Question

In a sample of rock, the ratio of ${}^{206}Pb$ to ${}^{238}U$ nuclei is found to be 0.5. The age of the rock is (given half-life of $U^{238}$ is $4.5\times {10}^9$ years)

• A. $2.25\times {10}^{9}year$
• B. $4.5\times {10}^9$ ln 3 year
• C. $4.5\times {10}^9\frac{ln\left(\frac{3}{2}\right)}{ln2}year$
• D. $2.25\times {10}^{9}ln\left(\frac{3}{2}\right)year$

Answer 1

The correct option is C $4.5\times {10}^9\frac{ln\left(\frac{3}{2}\right)}{ln2}year$

First order decay: $N\left(t\right)=N_o{e}^{-\lambda t}$
$\lambda =\frac{ln2}{t_{1/2}}$
$N\left(t\right)/N_o=(1+0.5)/0.5=3/2$
$1.5=e^{-\lambda t}$
$\lambda t=ln1.5$
$t=4.5\times {10}^9\frac{ln\left(3/2\right)}{ln2}$