Question

The ratio of the radii of hydrogen atom and its nucleus is $\sim {10}^5$. Assuming the atom and the nucleus to be spherical:
(i) What will be the ratio of their sizes?
(ii) If the atom is represented by planet earth ${}^{\prime }{R}_e^{\prime }=6.4\times {10}^{6}m,$ estimate the size of the nucleus.

Answer 1

(i) Volume of a sphere is given by $V=\frac{4}{3}\pi r^3$
Let "R" be the radius of the atom and "r" be that of the nucleus.
The ratio of the radii of hydrogen atom and its nucleus is $\sim {10}^5$
$\Rightarrow R={10}^{5}r$
Volume of the atom
$\begin{array}{c}=\frac{4}{3}\pi R^3=\frac{4}{3}\pi {\left({10}^{5}r\right)}^3(\because R={10}^{5}r)\\ =\frac{4}{3}\pi r^3\times {10}^{15}\end{array}$
Volume of the nucleus $=\frac{4}{3}\pi r^3$
Ratio of the size of the atom to that of the nucleus
$=\frac{\frac{4}{3}\times {10}^{15}\times \pi r^3}{\frac{4}{3}\pi r^3}={10}^{15}$
(ii) If the atom is represented by the planet Earth, then the radius of the nucleus would be given by :-
$\begin{array}{c}{r}_n=\frac{R_e}{{10}^5}\\ r_n=\frac{6.4\times {10}^{6}m}{{10}^5}(R_e=6.4\times {10}^{6}m)\\ =6.4\times 10m\\ =64m\end{array}$