Atomic spectra

NUST NET Physics · Physics · Modern and nuclear physics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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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

Bohr's condition: angular momentum mvr = nh/(2π)
Hydrogen: radius r_n = n² × 0.053 nm; energy E_n = −13.6/n² eV
Photon energy = E_upper − E_lower = hf = hc/λ
Wavelength of a line: 1/λ = R_H × (1/p² − 1/n²) for a fall from level n to level p, R_H = 1.097 × 10⁷ m⁻¹
Series: Lyman (to n = 1) ultraviolet; Balmer (to n = 2) visible; Paschen (3), Brackett (4) and Pfund (5) infrared
X-ray tube at voltage V: shortest wavelength λ_min = hc/(eV), that is λ_min (nm) = 1240 ÷ V (volts)
Laser: population inversion in a metastable state, then stimulated emission gives photons of the same frequency, phase and direction

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? (E_n = −13.6/n² eV)

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

Answer: About 12.1 eV

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Photons And Photoelectric

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