frac{2.303}{32} \log \left(\frac{100}{25}\right) $$

$$ k = \frac{2.303}{32} \log (4) $$

Now, for a first-order reaction, the half-life $ t_{1/2} $ is given by:

$$ t_{1/2} = \frac{0.693}{k} $$

Substituting the value of $ k $ from above:

$$ t_{1/2} = \frac{0.693 \times 32}{2.303 \times \log(4)} $$

Since $\log(4) = 0.602$, we have:

$$ t_{1/2} = \frac{0.693 \times 32}{2.303 \times 0.602} $$

$$ t_{1/2} = \frac{22.176}{1.387} $$

$$ t_{1/2} \approx 16 \text{ minutes} $$

Thus, the time for half completion of the reaction is 16 minutes, which corresponds to Option 3.

**Explanation for Incorrect Options:**

1. **Option 1 (4 minutes):** This option is incorrect because the calculated half-life of 16 minutes is significantly longer than 4 minutes. A half-life of 4 minutes would imply a much faster reaction rate than what is given.

2. **Option 2 (8 minutes):** This option is incorrect because, although closer than Option 1, it still underestimates the half-life. The calculation clearly shows that the half-life is 16 minutes, not 8 minutes.

3. **Option 4 (32 minutes):** This option is incorrect because 32 minutes is the time taken for 75% completion, not half completion. The half-life is specifically calculated to be 16 minutes, which is half of 32 minutes, indicating a slower progression to 50% completion compared to 75%.

Therefore, Option 3 is the correct answer, as it accurately reflects the calculated half-life of the reaction.