Electron Structure and Light

Note on topic scope: This topic (T12) covers electron structure and light — shells, orbitals, quantum numbers, electron configurations, and the connection to photons via the Bohr model and de Broglie wavelength. The companion topic T13: Periodic Properties and Trends covers atomic radius, ionization energy, electronegativity, electron affinity, ionic radius, isoelectronic species, and predicting bond type from periodic position. If you arrived here looking for periodic-trends content, follow the link to T13.

Understanding electron structure begins with light. The electromagnetic spectrum spans from low-energy radio waves to high-energy gamma rays. Two relationships connect a photon’s wavelength, frequency, and energy:

• c = λν — the speed of light c (2.998 × 10⁠⁸ m/s in vacuum) equals wavelength λ times frequency ν. Wavelength and frequency are inversely proportional.
• E = hν — the energy of one photon equals Planck’s constant h (6.626 × 10⁠^-34 J·s) times frequency. Higher frequency means higher energy.

The Bohr model explained the hydrogen emission spectrum by proposing that electrons occupy specific quantized energy levels (orbits). The electron does not radiate while in a stationary state, but it emits or absorbs a photon when it moves between levels: |ΔE| = hν. For a one-electron atom or hydrogen-like ion the orbit energies are E_n = −k Z⁠² / n⁠², where k = 2.179 × 10⁠^-18 J and Z is the nuclear charge (Z = 1 for H, Z = 2 for He⁺, …). The energy of a transition from n_i to n_f is ΔE = −k Z⁠² (1/n_f⁠² − 1/n_i⁠²). The model correctly reproduced the Rydberg formula and hydrogen’s spectral lines but failed for any atom with more than one electron, because Bohr did not yet account for electron-electron interactions.

Wave-particle duality resolves the apparent paradox: electrons (and other matter) behave as both particles and waves. The de Broglie wavelength λ = h/p = h/(mv) gives the wavelength of any moving particle of mass m and velocity v. Because h is so small, the wavelength is only appreciable for very low-mass and/or very fast objects: an electron at typical atomic speeds has a de Broglie wavelength on the order of atomic dimensions (about 10⁠^-10 m), which is why crystals diffract electrons. A 100-g softball at 35 m/s has a de Broglie wavelength of about 10⁠^-34 m, far below any measurable scale.

Modern quantum mechanics (Schrödinger’s wave equation, ca. 1926) replaced Bohr’s circular orbits with orbitals — three-dimensional probability regions where electrons are most likely found. Heisenberg’s uncertainty principle says the position and momentum of an electron cannot both be known precisely at the same time, so we describe an electron’s location probabilistically rather than as a definite trajectory.

The modern atomic model is hierarchical — four nested levels of organization, each with strict capacity rules:

1. Shells are labeled by the principal quantum number n = 1, 2, 3, … A shell sets the general region of space and the energy level for an electron occupying it. The shells of an atom can be thought of as concentric circles radiating out from the nucleus, with higher shell numbers corresponding to greater average distance from the nucleus and higher energy.
2. Subshells are labeled s, p, d, f within a shell, set by the angular-momentum quantum number l (with l = 0, 1, 2, 3 mapping to s, p, d, f). Shell n contains exactly n subshells: shell 1 has only 1s; shell 2 has 2s and 2p; shell 3 has 3s, 3p, 3d; and so on. The subshell determines the shape of the orbital.
3. Orbitals are individual three-dimensional probability regions within a subshell, set by the magnetic quantum number m_l. The number of orbitals in a subshell is 2l + 1: the s subshell has 1 orbital, p has 3, d has 5, f has 7. Each orbital has a distinct spatial orientation.
4. Electrons are the particles that occupy orbitals, distinguished within an orbital by the spin quantum number m_s. Each orbital holds at most 2 electrons (with opposite spins).

Capacity rules follow directly:

• Subshell capacity: s = 2, p = 6, d = 10, f = 14 (twice the orbital count, 2(2l + 1)).
• Shell capacity: 2n⁠² electrons. So shell 1 holds 2, shell 2 holds 8, shell 3 holds 18, shell 4 holds 32.

Every electron in an atom “lives” in exactly one orbital, which sits inside one subshell, which sits inside one shell. The four quantum numbers in the next section make this hierarchy precise.

Each electron in an atom is described by four quantum numbers:

• n (principal) — energy level / shell: 1, 2, 3, … Higher n means higher energy and larger orbital.
• l (angular momentum / subshell) — sublevel shape, with allowed values 0 to (n−1). Values 0, 1, 2, 3 correspond to s, p, d, f.
• m_l (magnetic / orbital) — orbital orientation, with allowed values −l to +l. For l = 1 (p), m_l = −1, 0, +1, giving three p orbitals. For l = 2 (d), m_l = −2, −1, 0, +1, +2 (five d orbitals).
• m_s (spin) — electron spin: +½ or −½.

The Pauli exclusion principle states that no two electrons in the same atom can have the same set of all four quantum numbers. As a consequence, each orbital holds at most 2 electrons (with opposite spins).

A valid quantum-number set respects all the allowed-value rules. Three checks decide validity:

1. n must be a positive integer (1, 2, 3, …).
2. l must be an integer in the range 0 ≤ l ≤ n−1.
3. m_l must be an integer in the range −l ≤ m_l ≤ +l. m_s must be +½ or −½.

Examples:

• (n, l, m_l, m_s) = (3, 2, −2, +½) → valid (3d orbital).
• (2, 2, 0, +½) → invalid: l cannot equal n.
• (3, 1, +2, −½) → invalid: m_l cannot exceed l.
• (4, 0, 0, +1) → invalid: m_s can only be ±½.

To assign a complete quantum-number set to a specific electron, locate the electron in the orbital diagram, then read off (n, l, m_l, m_s). For the last electron added to nitrogen (1s⁠² 2s⁠² 2p⁠³, last electron in 2p), a valid assignment is (n, l, m_l, m_s) = (2, 1, m_l, +½) where m_l can be −1, 0, or +1: n = 2 because the electron sits in the second shell, l = 1 because it is in a p subshell, and m_l can take any of the three allowed values for l = 1 because Hund’s rule (one electron in each degenerate orbital before any pairing) does not single out a specific m_l value. The m_s = +½ assignment for the third 2p electron follows from Hund’s rule itself, which says the three half-filled 2p electrons all carry parallel spins; conventionally the parallel-spin convention for unpaired electrons is m_s = +½. Many textbook orbital diagrams draw the three 2p boxes left-to-right as m_l = −1, 0, +1 and fill them in that order, but that diagrammatic ordering is a drawing convention — not a consequence of Hund’s rule.

An electron configuration lists the sublevels occupied by an atom’s electrons in order of energy. Three rules govern the filling order:

• Aufbau principle — fill lowest-energy orbitals first: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p.
• Pauli exclusion principle — maximum 2 electrons per orbital (opposite spins).
• Hund’s rule — within a sublevel, place one electron in each orbital before pairing any.

Example: Nitrogen (Z = 7) is 1s⁠² 2s⁠² 2p⁠³. The three 2p electrons occupy three separate orbitals (one each), all with parallel spins.

An orbital diagram uses boxes (or lines) for orbitals and arrows (↑↓) for electrons, visually showing Hund’s rule. A noble gas abbreviation replaces the inner-shell configuration with the preceding noble gas symbol in brackets: Na = [Ne] 3s⁠¹.

For ions: when a transition metal forms a cation, electrons are removed from the ns orbital before the (n−1)d orbital (even though ns filled first). For example, Fe⁠²⁺ = [Ar] 3d⁠⁶ — not [Ar] 4s⁠² 3d⁠⁴ — because both 4s electrons leave before any 3d electron does. Main-group cations and anions are simpler: remove the highest-energy electrons (cations) or add to the next available orbital (anions). Cl⁠^- = [Ne] 3s⁠² 3p⁠⁶ = [Ar].

A few elements have electron configurations that deviate from the standard filling order because a half-filled or fully filled d sublevel provides extra stability:

• Chromium (Z=24): Expected [Ar] 4s⁠² 3d⁠⁴, actual [Ar] 4s⁠¹ 3d⁠⁵ (half-filled d).
• Copper (Z=29): Expected [Ar] 4s⁠² 3d⁠⁹, actual [Ar] 4s⁠¹ 3d⁠¹⁰ (fully filled d).
• Similar exceptions occur with Mo, Ag, Au, and other transition metals.

The qualitative explanation: a half-filled or fully filled d subshell maximizes exchange energy (the stabilization that comes from electrons of parallel spin in different orbitals), and 4s and 3d are close enough in energy that promoting one 4s electron to 3d pays off.

Use this sequence when students get stuck:

1. Locate the element (period/group/block) first; this anchors expected valence behavior.
2. Write the configuration systematically (Aufbau order), then verify Pauli and Hund.
3. For ions, remove electrons from the highest principal quantum number first (for transition metals, remove 4s before 3d; main-group elements lose ns/np electrons before any (n−1)d).
4. Validate quantum-number sets against n > 0, 0 ≤ l ≤ n−1, −l ≤ m_l ≤ +l, m_s = ±½.

High-frequency errors: treating Cr/Cu exceptions as random, removing d electrons before 4s in cations, and forgetting the Hund-rule parallel-spin convention in orbital diagrams. A reliable final check is to confirm the total electron count matches the atomic number (or Z − charge for an ion).