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Question:

Out of the following which one is not a possible energy for a photon to be emitted by a hydrogen atom according to Bohr's atomic model?

11.1eV

1.9eV

13.6eV

0.65eV

Solution:

From Bohr's model, the energy of a photon emitted by a hydrogen atom is given by the difference in energy levels:

ΔE = E_i - E_f = -13.6eV * (1/n_f² - 1/n_i²)

where n_i and n_f are the initial and final principal quantum numbers (n_i > n_f). The possible energy differences are:

  • n_i = 2, n_f = 1: ΔE = -13.6eV * (1/1 - 1/4) = 10.2eV
  • n_i = 3, n_f = 1: ΔE = -13.6eV * (1/1 - 1/9) = 12.1eV
  • n_i = 3, n_f = 2: ΔE = -13.6eV * (1/4 - 1/9) = 1.89eV ≈ 1.9eV
  • n_i = 4, n_f = 1: ΔE = -13.6eV * (1/1 - 1/16) = 12.75eV
  • n_i = 4, n_f = 2: ΔE = -13.6eV * (1/4 - 1/16) = 2.55eV
  • n_i = 4, n_f = 3: ΔE = -13.6eV * (1/9 - 1/16) = 0.66eV ≈ 0.65eV
  • n_i = ∞, n_f = 1: ΔE = -13.6eV * (1/1 - 0) = 13.6eV

And so on. We see that 1.9eV, 13.6eV, and 0.65eV are possible energy differences. However, 11.1eV is not among the possible energy differences calculated from Bohr's model for hydrogen atom transitions.