Question

Derive an expression for the velocity of an electron revolving round the nucleus in the orbit of hydrogen atom.

Answer 1

From Bohr postulate, the centripetal force is provided by the electrostatic force between the nucleus and the electron, i.e.;

$\frac{m{v}^2}{r}=\frac{z{e}^2}{4\pi {ϵ}_0{r}^2}=\to$ (1)

z$\to$ atomic number

m$\to$ mass of electron

r$\to$ radius of orbit

and $mvr=\frac{nh}{2\pi }\to$ (2)

Dividing (1) by (2) :

$\Rightarrow v=\left(\frac{z{e}^2}{4\pi {ϵ}_0}\right)\frac{2\pi }{nh}=\frac{z{e}^2}{2{ϵ}_{0}nh}$

Thus, velocity of electron in the n^th permitted orbit is $v_n=\frac{z{e}^2}{2{ϵ}_{0}nh}$