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`.

Explanation

The decay constant for the first process is `lambda_(1) = (1n2)/(t_1)`
and for the second process it is `lambda_(2) = (In2)/(t_1)`. The probability that an active nucleus decays by the first process in a time interval it is `lambda_(1)` dt. Similarity, 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 second process is `lambda_(1) dt + lambda_(2)dt. If the effective decay constant is `lambda`, this probability is also equal to `lambda` dt. Thus
`lambda dt = lambda_(1)dt + lambda_(2)dt`
or , `" "lambda = lambda_(1) + lambda_(2)`
or, " "1/t = 1/(t_1) + 1/(t_2)` (To be remembered).