Question

Show that the mass of radium $\left({}_{88}{Ra}^{226}\right)$ with an activity of $1$ curie is almost $1$ gram

[$T_{\frac{1}{2}}=1600$ years].
[$1$ curie $=3.7\times {10}^{10}$ disintegrations per second].

Answer 1

$\lambda =\frac{0.6931}{3.8}$ per day
Amount of sample disintegrated $=60$%
Amount of sample present $=40$% and $T_{\frac{1}{2}}=3.8$ days
Let $N_0$ be the original amount of the sample present.
From law of disintegration,
$N=N_0{e}^{-\lambda t}$
Substituting for $N=40$% of $N_0$
$\frac{40}{100}{N}_0=N_0{e}^{-\lambda t}$
$\therefore e^{\lambda t}=\frac{10}{4}$
${\log }_{e}2.5=\lambda \times t$
$\therefore t=\frac{3.8}{0.6931}\times {\log }_{10}2.5\times 2.3026$
$=5.022$
$\therefore t=5.022$ days