Difficulty: Introductory to Intermediate | Prerequisites: Part 1 of these notes (Subatomic Particles, Mass, and the Bohr Model).
This picks up where the Bohr model leaves off. The quantum mechanical model replaces fixed orbits with probability clouds, and the four quantum numbers give you a precise address for every electron in an atom. Electron configuration, in turn, is how you predict an element's chemistry from its place on the periodic table. If you have not reviewed Part 1, do that first.
The quantum mechanical model describes electrons as probability clouds (orbitals) rather than fixed paths. Four quantum numbers (n, l, m_l, m_s) give each electron a unique address inside the atom. Electron configuration rules (Aufbau, Pauli exclusion, Hund's rule) tell you the order in which electrons fill those orbitals, which in turn predicts an element's chemical behaviour.
Orbital
A region of space around the nucleus where there is a high probability of finding an electron.
Think of it as a probability cloud, not a fixed path. You cannot say exactly where the electron is, only where it is most likely to be.
Heisenberg Uncertainty Principle
It is impossible to simultaneously determine both the exact momentum and the exact position of an electron.
In simple terms, the more precisely you know where an electron is, the less precisely you can know how fast it is moving, and vice versa.
Pauli Exclusion Principle
No two electrons in the same atom can have the same set of all four quantum numbers.
In simple terms, every electron in an atom has a unique address. Two electrons can share an orbital, but they must have opposite spins.
Principal quantum number (n)
An integer (1, 2, 3, ...) that indicates the main energy level (shell) of an electron. Larger n means higher energy and greater distance from the nucleus.
Think of it as the floor number in a building: higher floor, higher energy.
Azimuthal quantum number (l)
Also called the angular momentum quantum number. Ranges from 0 to (n - 1). It determines the shape of the orbital and the subshell.
In simple terms, l tells you the shape: l = 0 is spherical (s), l = 1 is dumbbell-shaped (p), l = 2 is cloverleaf (d), l = 3 is complex (f).
Magnetic quantum number (m_l)
Ranges from -l to +l. It specifies the orientation of the orbital within a subshell.
Think of it as pointing the orbital in a particular direction in 3D space.
Spin quantum number (m_s)
Either +1/2 or -1/2. It describes the intrinsic spin of the electron.
In simple terms, two electrons in the same orbital must spin in opposite directions.
Spectroscopic notation
A shorthand that combines the principal quantum number and the subshell letter: 1s, 2p, 3d, 4f, and so on.
Think of it as the postal code for an electron's subshell.
Aufbau principle
Electrons fill orbitals starting from the lowest available energy level before moving to higher ones.
In simple terms, electrons are lazy: they settle into the lowest-energy seat first.
n+l rule (Madelung rule)
Subshells fill in order of increasing (n + l). When two subshells have the same (n + l) sum, the one with the lower n fills first.
This is why 4s fills before 3d: 4 + 0 = 4, but 3 + 2 = 5.
Hund's rule
Within a subshell, electrons fill each orbital singly (with parallel spins) before any orbital gets a second electron.
Think of it as passengers on a bus: everyone takes their own seat before anyone doubles up.
Valence electrons
The electrons in the outermost shell of an atom. These are the electrons involved in chemical bonding and reactions.
Paramagnetic
A material with one or more unpaired electrons. These materials are attracted to an external magnetic field.
Diamagnetic
A material in which all electrons are paired. These materials are weakly repelled by an external magnetic field.
The Bohr model treated electrons as moving in fixed circular orbits. The quantum mechanical model replaces these with orbitals: three-dimensional regions of space where you are most likely to find an electron
You cannot pinpoint exactly where an electron is at any given moment. Instead, you describe the probability of finding it in a given region of space
It is impossible to simultaneously know both the exact position and the exact momentum of an electron
This is not a limitation of measuring instruments; it is a fundamental property of nature at the subatomic scale
Consequence: we describe electrons using probability distributions (orbitals), not precise locations
Every electron in an atom is described by a unique set of four quantum numbers. Together they specify the electron's energy level, orbital shape, orbital orientation, and spin direction.
Values: 1, 2, 3, 4, ...
Determines the main energy level (shell) and the approximate size of the orbital
Larger n = higher energy, greater distance from the nucleus
Maximum number of electrons in a shell = 2n²
The energy gap between shells decreases as n increases (the shells get closer together in energy the farther you go from the nucleus)
Values: 0 to (n - 1)
Determines the shape of the orbital and identifies the subshell
n tells you how many subshells are available: if n = 3, then l = 0, 1, 2 (three subshells)
Spectroscopic notation maps l to a letter:
l = 0: s subshell (spherical)
l = 1: p subshell (dumbbell)
l = 2: d subshell (cloverleaf)
l = 3: f subshell (complex)
Example: n = 4, l = 2 is the 4d subshell
Maximum number of electrons in a subshell = 4l + 2
Values: integers from -l to +l (including zero)
Specifies the orientation of the orbital within a subshell
Each distinct value of m_l corresponds to one orbital
Each orbital holds a maximum of 2 electrons
Example: for the p subshell (l = 1), m_l = -1, 0, +1, giving three orbitals (p_x, p_y, p_z)
s subshells are spherical (one orientation); p subshells are dumbbell-shaped (three orientations)
Values: +1/2 or -1/2
Describes the intrinsic spin of the electron
Two electrons sharing the same orbital must have opposite spins (one +1/2, one -1/2)
Quantum Number | Symbol | Possible Values | What It Tells You |
|---|---|---|---|
Principal | n | 1, 2, 3, 4, ... | Energy level (shell), size |
Azimuthal | l | 0, 1, 2, ..., (n-1) | Orbital shape, subshell |
Magnetic | m_l | -l, ..., -1, 0, 1, ..., +l | Orbital orientation |
Spin | m_s | +1/2, -1/2 | Electron spin direction |
Electron configuration uses spectroscopic notation: the principal quantum number, the subshell letter, and a superscript for the number of electrons in that subshell
Example: 2p⁴ means the second energy level, p subshell, containing 4 electrons
Electrons fill subshells starting from the lowest energy and working upward
Each subshell is completely filled before electrons begin entering the next one
Subshells fill in order of increasing (n + l) sum
When two subshells have the same (n + l) value, the one with the smaller n fills first
This explains the filling order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p
s: 2 electrons (1 orbital)
p: 6 electrons (3 orbitals)
d: 10 electrons (5 orbitals)
f: 14 electrons (7 orbitals)
Anion (negative ion): add electrons to the lowest-energy available subshell
Cation (positive ion): remove electrons from the subshell with the highest value of n first
This is a common exam point: when forming a cation from a transition metal, you remove the 4s electrons before the 3d electrons, even though 4s filled first
Within a subshell, electrons occupy each orbital singly (all with the same spin direction) before any orbital receives a second electron
Example: the 2p subshell has three orbitals (p_x, p_y, p_z). The first three electrons go one into each orbital, all with parallel spins. Only the fourth electron pairs up
Half-filled and fully filled subshells are more stable than partially filled ones
Chromium: expected configuration 4s² 3d⁴, actual configuration 4s¹ 3d⁵. The half-filled d subshell is more stable
Copper: expected 4s² 3d⁹, actual 4s¹ 3d¹⁰. The fully filled d subshell is more stable
This half-filled/full stability applies to d and f subshells, but never to p subshells. For p, the stability gain does not outweigh the cost of promoting an electron
Paramagnetic: the atom or material has at least one unpaired electron. An external magnetic field aligns the spins of unpaired electrons, causing attraction to the magnet
Diamagnetic: all electrons are paired. The material is weakly repelled by an external magnetic field
To determine which category an atom falls into, write its electron configuration and check whether any unpaired electrons remain
\text{Max electrons in shell } n = 2n^2Example: n = 3 holds up to 2(9) = 18 electrons.
\text{Max electrons in subshell } l = 4l + 2Example: d subshell (l = 2) holds up to 4(2) + 2 = 10 electrons.
n: 1, 2, 3, ... (positive integers)
l: 0, 1, 2, ..., (n - 1)
m_l: -l, -(l-1), ..., 0, ..., (l-1), l
m_s: +1/2 or -1/2
The number of orbitals equals (2l + 1). Each orbital holds 2 electrons.
Electron configuration determines how elements sit on the periodic table and why certain groups share chemical properties (the alkali metals all have one valence s electron, for instance). Paramagnetism and diamagnetism have practical uses: MRI machines exploit the magnetic behaviour of hydrogen nuclei and unpaired electrons in contrast agents, and magnetic susceptibility measurements help chemists confirm the electronic structure of transition-metal complexes.
Students often think electrons orbit the nucleus in neat circles (the Bohr picture). They do not. Orbitals are probability clouds; you never know the exact path of an electron.
Students often think the 4s subshell is always higher in energy than 3d. It is not always so. Once the 3d subshell begins filling, 4s can become higher in energy, which is why cations lose 4s electrons first.
Students often think Hund's rule means you should always promote an electron to get a half-filled subshell. You should not. The half-filled stability exception applies to d and f subshells (chromium, copper), never to p subshells.
Students often confuse the azimuthal quantum number (l) with the magnetic quantum number (m_l). Remember: l gives the shape of the subshell; m_l gives the orientation of a specific orbital within that subshell.
⚠️ Be able to write the full electron configuration for any element up to at least the fourth period. Know the filling order cold.
⚠️ Know the chromium (4s¹ 3d⁵) and copper (4s¹ 3d¹⁰) exceptions and be able to explain why they occur.
⚠️ When writing the electron configuration of a cation, remove electrons from the highest-n subshell first (e.g., remove 4s before 3d). This is a very common exam question for transition metals.
⚠️ Be able to list all four quantum numbers for a given electron and explain what each one describes.
⚠️ Know how to determine whether an atom or ion is paramagnetic or diamagnetic from its electron configuration.
True or false: An electron with quantum numbers n = 2, l = 2 is possible. (False. l can only go up to n - 1, so for n = 2, the maximum l is 1.)
Fill in the blank: The maximum number of electrons in a shell with n = 4 is ______. (32, because 2n² = 2 x 16 = 32.)
True or false: When forming Fe²⁺ from Fe, you remove the two 3d electrons first. (False. You remove the two 4s electrons first.)
Fill in the blank: Chromium's actual electron configuration is [Ar] ______ ______. (4s¹ 3d⁵.)
True or false: A diamagnetic atom has at least one unpaired electron. (False. Diamagnetic atoms have all electrons paired.)
Q: Write the full electron configuration of iron (Fe, Z = 26).
A: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶.
Q: Write the electron configuration of Fe²⁺.
A: 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁶. The two 4s electrons are removed first because 4s has the highest principal quantum number.
Q: What are the four quantum numbers for one of the 3p electrons in aluminium?
A: n = 3, l = 1, m_l = -1 (or 0 or +1), m_s = +1/2 (or -1/2). Any valid combination where l = 1 and m_l is within range is acceptable.
Q: Why does chromium have the configuration [Ar] 4s¹ 3d⁵ rather than [Ar] 4s² 3d⁴?
A: A half-filled d subshell (3d⁵) is unusually stable due to exchange energy. Promoting one 4s electron into 3d achieves this half-filled state, which lowers the overall energy.
Q: Is Cu²⁺ paramagnetic or diamagnetic? Explain.
A: Cu²⁺ has the configuration [Ar] 3d⁹, which contains one unpaired electron. It is paramagnetic.
Electron configuration connects directly to periodic trends (ionisation energy, electron affinity, electronegativity, atomic radius), because the way electrons fill orbitals determines each of those properties. It also connects to chemical bonding: the number and arrangement of valence electrons dictate whether an atom forms ionic, covalent, or metallic bonds. Understanding paramagnetism and diamagnetism is revisited in coordination chemistry and transition-metal chemistry later in the course.
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