Question

Obtain an expression for the radius of Bohr orbit for $H$-atom.

Answer 1

From Bohr's second postulate,

$L_n=m{v}_n{r}_n=\frac{nh}{2\pi }$---1

where $n$ is an integer, $r_n$ is the radius of nth possible orbit and $v_n$ is the speed of moving electron in the $n^{th}$ orbit.

the relation between $v_n$ and $r_n$ is

$v_n=\frac{e}{4\pi {ϵ}_{0}m{r}_n}$

Substituting value of $v_n$ from eqn 1 in above eqn, and solving for $r$ we get ,

$r=\frac{n^2{h}^2}{4{\pi }^{2}m{e}^2}\times \frac{1}{z}$

or, $r=0.529\times \frac{n^2}{z}\mathring{A}$

Angstroms, where $n=$ principal quantum number of orbit and $Z=$ atomic number.