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A piece of wood from the ruins of an ancient building was found to have a $^{14}C$ activity of $12$ disintegrations per minute per gram of its carbon content. The $^{14}C$ activity of the living wood is $16$ disintegrations per minute per gram. How long ago did the tree, from which the wooden sample came, die? Given half-life of $^{14}C$ is $5760$ years.
Solution
According to exponential law,
$\quad \mathrm{I}=\mathrm{I}_{0} e^{-\lambda t}$
$\therefore 12=16 e^{-\lambda t}$
$\therefore \frac{12}{16}=e^{-\lambda t}$
$\therefore e^{\lambda t}=\frac{16}{12}=1.333$
$\therefore \lambda t \ln e=\ln (1.333)$
$\therefore \lambda t(1)=2.303 \log (1.333)$
$\therefore t=\frac{(2.303)(0.1249)\left(\tau_{1 / 2}\right)}{0.693}$
$\therefore t=\frac{(2.303)(0.1249)(5760)}{0.693}$
$\therefore t \approx 2389 \text { years }$
