In a rock the number ratio of ²⁰⁶Pb to ²³⁸U nuclei is 0.5. Assume all the lead formed from uranium decay, with ²³⁸U half-life 4.5×10⁹ years. If the rock age is (18/n)×10⁹ × [ln(3/2)/ln2] years, find n.
Physics · Mixed Chapter
JEE Mains

Question

In a rock the number ratio of ²⁰⁶Pb to ²³⁸U nuclei is 0.5. Assume all the lead formed from uranium decay, with ²³⁸U half-life 4.5×10⁹ years. If the rock age is (18/n)×10⁹ × [ln(3/2)/ln2] years, find n.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Daughter–parent number relation: Since N(Pb) = N₀(U)−N(U), the stated ratio means (N₀−N)/N = 0.5. Thus N₀/N = 1.5 = 3/2. Step 2 — Exponential decay: N = N₀exp(−λt) ⇒ t = (1/λ)ln(N₀/N). Step 3 — Insert half-life: λ = ln2/(4.5×10⁹ yr), so t = 4.5×10⁹[ln(3/2)/ln2] yr. Step 4 — Match (18/n)×10⁹[ln(3/2)/ln2]: 18/n = 4.5 ⇒ n = 4. Answer: 4.