Question

A sample of uranium mineral was found to contain ${}^{206}Pb$ and ${}^{238}U$ in the ratio of $0.008:1$. Estimate the age of the mineral. (Half life of ${}^{238}U$ is $4.51\times {10}^9$ years)

Answer 1

We know that, $t=\frac{2.303t_{1/2}}{0.693}\log [1+\frac{{}^{206}Pb}{{}^{238}U}]$
Given, $t_{1/2}=4.51\times {10}^9$ years
Ratio by mass of ${}^{206}Pb{:}^{238}U=0.008:1$
Ratio by moles of ${}^{206}Pb{:}^{238}U=\frac{0.008}{206}:\frac{1}{238}=0.0092$
So, $t=\frac{2.303\times 4.51\times {10}^9}{0.693}\log [1+0.0092]$
$=\frac{2.303\times 4.51\times {10}^9}{0.693}\times 0.00397$
$=\frac{0.0412}{0.693}\times {10}^9=0.05945\times {10}^9$ years
Hence, age of the mineral is $5.945\times {10}^7$ years.