Question

Prove that time required for $99.9$% decay of a radioactive species is almost ten times its half life period.

Answer 1

We know that, $t=\frac{2.303}{\lambda }\log \frac{N_0}{N}$
$N_0=100$, $N=(100-99.9)=0.1$
So, Time required for $99.9$% decay, $t=\frac{2.303}{\lambda }\log \frac{100}{0.1}$$=\frac{2.303}{\lambda }\times 3$
Half life period $=\frac{0.693}{\lambda }$
So, $\frac{Timerequiredfor99.9decay}{Halflifeperiod}=\frac{2.303\times 3}{\lambda }\times \frac{\lambda }{0.693}$ $\approx 10$