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 (ml): Specifies the orientation of the orbital (ml = -l, ..., 0, ..., +l).

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

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

The Pauli Exclusion Principle

Fundamental Rule for Electron Arrangement

The Pauli Exclusion Principle states that no two electrons in the same atom can have the same set of four quantum numbers. This means:

• No more than two electrons can occupy the same orbital.

• If two electrons are in the same orbital, they must have opposite spins.

Electron Capacity of Shells and Subshells

Maximum Number of Electrons

• For n = 1: Only l = 0, ml = 0, ms = +1/2 or -1/2 → 2 electrons possible.

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

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

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

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-electron interactions and shielding effects.

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 that position electrons closer to the nucleus are lower in energy.

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

Writing Electron Configurations

Spectroscopic and Orbital Box Notation

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

• Orbital box notation: Uses boxes and arrows to represent orbitals and electron spins.

Example: Helium (He): 1s2

n	l	ml	ms
1	0	0	+1/2
1	0	0	-1/2

Hund's Rule

Electron Arrangement in Degenerate Orbitals

• When electrons occupy orbitals of equal energy (degenerate orbitals), they fill them singly with parallel spins before pairing up.

Example: Carbon (C): 1s2 2s2 2p2 (2p electrons occupy separate orbitals with parallel spins).

Electron Configurations and the Periodic Table

Relationship Between Electron Configuration and Element Position

• The periodic table is structured according to electron configurations.

• 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 3-12).

• f-block: Lanthanides and actinides.

Transition Metals and f-Block Elements

Special Electron Configurations

• Transition metals (d-block): Fill (n-1)d orbitals after ns orbitals.

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

Example: Cerium: [Xe] 6s2 5d1 4f1

Example: Uranium: [Rn] 7s2 6d1 5f3

Irregular Electron Configurations

Exceptions to Predicted 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: Chromium (Cr): Expected [Ar] 4s2 3d4, Actual [Ar] 4s1 3d5

Ion Configurations

Electron Removal and Addition

• For anions, add electrons to the next available (higher energy) orbital.

• For cations, remove electrons from the highest n orbital first. In 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.

Diamagnetic and paramagnetic properties can also be observed in molecules such as O2 and N2, based on their electron configurations.

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.