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

Let the decay constant for the first and second processes be `lambda_(1)` and `lambda_(2)` and the effective decay constant for the comined process be `lambda`. Then,
`lambda_(1)=(log_(e) 2)/(t_(1)) , lambda_(2)=(log_(e) 2)/(t_(2))` and` lambda=(log_(e) 2)/(t)`
Now, the probability for decay through first process in a small time interval dt is `lambda_(1) dt` and the probability for decay through second process in the same time interval dt is `lambda_(2) dt` . The probability for decay by the combined process in the same time interval dt is `lambda_(1) dt + lambda_(2) dt`.
But this is also equal to `lambda dt`.
` :. lambda dt = lambda_(1) dt + lambda_(2) dt`
`:. lambda=lambda_(1) + lambda_(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))`.