In moon rock sample the ratio of the number of stable argon-`40` atoms present to the number of radioactive potassium`-40` atoms is `7:1`. Assume that all the argon atoms were produced by the decay of potassium atoms, with a half-life of `2.5xx10^(9)yr`. The age of the rock is

Options

A. `2.5xx10^(9)yr`
B. `5.0xx10^(9) yr`
C. `7.5xx10^(9)yr`
D. `10^(10)yr`

Answer

C

Explanation

Let the number of radioactive Potassium atoms present initially `(t=0)` is `N_(0)` and the number of stable argon atoms at `t=0` is zero. After time `t` the number stable argon atoms is `m` and the radioactive potassium is `N_(0)-m` gives that
`(n_(0)-M)/(m)=(1)/(7), M=(7)/(8)N_(0)-m=(1)/(8)N_(0)`
Since after one half-life `N_(0)` reduces to `N_(0)//2` after `2` half-lives `N_(0)//4` and after `3` half-lives it reduces to `N_(0)//8`
`t= nT=3xx2.5xx10^(9)` years
Thus,
`=7.5xx10^(9)` year