Atomic energy levels and line spectra

Key ideas

• Electrons in an atom can only exist with certain discrete energy levels — not a continuous range. Each level has a specific allowed energy, often quoted in electron-volts (eV) or joules (J).
  • $1$ eV $= 1.60\times10^{-19}$ J (the energy gained by one electron accelerated through a potential difference of $1$ V).
• The ground state is the lowest (most negative) energy level, where an electron normally sits. Excited states are higher energy levels.
• Energy levels are usually written as negative values, with $0$ eV representing the electron just free of the atom (ionised).
• An electron can move between levels only by absorbing or emitting a photon whose energy exactly matches the gap between the two levels:

$hf = E_i - E_f$

• Each variable, with sub-bullets:
  • $h$ — the Planck constant
  • $f$ — the frequency of the absorbed or emitted photon
  • $E_i$ — the energy of the initial level (eV or J)
  • $E_f$ — the energy of the final level (eV or J)
  • if the electron moves down (higher level → lower level), $E_i > E_f$ and the atom emits a photon
  • if the electron moves up (lower level → higher level), it must absorb a photon of exactly that energy — no other photon energy will do, because there is no intermediate level to receive it

Emission and absorption spectra

• An emission spectrum is produced when excited atoms (e.g. in a hot gas) fall to lower energy levels, emitting photons at specific frequencies. Each possible downward transition produces one bright line at a wavelength set by $hf = E_i - E_f$.
• An absorption spectrum is produced when light with a continuous range of frequencies passes through a cooler gas. Only photons with exactly the right energy to lift an electron between two levels are absorbed, leaving dark lines at those same specific frequencies.
• Because each element has a unique set of energy levels, its line spectrum acts as a fingerprint — this is how the composition of stars (including our Sun) is determined from their absorption spectra.
• Ionisation energy is the energy needed to remove an electron completely from an atom — from its current level up to $0$ eV (the free, unbound state). For an electron in the ground state, this equals the magnitude of the ground-state energy.

Tips

• Every transition between the same two levels has the same energy, whether emission or absorption. A common Excellence question asks why absorption and emission lines occur at identical frequencies for the same pair of levels — because $|E_i - E_f|$ is the same either way, only the direction of photon travel differs.
• When a question gives energies in eV but asks for frequency (in Hz) or wavelength (in m), always convert to joules first — using $hf$ with eV values, without converting $h$ to its eV·s form, is a very common mistake.