Which of the following is the correct relationship between time required for completion of 99.9% of a first order reaction and its half-life?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(t_{99.9\%} = 10 × t_{1/2}\)
For a first order reaction the time for any fraction to react follows from the integrated rate equation:
\( \displaystyle t = \frac{2.303}{k}\log\frac{[R]_0}{[R]} \)
When 99.9% has reacted, 0.1% is left, so [R]0/[R] = 100/0.1 = 1000:
\( \displaystyle t_{99.9\%} = \frac{2.303}{k}\log 1000 = \frac{2.303 \times 3}{k} = \frac{6.909}{k} \)
\( \displaystyle t_{1/2} = \frac{2.303}{k}\log 2 = \frac{0.693}{k} \)
\( \displaystyle \frac{t_{99.9\%}}{t_{1/2}} = \frac{6.909}{0.693} \approx 10 \)
So t99.9% = 10 × t1/2.
Shortcut: 99.9% means 1/1000 is left, and 210 = 1024 ≈ 1000, so about 10 half-lives are needed.
Note: This question is from NCERT (2026-27). NTA sets numerical questions from NCERT too, so do not ignore NCERT while practising numericals.
NCERT (2026-27 edition):
Half-Life of a Reaction – “Show that in a first order reaction, time required for completion of 99.9% is 10 times of half-life (t1/2) of the reaction.”