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If the half-life ($t_{1/2}$) for a first order reaction is $1\text{ minute}$, then the time required for 99.9% completion of the reaction is closest to:
A
2 minutes
B
4 minutes
C
5 minutes
D
10 minutes
Explanation
For a first order reaction, $t_{99.9\%} \approx 10 \times t_{1/2} = 10 \times 1\text{ min} = 10\text{ minutes}$.
Detailed Solution
For a first order reaction: $k = \frac{0.693}{t_{1/2}} = \frac{0.693}{1\text{ min}} = 0.693\text{ min}^{-1}$. Time for 99.9% completion: $t = \frac{2.303}{k} \log\left(\frac{100}{100 - 99.9}\right) = \frac{2.303}{0.693} \log\left(\frac{100}{0.1}\right) = \frac{2.303}{0.693} \log(10^3) = \frac{2.303 \times 3}{0.693} = \frac{6.909}{0.693} \approx 10\text{ minutes}$.