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. `Kt-(K-lambda N_(0))/(lambda)e^(-lambda t)+K-lambda(N_(0))/(lambda)`
B. `Kt-(K-lambda N_(0))/(lambda)e^(-lambda t)+K-lambda(N_(0))/(lambda)`
C. `Kt+(K-lambda N_(0))/(lambda)e^(-lambda t)`
D. `Kt+(K-lambda N_(0))/(lambda)e^(-lambda t)`

Answer

B

Explanation

`(dN_X)/(dt)=K-lambda N_(X)`
N_(X)=(1)/(lambda)[K-K-lambda N_(0))e^(-lambda t)]`
`(dN_(Y))/(dt)=lambdaN_(X)`
`N_(Y)=Kt+(K-lambda N_0/lambda)e^(-lambda t) -(K-lambda N_(0))/(lambda)`.