Chemistry - Radioactivity Question with Solution | TestHub

Solution:

The decay of ${}_{90}{\mathrm{Th}}^{232}$ to ${}_{82}{\mathrm{Pb}}^{208}$ involves the emission of alpha particles (He nuclei).

The balanced nuclear reaction is:

$${}_{90}{\mathrm{Th}}^{232}\to {}_{82}{\mathrm{Pb}}^{208}+6{}_2{\mathrm{He}}^4+4{}_{-1}{\mathrm{e}}^0$$

This means 1 mole of ${\mathrm{Th}}^{232}$ produces 6 moles of He.

Given:

Volume of He = $8.4\times 10^{-5}\mathrm{mL}=8.4\times 10^{-8}\mathrm{L}$ at STP ($0^{\circ }\mathrm{C}$ and 1 atm).

Mass of ${\mathrm{Th}}^{232}$ remaining (N) = $5\times 10^{-7}\mathrm{g}$.

Half-life of ${\mathrm{Th}}^{232}$ ($t_{1/2}$) = $1.38\times 10^{10}$ years.

1. Calculate moles of He produced:

At STP, 1 mole of any gas occupies 22.4 L.

Moles of He = $\frac{8.4\times 10^{-8}\mathrm{L}}{22.4L/mol}=3.75\times 10^{-9}\mathrm{mol}$

2. Calculate moles of ${\mathrm{Th}}^{232}$ decayed:

Since 1 mole of ${\mathrm{Th}}^{232}$ produces 6 moles of He,

Moles of ${\mathrm{Th}}^{232}$ decayed = $\frac{MolesofHe}{6}=\frac{3.75\times 10^{-9}\mathrm{mol}}{6}=6.25\times 10^{-10}\mathrm{mol}$

3. Calculate moles of ${\mathrm{Th}}^{232}$ remaining:

Molar mass of ${\mathrm{Th}}^{232}$ = 232 g/mol.

Moles of ${\mathrm{Th}}^{232}$ remaining (N) = $\frac{5\times 10^{-7}\mathrm{g}}{232g/mol}=2.155\times 10^{-9}\mathrm{mol}$

4. Calculate initial moles of ${\mathrm{Th}}^{232}$ ($N_0$):

$N_0$ = Moles of ${\mathrm{Th}}^{232}$ remaining + Moles of ${\mathrm{Th}}^{232}$ decayed

$N_0=2.155\times 10^{-9}\mathrm{mol}+0.625\times 10^{-9}\mathrm{mol}=2.78\times 10^{-9}\mathrm{mol}$

5. Use the radioactive decay formula:

$$t=\frac{t_{1/2}}{0.693}\ln \left(\frac{N_0}{N}\right)$$

$$t=\frac{1.38\times 10^{10}\mathrm{yrs}}{0.693}\ln \left(\frac{2.78\times 10^{-9}\mathrm{mol}}{2.155\times 10^{-9}\mathrm{mol}}\right)$$

$$t=\frac{1.38\times 10^{10}}{0.693}\ln (1.29)=\frac{1.38\times 10^{10}}{0.693}\times 0.2547$$

$$t=5.06\times 10^9\mathrm{yrs}$$

The age of the ore sample is $5.06\times 10^9$ years.

Here, $x=5.06$ and $y=9$.

The required answer is $(x+y)=5.06+9=14.06$.