Lecture 4: Electron Configuration – Quantum Numbers, Aufbau Principle, and the Periodic Table

Electron Configuration

Introduction

Electron configuration describes the arrangement of electrons in an atom's orbitals. Understanding electron configuration is essential for explaining the chemical properties of elements, their placement in the periodic table, and the rules governing electron behavior.

Electron Spin: A Fourth Quantum Number

Electron Spin Quantum Number ($m_s$)

• Definition: The electron spin quantum number ($m_s$) describes the orientation of an electron's intrinsic angular momentum (spin).

• Possible Values: $m_s = +\frac{1}{2}$ or $m_s = -\frac{1}{2}$.

• Experimental Evidence: The Stern-Gerlach experiment (1920) demonstrated the existence of electron spin by splitting a beam of silver atoms in a magnetic field.

• Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers ($n$, $\ell$, $m_\ell$, $m_s$).

Example: In a 2p orbital, two electrons can occupy the same orbital only if they have opposite spins ($m_s = +\frac{1}{2}$ and $m_s = -\frac{1}{2}$).

Multielectron Atoms

Electron-Electron Repulsion and Energy Levels

• Electrons repel each other, causing them to stay apart and influencing the energy of orbitals.

• In hydrogen (single electron), energy depends only on principal quantum number $n$:

$E_n = -\frac{R_H}{n^2}$

• In multielectron atoms, orbitals with the same $n$ but different $\ell$ have different energies due to electron-electron repulsion and shielding effects.

• Lower $\ell$ (e.g., s-orbitals) are more stabilized because they penetrate closer to the nucleus.

Example: In sodium (Na, $Z=11$), the 3s orbital is lower in energy than the 3p and 3d orbitals.

Penetration and Shielding

Concepts of Penetration and Shielding

• Penetration: The ability of an electron to get close to the nucleus. s-orbitals have the highest penetration.

• Shielding: Electrons in inner shells repel outer electrons, reducing the effective nuclear charge ($Z_{\text{eff}}$) felt by outer electrons.

• Effective Nuclear Charge: The net positive charge experienced by an electron after accounting for shielding by other electrons.

$Z_{\text{eff}} = Z - S$

• Where $Z$ is the atomic number and $S$ is the number of shielding (core) electrons.

Example: 2s electrons in lithium experience a higher $Z_{\text{eff}}$ than 2p electrons due to greater penetration.

Electron Configuration

Rules for Electron Arrangement

• Electrons fill orbitals in a way that minimizes the atom's energy (Aufbau Principle).

• Order of filling: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, etc.

• Each orbital can hold a maximum of two electrons with opposite spins (Pauli Exclusion Principle).

• When orbitals of equal energy (degenerate) are available, electrons occupy them singly with parallel spins before pairing (Hund's Rule).

Shell	Orbitals	Maximum Electrons
n = 1	1 s orbital	2 electrons
n = 2	1 s orbital, 3 p orbitals	8 electrons
n = 3	1 s orbital, 3 p orbitals, 5 d orbitals	18 electrons

Example: The 2p subshell has three orbitals, each of which can hold two electrons, for a total of six electrons.

Electron Configuration of the Carbon Atom

Orbital Diagrams and Hund's Rule

• Carbon (Z = 6) has the electron configuration 1s2 2s2 2p2.

• In the 2p subshell, the two electrons occupy separate orbitals with parallel spins (Hund's Rule).

Example: The orbital diagram for carbon's 2p electrons:

• 2px: ↑

• 2py: ↑

• 2pz: (empty)

The Aufbau Process

Filling Order and Notation

• Electrons fill orbitals in order of increasing energy, following the diagonal rule (Aufbau diagram).

• Ground state electron configurations for the first 10 elements:

Element	Electron Configuration
H	1s1
He	1s2
Li	1s22s1
Be	1s22s2
B	1s22s22p1
C	1s22s22p2
N	1s22s22p3
O	1s22s22p4
F	1s22s22p5
Ne	1s22s22p6

Condensed (Noble Gas) Notation: Use the previous noble gas in brackets to represent core electrons. For example, carbon: [He] 2s22p2.

Writing Electron Configurations

Expanded, Condensed, and Orbital Diagram Notations

• Expanded spdf notation: Lists all occupied orbitals (e.g., 1s2 2s2 2p2 for carbon).

• Condensed notation: Uses noble gas core (e.g., [He] 2s22p2).

• Orbital diagrams: Show electrons as arrows in boxes representing orbitals, indicating spin.

Example: For carbon (Z=6):

• Expanded: 1s2 2s2 2p2

• Condensed: [He] 2s22p2

• Orbital diagram: 1s: ↑↓ 2s: ↑↓ 2p: ↑ ↑ _

Electron Configuration and the Periodic Table

Relationship to Periodic Trends

• Elements in the same group have similar valence electron configurations, leading to similar chemical properties.

• The periodic table is structured so that the filling of s, p, d, and f orbitals corresponds to the table's blocks.

Example: All Group 1 elements (alkali metals) have an ns1 valence configuration.

Special Stability of Half-Filled and Filled Subshells

Stability Patterns

• Subshells that are completely filled or exactly half-filled are particularly stable.

• This explains some exceptions to the predicted electron configurations (e.g., chromium and copper).

Example: Chromium (Cr) has the configuration [Ar] 4s13d5 instead of [Ar] 4s23d4.

Additional info: The notes reference quantum entanglement and the 2022 Nobel Prize, which is not typically covered in introductory electron configuration topics, but is related to advanced quantum mechanics.