Heat hydrogen until it glows, split the light, and you get four sharp coloured lines. Not a rainbow — four lines. Explaining those four lines took physics thirty years and gave chemistry the entire idea of energy levels.
📚 What you need to know
Electrons occupy fixed energy levels numbered by the principal quantum number, n.
Absorbing energy promotes an electron to a higher level (an excited state).
Falling back down emits a photon whose energy equals the gap between the levels.
Each line in the spectrum is one specific transition.
Transitions down to n = 1 give ultraviolet (Lyman), to n = 2 give visible (Balmer), to n = 3 give infrared (Paschen).
Lines converge at high frequency because the energy levels get closer together as n rises.
The convergence limit corresponds to n = ∞, which is the electron leaving the atom.
Absorption and emission
An electron sitting in its lowest available level is in the ground state. Give it exactly the right amount of energy and it jumps to a higher level. It will not stay there, and when it drops back it releases that energy again as a photon.
The electron cannot stop halfway. It either has enough energy for the full jump or it stays where it is, which is what makes the emitted frequencies so precise.
The hydrogen energy levels
Hydrogen has one electron, which makes its spectrum the simplest and the one every explanation starts from. The key feature of the levels is that they are not evenly spaced: they crowd together as n increases.
Look at the spacing on the left. The gap from n = 1 to n = 2 is enormous; the gaps between n = 4, 5 and 6 are almost invisible. That uneven spacing is what produces convergence.
Series
Transitions end at
Region
Relative energy
Lyman
n = 1
ultraviolet
highest
Balmer
n = 2
visible
medium
Paschen
n = 3
infrared
lowest of the three
There is a reliable way to answer any “which transition” question without memorising the series names. Ask where the electron lands, not where it starts. Landing on n = 1 is the biggest possible drop, so it releases the most energy, so it is ultraviolet. Landing on n = 2 is a smaller drop and comes out visible. The destination decides the region every time.
Convergence
Look at any emission series and the lines are not evenly spaced. They bunch up at the high-frequency end until they run together and stop. This is convergence, and it is a direct consequence of the energy levels crowding together as n increases.
The convergence limit is not the end of the spectrum for a boring reason. It is the point where the electron has escaped, and there is nothing left to make a line with.
The convergence limit of the Lyman series is the one that matters chemically. It represents an electron falling from n = ∞ all the way to n = 1 — and run in reverse, that is exactly the energy needed to remove the electron from a ground-state atom. In other words, it gives you the first ionisation energy.
WORKED EXAMPLE
Which of these transitions in a hydrogen atom produces a line in the visible region? (a) n = 1 to n = 3 (b) n = 4 to n = 2 (c) n = 3 to n = 1 (d) n = 2 to n = 4
First, which ones emit at allEmission requires the electron to fall. Options (a) and (d) go upwards, so they absorb energy rather than emitting it.Now check the destinationOption (c) lands on n = 1, the largest possible drop, so it is ultraviolet. Option (b) lands on n = 2.(b) n = 4 to n = 2, the Balmer seriesThe rule to carry awayVisible emission always ends at n = 2, whatever level the electron started from.
WORKED EXAMPLE
Explain why the lines in an emission series converge at high frequency.
Start with the energy levelsThe energy levels are not evenly spaced. As n increases the levels get closer and closer together.Link that to the transitionsEach line comes from a transition starting at a higher level. As the starting levels crowd together, the energy differences between successive transitions get smaller.Link the energy to the spectrumSince E = hf, smaller differences in energy mean smaller differences in frequency, so the lines appear closer together.converging levels give converging linesAnswer the question in that order — levels, then transitions, then frequencies. Jumping straight to “the lines get closer” describes the observation without explaining it.
WORKED EXAMPLE
A student says the hydrogen spectrum proves electrons orbit the nucleus in fixed circular paths. Comment on this claim.
What the spectrum does showOnly fixed frequencies are emitted, so electrons can only hold certain fixed energies. The evidence for quantised energy levels is solid.quantised energies: supportedWhat it does not showNothing in the spectrum says anything about the shape or path of the electron’s motion. Fixed energy is not the same claim as fixed orbit.circular orbits: not supportedWhere the model breaksThe simple orbit picture predicts the hydrogen spectrum well and then fails for every atom with more than one electron. That failure is what led to orbitals, which describe regions of probability rather than paths.
💡 Exam tip
For emission the electron falls; for absorption it is promoted. Check the direction first.
Identify the region from where the transition ends: n = 1 UV, n = 2 visible, n = 3 infrared.
Say the photon energy equals the difference between the levels.
Explain convergence by saying the energy levels get closer together as n increases.
State that the convergence limit corresponds to n = ∞, meaning the electron has left the atom.
Use the phrase quantised when explaining why the spectrum has lines rather than a continuum.
⚠️ Common mix-up
Saying emission happens when electrons jump up. It is the fall that releases the photon.
Thinking the energy levels are evenly spaced. They crowd together as n rises.
Naming the region from the starting level instead of the destination.
Treating convergence as the lines simply stopping, rather than the electron escaping.
Describing the Balmer series as “the spectrum”. It is only the visible part of it.
Claiming electrons travel in fixed circular orbits because the spectrum has lines.
Up next: Energy Levels, Sublevels and Orbitals — the simple picture of numbered shells explains hydrogen and then falls apart. The next page splits those shells into the structure that works for every other element.
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