Question

On analysis, a sample of uranium ore was found to contain $0.277g$ of ${}_{82}{Pb}^{206}$ and $1.667g$ of ${}_{92}{U}^{238}$. The half life period of $U^{238}$ is $4.51\times {10}^9$yrs. If all the lead were assumed to have come from decay of ${}_{92}{U}^{238}$, what is the age of the earth?

Answer 1

Moles of $U^{238}=\frac{1.667}{238}$
Moles of ${Pb}^{206}=\frac{0.277}{206}$
$N_0=\frac{1.667}{238}+\frac{0.277}{206}$
and $N=\frac{1.667}{238}$
$t=\frac{2.303}{\lambda }\log \frac{N_0}{N}$
$=\frac{2.303\times 4.51\times {10}^9}{0.693}{\log }_{10}\frac{\frac{1.667}{238}+\frac{0.277}{206}}{\frac{1.667}{238}}$
$=1.143\times {10}^9$ years