Quantum Numbers and Orbitals

What is it?

In quantum mechanics, an electron in an atom is described by an orbital: a region of space where the electron is likely to be found. Each orbital, and each electron in it, is labelled by quantum numbers, like an address:

Quantum number	Symbol	Allowed values	What it describes
Principal	$n$	1, 2, 3, …	The shell: size and main energy level
Angular momentum (azimuthal)	$l$	0 to $n - 1$	The subshell: shape of the orbital
Magnetic	$m_l$	$-l$ to $+l$ , in steps of 1	The orientation of the orbital in space
Spin	$m_s$	$+\tfrac{1}{2}$ or $-\tfrac{1}{2}$	The spin of the electron

The value of $l$ is usually given as a letter: $l = 0$ is s, $l = 1$ is p, $l = 2$ is d and $l = 3$ is f. So $n = 3, l = 1$ is the 3p subshell.

Key idea

The first three quantum numbers ( $n$ , $l$ , $m_l$ ) identify an orbital; the fourth ( $m_s$ ) identifies one of the two electrons in it. Pauli exclusion principle: no two electrons in an atom can have the same set of all four quantum numbers. That’s why each orbital holds at most two electrons, with opposite spins.

Why does it matter?

It explains the electron configuration rules. The number of $m_l$ values gives the number of orbitals per subshell (1, 3, 5, 7), and so the capacities 2, 6, 10 and 14 electrons that build the periodic table.
It explains the shape of the periodic table. The s, p, d and f blocks are 2, 6, 10 and 14 elements wide.
Orbital shapes explain bonding. The directions of p and d orbitals determine bond angles and molecular shapes.

How does it work?

1. Counting orbitals and electrons

For a given $l$ , $m_l$ takes $2l + 1$ values, so there are $2l + 1$ orbitals in the subshell:

Subshell	$l$	$m_l$ values	Orbitals	Max electrons
s	0	0	1	2
p	1	−1, 0, +1	3	6
d	2	−2, −1, 0, +1, +2	5	10
f	3	−3 to +3	7	14

Shell $n$ contains $n$ subshells, $n^2$ orbitals and at most $2n^2$ electrons. For example, the $n = 2$ shell has 2s and 2p: 4 orbitals, 8 electrons.

2. Orbital shapes

s orbitals are spherical. Larger $n$ gives a larger sphere.
p orbitals have two lobes (a dumbbell) on opposite sides of the nucleus, with a nodal plane between them. The three p orbitals point along the x, y and z axes: $p_x$ , $p_y$ , $p_z$ .
d orbitals mostly have four lobes (a cloverleaf); one ( $d_{z^2}$ ) has two lobes and a ring.

3. Nodes

A node is a surface where the chance of finding the electron is zero. An orbital has $n - 1$ nodes in total: $l$ of them are angular (planes or cones), and $n - l - 1$ are radial (spheres).

Think of it like this

The quantum numbers are like an address in a city. $n$ is the district (how far from the centre), $l$ is the type of street (its shape), $m_l$ is which way the street points, and $m_s$ is which of the two apartments in the building the electron lives in. No two electrons share the same full address.

More precisely

An orbital is a solution of the Schrödinger equation, a wavefunction, $\psi$ . The probability of finding the electron is proportional to $\psi^2$ . The usual drawings are boundary surfaces enclosing about 90 % of that probability. In hydrogen all subshells with the same $n$ have the same energy; in many-electron atoms, shielding makes s lower than p, and p lower than d, which leads to the Aufbau filling order. The two lobes of a p orbital have opposite phases (signs of $\psi$ ), which matters when orbitals overlap to form bonds.

Visualise it

Schematic orbital shapes. An s orbital is a sphere. The three p orbitals are dumbbells with two lobes along the x, y and z axes. A d orbital is drawn as a cloverleaf of four lobes between the axes. — Boundary surfaces of s, p and d orbitals (schematic). The two colours show the opposite phases of the lobes.

Worked example

Worked example: Allowed values in a shell

Question: List the allowed values of $l$ and $m_l$ for $n = 3$ . How many orbitals and electrons can the shell hold?

$l$ = 0, 1, 2 (3s, 3p, 3d).
$m_l$ : for $l = 0$ : 0; for $l = 1$ : −1, 0, +1; for $l = 2$ : −2, −1, 0, +1, +2.
Orbitals: 1 + 3 + 5 = 9 $= 3^2$ . Electrons: $2 \times 9 =$ 18 $= 2 \times 3^2$ .

Worked example: Which sets are allowed?

Question: Which of these sets $(n, l, m_l, m_s)$ are allowed? (a) $(2, 1, 0, +\tfrac{1}{2})$ (b) $(2, 2, 0, +\tfrac{1}{2})$ (c) $(3, 1, -2, -\tfrac{1}{2})$ (d) $(4, 0, 0, 0)$

(a) Allowed: $l$ is below $n$ , $m_l$ lies between $-l$ and $+l$ : a 2p electron.
(b) Not allowed: $l$ must be at most $n - 1 = 1$ .
(c) Not allowed: for $l = 1$ , $m_l$ can only be −1, 0 or +1.
(d) Not allowed: $m_s$ must be $+\tfrac{1}{2}$ or $-\tfrac{1}{2}$ .

Worked example: Naming a subshell

Question: Name the subshell with $n = 4$ , $l = 2$ , and state how many electrons it can hold.

$l = 2$ is d, so this is 4d. It has $2l + 1 = 5$ orbitals, holding 10 electrons.

Worked example: Quantum numbers of a valence electron

Question: Give a possible set of quantum numbers for the outermost electron of sodium (1s² 2s² 2p⁶ 3s¹).

The electron is in 3s: $n = 3$ , $l = 0$ , $m_l = 0$ , $m_s = +\tfrac{1}{2}$ (or $-\tfrac{1}{2}$ ).

Worked example: Counting nodes

Question: How many nodes of each kind does a 3p orbital have?

Total $= n - 1 = 2$ . Angular $= l = 1$ (a nodal plane). Radial $= n - l - 1 = 1$ (a nodal sphere).

Common mistake

Common mistake: Letting l equal n

$l$ runs from 0 to $n - 1$ . There is no 1p or 2d subshell: the first p subshell is 2p and the first d subshell is 3d.

Common mistake: Confusing orbitals with electrons

A p subshell has 3 orbitals but holds up to 6 electrons. Always check whether a question asks for orbitals or electrons.

Common mistake: Thinking an orbital is a path

An orbital is not an orbit the electron travels around. It is a region of probability; the electron has no definite path.

Notation note

$l$ is a lower-case L; some books write it $\ell$ .
A subshell is written number + letter (3p), and its electron count as a superscript (3p⁴).

Remember this

$n$ = 1, 2, 3…; $l$ = 0 to $n - 1$ (s, p, d, f); $m_l$ = $-l$ to $+l$ ; $m_s$ = $\pm\tfrac{1}{2}$ .
Orbitals per subshell: $2l + 1$ (1, 3, 5, 7). Per shell: $n^2$ orbitals, $2n^2$ electrons.
Pauli: no two electrons have the same four quantum numbers, so 2 electrons per orbital.
s spherical, p two lobes (x, y, z), d mostly four lobes.
Nodes: $n - 1$ total; $l$ angular; $n - l - 1$ radial.

Practise this topic with flashcards and a quiz at chemistryclarity.com/chemistry/quantum-numbers-and-orbitals/