4.1 · Atomic structure

Quantum numbers and orbitals

The address of every electron in an atom.

By the end you should be able to:

Use the four quantum numbers and relate them to orbitals and subshells
Describe the shapes of s, p and d orbitals and count radial and angular nodes

Key idea

The four quantum numbers

Symbol	Name	Allowed values	Describes
$n$	principal	1, 2, 3, …	the shell: size and (in hydrogen) energy
$l$	angular momentum	0 to $n - 1$	the subshell shape: s, p, d, f for $l$ = 0, 1, 2, 3
$m_l$	magnetic	integers from $-l$ to $+l$	the orientation of the orbital
$m_s$	spin	$+\tfrac{1}{2}$ or $-\tfrac{1}{2}$	the electron's spin

The first three label an orbital; all four label an electron. By the Pauli exclusion principle no two electrons in an atom share all four, so each orbital holds at most two electrons, with opposite spins.

Key idea

Allowed subshells and orbital counts

$n$	Subshells ( $l$ )	Orbitals per subshell	Orbitals in the shell	Maximum electrons
1	1s (0)	1	1	2
2	2s (0), 2p (1)	1, 3	4	8
3	3s (0), 3p (1), 3d (2)	1, 3, 5	9	18
4	4s, 4p, 4d, 4f (0 to 3)	1, 3, 5, 7	16	32

A subshell with quantum number $l$ has $2l + 1$ orbitals (one for each $m_l$ value). Shell $n$ has $n$ subshells, $n^2$ orbitals and room for $2n^2$ electrons.

Key idea

Orbital shapes

s ( $l = 0$ ): spherical; one orbital per shell.
p ( $l = 1$ ): two lobes on opposite sides of the nucleus, separated by a nodal plane through the nucleus; three orbitals, $p_x$ , $p_y$ and $p_z$ .
d ( $l = 2$ ): five orbitals. Four have four lobes (a cloverleaf) between two nodal planes; $d_{z^2}$ has two lobes along $z$ and a ring around the middle, with two conical nodes.

Orbitals grow with $n$ : 2s is larger than 1s, 3p larger than 2p. The sign (phase) of $\psi$ changes across every node, which matters for bonding in module 7.

Formula

Counting nodes

$\text{radial nodes} = n - l - 1 \qquad \text{angular nodes} = l \qquad \text{total} = n - 1$

Orbital	Radial	Angular	Total
1s	0	0	0
2s	1	0	1
2p	0	1	1
3s	2	0	2
3p	1	1	2
3d	0	2	2
4d	1	2	3

A radial node is a sphere around the nucleus where $\psi = 0$ ; an angular node is a plane or cone through the nucleus.

Method

Checking a set of quantum numbers

$n$ must be a positive integer: 1, 2, 3, …
$l$ must be an integer from 0 to $n - 1$ .
$m_l$ must be an integer from $-l$ to $+l$ .
$m_s$ must be $+\tfrac{1}{2}$ or $-\tfrac{1}{2}$ .
Name the orbital: the value of $n$ followed by the letter for $l$ (0 s, 1 p, 2 d, 3 f).

$(n, l, m_l, m_s) = (3, 2, -1, +\tfrac{1}{2})$ is allowed: a 3d electron. $(2, 2, 0, +\tfrac{1}{2})$ is not, because $l$ must be less than $n$ .

Common mistake

Quantum-number traps

Wrong: writing "2d" or "1p". Right: $l \le n - 1$ , so shell 1 has only 1s and shell 2 has only 2s and 2p.

Wrong: $(3, 1, -2, +\tfrac{1}{2})$ . Right: $|m_l|$ cannot exceed $l$ ; a p orbital ( $l = 1$ ) has $m_l$ = −1, 0 or +1 only.

Wrong: "3p has two radial nodes." Right: radial nodes $= n - l - 1 = 1$ , plus $l = 1$ angular node, for 2 nodes in total.

Wrong: $m_s = 0$ or $\pm 1$ . Right: only $+\tfrac{1}{2}$ or $-\tfrac{1}{2}$ .

Worked example

Worked example: counting orbitals, electrons and nodes

Orbitals with $n = 3$ : subshells 3s, 3p, 3d give $1 + 3 + 5$ = 9 orbitals (= $n^2$ ).
Electrons with $n = 4$ , $l = 2$ : the 4d subshell has 5 orbitals, so 10 electrons.
Electrons with $n = 3$ , $m_l = +1$ : $m_l = +1$ exists for $l = 1$ and $l = 2$ , so 2 orbitals and 4 electrons.
Nodes in a 4f orbital ( $n = 4$ , $l = 3$ ): radial $4 - 3 - 1 = 0$ , angular 3, 3 in total.

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