For the first order reaction, the time required for 99.9% completion of reaction is how many times that required for 50% completion?

Question

Accepted Answer

For a first-order reaction, the time required for a certain percentage of completion can be calculated using the concept of half-life and the integrated rate law for first-order reactions. The half-life ($t_{1/2}$) of a first-order reaction is constant and is given by the formula:

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

where $k$ is the rate constant.

1. **Time for 50% Completion:**
   - For a first-order reaction, the time required for 50% completion is equal to one half-life ($t_{1/2}$).

2. **Time for 99.9% Completion:**
   - The time required for 99.9% completion can be calculated using the integrated rate law for first-order reactions:
   $$ t = \frac{2.303}{k} \log \left(\frac{[A]_0}{[A]}\right) $$
   - For 99.9% completion, $[A] = 0.1\% \times [A]_0 = 0.001 \times [A]_0$.
   - Substituting these values, we get:
   $$ t_{99.9\%} = \frac{2.303}{k} \log \left(\frac{[A]_0}{0.001[A]_0}\right) = \frac{2.303}{k} \log (1000) $$
   - $\log (1000) = 3$, so:
   $$ t_{99.9\%} = \frac{2.303 \times 3}{k} = \frac{6.909}{k} $$

3. **Comparison:**
   - The ratio of the time required for 99.9% completion to the time required for 50% completion is:
   $$ \frac{t_{99.9\%}}{t_{1/2}} = \frac{\frac{6.909}{k}}{\frac{0.693}{k}} = \frac{6.909}{0.693} \approx 10 $$

Therefore, the time required for 99.9% completion of the reaction is approximately 10 times that required for 50% completion, which supports Option 2 as the correct answer.

**Explanation for Incorrect Options:**

- **Option 1 (50 times):** This option is incorrect because the calculation shows that the time for 99.9% completion is approximately 10 times the time for 50% completion, not 50 times.

- **Option 3 (5 times):** This option is incorrect because the ratio calculated is approximately 10, not 5.

- **Option 4 (2.5 times):** This option is incorrect because the ratio is significantly larger than 2.5, as shown by the calculation.

In conclusion, the correct answer is Option 2, as the time required for 99.9% completion is indeed 10 times that required for 50% completion in a first-order reaction.