lecture 10

Electron Configuration: How an Atom’s Electrons Occupy Orbitals

Introduction to Electron Configuration

Quantum-mechanical theory describes the behavior of electrons in atoms. Electrons exist in specific regions called orbitals, and the arrangement of electrons in these orbitals is known as the electron configuration.

• Electron configuration specifies the distribution of electrons among the available orbitals.

• Example: For hydrogen (H), the electron configuration is 1s1, indicating one electron in the 1s orbital.

Example: H: 1s1 (1 electron in the 1s orbital)

Electron Configuration and Quantum Theory

How Electrons Are Arranged Around the Atom’s Nucleus

The solution to Schrödinger’s equation for hydrogen shows that its single electron occupies the lowest energy orbital (1s). For multi-electron atoms, electron-electron interactions complicate the solution, but approximate methods show orbitals are hydrogen-like.

• Multi-electron atoms cannot be solved exactly due to electron-electron repulsion.

• Additional concepts for multi-electron atoms: spin quantum number (ms) and energy splitting of sublevels.

Electron Spin (ms)

Fundamental Property of Electrons

Electron spin is an intrinsic property of electrons, quantized to two possible orientations: up (+½) or down (−½). This adds a fourth quantum number to the description of electrons in an atom.

• All electrons have the same amount of spin.

• Spin quantum number (ms) can be +½ or −½.

• Electrons in an orbital can spin up or spin down.

Example: Helium (He): 1s2 (two electrons in the 1s orbital, with opposite spins)

Electron Spin and the Pauli Exclusion Principle

Pauli Exclusion Principle

No two electrons in an atom can have the same set of four quantum numbers. Therefore, each orbital can hold a maximum of two electrons, and these must have opposite spins.

• Orbital diagrams use squares for orbitals and half-arrows for electrons.

• Upward half-arrow: electron with spin up (+½).

• Downward half-arrow: electron with spin down (−½).

• Paired spins are diamagnetic (not attracted to a magnetic field).

Example: Helium orbital diagram:

n	l	ml	ms
1	0	0	+½
1	0	0	−½

Sublevel Energy Splitting in Multi-Electron Atoms

Degeneracy and Energy Splitting

In hydrogen and similar single-electron systems, all sublevels (s, p, d, f) in a principal energy shell have the same energy (are degenerate). In multi-electron atoms, sublevel energies split due to charge interaction, shielding, and penetration.

• Lower l quantum number (s orbital) means lower energy.

• General energy order: E(s orbital) < E(p orbital) < E(d orbital) < E(f orbital)

Coulomb’s Law: Attractive and Repulsive Forces Between Electrons

Understanding Coulomb’s Law

Coulomb’s law describes the forces between charged particles. The potential energy (E) depends on the charges and their separation.

• For like charges: E is positive and decreases as distance (r) increases.

• For opposite charges: E is negative and becomes more negative as r decreases.

• Strength of interaction increases with the magnitude of the charges.

Equation:

$E = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r}$

Example: Electrons are more strongly attracted to a nucleus with a 2+ charge than to one with a 1+ charge.

Shielding and Effective Nuclear Charge (Zeff)

Shielding in Multi-Electron Atoms

Each electron in a multi-electron atom experiences both attraction to the nucleus and repulsion from other electrons. Repulsions reduce the net attraction, a phenomenon called shielding.

• Electrons are shielded from the full nuclear charge by other electrons.

• The net attraction felt by an electron is called the effective nuclear charge (Zeff).

Penetration and Sublevel Splitting

Penetration and Shielding Effects

Penetration refers to how close an electron can get to the nucleus. Electrons in s orbitals penetrate more deeply and experience less shielding, resulting in lower energy compared to p, d, or f orbitals.

• Greater penetration means greater attraction to the nucleus and less shielding.

• Energy order: s < p < d < f (for a given principal quantum number).

Example: 2s electrons penetrate more than 2p electrons, so 2s is lower in energy.

General Energy Ordering of Orbitals in Multi-Electron Atoms

Order of Orbital Filling

Electrons fill atomic orbitals from lowest to highest energy, following the aufbau principle. The general order is:

• 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p

Each orbital can hold a maximum of two electrons with opposite spins (Pauli exclusion principle).

Hund’s Rule

Electron Filling in Degenerate Orbitals

When filling orbitals of equal energy (degenerate), electrons occupy them singly with parallel spins before pairing up. This minimizes electron repulsion and stabilizes the atom.

• Electrons fill each degenerate orbital singly before pairing (Hund’s rule).

• Once half-filled, electrons begin to pair.

Writing Electron Configurations

Steps for Writing Electron Configurations

To write the electron configuration for an element:

1. Locate the element on the periodic table and determine its atomic number (number of electrons).

2. Distribute electrons among orbitals in the order of increasing energy.

3. Remember the maximum number of electrons per sublevel:

   • s: 2 electrons

   • p: 6 electrons

   • d: 10 electrons

   • f: 14 electrons

Examples:

• Mg: 1s2 2s2 2p6 3s2 or [Ne] 3s2

• P: 1s2 2s2 2p6 3s2 3p3 or [Ne] 3s2 3p3

• Br: 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p5 or [Ar] 4s2 3d10 4p5

• Al: 1s2 2s2 2p6 3s2 3p1 or [Ne] 3s2 3p1

Orbital Diagrams and Quantum Numbers

Orbital Diagrams

Orbital diagrams visually represent electron configurations, showing the distribution of electrons in orbitals and their spins.

• Each box represents an orbital; arrows indicate electrons and their spins.

• Hund’s rule and Pauli exclusion principle must be followed.

Example: Sulfur (S, atomic number 16) has two unpaired electrons in its 3p orbitals.

Quantum Numbers for Electrons

Each electron in an atom is described by four quantum numbers:

• n: Principal quantum number (energy level)

• l: Angular momentum quantum number (sublevel type: s, p, d, f)

• ml: Magnetic quantum number (specific orbital)

• ms: Spin quantum number (+½ or −½)

Example: For two electrons in a 4s orbital: n = 4, l = 0, ml = 0, ms = +½ and −½

Summary Table: Quantum Numbers for Helium Electrons

n	l	ml	ms
1	0	0	+½
1	0	0	−½

Additional info: The notes also include conceptual questions to reinforce understanding of Coulomb’s Law, shielding, penetration, and quantum numbers. These are useful for self-assessment and exam preparation.