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Atoms do not emit or absorb radiation at arbitrary frequencies. Instead, each element produces a characteristic line spectrum — a set of discrete frequencies corresponding to specific atomic transitions. The pattern of these lines is the fingerprint of the element, and the explanation of why atoms behave this way was one of the foundational triumphs of quantum mechanics. This lesson develops the energy-level picture of the atom, the relationship between transition energies and emitted/absorbed photons, and the experimental evidence from emission and absorption spectra.
Specification mapping. This lesson develops AQA A-Level Physics (7408) Particles and Radiation strand, sub-strand 3.2.2.2 (energy levels and photon emission; the Bohr atomic model in simplified form; line spectra as evidence for discrete energy levels) with cross-reference to 3.2.2.3 (ionisation and excitation energies). Refer to the official AQA 7408 specification document for the authoritative wording. The lesson is sequenced after the photoelectric-effect / wave-particle-duality lesson (order 7) so that students already have the E = hf quantum-of-light framework in place — line spectra apply that framework to bound atomic systems rather than free electrons.
Synoptic links. Three principal synoptic threads run through this content. First, the photoelectric effect (3.2.2.1, our order 7) — both the photoelectric effect and atomic line spectra are direct consequences of the quantisation of electromagnetic energy E = hf; in the photoelectric effect a photon ionises an electron from a metal, in line spectra a photon excites or de-excites an electron between bound atomic states. Second, electron-atom collisions and the fluorescent tube (our order 9) — the discrete excitation energies introduced here are the energies that accelerated electrons must supply to atoms in fluorescent-tube physics; absorption is the inverse process to electron-impact excitation. Third, astronomy and stellar spectroscopy (course 7, beyond core spec) — the same line spectra are the basis of stellar classification, Doppler-shift measurements of galactic recession (Hubble's law), and the discovery of helium in the Sun before its identification on Earth.
When the line spectrum of hydrogen was first measured in the 19th century (Balmer, 1885; Lyman, 1906), the regular pattern of the lines seemed inexplicable in classical physics. A classical electron orbiting a proton would radiate continuously (an accelerated charge emits electromagnetic radiation), spiralling into the nucleus within microseconds. The atom should be unstable and the spectrum should be continuous — both prediction directly contradicted by experiment.
The resolution came in 1913 with the Bohr model: electrons in an atom can occupy only certain discrete energy levels, and transitions between these levels emit or absorb photons of exactly the right energy. The model is now superseded by full quantum mechanics, but it remains an excellent pedagogical introduction to atomic energy levels and is the conceptual basis of the A-Level treatment.
Each atom has a discrete set of allowed electron energy states, conventionally labelled with the principal quantum number n = 1, 2, 3, ... The lowest energy level (n = 1) is the ground state; higher levels (n ≥ 2) are excited states. The convention at A-Level is that energy levels are quoted as negative numbers, with the ionisation level (electron just barely free of the atom) set to zero. So for hydrogen:
| n | Energy level (eV) | Description |
|---|---|---|
| 1 | −13.6 | Ground state |
| 2 | −3.40 | First excited state |
| 3 | −1.51 | Second excited state |
| 4 | −0.85 | Third excited state |
| 5 | −0.54 | Fourth excited state |
| ∞ | 0 | Ionisation (electron just free) |
The values get smaller in magnitude as n increases, approaching zero (the ionisation limit). The electron is most tightly bound in the n = 1 ground state.
A convention: an electron at infinite distance from the nucleus, at rest, has zero energy by definition. As the electron is bound closer to the nucleus, its (negative) potential energy outweighs its kinetic energy, giving a total energy that is negative. Energy levels closer to the nucleus (smaller n) are therefore more negative — the electron has had energy released as it fell into the bound state. To ionise the electron from level n requires supplying enough energy to reach the n = ∞ limit.
For hydrogen specifically, the energy levels follow the formula:
E_n = −13.6 / n² (eV)
This is the Bohr formula; it works exactly for hydrogen (and singly ionised helium, doubly ionised lithium etc.) but is only approximate for multi-electron atoms.
graph LR
A["n = ∞ (0 eV) Ionisation"] --> B["n = 4 (−0.85 eV)"]
B --> C["n = 3 (−1.51 eV)"]
C --> D["n = 2 (−3.40 eV) First excited"]
D --> E["n = 1 (−13.6 eV) Ground state"]
An electron can move between energy levels by absorbing or emitting a photon whose energy exactly matches the difference between the levels.
When an electron transitions from a higher level (energy E_high, more negative) to a lower level (energy E_low, less negative — wait, lower in number but more negative in value, so when we say "lower" we mean lower in n, i.e. more tightly bound), it emits a photon:
ΔE = E_high − E_low = hf_photon
Note: both energies are negative, and E_high is less negative (smaller in magnitude) than E_low. The difference ΔE is positive — the electron loses energy, the photon carries it away.
The reverse: an electron in a lower state absorbs an incoming photon of exactly the right energy to be promoted to a higher state. The atom must have a photon of energy matching one of its transition energies; photons of other energies pass through without interacting.
A hydrogen atom transitions from n = 3 to n = 2. Calculate the energy of the emitted photon in eV and joules, and its wavelength in nm.
E_3 = −1.51 eV; E_2 = −3.40 eV
ΔE = E_3 − E_2 = −1.51 − (−3.40) = 1.89 eV
In joules: 1.89 × 1.60 × 10⁻¹⁹ = 3.02 × 10⁻¹⁹ J
Wavelength: λ = hc/E = (6.63 × 10⁻³⁴)(3.00 × 10⁸) / (3.02 × 10⁻¹⁹) = 6.58 × 10⁻⁷ m ≈ 656 nm
This is the famous H-α (hydrogen alpha) line, in the red part of the visible spectrum. It is one of the brightest lines in the spectrum of hot hydrogen gas and is visible in the spectra of stars, nebulae and aurorae. (Approximate value 656 nm is the established figure to three significant figures; precise value is 656.3 nm.)
A hydrogen atom in the ground state absorbs a photon and is excited to the n = 3 state. What is the wavelength of the absorbed photon?
ΔE = E_3 − E_1 = −1.51 − (−13.6) = 12.09 eV = 1.93 × 10⁻¹⁸ J
λ = hc / ΔE = (6.63 × 10⁻³⁴)(3.00 × 10⁸) / (1.93 × 10⁻¹⁸) = 1.03 × 10⁻⁷ m ≈ 103 nm
This is in the ultraviolet — too short for the human eye but readily detectable with UV spectrometers. The line is part of the Lyman series (transitions ending at n = 1).
The transitions of hydrogen are organised into series by their lower level:
| Series | Transitions ending at | Spectral region |
|---|---|---|
| Lyman | n = 1 | Ultraviolet |
| Balmer | n = 2 | Visible / near-UV |
| Paschen | n = 3 | Infrared |
| Brackett | n = 4 | Mid-infrared |
The Balmer series is the one historically discovered first because it falls in the visible range. Hα (n=3 to n=2) at ~656 nm is red; Hβ (n=4 to n=2) at ~486 nm is blue-green; Hγ (n=5 to n=2) at ~434 nm is violet.
When atoms are excited (by heating, electric discharge, or collisions with electrons), they emit photons at the specific frequencies of their transitions. Passing this light through a prism or diffraction grating reveals an emission spectrum — a set of bright lines on a dark background, with each line corresponding to one transition.
This is the basis of spectroscopic identification: a glowing gas's emission spectrum reveals what elements it contains. Helium was actually discovered this way — Janssen and Lockyer observed an unidentified yellow line in the solar spectrum during the 1868 eclipse and named the responsible element after the Greek word for sun. Helium was not isolated on Earth until 1895.
The reverse phenomenon: when white light (containing all frequencies) passes through a cool gas, the atoms absorb photons at exactly their characteristic transition frequencies — promoting electrons to higher states. The emerging light, viewed through a spectrometer, shows dark absorption lines at the same positions where the emission spectrum shows bright lines.
The dark lines visible in the solar spectrum were discovered in 1814 by Joseph von Fraunhofer (who catalogued hundreds of them, lettered A through K). They are caused by absorption in the relatively cool outer layers of the solar atmosphere (the photosphere and chromosphere), where calcium, hydrogen, iron, sodium and other elements absorb specific wavelengths from the broad-spectrum continuum emitted by the deeper, hotter layers.
The Fraunhofer lines are how astronomers determine:
A cool gas absorbs photons of wavelength 589 nm from a continuous spectrum. Calculate the transition energy and identify a plausible source.
E = hc/λ = (6.63 × 10⁻³⁴)(3.00 × 10⁸) / (589 × 10⁻⁹) = 3.38 × 10⁻¹⁹ J = 2.11 eV
This 589 nm yellow line is the famous sodium D-line, produced by the 3p → 3s transition in atomic sodium. It is responsible for the orange-yellow colour of sodium street lamps and is one of the most easily observed absorption lines in the solar spectrum.
The ionisation energy is the energy required to remove the outermost electron entirely from an atom — to take it from its ground state to the n = ∞ ionised state.
For hydrogen in the ground state, ionisation energy = 0 − (−13.6) = 13.6 eV. This corresponds to a photon wavelength:
λ = hc / E = (6.63 × 10⁻³⁴)(3.00 × 10⁸) / (13.6 × 1.60 × 10⁻¹⁹) = 91.2 nm (extreme ultraviolet)
A photon of wavelength shorter than 91.2 nm has enough energy to ionise hydrogen from its ground state. Excited hydrogen atoms can be ionised by less energetic photons (any photon with energy exceeding |E_n| for the current state).
| Element | First ionisation energy (eV) | Comment |
|---|---|---|
| Hydrogen | 13.6 | The benchmark |
| Helium | 24.6 | Highest of any element |
| Lithium | 5.4 | Easily ionised |
| Sodium | 5.1 | Outer-shell electron loosely bound |
| Potassium | 4.3 | |
| Caesium | 3.9 | Lowest first IE of any stable element |
| Carbon | 11.3 | |
| Nitrogen | 14.5 | |
| Oxygen | 13.6 | (Coincidentally equal to hydrogen) |
Calculate the wavelength of the series limit (n = ∞ to n = 1) of the Lyman series for hydrogen.
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