Question

Lead is the final product formed by a series of changes in which the rate determining stage is the radioactive decay of uranium $ -238 $ with a half-life of $ 4.5\times {{10}^{9}}yr $ . What would be the age of a rock sample originally lead free in which the molar proportion of uranium to lead is now 1 : 3?

• A. $ 1.5\times {{10}^{9}}yr $
• B. $ 2.25\times {{10}^{9}}yr $
• C. $ 4.5\times {{10}^{8}}yr $
• D. $ 9.0\times {{10}^{9}}yr $
• E. $ 13.5\times {{10}^{9}}yr $

Answer 1

Answer: (D)

All radioactive processes follow first order kinetics. Initially sample is free from lead. If its proportion is $ x, $ the amount of sample at time T is $ 4x $
$ \Rightarrow $ $ \frac{0.693}{4.5\times {{10}^{9}}yr}=\frac{2.303}{T}\log \frac{4x}{x} $
$ T=2\times 4.5\times {{10}^{9}}yr=9.0\times {{10}^{9}}yr $