The radius of a nucleus is of the order of \( 10^{-15}\,\mathrm{m} \) and that of an atom is of the order of \( 10^{-10}\,\mathrm{m} \). What is the correct relation between the volume of the nucleus and that of the atom?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → The volume of a nucleus is about \( 10^{-15} \) times the volume of the atom.
Both the nucleus and the atom are taken as spheres, with volume \( \frac{4}{3}\pi r^{3} \). The ratio of their volumes is therefore the cube of the ratio of their radii.
\( \begin{aligned} \frac{V_{\mathrm{nucleus}}}{V_{\mathrm{atom}}} &= \left(\frac{r_{\mathrm{nucleus}}}{r_{\mathrm{atom}}}\right)^{3} \\ &= \left(\frac{10^{-15}\,\mathrm{m}}{10^{-10}\,\mathrm{m}}\right)^{3} \\ &= (10^{-5})^{3} \\ &= 10^{-15} \end{aligned} \)
Therefore, the volume of a nucleus is about \( 10^{-15} \) times the volume of the atom. So an atom has about \( 10^{15} \) times the volume of its nucleus. The factor \( 10^{-5} \) is only the ratio of the radii. The ratio of the volumes is its cube.