Question

A radioactive nucleus, X, disintegrates into another nucleus Y, which in turn, disintegrates to form a nucles z. The disintegration constants, of X and Y, are λ¹ and λ₂ respectively. At t = 0; starting with N₀ atoms of X, the number of atoms of X and Y, present at t = t, are N₁ and N₂ , respectively. The ratio, of the instantaneous rates of disintegration, of X and Y, is

(1) \(\frac{λ_1}{λ_2}\)

(2) \(\frac{λ_1N_1}{λ_2N_2}\)

(3) \((\frac{λ_2N_2}{λ_1N_1}-1)^{-1}\)

(4) \((\frac{λ_1N_1}{λ_2N_2}+1)^{-1}\)

Answer 1

 (3) \((\frac{λ_2N_2}{λ_1N_1}-1)^{-1}\)

R_X = The instantaneous rate of disintegration of X = λ₁ N₁ .....(1) 

When one atom of X disintegrates, one atom of Y is produced. At t = t; the rate of production Y, equals the rate of disintegration of X (λ₁ N₁) . 

The instantaneous rate of decrease of Y, due to its own decay into Z, is (-λ₂ N₂) . The net rate of change of the number of along of Y, is