Two radioactive substance `A` and `B` have decay constants `5 lambda` and `lambda` respectively. At `t=0` they have the same number of nuclei. The ratio of number of nuclei of nuclei of `A` to those of `B` will be `(1/e)^(2)` after a time interval

Options

A. `(1)/(4lamda)`
B. `4lamda`
C. `2lamda`
D. `(1)/(2lamda)`

Answer

D

Explanation

At time t =0 , the number of nuclei in A and B is `N_0`
Using `N/N_0 = e^(-lamdat)` For A and B we get
`N_1 = N_0 e^(-5lamdat)` For A and `N_2 = N_0e^(-lamdat)` for B
`therefore (N_1)/(N_2) = (e^(-5lamdat))/(e^(-lamdat)) = e^((-5lamda + lamda)t) = e^(-4lamdat) = (1)/(e^(4lamdat)) " "...(1)`
But it is given that after time t,
`N_1/N_2 = 1/e^2 " "...(2)`
`therefore` From (1) and (2), `(1)/(e^2) = (1)/(e^(4lamdat))`
`therefore 2 = 4lamdat therefore t = (2)/(4lamda) = (1)/(2lamda)`