Atoms And Nuclei Question 383
Question: In a sample of rock, the ratio of $ ^{206}Pb $ to $ ^{238}U $ nuclei is found to be 0.5. The age in year of the rock is (given half-life of $ ,U^{238} $ is $ 4.5\times 10^{9} $ years)
Options:
A) $ 2.25\times 10^{9} $
B) $ 4.5\times 10^{9}ln3 $
C) $ 4.5\times 10^{9}\frac{ln( \frac{3}{2} )}{ln2} $
D) $ 2.25\times 10^{9}ln( \frac{3}{2} ) $
Show Answer
Answer:
Correct Answer: C
Solution:
- Suppose an initial radionuclide I decays to a final product F with a half- life $ {T_{1/2}}. $
At any time, $ N_{1}=N_{0}{{e}^{-\lambda t}} $ Number of product nuclei $ =N_{F}=N_{0}-N_{I} $ $ \frac{N_{F}}{N_{I}}=\frac{N_{0}-N_{I}}{N_{I}}=( \frac{N_{0}}{N_{I}}-I ) $
$ \frac{N_{0}}{N_{I}}=( 1+\frac{N_{F}}{N_{I}} )=1+0.5=1.5 $
$ \therefore ,\frac{{T_{1/2}}In(1.5)}{In2}=4.5\times 10^{9}\frac{\ell n( \frac{3}{2} )}{\ell n2}year $
