The ratio of the radii of hydrogen atom and its

Question:

The ratio of the radii of hydrogen atom and its nucleus is $10^{5}$. Assuming the atom and the nucleus to be spherical,

(i) what will be the ratio of their sizes?

(ii) If atom is represented by planet earth ' $\mathrm{Re}^{\prime}=6.4 \times 10^{6} \mathrm{~m}$, estimate the radius of the nucleus.

Solution:

(i) For a sphere, the volume $=\frac{4}{3} \pi r^{3}$

Let ' $R$ ' be the radius of the atom and ' $r$ ' be that of the nucleus. According to available information $\mathrm{R}=10^{5} r$

Volume of the atom $=\frac{4}{3} \pi \mathrm{R}^{3}=\frac{4}{3} \pi\left(10^{5} r\right)^{3}$

Volume of the nucleus $=\frac{4}{3} \pi r^{3}$

Ratio of the size of the atom to that of nucleus $=\frac{\frac{4}{3} \pi\left(10^{5} r\right)^{3}}{\frac{4}{3} \pi r^{3}}=10^{15}$

(ii) If the atom is represented by the planet earth, with radius (Re) as given :

Radius of nucleus $=\frac{\text { Radius of earth }}{10^{5}}=\frac{\left(6.4 \times 10^{6} \mathrm{~m}\right)}{10^{5}}=6.4 \times 10 \mathrm{~m}=64 \mathrm{~m} .$