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Nuclei of a radioactive element X are being produced at a constant rate k and this element decays to a stable nucleus by with a decay constant h and half life T1/2. At time t = 0 there N nuclei of the element x. The number of nuclei of y at t=T1/2 is?

Nuclei of a radioactive element X are being produced at a constant rate k and this element decays to a stable nucleus by with a decay constant h and  half life T1/2. At time t = 0 there N nuclei of the element x. The number of nuclei of y at t=T1/2  is?

Grade:12th pass

1 Answers

Arun
25763 Points
2 years ago
(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 N
(ii) Substituting α = 2λ N0 and t ⟶ ∞ in equation (1), we get
N = α / λ = 2 N0

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