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

Answer

D

Explanation

`{:(, X , to , Y),("Initial No. of atoms", N_(0), , 0), (" No. of atoms after time t", N, , N _(0) - N):}`
As per question
`(N)/(N_(0)-N) =1/7 , = 7N = N _(0) - N,`
`8N = N _(0) , (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)`
`((1)/(2)) ^(3) = ((1)/(2)) ^(n)`
`therefore n =3`
`n = (t)/(T _(1//2)) or t = n T _(1//2) = 3 xx 20`years = 60 years
Hence, the age of rock is 60 years