Nuclei of a radioactive element A are being produced at a constant rate a. The element has a decay constant X. At time t = 0, there are N0 nuclei of the element. (a) Calculate the number A of nuclei of A at time t. (b) If α = 2N0λ, calculate the number of nuclei of A after one half-life of A and also the limiting value of N as tarrow ∞.

Question

Nuclei of a radioactive element A are being produced at a constant rate a. The element has a decay constant X. At time t = 0, there are N0 nuclei of the element. (a) Calculate the number A of nuclei of A at time t. (b) If α = 2N0λ, calculate the number of nuclei of A after one half-life of A and also the limiting value of N as tarrow ∞. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

(a) Let at time t, number of radioactive nuclei are N. Net rate of formation of nuclei of A

\frac{d N}{d t} = α - λ N

or

\frac{d N}{α - λ N} = d t

or

\int_{N_{0}}^{N} \frac{d N}{α - λ N} = \int_{0}^{1} d t

Solving this equation, we get

N = \frac{1}{λ} [α - (α λ N_{0}) e^{- λ t}]

..(i)
(b) (i) Substituting

α = 2 λ N_{0}

and

t = t_{1/2} = \frac{I n ( 2 )}{λ} in

Eq. (i) we get,

N = \frac{3}{2} N_{0}

(ii) Substituting

α = 2 λ N_{0} an d t \to \infty

in Eq. (i), we get

N = \frac{α}{λ} = 2 N_{0}

or

N = 2 N_{0}