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 $30 \,s$. The effective half-life of the nucleus is calculated to be $\frac{\alpha}{11}$ s. The value of $\alpha$ is _______

Answer 1

$\frac{ dN _1}{ dt }=-\lambda_1 N $
$\frac{ dN _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} $
$\Rightarrow t _{1 / 2}=\frac{300}{11}$