A radioactive element `A` with a half-value period of `2` hours decays giving a stable element `Y`. After a time `t` the ratio of `X` and `Y` atoms is `1 : 7` then `t` is :

Options

A. 4 hours
B. 14 hours
C. 6 hours
D. Between 4 and 6 hour

Answer

C

Explanation

After time t , let `N_x " and " N_y` be the number of undecayed atoms of the elements X and Y respectively .
`therefore (N_y)/(N_x) = 7 therefore (N_y)/(N_x) + 1 = 7 + 1 = 8`
`therefore (N_y + N_x)/(N_x) = 8`
`therefore (N_x)/(N_y + N_x) = 1/8 = (1/2)^3`
This is of the form ,
`(N)/(N_0) = (1/2)^n " but " (N)/(N_0) = (1/2)^(t/(T_(1//2))`
`therefore n=3 = (t)/(T_(1//2)) = t/2`
`therefore t = 6h`