Question

The ratio of radii of fourth to third orbits for an atom will be :

• A. $\frac{4}{3}$
• B. $\frac{16}{9}$
• C. $\frac{8}{3}$
• D. $\frac{16}{3}$

Answer 1

The correct option is B $\frac{16}{9}$

It is given that the quantisation rule followed in aliens' planet is as follows:
$mvr=\frac{nh}{4\pi }$
On rearranging the above expression, we get
$v=\frac{nh}{4\pi mr}$

Squaring both sides, we get
$v^2=\frac{n^2{h}^2}{16{\pi }^2{m}^2{r}^2}$ $.....\left(i\right)$
Also, we know $v^2=\frac{kZ{e}^2}{mr}$ $......\left(ii\right)$
Equating $\left(i\right)$ and $\left(ii\right)$, we get
$\frac{n^2{h}^2}{16{\pi }^2{m}^2{r}^2}=\frac{kZ{e}^2}{mr}$ $⟹r=\frac{n^2{h}^2}{16{\pi }^{2}mZ{e}^{2}k}$
Thus, $r\propto n^2$
The ratio of the radius of fourth to third orbit for an atom will be as follows:
$\frac{r_4}{r_3}=\frac{n_4^2}{n_3^2}=\frac{4^2}{3^2}=\frac{16}{9}$

Hence, the ratio is $\frac{16}{9}$.