Atomic spectra

NUST NET (Engineering) · Physics · Modern and nuclear physics. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

In the Bohr model the electron of a hydrogen atom can move only in certain allowed orbits, each with a fixed energy. The energies are negative because the electron is bound. The ground state (n = 1) is the lowest at −13.6 eV, and the levels crowd together towards zero as n rises.

When the electron drops to a lower level it emits one photon whose energy equals the gap between the levels. Each gap gives one spectral line, and the lines form series, one for each level the electron falls to.

X-rays come from fast electrons striking a heavy metal target. A laser gives intense coherent light by stimulated emission, which needs a population inversion: more atoms in an excited state than in the lower state.

Rules to remember

Common mistake

Working out the photon energy from the difference of the n values. Find the energy of each level first with −13.6/n2, then subtract the two energies.

Worked example

How much energy must a hydrogen atom in its ground state absorb to raise its electron to the level n = 3? (En = −13.6/n2 eV)

  1. E1 = −13.6/12 = −13.6 eV.
  2. E3 = −13.6/32 = −13.6/9 = −1.51 eV.
  3. Energy absorbed = E3 − E1 = −1.51 − (−13.6) = 12.09 eV ≈ 12.1 eV.

Answer: About 12.1 eV

Practice questions

Try each one, then open the answer.

1. An X-ray tube works at 25 kV. Taking hc = 1240 eV nm, the shortest wavelength of the X-rays it produces is about

  1. A
    0.50 nm
  2. B
    50 nm
  3. C
    0.050 nm
  4. D
    0.0050 nm
Show answer

Answer: C. The shortest wavelength arises when an electron gives all its energy, 25 keV, to one photon: λmin = hc/eV = 1240/25 000 ≈ 0.050 nm. Forgetting the kilo in kV gives 50 nm.

2. The Balmer series of hydrogen, produced by transitions to the n = 2 level, lies in the

  1. A
    ultraviolet region
  2. B
    visible region
  3. C
    infrared region
  4. D
    X-ray region
Show answer

Answer: B. Balmer lines, such as the red 656 nm line, are in the visible region. Transitions to n = 1 (Lyman series) give ultraviolet, and to n = 3 (Paschen series) infrared.

3. The energy of the electron in the n-th orbit of hydrogen is En = −13.6/n2 eV. Its energy in the orbit n = 2 is

  1. A
    −6.8 eV
  2. B
    +3.4 eV
  3. C
    −3.4 eV
  4. D
    −1.51 eV
Show answer

Answer: C. E2 = −13.6/22 = −3.4 eV. −6.8 eV divides by n instead of n2, and −1.51 eV is the n = 3 level. Bound-state energies are negative.

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