Question

Using Bohr's postulates of the atomic model, derive the expression for radius of nth electron orbit. Hence obtain the expression for Bohr's radius.

Answer 1

Let's assume $m$ to be the mass of an electron and $v_n$ be its speed in nth orbit of radius $r_n$. From Rutherford model, the centripetal force for revolution is produced by electrostatic attraction between the electron and the nucleus.

$\frac{m{v}_n^2}{r_n}=\frac{1}{4\pi {\epsilon }_0}\frac{Z{e}^2}{r_n^2}...\left(i\right)$

From Bohr's postulate for quantization of angular momentum of nth orbit.

$m{v}_n{r}_n=\frac{nh}{2\pi }\Rightarrow v_n=\frac{nh}{2\pi m{r}_n}$

Substituting this value in equation (i), we get

$\frac{m}{r_n}{\left[\frac{nh}{2\pi m{r}_n}\right]}^2=\frac{1}{4\pi {\epsilon }_0}\frac{Z{e}^2}{r_n^2}$

or $r_n=\frac{{\epsilon }_0{n}^2{h}^2}{\pi mZ{e}^2}$

For Bohr's radius $n=1$,

$r_1=\frac{{\epsilon }_0{h}^2}{\pi mZ{e}^2}$

This is the expression for Bohr's radius.