Atoms And Nuclei Question 406
Question: In an ore containing uranium, the ratio of $ U^{238} $ to $ Pb^{206} $ nuclei is 3. Calculate the age of the ore, assuming that all the lead present in the ore in the final stable product of $ U^{238} $ . Take the half-life of $ U^{238} $ to be $ 4.5\times 10^{9} $ ear.
Options:
A) $ 1.8\times 10^{9} $ year
B) $ 2.3\times 10^{10} $ year
C) $ 5.1\times 10^{7} $ year
D) $ 6.2\times 10^{6} $ year
Show Answer
Answer:
Correct Answer: A
Solution:
- Suppose x is the number of $ Pb^{206} $ nulei.
The number of $ U^{238} $ nuclei will be 3x, Thus $ 3x+x=N_{0} $ We know that
$ N=N_{0}{{e}^{-\lambda t}} $ or $ 3x=4x{{e}^{-\lambda t}} $
$ \therefore {{e}^{\lambda t}}=\frac{4}{3} $
or $ t=\frac{In,4/3}{\lambda }=\frac{Im4/3}{(0.693/{t_{1/2}})} $ $ =\frac{In4/3}{(0.693/4.5\times 10^{9})}=1.868\times 10^{9} $ years.
