If time taken for a first order reaction to get 90% complete is 24 \…

The correct answer is Option (4) → 72 min

For a first order reaction the time to reach a given extent of reaction is:

    \( \displaystyle t = \frac{2.303}{k}\log\frac{[\mathrm{R}]_0}{[\mathrm{R}]} \)

For 90% completion, [R] = 0.1[R]₀; for 99.9% completion, [R] = 0.001[R]₀:

    \( \displaystyle t_{90\%} = \frac{2.303}{k}\log 10 = \frac{2.303}{k} \times 1 = 24\ \mathrm{min} \)

    \( \displaystyle t_{99.9\%} = \frac{2.303}{k}\log 1000 = \frac{2.303}{k} \times 3 \)

    \( \displaystyle \frac{t_{99.9\%}}{t_{90\%}} = \frac{3}{1} \;\Rightarrow\; t_{99.9\%} = 3 \times 24 = 72\ \mathrm{min} \)

So the reaction is 99.9% complete in 72 min.