Electron Spin, Electron Configurations, and Orbital Energies

Quantum Mechanics and Electron Configurations

Electron Spin

Electrons possess a property called spin, which is a fundamental quantum characteristic. Spin is not a literal spinning motion but an intrinsic form of angular momentum unique to quantum particles.

• Spin Quantum Number (ms): The spin quantum number can have values of +1/2 or -1/2, often referred to as "spin up" and "spin down." This is denoted by arrows (↑ or ↓) in orbital diagrams.

• Pauli Exclusion Principle: No two electrons in the same atom can have the same set of four quantum numbers. Thus, an orbital can hold a maximum of two electrons, and they must have opposite spins.

• Notation: Arrows are used to indicate electron spin in orbital diagrams: ↑ for +1/2 and ↓ for -1/2.

Example: The two electrons in a helium atom (1s2) have opposite spins: one is ↑ and the other is ↓.

Electron Configurations

Electron configuration describes the arrangement of electrons in an atom's orbitals. The order in which electrons fill orbitals is governed by the Aufbau Principle, Pauli Exclusion Principle, and Hund's Rule.

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

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

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

Notation: Electron configurations are written using the format: 1s2 2s2 2p6, etc.

Example: The electron configuration for nitrogen (N, atomic number 7) is 1s2 2s2 2p3.

Degenerate Orbitals

Orbitals with the same energy are called degenerate. For example, the three 2p orbitals (2px, 2py, 2pz) are degenerate in a single atom.

• Each p subshell contains three degenerate orbitals.

• Each orbital can hold two electrons, so the p subshell can hold a total of six electrons.

Condensed Electron Configurations

To simplify electron configurations, chemists use the condensed (noble gas) notation. The configuration of the nearest preceding noble gas is written in brackets, followed by the remaining configuration.

• Example: Sodium (Na, atomic number 11): Full: 1s2 2s2 2p6 3s1 Condensed: [Ne] 3s1

• Example: Phosphorus (P, atomic number 15): Full: 1s2 2s2 2p6 3s2 3p3 Condensed: [Ne] 3s2 3p3

This notation is especially useful for larger atoms.

Orbital Energies and the Periodic Table

The periodic table is organized to reflect the order in which atomic orbitals are filled. The energy of orbitals increases in the order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p, etc.

• s-block: Groups 1 and 2 (and helium) – filling s orbitals

• p-block: Groups 13–18 – filling p orbitals

• d-block: Transition metals – filling d orbitals

• f-block: Lanthanides and actinides – filling f orbitals

The periodic table can be used as a guide to determine the order of orbital filling.

Block	Type of Orbital	Maximum Electrons	Groups in Periodic Table
s-block	s	2	1–2
p-block	p	6	13–18
d-block	d	10	3–12
f-block	f	14	Lanthanides/Actinides

Counting Orbitals

The number of orbitals of a given type in a shell is determined by the quantum number l:

• s-orbitals (l = 0): 1 orbital per shell

• p-orbitals (l = 1): 3 orbitals per shell

• d-orbitals (l = 2): 5 orbitals per shell

• f-orbitals (l = 3): 7 orbitals per shell

Each orbital can hold two electrons.

Key Equations and Principles

• Maximum electrons in a shell: $2n^2$

• Aufbau Principle (Order of Filling): Orbitals fill in order of increasing energy.

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

• Hund's Rule: Every orbital in a subshell is singly occupied before any orbital is doubly occupied.

Summary Table: Quantum Numbers

Quantum Number	Symbol	Possible Values	Describes
Principal	n	1, 2, 3, ...	Energy level (shell)
Angular Momentum	l	0 to n-1	Subshell (s, p, d, f)
Magnetic	ml	-l to +l	Orbital orientation
Spin	ms	+1/2, -1/2	Electron spin direction

Additional info:

• The periodic table's structure directly reflects the order of orbital filling, which is crucial for predicting chemical properties and reactivity.

• Understanding electron configurations is foundational for topics such as chemical bonding, periodic trends, and spectroscopy.