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A radioactive sample can decay by two different processes. The half-life for the first process is $T_{1}$ and that for the second process is $T_{2}$ . The effective half-life $T$ of the radioactive sample is
$T = T_{1} + T_{2}$
$\frac{1}{T} = \frac{1}{T_{1}} + \frac{1}{T_{2}}$
$T = \frac{T_{1} + T_{2}}{T_{1} T_{2}}$
$T = \frac{T_{1} - T_{2}}{T_{1} T_{2}}$

A

T = \frac{T_{1} + T_{2}}{T_{1} T_{2}}

B

T = \frac{T_{1} - T_{2}}{T_{1} T_{2}}

C

T = T_{1} + T_{2}

D

\frac{1}{T} = \frac{1}{T_{1}} + \frac{1}{T_{2}}

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Solution

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Radioactive decay rate is directly proportional to number of nucleus present at any time

$\frac{d N}{d t} = - λ N$ ....(i)

where $λ$ is a constant related to half life as $λ = \frac{ln 2}{T_{\frac{1}{2}}}$ where $T_{\frac{1}{2}}$ is half life time of radioactive decay

The total rate of decay is

$\frac{d N}{d t} = - (λ_{1} + λ_{2}) N = λ_{e f f e c t i v e} N$ ...(ii)

$λ_{1} = \frac{ln 2}{T_{1}}$

$λ_{2} = \frac{ln 2}{T_{2}}$

$λ_{e f f e c t i v e} = \frac{ln 2}{T_{e f f e c t i v e}}$

substituting in equation (ii)

$(\frac{ln 2}{T_{1}}$ $+ \frac{ln 2}{T_{2}}) N$ $= \frac{ln 2}{T_{e f f e c t i v e}} N$

$\frac{1}{T} = \frac{1}{T_{1}} + \frac{1}{T_{2}}$

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