The rate constant for a first order reaction is 6.93 × 1 0 − 2 per minutes. What is the time required for the completion of 99.9% of the reaction?

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For a first order reaction, the time required is given by:

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

Given $k = 6.93 \times 1 0^{- 2} min^{- 1}$ and for $99.9%$ completion, $x = 0.999 a$ .

Remaining amount $a - x = a - 0.999 a = 0.001 a$

$t = \frac{2.303}{6.93 \times 1 0 ^{- 2}} lo g (\frac{a}{0.001 a})$

$t = \frac{2.303}{6.93 \times 1 0 ^{- 2}} lo g (1 0^{3})$

$t = \frac{2.303 \times 3}{6.93 \times 1 0 ^{- 2}}$

$t = \frac{6.909}{0.0693} \approx 99.7 minutes$

Answer: $99.7$ minutes

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