Azimuthal Quantum Number (l)
In quantum mechanics, every electron in an atom is described by a unique set of four quantum numbers. The azimuthal quantum number, symbolised as l (also called the angular momentum quantum number or subsidiary quantum number), is the second of these four numbers. It specifies which subshell an electron occupies within a given principal energy level, and — crucially — it determines the three-dimensional shape of the orbital in which the electron is found. The value of l is always a non-negative integer that ranges from 0 up to (n − 1), where n is the principal quantum number. The four possible values of l most commonly encountered in chemistry are 0, 1, 2, and 3, corresponding to the subshells labelled s, p, d, and f respectively. These letter labels originated from spectroscopic observations made in the nineteenth century: s stands for 'sharp', p for 'principal', d for 'diffuse', and f for 'fundamental', describing the appearance of spectral line series. Today we understand these designations purely in terms of orbital shape and angular momentum, but the historical labels have been retained throughout modern chemistry and physics. The azimuthal quantum number is deeply connected to the physical concept of orbital angular momentum. An electron in an s subshell (l = 0) has zero orbital angular momentum and occupies a spherically symmetric orbital. Electrons in p subshells (l = 1) possess one unit of angular momentum and occupy dumbbell-shaped orbitals oriented along the x, y, or z axis. The d subshell (l = 2) contains five orbitals with more complex geometries, and the f subshell (l = 3) contains seven orbitals with even more intricate shapes. Understanding l is therefore essential for predicting molecular geometry, chemical bonding, and spectroscopic behaviour.
- Allowed Values of l
- Subshell Labels and Number of Orbitals
- Orbital Shapes and Angular Momentum
- Energy Within a Subshell and Electron Configuration
Allowed Values of l
For any electron in a principal shell n, the azimuthal quantum number l can take integer values from 0 to (n − 1) inclusive. For example, when n = 1, l can only be 0 (only an s subshell exists). When n = 2, l can be 0 or 1 (s and p subshells). When n = 3, l can be 0, 1, or 2 (s, p, and d subshells). When n = 4, l can be 0, 1, 2, or 3 (s, p, d, and f subshells). This constraint means that the 1p, 2d, or 3f subshells do not exist in nature — there is no combination of n and l values that would produce them within the allowed rules.
Subshell Labels and Number of Orbitals
Each value of l defines a subshell and determines how many individual orbitals it contains. The number of orbitals within a subshell is given by (2l + 1). Therefore: l = 0 (s subshell) → 1 orbital; l = 1 (p subshell) → 3 orbitals; l = 2 (d subshell) → 5 orbitals; l = 3 (f subshell) → 7 orbitals. Since each orbital can hold a maximum of 2 electrons (by the Pauli exclusion principle), the s, p, d, and f subshells can hold 2, 6, 10, and 14 electrons respectively. This directly explains the block structure of the periodic table.
Orbital Shapes and Angular Momentum
The azimuthal quantum number quantises the magnitude of the orbital angular momentum L of the electron according to the relationship L = ℏ√(l(l+1)), where ℏ is the reduced Planck constant. For l = 0, L = 0 and the orbital is spherically symmetric (s orbital). For l = 1, L = ℏ√2 and the p orbitals are dumbbell-shaped, pointing along the three Cartesian axes. For l = 2, L = ℏ√6 and the d orbitals have more complex cloverleaf and dumbbell-with-ring shapes. These shapes are derived mathematically from the angular part of the Schrödinger wave equation, specifically from the spherical harmonic functions Y(θ, φ).
Energy Within a Subshell and Electron Configuration
In a hydrogen-like (one-electron) atom, energy depends only on n, so all subshells with the same n are degenerate (have the same energy). In multi-electron atoms, however, electron–electron repulsion causes the energy to depend on both n and l. Subshells with lower l values experience greater penetration toward the nucleus (their orbitals have more probability density close to the nucleus) and are therefore lower in energy. This is why the 2s orbital is lower in energy than the 2p orbital, and why the 4s orbital fills before the 3d orbital. The (n + l) rule (Madelung's rule) uses the sum of n and l to predict filling order: subshells with lower (n + l) fill first; when (n + l) is equal, the subshell with lower n fills first.
The Key Equation
- L — Orbital angular momentum (kg·m²·s⁻¹ (J·s)): The magnitude of the quantum mechanical angular momentum associated with an electron's orbital motion around the nucleus.
- \hbar — Reduced Planck constant (h-bar) (J·s): Equal to h/(2π) = 1.055 × 10⁻³⁴ J·s; the fundamental quantum of angular momentum.
- l — Azimuthal quantum number (dimensionless (integer)): Determines the subshell and orbital shape; takes integer values from 0 to (n − 1).
- n — Principal quantum number (dimensionless (positive integer)): Defines the principal energy level (shell) and limits the maximum value of l.
Worked Example
Problem. An electron is in the 3d subshell of a transition metal atom. (a) State the values of n and l for this electron. (b) How many orbitals are in the 3d subshell? (c) What is the maximum number of electrons the 3d subshell can hold? (d) Calculate the magnitude of the orbital angular momentum for a 3d electron, giving your answer in terms of ℏ.
Answer. n = 3, l = 2; 5 orbitals; maximum 10 electrons; orbital angular momentum L = ℏ√6 ≈ 2.58 × 10⁻³⁴ J·s.
Origins · 1 minArnold Sommerfeld and how this was discovered
Arnold Sommerfeld introduced the azimuthal quantum number in 1916 as part of his extension of Bohr's atomic model to include elliptical orbits. Bohr's original model (1913) used only the principal quantum number n and circular orbits. Sommerfeld realised that elliptical orbits were also possible and required a second quantum number — the azimuthal quantum number k (later relabelled l) — to quantise the shape of the orbit. This 'Bohr–Sommerfeld model' successfully explained the fine structure of hydrogen's spectral lines, which showed that what appeared to be single spectral lines were actually clusters of closely spaced lines corresponding to different orbital shapes at the same energy level. Sommerfeld supervised more Nobel Prize winners than almost any other physicist in history, including Werner Heisenberg, Wolfgang Pauli, Peter Debye, and Hans Bethe — yet he himself was nominated for the Nobel Prize 81 times without ever winning it, making him one of the most decorated scientists never to receive the honour.
Related Concepts
- Principal Quantum Number (n) — The principal quantum number n defines the energy shell and directly sets the upper limit for l, since l can only take values from 0 to (n − 1). Understanding n is a prerequisite for understanding l.
- Magnetic Quantum Number (mₗ) — The magnetic quantum number mₗ depends on l, taking integer values from −l to +l. It specifies the orientation of an orbital in space and determines the number of individual orbitals within each subshell.
- Electron Configuration — Writing electron configurations requires knowing which subshells (s, p, d, f) exist for each energy level — information encoded entirely by the allowed values of l for each n.
- Orbital Shapes and Hybridisation — The characteristic shapes of s, p, and d orbitals — determined by l — are the basis for understanding orbital hybridisation (sp, sp², sp³) and the geometry of covalent bonds.
- Aufbau Principle and Madelung's Rule — The order in which subshells fill with electrons is predicted by Madelung's rule, which uses the sum (n + l). A solid grasp of l is essential for correctly applying this filling rule.