A radioactive nucleus can decay by two different processes. The mean value period for the First process is ${\mathrm{t}}_1$and that for the second process is ${\mathrm{t}}_2$ . The effective mean value period for the two processes is:

detailed solution

Correct option is D

Let the decay constants for the first and second processes be λ1 and λ2 and the effective decay constant for the combined process be λ . Then λ1=loge2t1 and λ2=loge2t2  and λ=loge2tNow, the probability for decay through first process in a small time interval dt is λ1 dt  and the probability for decay through second process in the same time interval dt is λ2 dtThe probability for decay by the combined process in the same time interval dt is λ1 dt+λ2 dt.But this is also equal to λ dt∴λ dt=λ1 dt+λ2 dt ∴λ =λ1+λ2 or loge2t=loge2t1+loge2t2or 1t=1t1+1t2 or t=t1t1t1+t2