Quantum Numbers, Electron Configuration, and Periodic Trends in Multi-Electron Atoms

Quantum Numbers and Electron Spin

The Fourth Quantum Number: Electron Spin

In quantum mechanics, each electron in an atom is described by four quantum numbers: principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (ml), and spin quantum number (ms). The fourth quantum number, electron spin, distinguishes electrons in the same orbital.

• Electron spin (ms): Can be +1/2 (spin up) or -1/2 (spin down).

• Stern-Gerlach experiment: Demonstrates electron spin by splitting a beam of hydrogen atoms in a magnetic field, resulting in two distinct paths corresponding to the two spin states.

Example: In the Stern-Gerlach experiment, a beam of hydrogen atoms is separated into two lines, showing the two possible spin states.

Pauli Exclusion Principle

Arrangement of Electrons in Multi-Electron Species

The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of four quantum numbers. This principle governs the population of orbitals by electrons.

Electron	n	l	ml	ms	Orbital
1	1	0	0	+1/2	1s
2	1	0	0	-1/2	1s
3	2	0	0	+1/2	2s

Example: In the ground state of lithium (Li), the electron configuration is 1s22s1, with two electrons in the 1s orbital (opposite spins) and one in the 2s orbital.

Electron Configuration and Orbital Diagrams

Ground State and Excited State Configurations

Electron configuration describes the arrangement of electrons in an atom's orbitals. The ground state is the lowest energy configuration, while excited states occur when electrons occupy higher energy orbitals.

• Ground state configuration: Electrons fill orbitals in order of increasing energy.

• Excited state configuration: One or more electrons are promoted to higher energy orbitals.

• Orbital diagram: Visual representation showing electron spins in each orbital.

Example: Lithium (Li): Orbital diagram: ↑↓ (1s) ↑ (2s) Electron configuration: 1s22s1

Rules for Electron Arrangement

Aufbau Principle, Hund's Rule, and Degeneracy

Three main rules guide electron arrangement in multi-electron atoms:

• Aufbau Principle: Electrons fill the lowest energy orbitals first.

• Hund's Rule: Electrons occupy degenerate (equal energy) orbitals singly before pairing.

• Degeneracy: Orbitals with the same energy level (e.g., all three 2p orbitals) are degenerate.

Example: For boron (B), the electron configuration is 1s22s22p1, with the single 2p electron occupying one of the three degenerate 2p orbitals.

Penetration and Effective Nuclear Charge

Orbital Penetration and Energy Trends

Penetration refers to how close an electron in a given orbital can approach the nucleus. The effective nuclear charge (Zeff) is the net positive charge experienced by an electron.

Orbital Type	Penetration	Zeff	Energy (E)
s	Good	High	Low
p	Moderate	Moderate	Moderate
d	Poor	Low	High
f	Poor	Low	High

Example: In a given atom, s orbitals penetrate closer to the nucleus than p, d, or f orbitals, resulting in lower energy and higher Zeff.

Valence Electrons

Definition and Importance

Valence electrons are the electrons in the outermost shell of an atom. They are responsible for chemical bonding and reactivity.

• Core electrons: Electrons in inner shells, not involved in bonding.

• Valence electrons: Electrons in the highest principal energy level.

Example: For sodium (Na), the electron configuration is 1s22s22p63s1. The single 3s electron is the valence electron.

Order of Orbital Filling

Qualitative Order of Orbital Energies

Electrons fill orbitals in a specific order based on energy levels, as predicted by the Aufbau principle.

• Order: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p → 7s → 5f → 6d → 7p

Example: The electron configuration for iron (Fe, Z=26) is [Ar]4s23d6.

Blocks of the Periodic Table

Procedure for Writing Electron Configurations

The periodic table is divided into blocks based on the type of atomic orbital being filled:

• s-block: Groups 1-2

• p-block: Groups 13-18

• d-block: Transition metals (Groups 3-12)

• f-block: Lanthanides and actinides

Example: For sulfur (S, Z=16): 1s22s22p63s23p4

Sample Electron Configurations

First Thirty Elements

Electron configurations for selected elements:

• Nitrogen (N, Z=7): 1s22s22p3

• Sulfur (S, Z=16): 1s22s22p63s23p4

• Iron (Fe, Z=26): [Ar]4s23d6

Example: For Fe, the shorthand configuration uses the noble gas core: [Ar]4s23d6.

Key Equations

Quantum Numbers and Electron Configuration

• Quantum numbers: $n, \, l, \, m_l, \, m_s$

• Electron configuration notation: $1s^2 2s^2 2p^6 \ldots$

• Effective nuclear charge: $Z_{eff} = Z - S$ (where S is the shielding constant)

Additional info: These notes expand on the provided slides by including definitions, examples, and equations for clarity and completeness.