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Half lives of two isotopes \mathrm{X} and \mathrm{Y} of a material are known to be 2 \times 10^9 years and 4 \times 10^9 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:

Option: 1

2 \times 10^9 years


Option: 2

4 \times 10^9 years


Option: 3

6 \times 10^9 years


Option: 4

8 \times 10^9 years


Answers (1)

Let total sum of number of isotopes at the formation of the planet was \mathrm{N}_0 and currently is N

\therefore \quad \mathrm{N}_{\mathrm{X}}=\left(\frac{\mathrm{N}_0}{2}\right) \mathrm{e}^{-\lambda_1 \mathrm{t}}=0.2 \mathrm{~N}                             (i)

and  \mathrm{N}_{\mathrm{Y}}=\left(\frac{\mathrm{N}_0}{2}\right) \mathrm{e}^{-\lambda_2 \mathrm{t}}=0.8 \mathrm{~N}                               (ii)

Dividing equation (ii) by (i)

\mathrm{e}^{\left(\lambda_1-\lambda_2\right) \mathrm{t}}=4 \quad \text { or } \quad\left(\lambda_1-\lambda_2\right) \mathrm{t}=\ln 4

\begin{aligned} & \therefore \frac{\ln 2}{10^9}\left[\frac{1}{2}-\frac{1}{4}\right] \mathrm{t}=2 \ln 2 \quad\left(\because \mathrm{T}_{1 / 2}=\frac{\ln 2}{\lambda}\right) \\ & \therefore \mathrm{t}=8 \times 10^9 \text { years } \end{aligned}

Posted by

Ramraj Saini

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