A radioactive nucleus can decay by two different processes. The mean value period for the first process is $t_{1}$ and that for the second process is $t_{2}$ . The effective mean value period for the two processes is

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Answer 1

Mean value:
$λ_{1} = \frac{1}{t _{1}}$
$λ_{2} = \frac{1}{t _{2}}$

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

$\frac{d N}{d t} = - (λ_{1} + λ_{2})$
$\frac{d N}{d t} = - (\frac{1}{t _{1}} + \frac{1}{t _{2}}) N$
$\frac{d N}{d t} = - (\frac{t _{1} + t _{2}}{t _{1} t _{2}}) N$
$\frac{d N}{d t} = - λ_{e f f} N$
$t_{e q} = \frac{1}{λ _{e f f}}$

Answer 2

Mean value:

λ_{1} = \frac{1}{t _{1}}

λ_{2} = \frac{1}{t _{2}}

First order decay:

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

\frac{d N}{d t} = - (λ_{1} + λ_{2})

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

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

\frac{d N}{d t} = - λ_{e f f} N

t_{e q} = \frac{1}{λ _{e f f}}