A radio isotope `X` with a half-life `1.4 xx 10^9` years decays of `Y` which is stable. A sample of the rock from a cave was found to contain `X` and `Y` in the ratio `1 : 7`. The age of the rock is.

Options

A. `4.20xx10^9` years
B. `8.40xx10^9` years
C. `1.90xx10^9` years
D. `3.92xx10^9` years

Answer

B

Explanation

After time t, and `N_y` be the number of undecayed atoms of the elements X and Y respectively.
It is given that `(N_x)/(N_y) = 1/7 " or " (N_y)/(N_x) = 7`
`therefore (N_y)/(N_x) + 1 = 7 +1 = 8`
`therefore (N_y + N_x)/(N_x) = 8/1`
`therefore (N_x)/(N_y - N_x) = 1/8 (1/2)^3 " "...(1)`
This is of the form , `(N/N_0) = (1/2)^n = (1/2)^(t//T_(1//2)) " "..(2)`
`therefore` From (1) and(2) , n =3 `=(t)/(T_(1//2))`
`therefore 3= (t)/(1.4xx10^9) therefore t = 3xx 1.4xx10^9 = 4.2xx10^9` years