Question

The radius of the orbit of an electron in a Hydrogen-like atom is $4.5a_0$ where $a_0$ is the Bohr radius. Its orbital angular momentum is $\frac{3h}{2\pi }$ It is given that h is Planck 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)

$L=3\left(\frac{h}{2\pi }\right)$
$\therefore n=3,asL=n\left(\frac{h}{2\pi }\right)$
$r_n\propto \frac{n^2}{Z}$
$r_3=4.5a_0$
$\therefore$ Z=2
$\frac{1}{{\lambda }_1}=R{z}^2(\frac{1}{2^2}-\frac{1}{3^2})=4R(\frac{1}{4}-\frac{1}{9})$
$\therefore {\lambda }_1=\frac{9}{5R}$
$\frac{1}{{\lambda }_1}=R{z}^2(\frac{1}{1^2}-\frac{1}{3^2})=4R(1-\frac{1}{9})$
$\Rightarrow {\lambda }_2=\frac{9}{32R}$
$\frac{1}{{\lambda }_3}=(\frac{1}{1^2}-\frac{1}{2^2})=4R(1-\frac{1}{4})$
$\Rightarrow =\frac{1}{3R}$