A radio isotope `X` with a half life `1.4xx10^(9)` yr 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. `1.96xx xx10^(9) yr`0
B. `3.92xx xx10^(9) yr`
C. `4.20xx xx10^(9) yr`
D. `8.40xx10^(9)` yr

Answer

C

Explanation

According to the question, ratio of X:Y is given = 1: 7
`implies(m_(x))/(m_(y))=(1)/(7)implies7m_(x)=m_(y)`
Let the initial total mass is m.
`impliesm_(x)+m_(y)=mimpliesm_(y)/(7)+m_(y)=m`
`implies(8m_(y))/(7)=mimpliesm_(y)=(7)/(8)m`
Only `(1)/(8)` part remains left undisintegrated.
Number of half-lives involve are three as
`1overset(T_(1//2))to(1)/(2)overset(T_(1//2))to(1)/(4)overset(T_(1//2))to(1)/(8)`
So, time taken to become `(1)/(8)` unstable part
`=3xxT_(1//2)=3xx1.4xx10^(9)=4.20xx10^(9)yr`