Question

Using the expression for the radius of orbit for the Hydrogen atom, show that the linear speed varies inversely to the principal quantum number n the angular speed varies inversely to the cube of principal quantum number n. 

Answer 1

According to Bohr’s second postulate,

mr_nv_n = ${\displaystyle \frac{\text{nh}}{2\pi }}$

∴ ${\displaystyle {\text{m}}^2{\text{v}}_{\text{n}}^2{\text{r}}_{\text{n}}^2=\frac{{\text{n}}^2{\text{h}}^2}{4{\pi }^2}}$ 

∴ ${\displaystyle {\text{v}}_{\text{n}}^2=\frac{{\text{n}}^2{\text{h}}^2}{4{\pi }^2{\text{m}}^2{\text{r}}_{\text{n}}^2}}$

Substituting, r_n = ${\displaystyle \frac{{\epsilon }_0{\text{h}}^2{\text{n}}^2}{\pi \text{m}{\text{Ze}}^2}}$ in above relation,

${\displaystyle {\text{v}}_{\text{n}}^2=\frac{{\text{n}}^2{\text{h}}^2}{4{\pi }^2{\text{m}}^2}\times {\left(\frac{\pi \text{m}{\text{Ze}}^2}{{\epsilon }_0{\text{h}}^2{\text{n}}^2}\right)}^2}$

= ${\displaystyle \frac{{\text{n}}^2{\text{h}}^2}{4{\pi }^2{\text{m}}^2}\times \frac{{\pi }^2{\text{m}}^2{\text{Z}}^2{\text{e}}^4}{{\epsilon }_0^2{\text{h}}^4{\text{n}}^4}}$

= ${\displaystyle \frac{{\text{Z}}^2{\text{e}}^4}{4{\epsilon }_0^2{\text{h}}^2{\text{n}}^2}}$

∴ ${\displaystyle {\text{v}}_{\text{n}}^2\propto \frac{1}{{\text{n}}^2}}$

⇒ ${\displaystyle {\text{v}}_{\text{n}}\propto \frac{1}{\text{n}}}$

Expression for angular speed:

Since, v_n = r_n ω and r_n = ${\displaystyle \frac{{\epsilon }_0{\text{h}}^2{\text{n}}^2}{\pi {\text{m}}_{\text{e}}{\text{e}}^2}}$

∴ ${\displaystyle \omega =\frac{{\text{v}}_{\text{n}}}{{\text{r}}_{\text{n}}}=\left(\frac{{\text{e}}^2}{2{\epsilon }_0\text{h}}\right)\frac{1}{\text{n}}/\frac{{\epsilon }_0{\text{h}}^2{\text{n}}^2}{\pi {\text{m}}_{\text{e}}{\text{e}}^2}}$

∴ ${\displaystyle \omega =\frac{{\text{e}}^2}{2\epsilon 0\text{hn}}\times \frac{\pi {\text{m}}_{\text{e}}{\text{e}}^2}{{\epsilon }_0{\text{h}}^2{\text{n}}^2}=\left(\frac{\pi {\text{m}}_{\text{e}}{\text{e}}^4}{2{\epsilon }_0^2{\text{h}}^3}\right)\frac{1}{{\text{n}}^3}}$

⇒ ${\displaystyle \omega \propto \frac{1}{{\text{n}}^3}}$