(a) Statement of Bohr's postulate
Explanation in terms of de Broglie hypothesis
(b) Finding the energy in the $\mathit{n}=4$ level
Estimating the frequency of the photon
(a) Bohr's postulate, for stable orbits, states "The electron, in an atom, revolves around the nucleus only in those orbits for which its angular momentum is an integral multiple
of $\frac{\mathit{h}}{2\ne }$ ( $\mathit{h}=$ Planck's constant)."
[Also accept mvr $=\mathit{n}\cdot \frac{\mathit{h}}{2\ne }\left(\mathit{n}=1,2,3,......\right)$]
As per de Broglie's hypothesis $\mathit{\lambda }=\frac{\mathit{h}}{\mathit{p}}=\frac{\mathit{h}}{\mathit{m}\mathit{v}}$
For a stable orbit, we must have circumference of the orbit $=\mathit{n}\mathit{\lambda }\left(\mathit{n}=1,2,3,......\right)$
$\therefore 2\mathit{\pi }\mathit{r}=\mathit{n}.\mathit{m}\mathit{v}$
or $\mathit{m}\mathit{v}\mathit{r}=\frac{\mathit{n}\mathit{h}}{2\ne }$
Thus de -Broglie showed that formation of stationary pattern for integral ' $\mathit{n}$ ' gives rise to stability of the atom.
This is nothing but the Bohr's postulate.
(b) Energy in the $\mathit{n}=4$ level $=\frac{-{\mathit{E}}_0}{4^2}=-\frac{{\mathit{E}}_0}{16}$
$\therefore$ Energy required to take the electron from the ground state, to the $\mathit{n}=4$ level
$=\left(-\frac{{\mathit{E}}_0}{16}\right)-\left(-{\mathit{E}}_0\right)$
$=\frac{-1+16}{16}{\mathit{E}}_0=\frac{15}{16}{\mathit{E}}_0$
$=\frac{15}{16}\times 13.6\times 1.6\times {10}^{-19}\mathrm{J}$
Let the frequency of the photon be $\mathit{v}$, we have
$\mathit{h}\mathit{v}=\frac{15}{16}\times 13.6\times 1.6\times {10}^{-19}$
$\therefore \mathit{v}=\frac{15\times 13.6\times 1.6\times {10}^{-19}}{16\times 6.63\times {10}^{-34}}\mathrm{Hz}$
$\simeq 3.1\times {10}^{15}\mathrm{Hz}$
(Also accept $3\times {10}^{15}\mathrm{Hz}$ )