Quantum Numbers, Electron Configurations, and Magnetism in Atoms

Quantum Numbers and Electron Spin

Quantum Numbers

Quantum numbers are used to describe the unique quantum state of an electron in an atom. Each electron is defined by a set of four quantum numbers:

• Principal Quantum Number (n): Indicates the main energy level or shell (n = 1, 2, 3, ...).

• Angular Momentum Quantum Number (l): Defines the subshell or orbital shape (l = 0, 1, ..., n-1).

• Magnetic Quantum Number (m_l): Specifies the orientation of the orbital (m_l = -l, ..., 0, ..., +l).

• Spin Quantum Number (m_s): Describes the spin of the electron (m_s = +1/2 or -1/2).

Electron spin is an intrinsic property of electrons, creating a magnetic field. The two possible values of m_s correspond to "spin up" (+1/2) and "spin down" (-1/2).

The Pauli Exclusion Principle

Fundamental Rule for Electron Arrangement

• No two electrons in the same atom can have the same set of all four quantum numbers.

• Each orbital can hold a maximum of two electrons, and these must have opposite spins.

This principle explains the structure of electron shells and subshells in atoms.

Electron Capacity of Shells and Subshells

Maximum Number of Electrons

• For n = 1: Only l = 0, m_l = 0, m_s = +1/2 or -1/2 → 2 electrons

• For n = 2: l = 0 (2e-) and l = 1 (6e-) → Total 8 electrons

• For n = 3: 3s (2e-), 3p (6e-), 3d (10e-) → Total 18 electrons

• For n = 4: 4s (2e-), 4p (6e-), 4d (10e-), 4f (14e-) → Total 32 electrons

General Rule: The nth shell can accommodate up to 2n2 electrons.

Assigning Electrons to Atoms

Order of Filling Orbitals

• Electrons fill orbitals of increasing energy.

• For hydrogen (and one-electron systems), energy depends only on n: $E = -\frac{Rhc}{n^2}$

• For many-electron atoms, energy depends on both n and l due to electron shielding and repulsion.

Energy Levels in Multi-Electron Atoms

Shielding and Subshell Energy

• Electrons farther from the nucleus are shielded by inner electrons, reducing the effective nuclear charge they experience.

• Subshells closer to the nucleus (lower n and l) are lower in energy.

• For subshells with the same n + l, the subshell with lower n is lower in energy.

Writing Electron Configurations

Notation and Rules

• Spectroscopic notation: Indicates the shell, subshell, and number of electrons (e.g., 1s2).

• Orbital box notation: Uses boxes and arrows to represent electrons and their spins.

• Hund's Rule: Electrons occupy degenerate (equal energy) orbitals singly and with parallel spins before pairing up.

• Noble gas notation: Uses the symbol of the previous noble gas to abbreviate core electrons (e.g., [Ne] 3s1 for Na).

Example: Sodium (Na, Z = 11): 1s2 2s2 2p6 3s1 or [Ne] 3s1

Electron Configurations and the Periodic Table

Relationship to Element Groups

• s-block: Groups 1A and 2A (alkali and alkaline earth metals)

• p-block: Groups 3A to 8A (main group elements)

• d-block: Transition metals (groups 3B to 2B)

• f-block: Lanthanides and actinides

The periodic table structure reflects the order of orbital filling.

Transition Metals and f-Block Elements

Special Considerations

• Transition metals (d-block): Electron configurations often written by shell, but filling order may differ (e.g., 4s fills before 3d).

• Lanthanides and actinides (f-block): Involve filling of 4f and 5f orbitals, respectively.

Example: Cerium: [Xe] 6s2 5d1 4f1; Uranium: [Rn] 7s2 6d1 5f3

Irregular Electron Configurations

• Some elements (e.g., Cr, Cu, Mo) have electron configurations that differ from the expected order due to extra stability of half-filled or fully filled subshells.

Example: Cr: Expected [Ar] 4s2 3d4, Actual [Ar] 4s1 3d5

Electron Configurations of Ions

Rules for Cations and Anions

• Anions: Add electrons to the next available (higher energy) orbital.

• Cations: Remove electrons from the highest n orbital first. For transition metals, remove ns electrons before (n-1)d electrons.

Example: Fe: [Ar] 4s2 3d6 → Fe2+: [Ar] 3d6

Paramagnetism and Diamagnetism

Magnetic Properties of Atoms and Ions

• Paramagnetic: Atoms/ions with unpaired electrons; attracted to a magnetic field.

• Diamagnetic: Atoms/ions with all electrons paired; not attracted to a magnetic field.

Example: Fe3+ has 5 unpaired electrons and is paramagnetic.

These properties can also be observed in molecules such as O2 (paramagnetic) and N2 (diamagnetic).

Tables

Orbital Box Diagrams for Transition Metals (Ca through Zn)

Element	Configuration	3d	4s
Ca	[Ar] 4s2		↑↓
Sc	[Ar] 3d1 4s2	↑	↑↓
Ti	[Ar] 3d2 4s2	↑ ↑	↑↓
V	[Ar] 3d3 4s2	↑ ↑ ↑	↑↓
Cr*	[Ar] 3d5 4s1	↑ ↑ ↑ ↑ ↑	↑
Mn	[Ar] 3d5 4s2	↑ ↑ ↑ ↑ ↑	↑↓
Fe	[Ar] 3d6 4s2	↑↓ ↑ ↑ ↑ ↑	↑↓
Co	[Ar] 3d7 4s2	↑↓ ↑↓ ↑ ↑ ↑	↑↓
Ni	[Ar] 3d8 4s2	↑↓ ↑↓ ↑↓ ↑ ↑	↑↓
Cu*	[Ar] 3d10 4s1	↑↓ ↑↓ ↑↓ ↑↓ ↑↓	↑
Zn	[Ar] 3d10 4s2	↑↓ ↑↓ ↑↓ ↑↓ ↑↓	↑↓

Additional info: *Cr and Cu have irregular configurations due to the stability of half-filled and fully filled d subshells.