Question

A radioactive isotope is being produced at a constant rate $\frac{dN}{dt}$ = R in an experiment. The isotope has a half-life $t_{\frac{1}{2}}$. Show that after a times t>> $t_{\frac{1}{2}}$, the number of active nuclei will become constant. Find the value of this constant.

Answer 1

Given : Half life period = $t_{\frac{1}{2}}$,

Rate of Radio active decay

= $\frac{dN}{dt}$ = R

$\Rightarrow R=\frac{dN}{dt}$

After time t>>$t_{\frac{1}{2}}$ the number of active nuclei will become constant

i.e. ${\left(\frac{dN}{dt}\right)}_{\text{present}}=R={\left(\frac{DN}{Dt}\right)}_{\text{decay}}$

$\therefore R={\left(\frac{DN}{Dt}\right)}_{\text{decay}}$

$\Rightarrow R=\lambda N$

[Where, $\lambda$ = Radioactive decay constant, N= constant number]

$\Rightarrow R=\frac{0.693}{t_{\frac{1}{2}}}$(N)

$\Rightarrow R{t}_{\frac{1}{2}}$ = 0.693 N

$\Rightarrow N=\frac{R{t}_{\frac{1}{2}}}{0.693}$