The half-life of a radioactive isotope `X` is `20` years. It decays to another element `Y` which is stable. The two elements `X` and `Y` were found to be in the ratio of `1 : 7` in a sample of a given rock. The age of the rock was estimated to be.

Options

A. 80 years
B. 100 years
C. 40 years
D. 60 years

Explanation

`{:(,X,rarr,Y),("Initial number of atoms",N_(0),,0),("Number of atoms after",N,,N_(0)-N):}`
time t
As per question
`(N)/(N_(0)-N)=(1)/(7)`
`7N=N_(0)-N or 8N=N_(0) or (N)/(N_(0))=(1)/(8)`
As `(N)/(N_(0))=((1)/(2))^(n)` where n is the no. of half lives
`therefore" "(1)/(8)=((1)/(2))^(n) or ((1)/(2))^(3)=((1)/(2))^(n)`
`therefore n =3`
`n=(t)/(T_(1//2)) or t = nT_(1//2)=3xx"20 years = 60 years"`
Hence, the age of rock is 60 years.