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 `underset(const.rate)overset(R) rarr A underset(decay)overset(lambda) rarrB`
Let `N` be the number of nuclei of A any time t.
`because (dN)/(dt)=R-lambdaN" "underset(0)overset(N)(int)(dN)/(R-lambdaN)=underset(0)overset(t)(int)dt`
On solving, we will get
`N = R//lambda (1-e^(-lambda t))`
(b) Number of nuclei of B at any tiem t,
`N_(B) =Rt -N_(A)`
`=R t-R//lambda(1-e^(-lambda t) )=R//lambda(lambda t -1 +e^(- lambda t))`
Maximum number of nuclei of ''A'' present at any time during its formation ` =R//lambda` .