Nuclei of radioactive element A are being produced at a constant rate. `alpha`. The element has a decay constant `lambda`. At time `t=0`, there are`N_(0)` nuclei of the element.
(a) Calculate the number N of nuclei of A at time t.
(b) IF `alpha =2 N_(0) lambda` , calculate the number of nuclei of A after one half-life time of A and also the limiting value of N at `t rarr oo`.

Explanation

(i)` XrarrArarrB` Net rate of formulation of A is` (dN)/(dt)=alpha -lambda N`, `int (dN)/(alpha-lambdaN)=int dt+C `where C is a constant of integration, -`(1)/(lambda) In (alpha -lambdaN)=t +C`,At `t=0,N=N_(0),C=-(1)/(lambda)In (alpha -lambdaN_(0))`, or `t=(1)/(lambda)In((alpha-lambda N_(0))/(alpha-lambda N)) or N=`(alpha-(alpha-lambdaN_(0))e ^(-mu))/(lambda)`(ii) When `alpha=2N_(0) lambda and t=T_((1)/(2))=(In2)/(lambda)`,` implies N=(2N_(0)lambda-(2N_(0)lambda-lambdaN_(0))e^(-In2)=(3N_(0))/(2)`, Also ,`t rarr infty, N=2N_(0)`