Quantum Numbers, Electron Configuration, and Principles of Atomic Structure

Quantum Numbers and Atomic Orbitals

Types of Quantum Numbers

Quantum numbers are used to describe the properties and locations of electrons in atoms. Each electron in an atom is defined by a unique set of four quantum numbers:

• Principal quantum number (n): Indicates the main energy level or shell of an electron.

• Angular momentum quantum number (l): Defines the shape of the orbital (s, p, d, f).

• Magnetic quantum number (ml): Specifies the orientation of the orbital in space.

• Spin quantum number (ms): Describes the spin direction of the electron (+1/2 or -1/2).

Example: For a 2p electron, n = 2, l = 1, ml = -1, 0, or +1, ms = +1/2 or -1/2.

Electron Configuration and Subshells

Electron Configuration Notation

Electron configuration describes the arrangement of electrons in an atom's orbitals. It follows the order of increasing energy levels and subshells:

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

• Pauli Exclusion Principle: No two electrons in the same atom can have the same set of four quantum numbers.

• Hund's Rule: Electrons occupy degenerate orbitals singly before pairing up.

Example: The electron configuration of sulfur (atomic number 16) is $1s^2 2s^2 2p^6 3s^2 3p^4$.

Subshell Electron Counts

Each subshell can hold a specific number of electrons:

• s subshell: 2 electrons

• p subshell: 6 electrons

• d subshell: 10 electrons

• f subshell: 14 electrons

Example: The 3p subshell in xenon (Xe) can contain up to 6 electrons.

Valence Electron Configuration

Identifying Valence Electrons

Valence electrons are those in the outermost shell and are important for chemical reactivity. For example, an element with a configuration ending in 2s2 is beryllium (Be).

Electron Configuration Diagrams and Principles

Orbital Box Diagrams

Box diagrams visually represent electron configurations, showing the arrangement of electrons in orbitals and their spins. Each box represents an orbital, and arrows indicate electrons and their spins.

Configuration	Pauli Exclusion Principle Violation?
Two electrons with same spin in same orbital	Yes
Electrons paired with opposite spins	No
Single electrons in degenerate orbitals (Hund's rule)	No

Example: A box diagram with two arrows pointing up in the same box violates the Pauli Exclusion Principle.

Key Atomic Principles

Pauli Exclusion Principle

Definition: No two electrons in the same atom can have identical sets of quantum numbers. This means each orbital can hold a maximum of two electrons with opposite spins.

Hund's Rule

Definition: When electrons occupy orbitals of equal energy (degenerate orbitals), one electron enters each orbital until all are half-full before any orbital gets a second electron.

Aufbau Principle

Definition: Electrons fill atomic orbitals in order of increasing energy, starting with the lowest energy orbital available.

Summary Table: Atomic Principles

Principle	Description	Example
Pauli Exclusion Principle	No two electrons in the same atom have the same set of quantum numbers.	Two electrons in a 1s orbital must have opposite spins.
Hund's Rule	Electrons fill degenerate orbitals singly before pairing.	Three electrons in 2p occupy separate orbitals.
Aufbau Principle	Electrons fill lowest energy orbitals first.	1s is filled before 2s.

Equations and Notation

• Electron configuration notation: $1s^2 2s^2 2p^6 3s^2 3p^4$

• Maximum electrons per orbital: $2n^2$ (where n is the principal quantum number)

Additional info:

• These principles are foundational for understanding atomic structure, periodic trends, and chemical bonding.

• Mastery of electron configuration is essential for predicting chemical properties and reactivity.