In simple terms
A friendly intro before the formal notes — no formulas yet.
Electrons Live on Steps, Not a Ramp
An electron in an atom cannot have just any energy. It is restricted to specific allowed energy levels — like standing on the steps of a staircase but never in between. The light atoms give out, and the energy needed to strip electrons away, both reveal these hidden steps.
Picture a staircase. You can stand on step 1, step 2 or step 3, but never hover halfway between two steps. To move up, you must absorb exactly the right amount of energy to reach the next step; when you drop back down, you release exactly that amount — no more, no less. Because only fixed jumps are possible, an atom only ever emits light of certain fixed energies, which is why we see sharp coloured lines instead of a smooth rainbow.
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Give a hydrogen atom energy and its electron jumps up to a higher energy level.
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The electron falls back down, releasing the energy difference as a single photon of light.
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Because only certain jumps exist, only certain photon energies (colours) appear — a line spectrum, not a continuous one.
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Fill the levels from the bottom up (Aufbau) to write the electron configuration; condense the inner electrons as the previous noble gas.
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Key formulas
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Full topic notes
Formal explanation with the rigour you need for the exam.
Evidence 1: the emission spectrum of hydrogen
When white light is passed through a prism it spreads into a continuous spectrum — every wavelength of the rainbow blending smoothly with no gaps. But when energy is supplied to hydrogen gas (for example in a discharge tube) and the light it emits is passed through a prism, something very different appears: only a few sharp, discrete coloured lines against a dark background. This is a line emission spectrum.
The Bohr / energy-level model explains this. An electron in a hydrogen atom can only occupy certain allowed energy levels, labelled n = 1, 2, 3, 4... When the atom is given energy an electron is promoted (excited) to a higher level. It is unstable there, so it falls back to a lower level, and the energy it loses is released as a single photon of light. The energy of that photon is fixed by the gap between the two levels.
E(photon) = E(higher level) - E(lower level) = hf
Because only certain energy gaps exist, only certain photon energies — and therefore only certain frequencies (colours) — can ever be emitted. That is exactly why we see discrete lines rather than a continuous band. A crucial detail: the lines are not evenly spaced. As we look to higher frequency (higher energy), the lines get closer together and converge towards a limit. This is because the energy levels themselves get closer together as n increases; the gaps shrink, so the lines crowd together. The convergence limit corresponds to the electron being removed entirely (ionization) from that level.
Continuous spectrum — all wavelengths present, merging smoothly (e.g. white light). No information about energy levels.
Line (emission) spectrum — only certain discrete frequencies appear as separate lines. Each element has its own characteristic pattern.
The fact that hydrogen emits only specific frequencies, not a continuum, is the key experimental clue.
Evidence 2: successive ionization energies
The second, independent line of evidence comes from removing electrons one at a time. The successive ionization energies are the energies needed to remove the 1st, 2nd, 3rd... electron from a gaseous atom. Each value is larger than the last, because after each removal the remaining electrons are held by the same nuclear charge but there are fewer of them — the ion becomes more positive and holds on more tightly. But the increase is not smooth: it shows sudden large jumps.
Successive ionization energies always increase across a single atom.
A large jump occurs when the next electron must be removed from a main energy level (shell) closer to the nucleus — less shielded and more strongly attracted, so much more energy is needed.
Counting how many electrons are removed before each big jump reveals how many electrons occupy each main energy level — direct evidence for shells.
Example: sodium removes 1 electron easily (outer n=3), then a large jump before the next 8 (n=2), then another large jump before the final 2 (n=1) — the 2,8,1 pattern.
Main energy levels and sub-levels
The energy-level model is refined by dividing each main energy level (n) into sub-levels. At SL you need three types: s, p and d. An s sub-level holds up to 2 electrons, a p sub-level up to 6, and a d sub-level up to 10. Which sub-levels exist depends on the main level: n=1 has only 1s; n=2 has 2s and 2p; n=3 has 3s, 3p and 3d; n=4 has 4s, 4p, 4d... This structure is what the finer detail of atomic spectra reveals.
Main level n=1: 1s (max 2 electrons).
Main level n=2: 2s, 2p (max 2 + 6 = 8 electrons).
Main level n=3: 3s, 3p, 3d (max 2 + 6 + 10 = 18 electrons).
Sub-level capacities: s = 2, p = 6, d = 10 electrons.
Writing electron configurations (Aufbau)
To write an electron configuration we place the atom's electrons (equal to the atomic number Z for a neutral atom) into sub-levels from lowest energy upwards — the Aufbau principle. The filling order is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p... Notice that 4s fills before 3d: although 4s is in a higher main level, it is slightly lower in energy. Each occupied sub-level is written as n followed by the letter and a superscript giving the number of electrons in it.
HL extension: orbitals and the 4s/3d nuance
HL only. At HL the sub-levels are described in terms of orbitals — regions of space where an electron is most likely to be found. An s sub-level is one spherical orbital; a p sub-level is three dumbbell-shaped orbitals; a d sub-level is five orbitals. Each orbital holds a maximum of two electrons. The orbital picture explains, more precisely than simple Bohr orbits, why the 4s sub-level is filled before the 3d: a 4s electron penetrates closer to the nucleus and is less shielded, making 4s slightly lower in energy than 3d. HL consequence: when transition metals such as iron form positive ions, the 4s electrons are removed before the 3d electrons (e.g. Fe -> Fe2+ is [Ar]3d6, not [Ar]4s2 3d4).
HL flag: the 4s-before-3d filling but 4s-removed-first-on-ionization pattern is a classic HL trap. SL students should still know the Aufbau order includes 4s before 3d; the orbital shapes and ion-formation nuance are examined at HL.
Common mistakes examiners penalise
Confusing a line spectrum with a continuous spectrum — the evidence for discrete levels is that only specific lines appear, NOT a smooth band. Omitting this distinction loses the key mark.
Saying the lines spread apart at high energy — they converge (get closer) because the energy levels get closer together as n increases.
Explaining an ionization-energy jump as just 'harder to remove' — you must say the electron comes from a new main level closer to the nucleus, less shielded and more strongly attracted.
Filling 3d before 4s — the Aufbau order is 3s, 3p, 4s, 3d. The 4s sub-level fills first.
Superscripts that don't add up to Z — always total the electrons; a miscount is the most common configuration error.
Vague answers like 'electrons jump around' — this earns nothing; the marking engine needs the discrete-levels / line-spectrum / E=hf points made explicitly.
HL: removing 3d before 4s when forming ions — 4s electrons are lost first from transition-metal atoms.
Worked examples
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Explain how the emission spectrum of hydrogen provides evidence that electrons occupy discrete energy levels. [3]
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Model answer: When energy is supplied to hydrogen atoms, electrons are excited to higher energy levels and then fall back to lower levels. As an electron falls, it emits a photon of light whose energy equals the difference between the two energy levels (E = hf). The spectrum observed is a line spectrum — only certain discrete frequencies (lines) are seen, not a continuous band of all frequencies. Since only specific photon energies are emitted, the differences between energy levels must be fixed, which means the electron can only occupy fixed, discrete energy levels. (The lines also converge at higher frequency, showing the levels get closer together as n increases.)
The graph below shows the successive ionization energies of an element X (all values in kJ/mol, plotted against the number of electrons removed):
| Electron removed | 1st | 2nd | 3rd | 4th | 5th |
|---|---|---|---|---|---|
| IE (kJ/mol) | 578 | 1817 | 2745 | 11580 | 14840 |
(a) Explain why the ionization energies increase from the 1st to the 3rd. (b) Explain the large jump between the 3rd and 4th values. (c) Deduce which main energy level (shell) the outer electrons occupy and hence identify the group of element X.
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(a) Each successive electron is removed from a species that is already more positively charged, while the nuclear charge stays the same. With fewer electrons but the same number of protons, the remaining electrons are held more strongly, so more energy is needed each time. The rise 578 -> 1817 -> 2745 reflects this steadily increasing attraction within the same main energy level.
Write the full and the condensed (noble-gas) electron configuration of a neutral calcium atom (Ca, Z = 20). [2]
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1. Count the electrons: a neutral atom has electrons equal to Z, so Ca has 20 electrons.
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Continuous spectrum vs line spectrum
A continuous spectrum shows all wavelengths merging smoothly (e.g. white light through a prism). A line (emission) spectrum shows only a few discrete coloured lines against a dark background — evidence that only certain electron energy jumps are allowed.
Key takeaways
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Continuous spectrum — all wavelengths present, merging smoothly (e.g. white light). No information about energy levels.
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Line (emission) spectrum — only certain discrete frequencies appear as separate lines. Each element has its own characteristic pattern.
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The fact that hydrogen emits only specific frequencies, not a continuum, is the key experimental clue.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
Get a Paper 2 answer marked: explain how hydrogen's emission spectrum shows discrete energy levels
Get a Paper 2 answer marked: explain how hydrogen's emission spectrum shows discrete energy levels
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