A freshly prepared radioactive sample of half life 1 hour emits radiations that are 128 times as intense as the permissible safe limit. What is the minimum time after which the sample can be safely used?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 7 h
The intensity of the radiation is proportional to the activity of the sample, which halves in every half-life. After n half-lives the intensity is \( I = I_0\left(\frac{1}{2}\right)^{n} \). The sample is safe when the intensity has fallen to the safe limit, so \( \frac{I_0}{I_{\mathrm{safe}}} = 128 \).
\( \displaystyle \frac{I_{\mathrm{safe}}}{I_0} = \frac{1}{128} = \left(\frac{1}{2}\right)^{n} \Rightarrow 2^{n} = 128 = 2^{7} \Rightarrow n = 7 \)
\( t = n\,T_{1/2} = 7\times1\,\mathrm{h} = 7\,\mathrm{h} \)