Question

A radioactive nucleus decays by two different process. The half life of the first process is $5$ minutes and that of the second process is $30s$. The effective half-life of the nucleus is calculated to be $\frac{\alpha }{11}$ s. The value of $\alpha$ is _______

Answer 1

Answer: 300

$\frac{d{N}_1}{dt}=-{\lambda }_{1}N$
$\frac{d{N}_2}{dt}=-{\lambda }_{2}N$
$\frac{dN}{dt}=-\left({\lambda }_1+{\lambda }_2\right)N$
$\Rightarrow {\lambda }_{eq}={\lambda }_1+{\lambda }_2$
$\Rightarrow \frac{1}{t_{1/2}}=\frac{1}{300}+\frac{1}{30}=\frac{11}{300}$
$\Rightarrow t_{1/2}=\frac{300}{11}$