A first order reaction takes 10 minutes for 90% completion of the reaction. Calculate the time required for completion of 99.9% of the reaction ?

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For a first order reaction, the rate constant $k$ is given by

$k = \frac{2.303}{t} lo g (\frac{a}{a - x})$

For $90%$ completion, $t = 10$ min and $x = 0.9 a$ .

$k = \frac{2.303}{10} lo g (\frac{a}{a - 0.9 a}) = \frac{2.303}{10} lo g 10 = \frac{2.303}{10}$ min $^{- 1}$

For $99.9%$ completion, $x = 0.999 a$ .

$t = \frac{2.303}{k} lo g (\frac{a}{a - 0.999 a})$

$t = \frac{2.303}{\frac{2.303}{10}} lo g (\frac{a}{0.001 a})$

$t = 10 lo g 1000 = 10 \times 3 = 30$ min.

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