Question

A radioactive isotope $X$ with half life $1.5\times {10}^9$yr decays into a stable nucleus $Y$. A rock sample contains both elements $X$ and $Y$ in ratio $1:15$. Find the age of the rock

Answer 1

Given: $T_{1/2}=1.5\times {10}^9$ yr

$X:Y=1:15$

$⟹Y=15X$

Let $X_o$ be the amount of radioactive isotope X present initially.

$\Rightarrow$$X+Y=X_o$

$\Rightarrow X+15X=X_o$

$\Rightarrow \frac{X_o}{X}=16$

Using, $\lambda t=2.303lo{g}_{10}\frac{X_o}{X}$

$\Rightarrow \frac{0.693}{1.5\times {10}^9}t=2.303lo{g}_{10}16$

$\Rightarrow \frac{0.693}{1.5\times {10}^9}t=2.303\times 1.204$

$\Rightarrow t=6\times {10}^9$ yr

Thus the rock is $6\times {10}^9$ yr old.