Electron Configurations: Aufbau Order, Hund's Rule, and Ions

Shells, subshells, and orbital capacity

Each electron in an atom is described by four quantum numbers. The principal quantum number n (1, 2, 3, …) sets the shell. The angular momentum quantum number l runs from 0 to n − 1 and sets the subshell: l = 0, 1, 2, and 3 are labeled s, p, d, and f. The magnetic quantum number mₗ runs from −l to +l, so a subshell contains 2l + 1 orbitals. The spin quantum number mₛ is +½ or −½.

Because each orbital holds at most two electrons, an s subshell holds 2, p holds 6, d holds 10, and f holds 14. A complete shell holds 2n² electrons: 2 for n = 1, 8 for n = 2, and 18 for n = 3. These capacities are why the blocks of the periodic table are 2, 6, 10, and 14 columns wide.

Aufbau filling order and the periodic table

In the ground state, electrons occupy the lowest-energy subshells available; this is the aufbau principle. In a many-electron atom, subshell energy depends on l as well as n, because an s electron penetrates closer to the nucleus than a p or d electron of similar n and is less shielded by inner electrons. The resulting order for neutral atoms is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p.

The n + l rule reproduces this order: fill subshells in order of increasing n + l, and when two subshells tie, fill the one with lower n first. So 4s (4 + 0 = 4) fills before 3d (3 + 2 = 5). You can also read the order from the periodic table. Moving across a period, you pass through the s block, then the f block if present, then the d block, then the p block, with the d subshell one shell behind the period number and the f subshell two behind.

The order tells you which subshell receives the next electron from one element to the next. It does not mean 4s is always lower in energy than 3d. Once 3d is occupied, its energy falls below that of 4s, which is why transition-metal cations lose their 4s electrons first.

Pauli exclusion, Hund's rule, and orbital diagrams

The Pauli exclusion principle states that no two electrons in an atom share the same four quantum numbers. Two electrons in one orbital already share n, l, and mₗ, so they must have opposite spins, and no orbital can hold a third.

Hund's rule applies to degenerate orbitals, those of equal energy within a subshell. Electrons occupy them singly, with parallel spins, before any pairing begins. Pairing raises the energy because two electrons in one orbital repel more strongly, while parallel spins in separate orbitals gain exchange stabilization. Nitrogen's 2p³ therefore has three unpaired electrons, and oxygen's 2p⁴ has one pair and two unpaired electrons.

In an orbital diagram, draw one box per orbital, fill subshells in aufbau order, and within a subshell place one arrow in every box before adding a second. Atoms with unpaired electrons are paramagnetic and are drawn into a magnetic field; atoms with all electrons paired are diamagnetic.

Worked examples: selenium and zirconium

Example 1: selenium, Z = 34. Fill in aufbau order until 34 electrons are placed: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁴. Check the total: 2 + 2 + 6 + 2 + 6 + 2 + 10 + 4 = 34. For noble-gas notation, replace the first 18 electrons with [Ar], the noble gas that ends the previous period: [Ar] 3d¹⁰ 4s² 4p⁴. Listing subshells by shell number, as here, is a common convention; [Ar] 4s² 3d¹⁰ 4p⁴ describes the same configuration in filling order.

Selenium's four 4p electrons spread over three orbitals. Hund's rule puts one in each and pairs the fourth, leaving two unpaired electrons. Its six valence electrons, 4s² 4p⁴, match the outer pattern of oxygen and sulfur in group 16, which is why selenium forms the selenide ion, Se²⁻, isoelectronic with krypton.

Example 2: zirconium, Z = 40. After [Kr], which accounts for 36 electrons, four remain. The next subshells are 5s and then 4d, giving [Kr] 4d² 5s², which agrees with the NIST ground-state data. The two 4d electrons occupy separate orbitals with parallel spins, so zirconium has two unpaired electrons.

Chromium, copper, and exceptions that resist simple rules

The aufbau order predicts [Ar] 3d⁴ 4s² for chromium and [Ar] 3d⁹ 4s² for copper. Spectroscopy shows [Ar] 3d⁵ 4s¹ and [Ar] 3d¹⁰ 4s¹. Because 4s and 3d are close in energy in these atoms, moving one electron into 3d lowers the total energy, helped by the exchange stabilization of a half-filled or filled d subshell. Molybdenum, [Kr] 4d⁵ 5s¹, and silver, [Kr] 4d¹⁰ 5s¹, follow the same pattern as the elements above them.

The half-filled-subshell explanation is not a general rule. Niobium is [Kr] 4d⁴ 5s¹ although its 4d subshell ends up neither half filled nor filled; palladium is [Kr] 4d¹⁰ with no 5s electrons; and tungsten, directly below molybdenum, is [Xe] 4f¹⁴ 5d⁴ 6s². Learn the exceptions your course requires from data rather than deriving them from a rule that does not generalize.

Configurations of monatomic ions

For an anion, add electrons to the next available subshell: chlorine, [Ne] 3s² 3p⁵, becomes Cl⁻, [Ne] 3s² 3p⁶, isoelectronic with argon. For a cation, remove electrons from the subshell with the highest n first, and within that shell from p before s. For a first-row transition metal, that takes the 4s electrons before any 3d electrons.

Example 3: iron ions. Iron is [Ar] 3d⁶ 4s². Fe²⁺ loses both 4s electrons to give [Ar] 3d⁶, and Fe³⁺ loses one more, from 3d, to give [Ar] 3d⁵. Check the counts: iron has 26 electrons, Fe²⁺ has 18 + 6 = 24, and Fe³⁺ has 18 + 5 = 23. The free Fe³⁺ ion has five unpaired electrons, one in each 3d orbital.

The same logic applies in the p block. Tin is [Kr] 4d¹⁰ 5s² 5p², and Sn²⁺ loses its two 5p electrons to become [Kr] 4d¹⁰ 5s². Copper(I) is [Ar] 3d¹⁰, because copper's single 4s electron leaves first.

Common mistakes when writing configurations

Most configuration errors fall into a few patterns, and each can be caught with a quick check before you move on.

When a question asks for unpaired electrons, draw the orbital diagram for the partly filled subshell instead of judging from the configuration string. When it asks about an ion, write the neutral atom first, then add or remove electrons.

• Removing 3d electrons before 4s: Fe²⁺ is [Ar] 3d⁶, not [Ar] 3d⁴ 4s². Common first-row cations such as Fe²⁺, Co²⁺, and Cu⁺ have no 4s electrons.
• Using the wrong noble-gas core: the core is the noble gas from the previous period, [Ar] for elements 19 to 36.
• Miscounting electrons: the superscripts must add up to Z for an atom and to Z minus the charge for an ion.
• Overfilling a subshell, such as 2p⁷ or 3d¹¹.
• Pairing electrons in a p or d subshell before every orbital holds one.
• Placing the d block in the wrong shell: period 4 fills 3d, not 4d.
• Applying the Cr and Cu pattern to every element below them without checking data.