Imagine an atom made up of a proton and a hypothetical particle of double the mass of the electron but having the same charge as the 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 λ (given in terms of the Rydberg constant R for the hydrogen atom) equal to

In hydrogen atom En=−Rhcn2Also En∝m;where m is the mass of the electron. Here the electron has been replaced by a particle whose mass is double of an electron. Therefore, for this hypothetical atom energy in nth orbit will begiven by En=−2Rhcn2The longest wavelength λmax(or minimum energy) photon will correspond to the transition of particle fromn=3  to  n=2⇒hcλmax=E3−E2=2Rhc122−132This  gives  λmax=185R