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 $4x$
$\Rightarrow$ $\frac{0.693}{4.5\times {10}^{9}yr}=\frac{2.303}{T}\log \frac{4x}{x}$
$T=2\times 4.5\times {10}^{9}yr=9.0\times {10}^{9}yr$