Q. Imagine an atom made up of proton and a hypothetical particle of double the mass of electron, but having the same charge as that of electron. Apply the Bohr atom model and consider all possible transitions of this hypothetical particle to the first excited level. The longest wavelength photon that will be emitted has wavelength $\lambda,$(given in terms of Rydberg constant $R$ for hydrogen atom) equal to
Explanation
(c) In hydrogen atom, $E_{n} = \frac{Rhc}{n^{2}}$
Also, $E_{n} \propto m$, where $m$ is the mass of the electron. Here, the electron has been replaced by a particle, whose mass is double the mass of an electron. Therefore, this hypothetical atom, energy is $n$th orbit will be given by
$E_{n} = - \frac{2Rhc}{n^{2}}$
The longest wavelength (or minimum energy) photon will correspond to the transition of particle from $n = 3$to $n = 2$
$\Rightarrow \ \frac{hc}{\lambda_{\max}} = E_{3} - E_{2} = 2Rhc\left\lbrack \frac{1}{2^{2}} - \frac{1}{3^{2}} \right\rbrack = 2Rhc \times \frac{5}{36}$
$\therefore\ \ \lambda_{\max} = \frac{hc}{\frac{5}{18}Rhc} = \frac{18}{5R}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.