Question

A sample of uraninite, a uranium containing mineral, was found on analysis to contain 0.214 g of lead-206 for every g uranium. If all the lead came from the radioactive disintegration of the uranium and assume that all isotopes of uranium other than ${}^{238}U$ can be neglected, estimate the date when the mineral was formed in the earth's crust? (The half-life of ${}^{238}U$ is $4.5\times {10}^9$ years).
Use $1+\frac{\left[Pb\right]}{\left[U\right]}=2^y$, where $y=\frac{t}{T_{50}}$

• A. $1.4344\times {10}^9$ years
• B. $6.4344\times {10}^9$ years
• C. $4.4344\times {10}^9$ years
• D. None of these

Answer 1

The correct option is A $1.4344\times {10}^9$ years

If 0.214 g of ${}^{206}Pb$ is present, then 1 g of ${}^{238}U$ is also present.
$\therefore \left[Pb\right]=\frac{0.214}{206}=0.00104$ mol,
$\left[U\right]=\frac{1}{238}=0.0042$ mol
$\therefore \frac{\left[Pb\right]}{\left[U\right]}=0.247$
$\therefore$ For 'rock dating' analysis, we use
$[1+\frac{\left[Pb\right]}{\left[U\right]}]=2^y$ where $y=\frac{t}{T_{50}}$
$[1+0.247]=(2{)}^y$
Taking log, we get $y=0.3188$
$\therefore \frac{t}{T_{50}}=0.3188$
$\therefore t=0.3188\times T_{50}$
$=0.3188\times 4.5\times {10}^9$
$=1.4344\times {10}^9$ years