Question

In an ore containing uranium, the ratio of \(^{238}\text{U}\) to \(^{206}\text{Pb}\) is \(3:1\). Calculate the age of the ore, assuming that all the lead present in the ore is the final stable product of \(^{238}\text{U}\). Take the half-life of \(^{238}\text{U}\) to be \(4.5\times10^9\) yr.

• A. \(1.6\times10^3\) yr
• B. \(1.5\times10^4\) yr
• C. \(1.867\times10^9\) yr
• D. \(2\times10^5\) yr

Hint:

For radioactive dating: \[ \frac{D}{N}=e^{\lambda t}-1 \] Always check whether ratio given is parent:daughter or vice versa.

Answer 1

The Correct Option is C

Step 1: Given ratio \(U:Pb = 3:1 \Rightarrow \dfrac{Pb}{U}=\dfrac{1}{3}\).
Step 2: Radioactive decay relation: \[ \frac{Pb}{U}=e^{\lambda t}-1 \]
Step 3: Decay constant: \[ \lambda=\frac{0.693}{T_{1/2}}=\frac{0.693}{4.5\times10^9} \]
Step 4: Substitute: \[ \frac{1}{3}=e^{\lambda t}-1 \Rightarrow e^{\lambda t}=\frac{4}{3} \] \[ t=\frac{1}{\lambda}\ln\!\left(\frac{4}{3}\right)\approx1.867\times10^9\text{ yr} \]