Nuclei of a radioactive element `X` are being produced at a constant rate `K` and this element decays to a stable nucleus `Y` with a decay constant `lambda` and half-life `T_(1//3)`. At the time `t=0`, there are `N_(0) `nuclei of the element X.
The number `N_(Y)` of nuclei of `Y` at time `t` is .

Options

A. `qt-((q-lambdaN_(0))/(lambda))e^(-lambda t)+(q-lambdaN_(0))/(lambda)`
B. `qt+((q-lambdaN_(0))/(lambda))e^(-lambda t)`
C. `qt+((q-lambdaN_(0))/(lambda))e^(-lambda t)+(1-lambdaN)/(lambda)`
D. `qt-((q-lambdaN_(0))/(lambda))e^(-lambda t)`

Answer

C

Explanation

`(dN_(Y))/(dt)=lambdaN_(X)=lambda(1)/(lambda)[q-(q-lambdaN_(0))e^(-lambda t)]`
`rArr N_(Y)=qt+((1-lambdaN_(0))/(lambda))e^(-lambda t)- (q-lambdaN_(0))/(lambda)`