A radioactive substance is being produced at a constant rate of 200 muclei/s The decay constant of the substance is `1 s^(-1)` After what tiem the number of radioactive nuclie will become 100 ? Initially there are no nuclei present.

Options

A. `1 s`
B. `(1)/(1 n(2))s`
C. `ln(2)s`
D. `2s`

Answer

c

Explanation

Let N be the number of nuclei at any time t. Then,
`(dN)/(dt) =200 -lambda N`
`:.int_(0)^(N) (dN)/((200 -lambda N)) = int_(0)^(t)dt`
or N=(200)/(lambda) (1-e^(-lambda t))`
Given: `N =100 and lambda =1 s^(-1)`
`:. 100=200 (1-e^(-t))`
or `e^(-t)=((1)/(2)) :. t=In(2) s`.