A sample of pitch blend is found to contain 50% Uranium and 2.425% Lead. Of this lead, only 93% was — Nuclear Chemistry and Radioactivity Chemistry Question

💡 Solution & Explanation

Step 1 - Set Up the Problem In 100 g of pitch blend: - Mass of $\ce{^{238}U}$ = 50 g → $n_U = 50/238 = 0.2101$ mol - Mass of Pb = 2.425 g, of which 93% is Pb-206 = $2.425 \times 0.93 = 2.255$ g → $n_{Pb} = 2.255/206 = 0.01094$ mol Step 2 - Initial Moles of U-238 All $\ce{^{206}Pb}$ came from $\ce{^{238}U}$ decay (mole for mole): $$N_0 = n_U + n_{Pb} = 0.2101 + 0.01094 = 0.2210\ \text{mol}$$ Step 3 - Apply First-Order Decay Equation Given: $\lambda = 1.52 \times 10^{-10}\ \text{yr}^{-1}$ $$\frac{N_0}{N} = \frac{0.2210}{0.2101} = 1.0519$$ $$t = \frac{1}{\lambda} \ln\!\left(\frac{N_0}{N}\right) = \frac{1}{1.52 \times 10^{-10}} \times \ln(1.0519) = \frac{0.05060}{1.52 \times 10^{-10}}$$ $$t = 3.33 \times 10^8\ \text{yr} \approx \boxed{3.34 \times 10^8\ \text{yr}}$$ Step 4 - Evaluate Options - **(A) $3.34 \times 10^8$ years**: Matches. **Correct.** - **(B) $1.67 \times 10^8$ years**: Half the correct answer. Incorrect. - **(C) $6.68 \times 10^8$ years**: Twice the correct answer. Incorrect. - **(D) $5.01 \times 10^8$ years**: Incorrect. $$\boxed{\text{Answer: A — }3.34 \times 10^8\ \text{years}}$$