Half lives of two isotopes X and Y of a material are known to be $2 \times 10^{9}$ years and $4 \times 10^{9}$ years respectively. If a planet forms with equal number of these isotopes , estimate the current age of the planet, given that currently the material has 20% of X and 80% of Y by number

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Solution

The correct option is D

Let total sum of number of isotopes at the formation of the planet was $N_{0}$ and currently is N

$∴ N_{x} = (\frac{N_{0}}{2}) e^{- λ_{1} t} = 0.2 N \dots (i) N_{y} = (\frac{N_{0}}{2}) e^{- λ_{2} t} = 0.8 N \dots (i i) Dividing equation(ii) by (i) ∴ e^{(λ_{1} - λ_{2}) t} = 4 =∴ (λ_{1} - λ_{2}) t = In 4 ∴ \frac{In 2}{10^{9}} [\frac{1}{2} - \frac{1}{4}] t = 2 In 2 ∴ t = 8 \times 10^{9} years$

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