Nucleus A changes into a stable nucleus C by emission of two α particles and three β particles. Nucleus B changes into C by emission of one α particle and five β particles. At t = 0, numbers of A and B nuclei are 4N₀ and N₀. Their respective half-lives are 1 min and 2 min. Find the time (in minutes) when their disintegration rates become equal.
Physics · Mixed Chapter
JEE Mains

Question

Nucleus A changes into a stable nucleus C by emission of two α particles and three β particles. Nucleus B changes into C by emission of one α particle and five β particles. At t = 0, numbers of A and B nuclei are 4N₀ and N₀. Their respective half-lives are 1 min and 2 min. Find the time (in minutes) when their disintegration rates become equal.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — The decay rates (activities) are Aₐ(t) = λₐ Nₐ(t) and Aᵦ(t) = λᵦ Nᵦ(t). Step 2 — From the two half-lives, λₐ = ln2/(1 min) and λᵦ = ln2/(2 min); therefore λₐ/λᵦ = 2. Step 3 — Starting numbers of nuclei are Nₐ(0) = 4N₀ and Nᵦ(0) = N₀. At time t: Nₐ(t) = 4N₀ exp(−λₐt) and Nᵦ(t) = N₀ exp(−λᵦt). Step 4 — Set decay rates equal: 4λₐN₀ exp(−λₐt) = λᵦN₀ exp(−λᵦt). Step 5 — Rearranging gives exp[(λₐ − λᵦ)t] = 4λₐ/λᵦ = 8. Since λₐ − λᵦ = ln2/2 min⁻¹, t = ln8/(ln2/2) = 3ln2 × 2/ln2 = 6 min. Step…