Question

Some amount of a radioactive substance (half life $=10$ days ) is spread inside a room and consequently the level of radiation becomes $50$ times permissible level for normal occupancy of the room. After how many days the room will be safe for occupation?
[Use, $\ln 50=3.9120$]

Answer 1

Given, $t_{\frac{1}{2}}=10\text{days}$

If $N_0$ is the initial level of radiation and $N$ is the permissible level, then

$N=\frac{N_0}{50}$

We know that,

$N=N_0{e}^{-\lambda t}.....\left(1\right)$

where $\lambda =\frac{0.693}{t_{\frac{1}{2}}}$

$\frac{N_0}{50}=N_0{e}^{-\lambda t}$

$\Rightarrow 50=e^{-\lambda t}$

$\Rightarrow \ln 50=\lambda t$

$\Rightarrow t=\frac{\ln 50}{\lambda }=\frac{3.9120}{0.693}\times 10$

$\Rightarrow t=56.45\text{days}$