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. Because electrons are extremely small particles, their behavior is best described using the principles of quantum mechanics, which often involve wave-like properties.

• Wave functions (or orbitals) describe the probable locations and energies of electrons.

• Each orbital represents a spatial distribution of electron density.

• Orbitals are characterized by a set of three quantum numbers.

Quantum Numbers and Atomic Orbitals

Principal Quantum Number (n)

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

• Allowed values: integers ≥ 1 (i.e., 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).

• 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 0.

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

Electron Shells and Subshells

All orbitals with the same value of n form an electron shell. Different types of orbitals within a shell (different l values) are called subshells.

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

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

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 (e.g., 1s, 2s, 3s).

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 (for n ≥ 2), corresponding to ml = -1, 0, +1.

d Orbitals

d orbitals have l = 2. Four of the five d orbitals have four lobes, while the fifth (dz2) has a unique shape 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 but different l values. Sublevels within the same shell are still degenerate, but energy levels start to overlap as n increases.

• As the number of electrons increases, so does repulsion between them.

• Not all orbitals on the same energy level are degenerate in many-electron atoms.

Spin Quantum Number (ms)

The spin quantum number, ms, describes the intrinsic spin of the electron, which affects its magnetic properties.

• Allowed values: +1/2 or -1/2.

Pauli Exclusion Principle

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

• This means each electron in an atom must differ by at least one quantum number.

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.

• Example: $4p^5$ means five electrons in the 4p orbitals.

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

Example: Electron Configuration

• For fluorine (atomic number 9): $1s^2 2s^2 2p^5$

• For sodium (atomic number 11): $1s^2 2s^2 2p^6 3s^1$

Additional info: The notes above are based on standard quantum mechanical models and the Aufbau principle for electron filling order.