The half-life of a radioactive isotope `X` is `50` 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 : 15` in a sample of a given rock. The age of the rock was estimated to be

Options

A. `150yr`
B. `200yr`
C. `250yr`
D. `100yr`

Answer

B

Explanation

Let at a time t, `N_(X)` be the number of X in an element and `N_(Y)` be the number of Y in that element.
Given, `(N_(X))/(N_(Y))=1/15 orN_(X)=15N_(Y)`
Part of `N_(X)=1/16[N_(X)+N_(Y)]`
or `(N_(X))/(N_(X)+N_(Y))=1/16=(1/2)^(4)`
In n is the number of half lives of isotope X.
Then `(N_(X))/(N_(X)+N_(Y))=(1/2)^(n)=(1/2)^(4) or n=4`
If t is the age of rock, and T is the half life period of X, then `t=nT=4xx50=200yrs`