Orbitals and Quantum Numbers – Electron Configurations of Atoms and Ions (Senior 4 Chemistry) | Ganzaa

Electron Configurations of Atoms and Ions

Lessons

Senior 4 · Chemistry

Electron Configurations of Atoms and Ions

5 lessons

Lesson 3 of 5

Orbitals and Quantum Numbers

25 min

A blur, not a track

Photograph a spinning fan with a slow shutter and you do not see the blades; you see a blurred disc showing where the blades spend their time. The modern picture of an electron is like that blur. In the 1920s experiments showed that electrons behave as waves as well as particles: a beam of electrons is diffracted by a crystal, just as light is diffracted by a grating. A wave is spread out, so an electron in an atom cannot follow a sharp track like Bohr's orbit. In 1926 Erwin Schrödinger wrote an equation for the electron wave. Its solutions describe the regions where an electron is likely to be found, called orbitals, and each is labelled by a set of quantum numbers.

Definition

Atomic orbital: a region of space around the nucleus where there is a high probability (usually taken as 90%) of finding an electron. Each orbital holds at most two electrons.

Orbit and orbital are not the same. An orbit (Bohr) is a fixed circular path; an orbital is a three-dimensional region of probability, with a definite shape and energy.

The four quantum numbers

Every electron in an atom is described by four quantum numbers. The first three describe the orbital; the fourth describes the electron itself.

Table 3.1: The quantum numbers

Quantum number	Symbol and allowed values	What it tells you
principal	$n = 1, 2, 3, \dots$	the shell (main energy level): its size and most of its energy
azimuthal (subsidiary)	$l = 0, 1, \dots, n - 1$	the subshell and the shape of its orbitals: $l = 0$ s, $l = 1$ p, $l = 2$ d, $l = 3$ f
magnetic	$m_l = -l, \dots, 0, \dots, +l$	the orientation of the orbital in space; a subshell has $2 l + 1$ orbitals
spin	$m_s = +\frac{1}{2}$ or $-\frac{1}{2}$	the direction of the electron's spin, often drawn as ↑ or ↓

Definition

Pauli exclusion principle: no two electrons in the same atom can have the same set of four quantum numbers. Two electrons in one orbital already share $n$ , $l$ and $m_{l}$ , so they must differ in $m_{s}$ : an orbital holds at most two electrons, with opposite spins.

Shells, subshells and orbitals

Combining the rules gives the structure of each shell (Table 3.2). Shell $n$ has $n$ subshells and $n^{2}$ orbitals, so it holds at most $2 n^{2}$ electrons.

Table 3.2: Shells n = 1 to 4

Shell $n$	Values of $l$	Subshells	Orbitals in each subshell	Total orbitals $n^{2}$	Maximum electrons $2 n^{2}$
1	0	1s	1	1	2
2	0, 1	2s, 2p	1, 3	4	8
3	0, 1, 2	3s, 3p, 3d	1, 3, 5	9	18
4	0, 1, 2, 3	4s, 4p, 4d, 4f	1, 3, 5, 7	16	32

Because $l$ can be at most $n - 1$ , there is no 1p, 2d or 3f subshell.

The shapes of the orbitals

s orbitals ( $l = 0$ ) are spherical, centred on the nucleus. There is one s orbital in every shell, and it gets bigger as $n$ increases: 2s is larger than 1s.

p orbitals ( $l = 1$ ) are dumbbell-shaped, with two lobes on opposite sides of the nucleus and zero probability at the nucleus itself. There are three in each p subshell, at right angles to each other, along the x, y and z axes: $p_{x}$ , $p_{y}$ , $p_{z}$ . They have the same energy.

d orbitals ( $l = 2$ ) come in sets of five. Three have four lobes lying between the axes ( $d_{xy}$ , $d_{xz}$ , $d_{yz}$ ); $d_{x^2-y^2}$ has four lobes along the x and y axes; and $d_{z^2}$ has two lobes along z with a ring around its middle. In an isolated atom all five have the same energy. f orbitals ( $l = 3$ ), seven per subshell, have more complicated shapes that you do not need to draw.

Worked example 1: which sets are allowed?

Say whether each set $(n, l, m_{l}, m_{s})$ is allowed. If it is, name the subshell. (a) $(2, 1, 0, +\frac{1}{2})$ (b) $(2, 2, 1, -\frac{1}{2})$ (c) $(3, 1, -2, +\frac{1}{2})$ (d) $(4, 3, -3, -\frac{1}{2})$

Solution. Check $l \le n - 1$ , then $-l \le m_l \le +l$ , then $m_s = \pm\frac{1}{2}$ .

(a) $l = 1 \le 1$ ; $m_{l} = 0$ is within $\pm 1$ : allowed, a 2p electron.
(b) $l = 2$ is greater than $n - 1 = 1$ : not allowed (there is no 2d).
(c) $l = 1$ , so $m_{l}$ can only be $- 1, 0, + 1$ : not allowed.
(d) $l = 3 \le 3$ ; $m_{l} = - 3$ is within $\pm 3$ : allowed, a 4f electron.

Worked example 2: counting electrons

How many electrons in an atom can have (a) $n = 3$ ; (b) $n = 3$ and $l = 2$ ; (c) $n = 4$ , $l = 1$ and $m_{l} = 0$ ; (d) $n = 2$ and $m_s = +\frac{1}{2}$ ?

Solution.

(a) $2n^2 = 2 \times 9 = 18$ .
(b) The 3d subshell: $2 l + 1 = 5$ orbitals, $5 \times 2 = 10$ .
(c) One orbital (a 4p orbital): 2 electrons.
(d) Shell 2 has 4 orbitals; one electron in each can have spin $+\frac{1}{2}$ : 4 electrons.

Warning

Common mistakes. (1) $l$ runs from 0 to $n - 1$ , never up to $n$ . (2) The number of orbitals in a subshell is $2 l + 1$ , but the number of electrons is twice that. (3) Do not call an orbital a "path": the electron's position is described only by probability. (4) The three p orbitals of one subshell have the same energy; they differ only in direction.

Tip

Remember s p d f → 1 3 5 7 orbitals → 2 6 10 14 electrons. Shell $n$ holds $2 n^{2}$ : 2, 8, 18, 32.

Professor Dave Explains: Quantum Numbers, Atomic Orbitals, and Electron Configurations (9 min). Watch the first part for what each quantum number describes and how the s, p and d orbital shapes follow from them.

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Practice

Name the four quantum numbers and give the allowed values of each.
Give the values of $n$ and $l$ for: 3p, 4s, 5d, 4f.
Which of these subshells cannot exist: 2p, 3f, 1s, 2d, 4d? Explain.
How many orbitals are there in (a) a 4p subshell, (b) the $n = 4$ shell? How many electrons can (c) a 5f subshell hold?
Write a possible set of four quantum numbers for the outermost electron of a sodium atom (configuration ending $3 s^{1}$ ).
Sketch a $2 p_{z}$ orbital on x, y and z axes. Where is the probability of finding the electron zero?
Use the Pauli principle to explain why the 2p subshell can hold six electrons but no more.

Show answersHide answers

Principal $n = 1, 2, 3, \dots$ ; azimuthal $l = 0$ to $n - 1$ ; magnetic $m_{l} = - l$ to $+ l$ ; spin $m_s = \pm\frac{1}{2}$ .
3p: $n = 3, l = 1$ ; 4s: $n = 4, l = 0$ ; 5d: $n = 5, l = 2$ ; 4f: $n = 4, l = 3$ .
3f ( $l = 3$ needs $n \ge 4$ ) and 2d ( $l = 2$ needs $n \ge 3$ ).
(a) 3; (b) $n^{2} = 16$ ; (c) $7 \times 2 = 14$ .
$n = 3, l = 0, m_l = 0, m_s = +\frac{1}{2}$ (or $-\frac{1}{2}$ ).
Two lobes along the z axis, one above and one below the nucleus; the probability is zero at the nucleus, in the xy plane (the node between the lobes).
2p has three orbitals ( $m_{l} = - 1, 0, + 1$ ). In each, the two electrons must have opposite spins, so each orbital holds two: $3 \times 2 = 6$ . A seventh electron would have to repeat a set of four quantum numbers.

Quick check

Key points

An orbital is a region of high probability of finding an electron; it holds at most two electrons, with opposite spins (Pauli).
$n$ gives the shell; $l = 0$ to $n - 1$ gives the subshell (s, p, d, f); $m_{l} = - l$ to $+ l$ gives the orbital's orientation; $m_s = \pm\frac{1}{2}$ gives the spin.
s, p, d, f subshells have 1, 3, 5, 7 orbitals; shell $n$ has $n^{2}$ orbitals and holds $2 n^{2}$ electrons.
s orbitals are spheres; p orbitals are dumbbells along x, y and z; d orbitals have four lobes, except $d_{z^2}$ .

Sources and credits

Written by Ganzaa.
Video: Quantum Numbers, Atomic Orbitals, and Electron Configurations, Professor Dave Explains, embedded from YouTube.
The orbital diagrams, tables, worked examples and Practice items were written and drawn by Ganzaa.

1A blur, not a track

2The four quantum numbers

3Shells, subshells and orbitals

4The shapes of the orbitals

A blur, not a track

The four quantum numbers

Shells, subshells and orbitals

The shapes of the orbitals