A radioactive sample 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` Find the effective half - life T of the radioactive sample.

Explanation

Let N be the total number of atoms of the radioactive sample initially ,
Let `(dN_1)/(dt) and (dN_2)/(dt)`
be the initial rates of disintegrations of the radioactive sample by the two processes respectively . Then
`(dN_1)/(dt) = lamda_(1) N and (dN_2)/(dt) = lamda_(2)N`
where `lamda_(1) and lamda_(2)` are the decay constants for the first and second processes respectively .
The initial rate of disintegrations of the radioactive sample by both the processes
`= (dN_1)/(dt) +(dN_2)/(dt) = lamda_(1) N + lamda_(2)N = (lamda_(1)+lamda_(2) N = (lamda_(1)+lamda_(2))N`.
If `lamda` is the effective decay constant of the radioactive sample , its initial rate of disintegration ,
`(dN)/(dt) = lamdaN " But " (dN)/(dt) = (dN_1)/(dt)+(dN_2)/(dt)`
`lamdaN=(lamda_(1)+lamda_(2))N implies lamda= lamda_(1)+lamda_2`
`(0.693)/T_1+(0.693)/T_2=(0.693)/Timplies1/T=1/T_1+1/T_2, T = (T_1T_2)/(T_1+T_2)`