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A radioactive nucleus can decay by two d...

A radioactive nucleus 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`. Show that the effective half-life `t` of the nucleus is given by
`1/t = 1/t_1 + 1/t_2`.

Text Solution

Verified by Experts

The decay constant for the first process is `lambda_1 = (1n 2)/t_1` and for the second process it is `lambda_2 = (1n 2)/t_1`. The probability that an active nucleus decays by the first process in a time interval `dt` is `lambda_1 dt`. Similarly, the probability that it decays by the second process is `lambda_2 dt`. The probability that it either decays by the first process or by the second process is `lambda_1 dt + lambda_2 dt`. If the effective decay constant is `lambda`, this probability is also equal to `lambdadt`. Thus,
`lambdadt = lambda_1 dt + lambda_2 dt`
or, `lambda = lambda_1 + lambda_2`
or, `1/t = 1/t_1 + 1/t_2`.
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