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 `4N_0` and that of Q are `N_0`. Half-life of P (for conversion to R) is 1 min whereas that of Q is 2min. 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. `2N_0`
B. `3N_0`
C. `(9N_0)/2`
D. `(5N_0)/2`

Answer

C

Explanation

At t=0, number of nucleus of P and Q are respectively `4N_0` and `N_0`.
Let at t=t, number of nucleus of P and Q are respectively `N_P` and `N_Q`.
`therefore N_P=4N_0(1/2)^t, N_Q=N_0(1/2)^(t//2)`
`because N_P=N_Q`
`therefore 4N_0(1/2)^t=N_0(1/2)^(t//2)`
So, `4/(2)^t = 1/(2^(t//2)) therefore ` t= 4 min
`therefore` After 4 min number of atoms of both type is same .
Number of atoms after 4 min,
`(4N_0-N_0/4 ) +(N_0-N_0/4)=(9N_0)/2`