A radioactive nucleus can decay by two differnet processess. The mean value period for the first process is `t_(1)` and that the second process is `t_(2)` .The effective mean value period for the two processes is .

Options

A. `(t_1 + t_2)/(2)`
B. `t_1 + t_2`
C. `sqrt(t_1 t_2)`
D. `(t_1 t_2)/(t_1 + t_2)`

Answer

D

Explanation

(d) Let the decay constant for the first and second processes be `lamda_1` and `lamda_2` and the effective decay constant for the combined process be `lamda`. Then
`lamda_1 = (log_e 2)/(t_1)` and `lamda_2 = (log_e 2)/(t_2)`
and `lamda = (log_e 2)/(t)`
Now, the probability for decay through first process in a small time interval `dt` is `lamda_1 dt` and the probability for decay through second process in the same time interval `dt` is `lamda_2 dt`.
The probability for decay by the combined process in the same time interval `dt` is `lamda_1 dt + lamda_2 dt`.
But this is also equal to `lamda dt`.
`:. lamda dt = lamda_1 dt + lamda_2 dt`
`:. lamda = lamda_1 + lamda_2`
or `(log_e 2)/(t) = (log_e 2)/(t_1) + (log_e 2)/(t_2)`
or `(1)/(t) = (1)/(t_1) + (1)/(t_2)` or `t = (t_1 t_2)/(t_1 + t_2)`.