How to Count Unpaired Electrons in Atoms and Ions

“How many unpaired electrons does Fe³⁺ have?” is a classic chemistry question, and the answer tells you a lot: whether the ion is magnetic, how strongly, and often something about its colour and stability. Counting unpaired electrons is quick once you know which subshell to look at and how electrons fill it. This guide gives a step-by-step method, shortcuts, and plenty of examples.

Why unpaired electrons matter

• Magnetism: any unpaired electrons make a species paramagnetic; none makes it diamagnetic. See paramagnetic vs diamagnetic.
• Reactivity: species with unpaired electrons, called radicals when they’re molecules, are often very reactive.
• Bonding: the number of unpaired electrons in an atom often relates to the number of covalent bonds it forms (nitrogen, with three unpaired electrons, typically forms three bonds).
• Stability: half-filled subshells (with the maximum number of unpaired electrons) have extra stability.

The method in five steps

1. Find the number of electrons. Neutral atom: atomic number. Ion: atomic number minus the charge.
2. Write the electron configuration in noble gas shorthand. For transition metal ions, remove 4s electrons before 3d. See electron configurations of ions.
3. Ignore full subshells. Filled s² , p⁶, d¹⁰ and f¹⁴ subshells, and the noble gas core, have no unpaired electrons.
4. Look only at the partly filled subshell(s). Draw the right number of boxes: 3 for p, 5 for d, 7 for f.
5. Fill using Hund’s rule: one electron per box first, then pair up. Count the boxes containing a single electron. See Hund’s rule.

The shortcut rule

For a subshell with k orbitals (p: 3, d: 5, f: 7) containing x electrons:

• If x ≤ k (half-full or less): unpaired = x
• If x > k (more than half-full): unpaired = 2k − x

Examples:

• p⁴: 3 orbitals, 4 electrons (more than half) → 2(3) − 4 = 2
• d⁷: 5 orbitals, 7 electrons → 10 − 7 = 3
• f⁹: 7 orbitals, 9 electrons → 14 − 9 = 5

Quick-reference table for d electrons

d electrons	Box diagram	Unpaired
d¹	↑ _ _ _ _	1
d²	↑ ↑ _ _ _	2
d³	↑ ↑ ↑ _ _	3
d⁴	↑ ↑ ↑ ↑ _	4
d⁵	↑ ↑ ↑ ↑ ↑	5
d⁶	↑↓ ↑ ↑ ↑ ↑	4
d⁷	↑↓ ↑↓ ↑ ↑ ↑	3
d⁸	↑↓ ↑↓ ↑↓ ↑ ↑	2
d⁹	↑↓ ↑↓ ↑↓ ↑↓ ↑	1
d¹⁰	↑↓ ↑↓ ↑↓ ↑↓ ↑↓	0

Notice the symmetry: d⁴ and d⁶ both have 4; d³ and d⁷ both have 3. The maximum, 5, is at d⁵.

(For transition metal complexes with strongly interacting ligands, electrons may pair up earlier, giving “low-spin” configurations with fewer unpaired electrons. The table shows the “high-spin” or free-ion case, which is what most introductory questions expect.)

Worked examples: main-group atoms

Oxygen (Z = 8)

• Configuration: 1s² 2s² 2p⁴
• Partly filled subshell: 2p⁴ → ↑↓ ↑ ↑
• 2 unpaired electrons

Phosphorus (Z = 15)

• [Ne] 3s² 3p³ → ↑ ↑ ↑
• 3 unpaired

Chlorine (Z = 17)

• [Ne] 3s² 3p⁵ → ↑↓ ↑↓ ↑
• 1 unpaired

Argon (Z = 18)

• [Ne] 3s² 3p⁶: all subshells full
• 0 unpaired

Pattern for p-block atoms: groups 13 to 18 have 1, 2, 3, 2, 1, 0 unpaired electrons.

Worked examples: transition metal atoms and ions

Iron atom (Z = 26)

• [Ar] 3d⁶ 4s²
• 4s² is full; look at 3d⁶ → 10 − 6 = 4 unpaired

Fe³⁺

• Fe: [Ar] 3d⁶ 4s² → remove 2 from 4s and 1 from 3d → [Ar] 3d⁵
• 5 unpaired

Co²⁺ (Z = 27)

• Co: [Ar] 3d⁷ 4s² → Co²⁺: [Ar] 3d⁷
• 10 − 7 = 3 unpaired

Cu²⁺ (Z = 29)

• Cu: [Ar] 3d¹⁰ 4s¹ → Cu²⁺: [Ar] 3d⁹
• 1 unpaired

Chromium atom (Z = 24)

• Actual configuration: [Ar] 3d⁵ 4s¹ (an exception; see electron configuration exceptions)
• 3d⁵ gives 5 unpaired, and 4s¹ gives 1 more
• 6 unpaired (two partly filled subshells, so count both)

Sc³⁺ (Z = 21)

• Sc: [Ar] 3d¹ 4s² → Sc³⁺: [Ar]
• 0 unpaired

Worked examples: f-block

Gadolinium, Gd³⁺ (Z = 64)

• Gd: [Xe] 4f⁷ 5d¹ 6s² → Gd³⁺: [Xe] 4f⁷
• 7 electrons in 7 orbitals: 7 unpaired, the maximum for any common ion. That’s why gadolinium compounds are used as MRI contrast agents.

Er³⁺ (Z = 68)

• Er³⁺: [Xe] 4f¹¹
• 14 − 11 = 3 unpaired

Common mistakes

Mistake	Example	Fix
Removing 3d electrons before 4s in ions	Fe²⁺ as 3d⁴ 4s² (4 unpaired by luck, but wrong reasoning)	remove 4s first: 3d⁶
Pairing before filling every box	p³ drawn as ↑↓ ↑ _ (1 unpaired)	↑ ↑ ↑ (3 unpaired)
Forgetting a second partly filled subshell	Cr counted as 5	3d⁵ 4s¹ → 6
Using the wrong number of boxes	d⁷ in 4 boxes	d always has 5 boxes
Counting electrons in full subshells	counting the 2 in 4s²	full subshells contribute 0

Practice questions

Find the number of unpaired electrons.

1. Nitrogen (Z = 7)
2. Sulfur (Z = 16)
3. Mn²⁺ (Z = 25)
4. Ni²⁺ (Z = 28)
5. Ti²⁺ (Z = 22)
6. V (Z = 23)
7. Zn (Z = 30)
8. Ce³⁺ (Z = 58; Ce³⁺ is [Xe] 4f¹)

Answers:

1. 2p³ → 3
2. 3p⁴ → 2
3. [Ar] 3d⁵ → 5
4. [Ar] 3d⁸ → 2
5. [Ar] 3d² → 2
6. [Ar] 3d³ 4s² → 3
7. [Ar] 3d¹⁰ 4s² → 0
8. 4f¹ → 1

Check atoms with the electron configuration tool, which shows orbital boxes.

Unpaired electrons in molecules: radicals

The same counting logic applies to molecules and molecular fragments. A species with an unpaired electron is called a radical (or free radical). Examples include:

• Nitrogen monoxide, NO: 11 valence electrons, so one must be unpaired. This makes NO reactive; it quickly combines with oxygen in air to form brown NO₂, and it acts as a signalling molecule in the human body.
• Chlorine atoms, Cl•: formed when ultraviolet light breaks chlorine-containing molecules in the upper atmosphere. Each chlorine radical can destroy thousands of ozone molecules in a chain reaction.
• The hydroxyl radical, •OH: extremely reactive, it cleans the lower atmosphere by breaking down many pollutants.

A quick check for small molecules: if the total number of valence electrons is odd, there must be at least one unpaired electron. An even total usually (but not always, as oxygen shows) means all electrons are paired. See paramagnetic vs diamagnetic.

Why half-filled subshells are special

Counting unpaired electrons also reveals why half-filled subshells (p³, d⁵, f⁷) have extra stability. In these arrangements, every orbital holds exactly one electron with the same spin, which minimises electron repulsion and maximises the stabilising exchange energy. This explains nitrogen’s relatively high first ionisation energy, chromium’s unusual configuration and the stability of Fe³⁺ and Mn²⁺.

Connecting to measurements

Chemists can measure the number of unpaired electrons by determining a substance’s magnetic moment. For first-row transition metal ions, the spin-only formula μ = √[n(n + 2)] Bohr magnetons links the measured moment to n, the number of unpaired electrons. A measured moment of about 3.9 suggests n = 3; about 5.9 suggests n = 5. This is how chemists confirm oxidation states and spin states in new compounds.

Key takeaways

• Write the configuration, remove 4s before 3d for transition metal ions, and focus only on partly filled subshells.
• Fill p (3 boxes), d (5) and f (7) subshells singly before pairing.
• Shortcut: unpaired = x if the subshell is half-full or less; otherwise 2k − x.
• d⁵ and f⁷ give the maximum unpaired electrons (5 and 7).
• Unpaired electrons determine paramagnetism and can be measured through magnetic moments.