Physics 9702 · Paper 4

Energy Levels & Line Spectra

A hot gas glows in only a few sharp colours, and a cool gas bites the same colours out of white light. Both are fingerprints of one idea: electrons in atoms have only certain allowed energies.

Discrete levels, one photon per jump

The hydrogen ladder (real values): levels crowd toward zero at the top. An electron falling from a higher level to a lower one emits one photon whose energy is exactly the gap, hf = E₁ − E₂ — so a bigger drop makes a more energetic, shorter-wavelength photon.

Electrons in an atom sit only on fixed — negative because the electron is bound, crowding toward zero (free) at the top. When an electron drops from a higher level to a lower one it emits one photon whose energy is exactly the gap:

hf = E_1 - E_2

Symbols

$hf$ = energy of the emitted (or absorbed) photon (J)
$E_1, E_2$ = the higher and lower energy levels (J)

The logic runs both ways, and 2025 wanted the full chain (9702/42/O/N/25 Q8(a)): the spectrum shows discrete frequencies, so the energy differences must be discrete, so the levels themselves must be discrete. A bigger drop makes a more energetic, shorter-wavelength photon.
Because $\lambda$ is inversely related to the gap, the longest wavelength (or lowest frequency) comes from the smallest jump — a recurring trap when a table lists several transitions (9702/42/F/M/23 Q7(b)) (9702/42/O/N/25 Q8(b)(ii)).

Worked example

Filling in a level (2024 paper)

A 488 nm emission line ends on the −3.40 eV level. Which level did the electron fall from (9702/42/M/J/24 Q8(b)(ii))?

Photon energy: $E = \dfrac{hc}{\lambda} = 4.08\times10^{-19}\ \text{J} = 2.55\ \text{eV}$ .
The starting level is that much above the ending one: $-3.40 + 2.55 = -0.85\ \text{eV}$ .

Answer

-0.85\ \text{eV}

Check the signs: you add the gap to climb from a deeper (more negative) level to a shallower one.

Emission vs absorption

	Emission line spectrum	Absorption line spectrum
seen from	a hot gas	white light through a cool gas
looks like	bright lines on a dark background	dark lines on a continuous spectrum
what electrons do	fall to lower levels, emit photons	absorb matching photons, jump up
which photons	energy = a level gap, emitted any direction	only those whose energy exactly fits a gap

The absorption story is the emission story reversed: a photon is only swallowed if its energy exactly matches a gap; the electron is excited, then later de-excites and re-emits in any direction, so those wavelengths go missing from the forward beam (9702/42/F/M/23 Q7(a)). Same set of gaps, so the dark lines line up with the bright ones — which is how starlight reveals what stars are made of.

Discrete lines ⇒ discrete gaps ⇒ discrete levels. $hf = E_1 - E_2$ : biggest jump → shortest wavelength. Emission = falling and glowing; absorption = matching photons removed from white light.

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