Question

According to Bohr's model of hydrogen atom, relation between principal quantum number n and radius of stable orbit is

• A. $r\propto \frac{1}{n}$
• B. $r\propto n$
• C. $r\propto \frac{1}{n^2}$
• D. $r\propto n^2$

Answer 1

The correct option is D $r\propto n^2$

Let an electron (mass m), is revolving around nucleus of a hydrogen atom in an orbit of radius $r$ with velocity $v$. Electron gets centripetal force to revolve, from the electrostatic attraction force between electron and nucleus,

$\frac{m{v}^2}{r}=\frac{e\times e}{4\pi {\epsilon }_0{r}^2}$ (for H- atom, $Z=1$ hence charge on nucleus is $=e$)

$m{v}^2=\frac{e^2}{4\pi {\epsilon }_{0}r}$ ...................eq1

According to Bohr's model, angular momentum of electron :

$mvr=\frac{nh}{2\pi }$ ..................eq2

Now, squaring eq2 and dividing by eq1, we get

$\frac{m{v}^2{r}^2}{m{v}^2}=\frac{n^2{h}^{2}4\pi {\epsilon }_{0}r}{4{\pi }^2{e}^2}$

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

which gives $r\propto n^2$