Question

Imagine an atom made of a nucleus of charge $Ze$ and a hypothetical particle of the same mass but double the charge of the electron. Apply the Bohr atom model and consider all possible transitions of this hypothetical particle to the ground state. The longest wavelength of a photon that will be emitted has wavelength $\lambda $ (Given in terms of Rydberg constant $R_{H}$ of the hydrogen atom) equal to

• A. $\frac{1}{4 Z^{2} R_{H}}$
• B. $\frac{1}{3 Z^{2} R_{H}}$
• C. $\frac{4}{3 Z^{2} R_{H}}$
• D. $\frac{4}{Z^{2} R_{H}}$

Answer 1

Answer: (B)

Longest wavelength $n=2$ to $n=1$
$\frac{m v^{2}}{r}=\frac{k . Z e \left(\right. 2 e \left.\right)}{r^{2}}$
$mvr=\frac{n h}{2 \pi }$
Kinetic energy, potential energy and total energy becomes four times
$\therefore \, \Delta E$ becomes four times
$\therefore \, \Delta E=4Z^{2}R_{H}ch\left(\frac{1}{1^{2}} - \frac{1}{2^{2}}\right)$ ( $c$ is the speed of light)
$\Delta E=3R_{H}chZ^{2}$
$\frac{h c}{\lambda }=3R_{H}chZ^{2}$
$\lambda =\frac{1}{3 Z^{2} R_{H}}$