A radioactive sample undergoes competing α decay and β decay with partial half-lives 3 h and 6 h, respectively. Find the ratio of the initial number of radioactive nuclei to the number remaining after 6 h.
Physics · Mixed Chapter
JEE Mains

Question

A radioactive sample undergoes competing α decay and β decay with partial half-lives 3 h and 6 h, respectively. Find the ratio of the initial number of radioactive nuclei to the number remaining after 6 h.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — For independent competing decay channels the total probability of decay per unit time is the sum of their partial decay constants: λ = λₐ + λᵦ. Step 2 — Their partial half-lives give λₐ = ln(2)/3 h⁻¹ and λᵦ = ln(2)/6 h⁻¹. Step 3 — Thus λ = ln(2)(⅓ + ⅙) = ln(2)/2 h⁻¹. Step 4 — The survival law is N(t) = N₀ exp(−λt). At t = 6 h, N(6)/N₀ = exp[−(ln2/2)×6] = exp(−3 ln2) = 2⁻³ = ⅛. Step 5 — Therefore N₀/N(6) = 8. Answer: 8.