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Half lives of two isotopes X and Y of material are known to be 2 × 10⁹ years and 4 × 10⁹ years respectively. If a planet was formed with equal number of these isotopes, then the current age of planet, given that currently the material has 20% of X and 80% of Y by number, will be

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2 × 10⁹ years

4 × 10⁹ years

6 × 10⁹ years

8 × 10⁹ years

answer is D.

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Detailed Solution

Given

$\begin{aligned} t_{X} = 2 \times 10^{9} Years \\ t_{Y} = 4 \times 10^{9} Years \end{aligned}$

$N_{x} = \frac{N_{0}}{2} \cdot e^{- λ_{1} t} = 0.2 N_{0} - - - - - - - - (1)$

$N_{y} = \frac{N_{0}}{2} \cdot e^{- λ_{2} t} = 0.8 N_{0} . --------(2)$

Ratio of both equation

$e^{t (λ_{1} - λ_{2})} = 4 \Rightarrow t = 8 \times 10^{9} years$

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