Question

The radius of the first Bohr orbit is $a_o$. The $n^{th}$ orbit has a radius:

• A. $n{a}_o$
• B. $a_o/n$
• C. $n^2{a}_o$
• D. $a_o/n^2$

Answer 1

Answer: (C)

$n^2{a}_o$

Given :- The radius of the first bohr orbit $=a_0$

according to $3rd$ Postulate of bohr atomic model

$L=mvr=\frac{rh}{2\pi }$; where $n=1,2,3$

$\Rightarrow$ For $n^{th}$ orbit

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

and $n$ corresponds to a permitted value of orbit radius.

According to newtons second Law :-

a radially inward centripetal force,

$F=\frac{m{v}^2}{r_n}$ and needed

Centripetal force is balanced by electrical attraction by $e^{-}$ and nucleus

$\therefore \frac{m{v}_r^2}{r_n}=\frac{1}{4\pi {\epsilon }_0}\frac{e^2}{r_n}$--------(2)

From (1) and (2)

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

$\Rightarrow r_n=a_0{n}^2$

where $a_0=$ radius of first orbit

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

$\therefore$ $\left(C\right)$ is the solution