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

Answer 1

Answer: (B), (D)

$\frac{9}{32R}$

$n\times \frac{h}{2\pi }=\frac{3h}{2\pi }$
$\Rightarrow n=3$
$4.5a_0=a_0\frac{n^2}{Z}$
$\Rightarrow Z=2$
Possible transitions are $3\to 2,3\to 1$and $2\to 1$
For $3\to 2:\frac{1}{\lambda }=R(2{)}^2[\frac{1}{4}-\frac{1}{9}]=4R\left[\frac{9-4}{36}\right]=\frac{5R}{9}$
$\Rightarrow \lambda =\frac{9}{5R}$
$3\to 1:\frac{1}{\lambda }=R(2{)}^2[1-\frac{1}{9}]=4R\left(\frac{8}{9}\right)=\frac{32R}{9}$
$\Rightarrow \lambda =\frac{9}{32R}$
$2\to 1:\frac{1}{\lambda }=R(2{)}^2[1-\frac{1}{4}]=4R\left(\frac{3}{4}\right)=3R$
$\lambda =\frac{1}{3R}$