Question

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

• A.
• B.
• C.
• D.

Answer 1

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

$\therefore N_x=\left(\frac{N_0}{2}\right)e^{-{\lambda }_{1}t}=0.2N\cdots \left(i\right)N_y=\left(\frac{N_0}{2}\right)e^{-{\lambda }_{2}t}=0.8N\cdots \left(ii\right)\text{Dividing equation(ii) by (i)}\therefore e^{({\lambda }_1-{\lambda }_2)t}=4=\therefore ({\lambda }_1-{\lambda }_2)t=\text{In}4\therefore \frac{\text{In}2}{{10}^9}[\frac{1}{2}-\frac{1}{4}]t=2\text{In}2\therefore t=8\times {10}^9\text{years}$