Question

Nuclei of radioactive element a are produced at rate ' t2 ' at any time t. The element a has decay constant . Let n be the number of nuclei of element a at any time t. At time t = t0, dndt is minimum. Then the number of nuclei of element a at time t = t0 is

Answer 1

a) Let at time ‘t’ number of radioactive are N.

Net rate of formation of nuclei of A.

dN /dt = α – λ N or dN / α – λN = dt

or \int_{N_0}^{N}\frac{dN}{\alpha - \lambda N } = \int_{0}^{t} dt

Solving this equation, we get

N = 1 / λ [α – (α – λN0)e-λt] …..(1)

(b) Substituting α = 2λN0 and

t = t1/2 = 1n(2) / λ in equation (1),

we get, N = 3/2 N0­

(ii) Substituting α = 2λ N0 and t ⟶ ∞ in equation (1), we get

N = α / λ = 2 N0