Search any question & find its solution
Question:
Answered & Verified by Expert
In an ore containing uranium, the ratio of $U^{238}$ to $\mathrm{Pb}^{206}$ is 3. Calculate the àge of the ore, assuming that all the lead present in the ore is the final stable product of $U^{238}$. Take the half-life of $U^{238}$ to be $4.5 \times 10^{9} \mathrm{yr}$.
Options:
Solution:
2673 Upvotes
Verified Answer
The correct answer is:
$1.867 \times 10^{9} \mathrm{yr}$
Let the initial mass of uranium be $\mathrm{M}_{0}$ Final mass of uranium after time $\mathrm{t}$
$$
\mathrm{M}=\frac{3}{4} \mathrm{M}_{0}
$$
According to the law of radioactive disintegration.
$$
\begin{array}{l}
\frac{\mathrm{M}}{\mathrm{M}_{0}}=\left(\frac{1}{2}\right)^{\mathrm{t} / \mathrm{T}} \Rightarrow \frac{\mathrm{M}_{0}}{\mathrm{M}}=(2)^{\mathrm{t} / \mathrm{T}} \\
\therefore \log _{10}\left(\frac{\mathrm{M}_{0}}{\mathrm{M}}\right)=\frac{\mathrm{t}}{\mathrm{T}} \log _{10}(2) \\
\mathrm{t}=\mathrm{T} \frac{\log _{10}\left(\frac{\mathrm{M}_{0}}{\mathrm{M}}\right)}{\log _{10}(2)}=\frac{\mathrm{T} \log _{10}\left(\frac{4}{3}\right)}{\log _{10}(2)} \\
=\frac{\mathrm{T} \log _{10}(1.333)}{\log _{10}(2)}=4.5 \times 10^{9}\left(\frac{0.1249}{0.3010}\right) \\
\Rightarrow \mathrm{t}=1.867 \times 10^{9} \mathrm{yr} .
\end{array}
$$
$$
\mathrm{M}=\frac{3}{4} \mathrm{M}_{0}
$$
According to the law of radioactive disintegration.
$$
\begin{array}{l}
\frac{\mathrm{M}}{\mathrm{M}_{0}}=\left(\frac{1}{2}\right)^{\mathrm{t} / \mathrm{T}} \Rightarrow \frac{\mathrm{M}_{0}}{\mathrm{M}}=(2)^{\mathrm{t} / \mathrm{T}} \\
\therefore \log _{10}\left(\frac{\mathrm{M}_{0}}{\mathrm{M}}\right)=\frac{\mathrm{t}}{\mathrm{T}} \log _{10}(2) \\
\mathrm{t}=\mathrm{T} \frac{\log _{10}\left(\frac{\mathrm{M}_{0}}{\mathrm{M}}\right)}{\log _{10}(2)}=\frac{\mathrm{T} \log _{10}\left(\frac{4}{3}\right)}{\log _{10}(2)} \\
=\frac{\mathrm{T} \log _{10}(1.333)}{\log _{10}(2)}=4.5 \times 10^{9}\left(\frac{0.1249}{0.3010}\right) \\
\Rightarrow \mathrm{t}=1.867 \times 10^{9} \mathrm{yr} .
\end{array}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.