A doubly ionized lithium atom is in the ground state. It absorbs energy and make

Question

A doubly ionized lithium atom is in the ground state. It absorbs energy and makes a transition to the n = 3 excited state. The ion returns to the ground state by emitting two photons. What are their wavelengths?
_____________ m (lower energy photon)
______________m (higher energy photon)

Explanation / Answer

the other solution was too small i think : Lithium has an atomic number of 3, so the neutral atom has three protons (Z=3) and 3 electrions. The Li2+ ion, therefore, has only one electron, and would be called a "hydrogen-like" ion. If we ignore effects such as spin-orbit coupling, the Lamb shift, and hyperfine splitting, the energy of an electronic orbital in a hydrogen-like ion depends only on the value of the principal quantum number, n. The excited electron in this case has n = 3, and the only way it can emit two photons as it decays to the ground state (n=1) is if it does this in two steps: n=3 -> n=2 then n=2 -> n=1 The energy (or equivalently, the wavelength) of the photon emitted by the transition between two electronic energy levels in a hydrgen-like ion or atom is given by the Rydberg formula: 1/wavelength = R*(Z^2)*(1/(n_1)^2 - 1/(n_2)^2) where Z is the atomic number of the ion, R is the Rydberg constant for that element, and the wavelength is of the photon emitted by the transition from the state n_2 -> n_1 (n_2 > n_1). The Rydberg constant for a given element is equal to: R = R_i/(1 + (me)/M) where R_i is the "infinite" Rydberg constant = 1.0974*10^7 meters^-1, (me) is the mass of an electron = 9.10938*10^-31 kg M is the mass of all the protons in the nucleus of the ion. Here, Z = 3, and the mass of a proton = 1.67262*10^-27 kg, so M = 5.01786*10^-27 kg (neglecting the mass defect due to the binding energy of the nucleons in the Li nucleus) Putting these together gives R_Li = (1.0974*10^7 meters^-1)/(1 + (9.10938*10^-31)/(5.019*10^-27)) = 1.0972 * 10^7 meters^-1 We then have for Li2+ 1/wavelength = (3^2)* (1.0972 * 10^7 meters^-1)*(1/(n_1)^2 - 1/(n_2)^2) 1/wavelength = (9.87481*10^7 meters^-1)*)*(1/(n_1)^2 - 1/(n_2)^2) Plug in n_1 = 2 and n_2 = 3, for the first transition to get: wavelength = 72.91 nm Plug in n_1 = 1 and n_2=2 for the second transition to get: wavelength = 13.50 nm

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