Lesson 3.4: Quantum Numbers and Atomic Orbitals

Quantum Numbers

Introduction to Quantum Numbers

Quantum numbers are a set of four values that describe the unique quantum state of an electron in an atom. These numbers arise from the solutions to Schrödinger’s wave equation and are essential for understanding the arrangement and behavior of electrons within atoms.

• Quantum numbers specify the energy, shape, orientation, and spin of atomic orbitals.

• Each electron in an atom is described by a unique set of four quantum numbers.

The Principal Quantum Number (n)

The principal quantum number (n) describes the main energy level or shell that an electron occupies in an atom. It determines the size and energy of the orbital.

• Values: Positive integers (1, 2, 3, ...).

• Interpretation: As n increases, the orbital becomes larger and the electron is, on average, farther from the nucleus.

• Energy: Higher n means higher energy and less tightly bound electrons.

• Shells: Each value of n corresponds to a shell (e.g., n = 1 is the first shell).

• Example: For n = 3, the electron is in the third shell, which is larger and higher in energy than n = 1 or n = 2.

The Secondary Quantum Number (l)

The secondary quantum number (l), also called the angular momentum quantum number, describes the shape of the atomic orbital (subshell) and its energy within a shell.

• Values: Integers from 0 to n – 1 for each value of n.

• Subshells: Each value of l corresponds to a subshell, designated by letters:

l	Letter	Name
0	s	sharp
1	p	principal
2	d	diffuse
3	f	fundamental
4	g	(no stable elements use this in ground state)

• Example: For n = 3, possible l values are 0 (3s), 1 (3p), and 2 (3d).

The Magnetic Quantum Number (ml)

The magnetic quantum number (m_l) describes the orientation of the orbital in space relative to the other orbitals in the atom.

• Values: Integers from –l to +l, including zero.

• Number of Orbitals: For each l, there are (2l + 1) possible values of ml, corresponding to the number of orbitals in a subshell.

• Example: For l = 1 (p orbitals), ml = –1, 0, +1 (three p orbitals: px, py, pz).

The Spin Quantum Number (ms)

The spin quantum number (m_s) describes the intrinsic spin of the electron, which gives rise to its magnetic properties.

• Values: +1/2 or –1/2.

• Interpretation: Each orbital can hold a maximum of two electrons, which must have opposite spins.

Summary Table: Quantum Numbers for Hydrogen Orbitals

n	l	Sublevel	ml	Number of Orbitals	Max Electrons (2n2)
1	0	1s	0	1	2
2	0	2s	0	1	8
2	1	2p	–1, 0, 1	3
3	0	3s	0	1	18
3	1	3p	–1, 0, 1	3
3	2	3d	–2, –1, 0, 1, 2	5

Shapes and Orientations of Orbitals

Each orbital has a unique probability distribution, shape, and orientation:

• s orbitals: Spherical shape; size increases with n.

• p orbitals: Dumbbell-shaped with two lobes; oriented along x, y, or z axes.

• d orbitals: More complex shapes, typically with four lobes (except dz2).

• Nodes: Areas of zero probability within orbitals, increasing with n.

The 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 orbital can hold a maximum of two electrons with opposite spins.

Applications and Examples

• Allowed quantum numbers: For n = 5, l can be 0, 1, 2, 3, or 4; for each l, ml ranges from –l to +l.

• Orbital designations: Not all combinations exist (e.g., 1p, 2d, 3f are not allowed).

• Electron arrangements: The quantum numbers determine the arrangement of electrons in the periodic table and chemical properties of elements.

Summary of Key Points

• Four quantum numbers (n, l, ml, ms) uniquely describe each electron in an atom.

• Principal quantum number (n): main energy level (shell).

• Secondary quantum number (l): subshell (shape).

• Magnetic quantum number (ml): orientation of orbital.

• Spin quantum number (ms): electron spin (+1/2 or –1/2).

• Pauli exclusion principle: no two electrons in an atom can have the same set of quantum numbers.