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Modern Physics

Atomic Spectra & Energy Levels

Why a hot gas gives a handful of razor-thin lines instead of a rainbow, and why the ladder is drawn wrong in almost every textbook.

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Atomic Spectra & Energy Levels

Why a hot gas gives a handful of razor-thin lines instead of a rainbow, and why the ladder is drawn wrong in almost every textbook.

26 min4 checkpoints5 simulation tasks

Introduction

Look at a street lamp through a cheap diffraction grating and you see a few thin coloured lines with darkness between them. A hot solid glows with every colour. Why does a hot gas refuse to?

Predict

You look at a low-pressure hydrogen discharge tube (the pink glow of a school lab) through a diffraction grating. What do you see?

One atom, one line

Try it

Each run, one electron drops to n = 2 and one photon lands on the strip. Set the jump that makes hydrogen's red line, at 656 nm.

GoalHydrogen's red line, at 656 nm.

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"Hot things glow, and glowing things make rainbows." Both halves hold for a solid. A thin gas's atoms have a fixed set of allowed energies, and a photon can only carry away a difference between two of them. No transition produces 1.95 eV, so there is no line at 636 nm: darkness, not dim light.

The real spacing of the ladder

Almost every energy-level diagram you have seen draws the levels evenly spaced, because that is easy to draw.

Predict

Draw a true energy axis for hydrogen: the ground state n = 1 at the bottom, E = 0 at the top. Whereabouts on that axis does n = 2 sit?

The real spacing is

En=−13.6 eVn2E_n = -\frac{13.6\ \text{eV}}{n^2}

The minus sign says bound: zero is what the electron has once it is out. A photon carries off the difference between two levels, so a line belongs to a pair of levels, never to one:

λ=hc∣Ei−Ef∣\lambda = \frac{hc}{|E_i - E_f|}
Try it

The electron still lands on n = 2. Choose where it falls from so the photon carries 2.55 eV.

GoalA photon carrying 2.55 eV.

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Lines you cannot see are still lines

The Balmer series, the jumps down to n = 2, is famous for an unfair reason: it is the only one of the three that lands in the narrow band a human eye responds to.

Predict

Lyman α carries 10.2 eV, five times the energy of the red Balmer α photon. What colour is it?

Try it

Hydrogen's n = 4 → 3 line sits at 1875 nm, deep in the infrared. Give the nucleus more charge, keep the jump, and bring the line into view.

GoalThe same n = 4 to n = 3 jump, landing in the visible band.

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Nothing is used up

Read the graph

Run the n = 3 → 2 jump. Once the photon has gone, what energy does the electron have?

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Run the simulation first, then read the value off it.

"The photon comes out of the electron, so the electron is spent." The photon is made of the difference between two levels. That is why an atom can emit line after line as it cascades down and still have the one electron it started with.

Practice

A hydrogen electron falls to n = 2 from levels above it. What energy does the photon carry?

eV

The same lines, in reverse

Now run the experiment the other way. Shine white light, every wavelength at once, through the same gas, cold this time, and look at what comes out the far side.

Try it

In 1814 Fraunhofer mapped dark lines across the Sun's spectrum and lettered the strongest. His F line, at 486 nm, is hydrogen. Make the atom on n = 2 take it.

GoalA dark line at 486 nm.

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Predict

A hydrogen atom sitting in n = 2 is bathed in white light. A photon carrying 2.00 eV comes past. What happens?

What the ladder does not claim

Nothing on this canvas draws an orbit, on purpose: the atom here is its ladder of allowed energies, because a spectrum is evidence about energies. The Bohr model, with its electron on allowed circular orbits, gets these energies right for one-electron atoms. But the electron in a real atom has no path at all. What it has instead, a wave with a set of allowed standing patterns that hand back these energies, is where this unit goes next.

Ideas this lesson took on

Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.

  • “Hot things glow, and glowing things make rainbows”
  • “The energy levels are evenly spaced, like rungs on a ladder”
  • “More energy means bluer, and violet is as blue as light gets”
  • “The electron is spent once it has emitted a photon”

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Research behind this lesson

  • Ivanjek et al. 2015: A four-year investigation following more than 1000 students found serious and persistent difficulties in connecting a discrete line spectrum with a set of atomic energy levels: the step from knowing that atoms have levels to explaining why a spectroscope shows lines rather than a continuum.
  • McKagan et al. 2008: Examining student descriptions of atoms on final exams across several versions of a transformed modern-physics curriculum, the authors found that where the curriculum does not explicitly connect the different models of the atom, many students keep a Bohr-like planetary picture instead of the Schrodinger one.
  • Savall-Alemany et al. 2016: High-school students, university students and their own teachers share the same difficulties in explaining atomic spectra, and those difficulties concern three things at once: the model of the atom, the model of the radiation, and the model of the interaction between them.