Nuclear Chemistry Question 27
Question: The radium and uranium atoms in a sample of uranium mineral are in the ratio of $ 1:2.8\times 10^{6} $ . If half-life period of radium is 1620 years, the half-life period of uranium will be
[MP PMT 1999]
Options:
A) $ 45.3\times 10^{9} $ years
B) $ 45.3\times 10^{10} $ years
C) $ 4.53\times 10^{9} $ years
D) $ 4.53\times 10^{10} $ years
Show Answer
Answer:
Correct Answer: C
Solution:
According to radioactive equilibrium $ {\lambda _{A}}N _{A}={\lambda _{B}}N _{B} $ or $ \frac{0.693\times N _{A}}{{t _{1/2}}(A)}=\frac{0.693\times N _{B}}{{t _{1/2}}(B)}[ \lambda =\frac{0.693}{{t _{1/2}}} ] $ Where $ {t _{1/2}}(A) $ and $ {t _{1/2}}(B) $ are half periods of A and B respectively
$ \therefore \frac{N _{A}}{{t _{1/2}}(A)}=\frac{N _{B}}{{t _{1/2}}(B)}or\frac{N _{A}}{N _{B}}=\frac{{t _{1/2}}(A)}{{t _{1/2}}(B)} $
$ \therefore $ At equilibrium A and B are present in the ratio of their half lives $ \frac{1}{2.8\times 10^{6}}=\frac{1620}{Halflifeofuranium} $
$ \therefore $ Half-life of uranium = $ 2.8\times 10^{6}\times 1620=4.53\times 10^{9} $ years.
BETA
