Atomic spectra and the photoelectric effect: test quantized energy

MCAT content library / Chemistry & physics / Atomic spectra and the photoelectric effect

Use line spectra and electron-emission thresholds to distinguish photon energy from photon number and to test quantized-energy models.

About 16 minutes, including a hydrogen transition and photoelectric graph example.

Spectral lines report energy differences

Atoms do not emit or absorb every photon energy continuously. Bound electrons occupy quantized energy states. When an atom changes from a higher-energy state to a lower-energy state, it can emit a photon whose energy equals the difference between those states. Absorption requires a photon that matches an allowed upward transition.

This is why atomic spectra contain discrete lines. Each line corresponds to a particular energy difference rather than to an arbitrary color chosen by the atom.

Concept: photon energy is set by frequency

Photon energy is

E = hf = hc/λ.

Higher frequency means higher photon energy. Longer wavelength means lower photon energy. Changing the number of photons without changing their frequency changes total delivered energy but not the energy carried by each photon.

Worked application: a hydrogen emission line

An original calculation uses hydrogen energy levels E₁ = −13.6 eV and E₂ = −3.4 eV. An electron drops from n = 2 to n = 1.

The emitted photon energy is:

ΔE = (−3.4 eV) − (−13.6 eV) = 10.2 eV.

Using the supplied approximation hc = 1240 eV·nm:

λ = 1240/10.2 ≈ 122 nm.

The transition therefore produces an ultraviolet photon. The direction matters: the downward transition emits the photon; an upward n = 1 to n = 2 transition would require absorption of a photon with the same energy under the idealized model.

Pattern: photoelectric thresholds separate energy from intensity

In the photoelectric effect, electrons are emitted from a material only when individual photons carry enough energy to overcome the material’s work function φ. For a simple model,

K_max = hf − φ.

The threshold frequency is therefore

f₀ = φ/h.

If f is below f₀, increasing light intensity does not make each photon energetic enough to eject electrons. If f is above threshold, increasing frequency raises the maximum electron kinetic energy.

For example, if the work function is 2.0 eV and incoming photons carry 3.0 eV each, the maximum emitted-electron kinetic energy is 1.0 eV. Doubling intensity at the same 3.0-eV photon energy does not double K_max; it increases the photon flux.

Rule: intensity changes photon number, not energy per photon

This distinction links the spectrum and photoelectric examples. Frequency sets photon energy. Intensity describes energy delivered per area per time and can be increased by sending more photons of the same energy.

A brighter beam of subthreshold photons remains subthreshold in the simple one-photon photoelectric model. A dimmer beam of sufficiently high-frequency photons can still eject electrons, though fewer photons per time can mean fewer emission events.

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Continue learning

• Electromagnetic spectrum and photons
• Electron configurations and magnetic behavior
• UV–visible absorption and color
• Browse Chemistry & Physics MCAT lessons

Sources

1. AAMC. Content Category 4E.
2. OpenStax Physics. Einstein and the Photoelectric Effect.