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. `4lambda`
B. `2lambda`
C. `1//2lambda`
D. `1//4lambda`

Answer

C

Explanation

Given : `lambda_A=5 lambda, lambda_B=lambda`
At t=0, `(N_0)_A =(N_0)_B`
At time t, `N_A/N_B=(1/e)^2`
According to radioactive decay, `N/N_0=e^(-lambdat)`
`therefore N_A/((N_0)_A)=e^(-lambda_A t)`…(i)
and `N_B/((N_0)_B )=e^(-lambda_B t)` ....(ii)
Divide (i) by (ii) , we get
`N_A/N_B=e^(-(lambda_A-lambda_B)t)` or `N_A/N_B=e^(-(5lambda-lambda)t)`
or `(1/e)^2 = e^(-4 lambdat) ` or `(1/e)^2 =(1/e)^(4 lambdat)`
`rArr 4lambdat=2 ` or `t=2/(4lambda)=1/(2lambda)`