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 that of 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 Rydberg constant R for hydrogen atom) equal to

Step-by-step solution

In hydrogen atom, E n = R h c n 2 Also, E n ∝ 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 Here, the electron has been replaced by a particle, whose mass is double the mass of an electron.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, for this hypothetical atom, energy in n th orbit will be given by E n = - 2 R h c n 2 The longest wavelength (or minimum energy) photon will correspond to the transition of the particle from n = 3 to n = 2 ⇒ h c λ m a x = E 3 - E 2 = 2 R h c 1 2 2 - 1 3 2 = 2 R h c × 5 36 ∴ λ m a x = h c 5 18 R h c = 18 5 R