Question

Use de-Broglie's hypothesis to write the relation for the $n^{th}$ radius of Bohr orbit in terms of Bohr quantisation conditions of orbital angular momentum.

Answer 1

According to de-Broglie's hypothesis wavelength is $\lambda =\frac{h}{mv}$

Bohr proposed that the angular momentum of the electrons orbiting the atom is quantized. i.e.

$mvr=\frac{nh}{2\pi }$

$v=\frac{nh}{2\pi mr}..........\left(1\right)$

The circular motion is balanced by electrostatic forces.

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

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

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

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

$r=a_0{n}^2$

Where $a_o$ is bohr’s radius $a_0=\frac{h^2}{4{\pi }^{2}mkz{e}^2}$