Worked examples — Hund's rule of maximum multiplicity

This page is the "drill ground" for Hund's rule. The parent note told you the rule; here we hammer it against every kind of case you can meet — empty seats, half-full seats, over-full seats, the tricky Cr/Cu exceptions, ions where you remove electrons, and an exam twist that tries to fool you. If a scenario exists, it has a worked example below.

Before we start, one reminder of the vocabulary we will use again and again — no symbol will appear until it means something to you.

We draw orbitals as boxes and electrons as arrows: ↑ = spin up, ↓ = spin down. A box with ↑↓ is paired; a box with a single arrow is unpaired.

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The scenario matrix

Every case Hund's rule can throw at you falls into one of these cells. The examples below are tagged with the cell(s) they cover, so you can see the whole space is filled.

Cell	What makes it special	Covered by
A. Empty → part-filled, below half	fewer electrons than orbitals; all stay unpaired	Ex 1 (Carbon $2p^2$)
B. Exactly half-filled	one electron per orbital, max unpaired	Ex 2 (Nitrogen $2p^3$)
C. Above half → pairing begins	more electrons than orbitals; forced pairs	Ex 3 (Oxygen $2p^4$)
D. Fully filled (degenerate zero-spin)	every box ↑↓; $n=0$	Ex 4 ($2p^6$ / Ne)
E. Exception by exchange (half-full $d$)	Aufbau "prediction" beaten by Hund/exchange	Ex 5 (Chromium)
F. Exception by exchange (full $d$)	$d^{10}$ steals from $4s$	Ex 6 (Copper)
G. Ion — remove electrons	strip $4s$ before $3d$; recount	Ex 7 ($F{e}^{2+}$, $M{n}^{2+}$)
H. Real-world / magnetism word problem	go from a measured $\mu$ back to $n$	Ex 8 (unknown ion)
I. Exam twist — the trap	"more electrons = more multiplicity?"	Ex 9 ($2p^6$ vs $2p^3$)

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A → D: the second-period subshell, filled one electron at a time

The clearest way to see cells A, B, C, D is to watch the three $2p$ boxes fill up from empty.

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E, F: the two famous exceptions — exchange energy beats naïve Aufbau

A pure Aufbau count would predict Chromium as $3d^{4}4s^2$ and Copper as $3d^{9}4s^2$. Nature disagrees, because promoting one $4s$ electron into $3d$ creates a half-filled or fully-filled $d$ shell whose extra exchange energy outweighs the small cost of moving an electron. The figure counts the exchange pairs on both sides.

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G: ions — remove electrons in the right order, then recount

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H: a magnetism word problem — running the logic backwards

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I: the exam twist that catches everyone

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Active Recall

Recall Which cell forces the first pairing in

$p$ orbitals? Cell C — the 4th $p$ electron ($p^4$, e.g. oxygen), since only 3 boxes exist. ::: Answer above. Multiplicity of oxygen $2p^4$? ::: 3 (two unpaired).

Recall Why is Cr

$3d^{5}4s^1$ and not $3d^{4}4s^2$? Going to $3d^5$ keeps all five spins parallel, raising exchange pairs from 6 to 10 ($+4K$), which outweighs the promotion cost. ::: See Ex 5.

Recall From which orbital do you remove electrons first in

$Fe\to F{e}^{2+}$? The $4s$ orbital (highest $n$), not $3d$. ::: Ex 7.

Recall Given

$\mu \approx 5.92{\mu }_B$, how many unpaired electrons? Solve $n(n+2)=35\Rightarrow n=5$. ::: Ex 7 ($M{n}^{2+}$).

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Connections

• Aufbau Principle — sets which subshell fills and the $4s$-before-$3d$ order that Ex 7 unwinds.
• Pauli Exclusion Principle — why pairs must be opposite-spin (Ex 3).
• Electron Spin Quantum Number — the $+\frac{1}{2}$ each lone electron contributes to $S$.
• Exchange Energy and Half-filled Stability — the deep "why" of Ex 5 & 6.
• Paramagnetism and Diamagnetism — unpaired ⇒ paramagnetic (Ex 1, 3, 9).
• Magnetic Moment of Atoms — $\mu =\sqrt{n(n+2)}$, used to close every example.
• Hund's rule of maximum multiplicity — the parent statement of the rule.