Electronic Structure of Atoms: Quantum Numbers, Orbitals, and Electron Configurations

Electronic Structure of Atoms

Introduction to Electronic Structure

The electronic structure of an atom refers to the arrangement and energy of electrons within the atom. Understanding electronic structure is fundamental to explaining chemical properties and behaviors. The wave-like nature of extremely small particles, such as electrons, requires a quantum mechanical approach to describe their behavior.

• Wave functions (or orbitals) describe the spatial distribution and energy of electrons.

• Each orbital is characterized by a set of quantum numbers.

Quantum Numbers and Atomic Orbitals

Principal Quantum Number (n)

The principal quantum number, denoted as n, specifies the energy level (or shell) of an electron in an atom.

• Allowed values: integers ≥ 1 (i.e., n = 1, 2, 3, ...).

• Corresponds to the energy levels in the Bohr model.

Angular Momentum Quantum Number (l)

The angular momentum quantum number, l, defines the shape of the orbital.

• Allowed values: integers from 0 to n-1 for each value of n.

• Each value of l is designated by a letter:

Value of l	0	1	2	3
Letter used	s	p	d	f

Magnetic Quantum Number (ml)

The magnetic quantum number, ml, describes the three-dimensional orientation of the orbital.

• Allowed values: integers from -l to +l, including zero.

• For each value of l, there are (2l + 1) possible orientations.

Electron Shells and Subshells

All orbitals with the same value of n form an electron shell. Different orbital types (s, p, d, f) within a shell are called subshells.

Summary Table: Relationship Among Quantum Numbers (n = 4 Example)

n	l	Subshell Designation	Possible ml Values	Number of Orbitals in Subshell	Total Number of Orbitals in Shell
4	0	4s	0	1	16
4	1	4p	-1, 0, +1	3
4	2	4d	-2, -1, 0, +1, +2	5
4	3	4f	-3, -2, -1, 0, +1, +2, +3	7

Types of Atomic Orbitals

s Orbitals

s orbitals have l = 0 and are spherical in shape. The size of the s orbital increases with increasing n (1s, 2s, 3s, etc.).

p Orbitals

p orbitals have l = 1 and are dumbbell-shaped, with two lobes and a node at the nucleus. There are three p orbitals per energy level (px, py, pz).

d Orbitals

d orbitals have l = 2. Four of the five d orbitals have four lobes; the fifth (dz2) has a unique shape resembling a p orbital with a doughnut around the center.

Degenerate Orbitals

In a hydrogen atom (single electron), all orbitals with the same n have the same energy and are called degenerate orbitals.

Energies of Orbitals in Many-Electron Atoms

In atoms with more than one electron, electron-electron repulsion causes energy differences among orbitals with the same n. Sublevels (s, p, d, f) within a shell are no longer degenerate.

• Within a sublevel, orbitals remain degenerate.

• Energy levels start to overlap in higher shells (e.g., 4s fills before 3d).

Spin Quantum Number (ms)

The spin quantum number, ms, describes the intrinsic spin of the electron, which creates a magnetic field.

• Allowed values: +1/2 or -1/2.

• No two electrons in the same orbital can have the same set of four quantum numbers.

Pauli Exclusion Principle

The Pauli Exclusion Principle states that no two electrons in the same atom can have identical sets of all four quantum numbers (n, l, ml, ms).

• This ensures that each electron in an atom has a unique quantum state.

Electron Configurations

Definition and Notation

The electron configuration of an atom describes the distribution of electrons among the available orbitals. The most stable arrangement is called the ground state.

• Notation: a number for the energy level, a letter for the type of orbital, and a superscript for the number of electrons in those orbitals (e.g., 4p5).

Example

• Chlorine (Cl, atomic number 17): $1s^2 2s^2 2p^6 3s^2 3p^5$

Summary Table: Quantum Numbers and Orbitals

Quantum Number	Symbol	What it Specifies	Allowed Values
Principal	n	Energy level (shell)	1, 2, 3, ...
Angular Momentum	l	Subshell (shape)	0 to n-1
Magnetic	ml	Orbital orientation	-l to +l
Spin	ms	Electron spin	+1/2, -1/2

Additional info:

• Hund's Rule: For degenerate orbitals, electrons fill each orbital singly before pairing, and all unpaired electrons have the same spin.

• Aufbau Principle: Electrons occupy the lowest energy orbitals available.