a freshly prepared radioactive source of half life 2 h emits radiation of intensity which is 64 times the permissible safe level . The minimum time after which it would be possible to work safely with this source is

Dear Student,

Please find below the solution to the asked query:

According to the law of radioactivity,

$N = N_0 e^{-\lambda t} \Rightarrow N = N_0 e^{-\frac{\ln \left(2\right)}{T_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}t}$
Where,
N = intensity of radiation at any time.
N₀ = intensity of radiation at time t = 0 sec.

Given that,

$N_0 = 64 N_{safe}$ and $T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} = 2 hrs$. Therefore, Let after a time t, the intensity of radiation reaches the safe level. Then,

$$\begin{gathered} N_s = 64 N_s e^{-\frac{ln\left(2\right)}{2}t} \Rightarrow e^{-\frac{ln\left(2\right)}{2}t} =\frac{1}{64} \Rightarrow e^{\frac{ln\left(2\right)}{2}t} =64 \\ \Rightarrow \frac{ln\left(2\right)}{2}t = ln\left(2^6\right) \Rightarrow \frac{ln\left(2\right)}{2}t = 6 ln\left(2\right) \\ \Rightarrow t = 12 hrs \end{gathered}$$

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