Question

The radius of the orbit of an electron in a Hydrogen-like atom is $4.5$$a_0$ where $a_0$ is the Bohr radius. Its orbital angular momentum is $\frac{3h}{2\pi }$ It is given that $h$ is Planck's constant and $R$ is Rydberg constant. The possible wavelength(s), when the atom de-excites, is (are)

• A. $\frac{9}{32R}$
• B. $\frac{9}{16R}$
• C. $\frac{9}{5R}$
• D. $\frac{4}{3R}$

Answer 1

Answer: (A), (C)

$\frac{9}{5R}$
$\frac{9}{32R}$

Given data:
$4.5a_0=a_0\frac{n^2}{Z}$........... ($i$)

$\frac{nh}{2\pi }=\frac{3h}{2\pi }$ ..........(ii)

So, $n=3$ and $z=2$
So, possible wavelength are:
$\frac{1}{{\lambda }_1}=R{Z}^2[\frac{1}{1^2}-\frac{1}{3^2}]\Rightarrow {\lambda }_1=\frac{9}{32R}$

$\frac{1}{{\lambda }_2}=R{Z}^2[\frac{1}{1^2}-\frac{1}{2^2}]\Rightarrow {\lambda }_2=\frac{1}{3R}$

$\frac{1}{{\lambda }_3}=R{Z}^2[\frac{1}{2^2}-\frac{1}{3^2}]\Rightarrow {\lambda }_3=\frac{9}{5R}$