A radioactive isotope having a half - life period of $3$ days was received after $12$ days. If $3…
A radioactive isotope having a half - life period of $3$ days was received after $12$ days. If $3 \mathrm{~g}$ of the isotope is left in the container, what would be the initial mass of the isotope ?
$12 \mathrm{~g}$
$36 \mathrm{~g}$
$48 \mathrm{~g}$
$24 \mathrm{~g}$
Solution
Given $t_{1 / 2}=3$
Total time $\mathrm{T}=12$
No. of half lives $(n)=\frac{12}{3}=4$
$\left(\frac{1}{2}\right)^n=\frac{\mathrm{N}}{\mathrm{N}_{\mathrm{o}}}$
$\therefore \left(\frac{1}{2}\right)^4=\frac{3}{\mathrm{~N}_{\mathrm{o}}}$
$\frac{3}{\mathrm{~N}_{\mathrm{o}}}=\frac{1}{16}$
$\mathrm{~N}_{\mathrm{o}}=48 \mathrm{~g}$