Question

Show that the radius of the orbit in hydrogen atom varies as $n^2$. Where $n$ is the principal quantum number of the atom

Answer 1

According to Bohr's second postulate, $mvr=\frac{nh}{2\pi }$

where m be the mass of an electron, v be the velocity and r be the radius of orbit in which the electron revolves around the nucleus.

So, $v=\frac{nh}{2\pi mr}$

Here, the centripetal force is provided by the electrostatic force of attraction between electron and nucleus.

Thus, $\frac{m{v}^2}{r}=\frac{k\left(Ze\right)\left(e\right)}{r^2}$

$\Rightarrow r=\frac{kZ{e}^2}{m{v}^2}$

Substituting the value of $v$:

$r=\frac{n^2{h}^2}{4{\pi }^{2}mkZ{e}^2}$

Thus, $r\propto n^2$