If the half-life for a first order reaction is 1 minute, then the time required for 99.9% completion of the reaction is closest to [2025]

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Solution

Q.
If the half-life $(t_{1 / 2})$ for a first order reaction is 1 minute, then the time required for 99.9% completion of the reaction is closest to [2025]

1 5 minutes

2 10 minutes

3 2 minutes

4 4 minutes

Ans.
(2)

For a first order reaction,

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

or,
$k = \frac{0.693}{t_{1 / 2}} = \frac{0.693}{1} = 0.693$ ${min}^{- 1}$ $(∵ t_{1 / 2} = 1 minute)$

For 99.9% completion of reaction, $k = \frac{2.303}{t} \log \frac{a}{a - x}$

or, $t = \frac{2.303}{k} \log \frac{a}{a - x} = \frac{2.303}{0.693} \log \frac{100}{100 - 99.9}$

or, $t = \frac{2.303}{0.693} \times 3 = 9.97$ $min \approx 10$ $min$

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