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 $N_0$ nuclei of the element.
(a) Calculate the number A of nuclei of A at time t.
(b) If $\alpha = 2N_0\lambda,$ calculate the number of nuclei of A after one half-life of A and also the limiting value of N as $t\rightarrow \infty$.

Answer 1

(a) Let at time t, number of radioactive nuclei are N. Net rate of formation of nuclei of A
$\frac{dN}{dt}=\alpha-\lambda N$
or $\frac{dN}{\alpha-\lambda N}=dt$
or $\int^N_{N_0}\frac{dN}{\alpha-\lambda N}=\int^1_0dt$
Solving this equation, we get
$N=\frac{1}{\lambda}[\alpha-(\alpha\lambda N_0)e^{-\lambda t}]$..(i)
(b) (i) Substituting $\alpha=2\lambda N_0$ and $t=t_{1/2}=\frac{In(2)}{\lambda}in$
Eq. (i) we get, $N=\frac{3}{2}N_0$
(ii) Substituting $\alpha=2\lambda N_0 and t\rightarrow \infty$ in Eq. (i), we get
$N=\frac{\alpha}{\lambda}=2N_0$
or $N=2N_0$