Electron Spin and the Pauli Exclusion Principle in Atomic Orbitals

Atomic Orbitals and Electron Spin

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental rule in chemistry and quantum mechanics that governs the arrangement of electrons in atomic orbitals. It states that an orbital can hold a maximum of 2 electrons, and these electrons must have opposite spins.

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

• Application: Within a single orbital, one electron will have spin up ($+1/2$) and the other will have spin down ($-1/2$).

Example: In a 1s orbital, two electrons can be present, but one must be spin up and the other spin down.

Electron Spin

Electron spin refers to the intrinsic angular momentum of an electron within an atomic orbital. It is described by the spin quantum number ($m_s$), which can have values of $+1/2$ (spin up) or $-1/2$ (spin down).

• Spin Quantum Number ($m_s$): Indicates the direction of electron spin.

• Spin Up ($+1/2$): Electron is represented with an upward arrow (↑).

• Spin Down ($-1/2$): Electron is represented with a downward arrow (↓).

• Electrons are added to orbitals starting with spin up, then spin down.

Visual Representation: Paired electrons in an orbital are shown as one arrow up and one arrow down.

Quantum Numbers and Electron Configuration

Principal Quantum Number ($n$)

The principal quantum number ($n$) indicates the main energy level or shell of an electron. It can be any positive integer (1, 2, 3, ...).

• Example: $n = 3$ refers to the third energy level.

Subshell Letter ($l$)

The azimuthal quantum number ($l$) defines the subshell or shape of the orbital:

• s ($l = 0$)

• p ($l = 1$)

• d ($l = 2$)

• f ($l = 3$)

Electron Spin Quantum Number ($m_s$)

The spin quantum number ($m_s$) can be either $+1/2$ or $-1/2$.

• Clockwise spin: $m_s = +1/2$

• Counterclockwise spin: $m_s = -1/2$

Examples and Practice Problems

Example: Identifying Quantum Numbers

Given an electron in the third principal level ($n = 3$), in a p subshell ($l = 1$), with spin down ($m_s = -1/2$):

• n = 3

• Subshell = p ($l = 1$)

• Electron spin = -1/2

Application: This information is used to specify the exact state of an electron in an atom.

Practice: Identifying Electron Spin in 5d Orbitals

Which of the following can represent the highlighted electron in a set of 5d orbitals?

• n = 5, subshell = d, electron spin = -1/2

Answer: The correct quantum numbers for the highlighted electron are $n = 5$, subshell $l = 2$ (d), electron spin $m_s = -1/2$.

Summary Table: Quantum Numbers and Electron Spin

Quantum Number	Symbol	Possible Values	Description
Principal	n	1, 2, 3, ...	Main energy level
Azimuthal (Subshell)	l	0 (s), 1 (p), 2 (d), 3 (f)	Orbital shape/subshell
Spin	m_s	+1/2, -1/2	Electron spin direction

Key Equations

• Maximum electrons per orbital: $2$

• Spin quantum number: $m_s = +1/2$ or $m_s = -1/2$

Additional info: The notes and questions are directly relevant to GOB Chemistry, focusing on atomic structure, quantum numbers, and electron configuration, which are foundational topics in the course.