Two radioactive nuclei `P` and `Q`, in a given sample decay into a stable nucleus `R`. At time `t = 0`, number of `P` species are `4 N_0` and that of `Q` are `N_0`. Half-life of `P` (for conversation to `R`) is `1mm` whereas that of `Q` is `2 min`. Initially there are no nuclei of `R` present in the sample. When number of nuclei of `P` and `Q` are equal, the number of nuclei of `R` present in the sample would be :

Options

A. `3 N_0`
B. `(9 N_0)/(2)`
C. `(5 N_0)/(2)`
D. `2 N_0`

Answer

B

Explanation

(b) Initially `p rarr 4 N_0`
`Q rarr N_0`
Half-life `T_p rarr 1 min`
`T_Q rarr 2 min`
Let after time `t` number of nuclei of `P` and `Q` are equal,
i.e., `(4 N_0)/(2^(t//1)) = (N_0)/(2^(t//2))`
`4 = 2^(t//2)`
`2^2 = 2^(t//2) rArr (t)/(2) = 2`
`t = 4 min`
Deactivate nucleus or nuclei of `R`
=`(4 N_0 - (4 N_0)/(2^4)) + (N_0 - (N_0)/(2^2))`
=`4 N_0 (N_0)/(4) + N_0 - (N_0)/(4) = 5 N_0 - (N_0)/(2) = (9)/(2) N_0`.