State Bohr's second postulate for his theory of hydrogen atom. Prove that the radius of Bohr's orbit of a hydrogen atom is directly proportional to $\displaystyle \mathrm{n}^{2}$, where n is the principal quantum number.

Statement of Bohr's second postulates \(\displaystyle \frac{1}{2}\) Derivation of \(\displaystyle \mathrm{r}_{\mathrm{n}} \alpha \mathrm{n}^{2}\) Bohr's second postulate Electron revolves around the nucleus only in those orbits for which the angular momentum in some integral multiple of \(\displaystyle \mathrm{h} / 2 \pi\). Electrostatic force between revolving electron & nucleus provide requisite centripetal force \[\frac{m v_{\mathrm{n}}^{2}}{\mathrm{r}_{\mathrm{n}}}=\frac{1}{4 \pi \varepsilon_{0}} \frac{e^{2}}{\mathrm{r}_{\mathrm{n}}^{2}} \] \(\displaystyle v_{\mathrm{n}}=\frac{e}{\sqrt{4 \pi \varepsilon_{0} m \mathrm{r}_{\mathrm{n}}}}\) \[m v_{\mathrm{n}} \mathrm{r}_{\mathrm{n}}=\frac{\mathrm{n} \mathrm{h}}{2 \pi} \] From eqn. (i) and (ii) \[\mathrm{r}_{\mathrm{n}}=\left(\frac{\mathrm{n}^{2}}{m}\right)\left(\frac{\mathrm{h}}{2 \pi}\right)^{2} \frac{4 \pi \varepsilon_{0}}{e^{2}} \] \(\displaystyle \mathrm{r}_{\mathrm{n}} \propto \mathrm{n}^{2}\)