A hydrogen atom (Z = 1) is in the third excited state, and a photon is either em

Question

A hydrogen atom (Z = 1) is in the third excited state, and a photon is either emitted or absorbed.

(a) Determine the quantum number nf of the final state and the energy of the photon (eV) when the photon is emitted with the shortest possible wavelength.

(b) Determine the quantum number nf of the final state and the energy of the photon (eV) when the photon is emitted with the longest possible wavelength.

(c) Determine the quantum number nf of the final state and the energy of the photon (eV) when the photon is absorbed with the longest possible wavelength.

Explanation / Answer

Given
Hydrogen Atom, Z = 1, in third excited state, n = 4
a) For emitted photon with shortest wavelength, the frequency has to be the highest ( highest energy)
   so, the transition is from n = 4 to n = 1
   final excited state = ground state
   Energy of the emitted photon = -13.6[1/n1^2 - 1/n2^2] = -13.6[1/16 - 1] = 12.75 eV
b) For emitted photon with longest wavelength, the frequency has to be the smallest ( lowest energy)
   so, the transition is from n = 4 to n = 3
   final excited state = 2nd excited state
   Energy of the emitted photon = -13.6[1/n1^2 - 1/n2^2] = -13.6[1/16 - 1/9] = 0.6611 eV
c) For absorbed photon with longest wavelength, the frequency has to be the smallest ( lowest energy)
   so, the transition is from n = 4 to n = 5
   final excited state = 4rth excited state
   Energy of the absorbed photon = 13.6[1/n1^2 - 1/n2^2] = -13.6[1/16 - 1/25] = 0.0.306 eV

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