A radioactive substance decays with a rate constant of \( 9.2 × 10^{-3} min^{-1} \). Time taken for 75% of the sample to decompose is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 150.7 min
Radioactive decay is a first order process, so the integrated rate equation applies:
\( \displaystyle t = \frac{2.303}{k}\log\frac{[\mathrm{R}]_0}{[\mathrm{R}]} \)
When 75% of the sample has decayed, 25% is left, so \( [\mathrm{R}] = 0.25\,[\mathrm{R}]_0 \):
\( \displaystyle \frac{[\mathrm{R}]_0}{[\mathrm{R}]} = \frac{100}{25} = 4 \)
\( \displaystyle t = \frac{2.303}{9.2 \times 10^{-3}} \times \log 4 = 250.3 \times 0.6021 \approx 150.7\ \mathrm{min} \)
So 150.7 min is needed.
Shortcut: 75% decay means two half-lives (100% → 50% → 25%). \( t_{1/2} = 0.693/(9.2 \times 10^{-3}) \approx 75.3 \) min, and twice this is about 150.7 min.