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 . The value of is _________. [2023]

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Solution

Q.
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 $\frac{α}{11} s$ . The value of $α$ is _________. [2023]

Ans.
(300)

$\frac{d N_{1}}{d t} = - λ_{1} N$

$\frac{d N_{2}}{d t} = - λ_{2} N$

$\frac{d N}{d t} = - (λ_{1} + λ_{2}) N$

$\Rightarrow λ_{eq} = λ_{1} + λ_{2}$

$\Rightarrow \frac{1}{t_{1 / 2}} = \frac{1}{300} + \frac{1}{30} = \frac{11}{300}$

$\Rightarrow t_{1 / 2} = \frac{300}{11}$

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