Half lives of two isotopes X and Y of a material are known to be 2 × 1 0 9 years and 4 × 1 0 9 years respectively. If a planet was formed with equal number of these isotopes, estimate the current age

Question

Half lives of two isotopes X and Y of a material are known to be 2 × 1 0 9 years and 4 × 1 0 9 years respectively. If a planet was formed 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.

TestHub · Accepted Answer

Let total sum of number of isotopes at the formation of the planet was No and currently is N

$∴ N_{x} = (\frac{N _{0}}{2}) e^{- λ_{1} t} = 0.2 N N_{y} = (\frac{N _{0}}{2}) e^{- λ_{2} t} = 0.1 N$

Dividing equation (ii) by (i)

$∴ e^{(λ_{1} - λ_{2}) t} = 4 ∴ (λ_{1} - λ_{2}) t = ln 4 ∴ \frac{l n 2}{1 0 ^{9}} [\frac{1}{2} - \frac{1}{4}] t = 2 ln 2 ∴ t = 8 \times 1 0^{9} years$

[RT-10/7 11-02-2007]

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