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13.Nuclei
medium
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 $..............$
A
$301$
B
$302$
C
$300$
D
$303$
(JEE MAIN-2023)
Solution
$\frac{ dN _1}{ dt }=-\lambda_1 N \quad \frac{ dN _2}{ dt }=-\lambda_2 N$
$\frac{ dN }{ dt }=-\left(\lambda_1+\lambda_2\right) N$
$\Rightarrow \lambda_{\text {eq }}=\lambda_1+\lambda_2$
$\Rightarrow \frac{1}{t_{1 / 2}}=\frac{1}{300}+\frac{1}{30}=\frac{11}{300}$
$\Rightarrow t _{1 / 2}=\frac{300}{11}$
Standard 12
Physics
