A factory produces a radioactive substance A at a constant rate R which decays with a decay constant ` lambda` to form a stable substance. Find (a) the number of nuclei of A and (b) number of nuclei of B, at any time t assuming the production of A starts at `t=0`. (c ) Also, find out the maximum number of nuclei of `A` present at any time during its formation.

Explanation

Factory `overset(R )underset("constrate")rarrAoverset(lambda)underset("decay")rarrB`
Let `N` be the number of nuclei of `A` at time `t`
`:. (dN)/(dt)=R-lambdaN" " overset(N)underset(0)int(dN)/(R-lambdaN)=overset(t)underset(0)intdt`
On solving we will get
`N=R//lambda(1-e^(-lambdat))`
(i) Number of nuclei of `B` at any time `t, N_(B)=Rt-N_(A)-R//lambda(lambdat-1+e^(-lambdat))`
(iii) Maximum number of nuclei of `'A'` present at any time during its formation `=R//lambda`