A radioactive substance (A) is not produced at a constant rate which decays with a decay constant `lambda` to form stable substance (B).If production of A starts at t=0 . Find (i)The number of nuclei of A and (ii)Number of nuclei at any time t.

Explanation

(i)Let N be the number of nuclei of A at any time t
`"dN"/"dt"=alpha-lambdaN`
`int_0^N"dN"/(alpha-lambdaN)=int_0^t"dt"`
On solving , we will get
`N=alpha/lambda(1-e^(-lambdat))`
(ii) Number of nuclei of B at any time t
`N_B=alphat-N_A`
`=alphat-alpha/lambda(1-e^(-lambdat))`
`=alpha/lambda(lambdat-1+e^(-lambdat))`