A monochromatic beam of light is absorbed by a collection ofground-state hydrogen atoms in such a way that six differentwavelengths are observed when the hydrogen relaxes back to theground state.
(a) What is the wavelength of the incident beam? Explain the stepsin your solution.
(b) What is the longest wavelength in the emission spectrum ofthese atoms? To what portion of the electromagnetic spectrum and towhat series does it belong?
(c) What is the shortest wavelength? To what series does itbelong?

Answer Key:
(A) The atoms must be excited to energy level n = 4, to emit sixdifferent photon
energies in the downward transitions 4→3, 4→2,4→1, 3→2, 3→1, and 2→1. The
photon energy absorbed in the 1→4 transition is 12.8 eV,making the wavelength 97.4
nm.
(B) 1.88 μm, infrared, Paschen
(C) 97.4 nm, ultraviolet, Lyman

Question

09-12-2022
Physics

Answered

A monochromatic beam of light is absorbed by a collection ofground-state hydrogen atoms in such a way that six differentwavelengths are observed when the hydrogen relaxes back to theground state.
(a) What is the wavelength of the incident beam? Explain the stepsin your solution.
(b) What is the longest wavelength in the emission spectrum ofthese atoms? To what portion of the electromagnetic spectrum and towhat series does it belong?
(c) What is the shortest wavelength? To what series does itbelong?

Answer Key:
(A) The atoms must be excited to energy level n = 4, to emit sixdifferent photon
energies in the downward transitions 4→3, 4→2,4→1, 3→2, 3→1, and 2→1. The
photon energy absorbed in the 1→4 transition is 12.8 eV,making the wavelength 97.4
nm.
(B) 1.88 μm, infrared, Paschen
(C) 97.4 nm, ultraviolet, Lyman

Answer :

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The incident beam of light has a single, monochromatic wavelength. When this beam is absorbed by the hydrogen atoms, the atoms become excited and eventually relax back to the ground state, emitting light at several different wavelengths.

(a) It is not possible to determine the wavelength of the incident beam based on the information provided.

(b) The longest wavelength in the emission spectrum of the hydrogen atoms corresponds to the transition between the lowest energy levels in the atom. In the case of hydrogen, the ground state is the lowest energy level, so the longest wavelength would correspond to a transition from an excited state to the ground state.

(c) The portion of the electromagnetic spectrum to which the longest wavelength belongs depends on the specific value of the wavelength. The electromagnetic spectrum includes radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays.

(d) The series to which the longest wavelength belongs can be determined by the values of the energy levels involved in the transition. In the case of hydrogen, there are three series: the Lyman series, the Balmer series, and the Paschen series. The Lyman series corresponds to transitions from higher energy levels to the ground state, the Balmer series corresponds to transitions from intermediate energy levels to the ground state, and the Paschen series corresponds to transitions from lower energy levels to the ground state.

(e) The shortest wavelength in the emission spectrum corresponds to the transition between the highest energy levels in the atom. In the case of hydrogen, the shortest wavelength would correspond to a transition from an excited state to a higher energy level.

(f) The portion of the electromagnetic spectrum to which the shortest wavelength belongs depends on the specific value of the wavelength.

(g) The series to which the shortest wavelength belongs can be determined by the values of the energy levels involved in the transition. In the case of hydrogen, the shortest wavelength would belong to the Lyman series, since it corresponds to a transition from an excited state to a higher energy level

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