Question

Nuclei of a radioactive element X are being produced at a constant rate $\alpha$ and this element dacays to a stable nucleus Y with a decay constant $\lambda$ and half-life $T_{1/2}$. At time t=0, there are $N_0$ nuclei of the element X.
The number $N_Y$ of nuclei of Y at t=$T_{1/2}$ is

Answer 1

Let at time 't' number of Radioactive are $a$
$\therefore$ Net rate of formation of Nuclei of A

$\therefore$ $\frac{dN}{dT}=\alpha -\lambda NOR$

$\frac{dN}{\alpha -\lambda N}=dt$

$Q\underset{N_0}{\overset{N}{\int }}\frac{dN}{\alpha -\lambda N}=\underset{0}{\overset{t}{\int }}dt$

$\Rightarrow N=\frac{1}{\lambda }[\alpha -[\alpha -\lambda N_0)e^{\lambda }$

$Puttingt=t_{1/2}$

$andweget$

$N=\frac{1}{\lambda }[\alpha -[\alpha -\lambda N_0]e^{{\lambda }^{t_{1/2}}}$