Quantum Numbers, Atomic Orbitals, and Electron Configurations

Quantum Numbers and Atomic Orbitals

Wave Functions and Atomic Orbitals

The behavior of electrons in atoms is described by quantum mechanics. By solving the Schrödinger equation ($H\psi = E\psi$), we obtain wave functions ($\psi$), which represent the probability distribution of electrons in an atom. These wave functions are called atomic orbitals, and they define regions in space where electrons are most likely to be found.

• Energy changes in atoms occur when electrons transition between orbitals, often accompanied by the absorption or emission of photons.

• Each electron is described by four quantum numbers that specify its energy, location, and spin.

The Four Quantum Numbers

Quantum numbers are used to uniquely identify the state of each electron in an atom.

• Principal Quantum Number (n): Specifies the energy level and size of the orbital. $n = 1, 2, 3, \ldots$

• Angular Momentum Quantum Number (l): Specifies the shape of the orbital. $l = 0, 1, \ldots, n-1$

• Magnetic Quantum Number (m_l): Specifies the orientation of the orbital in space. $m_l = -l, \ldots, 0, \ldots, +l$

• Spin Quantum Number (m_s): Specifies the spin direction of the electron. $m_s = +\frac{1}{2}$ or $-\frac{1}{2}$

Principal Quantum Number (n)

• Defines the shell or energy level of the electron.

• All orbitals with the same n are in the same shell.

• The number of orbitals in a shell is $n^2$.

• Example: For hydrogen, $n=1$ is the ground state; $n=2$ is an excited state.

Angular Momentum Quantum Number (l)

• Defines the subshell and the shape of the orbital.

• Each value of l corresponds to a letter code:

l	Letter
0	s
1	p
2	d
3	f
4	g
5	h
...	...

• Subshells are labeled by n and l (e.g., 2p, 3s).

• Energy increases with l: s < p < d < f.

Magnetic Quantum Number (m_l)

• Specifies the orientation of the orbital.

• For each l, $m_l$ ranges from $-l$ to $+l$.

• The number of orbitals in a subshell is $2l+1$.

• Example: s subshell (l=0) has 1 orbital; p subshell (l=1) has 3 orbitals.

Spin Quantum Number (m_s)

• Specifies the spin direction of the electron: $+\frac{1}{2}$ (up) or $-\frac{1}{2}$ (down).

• Each orbital can hold two electrons with opposite spins.

Pauli Exclusion Principle

The Pauli exclusion principle states that no two electrons in the same atom can have identical values for all four quantum numbers. This limits each orbital to a maximum of two electrons with opposite spins.

• Electrons with paired spins are diamagnetic (not attracted to magnets).

• Atoms with unpaired electrons are paramagnetic (weakly attracted to magnets).

Table of Allowed Quantum Numbers

This table summarizes the allowed quantum numbers, orbitals, and electron capacities for the first few shells:

n	l	m_l	Number of Orbitals	Name	Number of Electrons
1	0	0	1	1s	2
2	0	0	1	2s	2
2	1	-1, 0, +1	3	2p	6
3	0	0	1	3s	2
3	1	-1, 0, +1	3	3p	6
3	2	-2, -1, 0, +1, +2	5	3d	10
4	0	0	1	4s	2
4	1	-1, 0, +1	3	4p	6
4	2	-2, -1, 0, +1, +2	5	4d	10
4	3	-3, ..., +3	7	4f	14

Additional info: Table entries for n=3, n=4, and higher shells have been logically completed based on quantum number rules.

Writing Electron Configurations

Electron Configuration Principles

The electron configuration describes the distribution of electrons among the orbitals of an atom. Electrons fill orbitals according to the Aufbau principle (building-up), which generally follows increasing subshell energy.

• Order of filling: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f

• The order can be determined from the periodic table.

• Electron configurations are written with the orbital and a superscript for the number of electrons (e.g., H: 1s1).

• Orbital diagrams use squares or circles for orbitals and arrows for electrons, indicating spin.

• Hund's rule: Electrons occupy orbitals singly as much as possible before pairing.

• Ground state: All electrons are in the lowest possible energy levels.

• Excited state: Electrons occupy higher energy orbitals after absorbing energy.

Properties of Monatomic Ions

Valence and Core Electrons

The valence shell contains the electrons with the highest principal quantum number (n) and is most involved in chemical bonding. Core electrons are inner electrons and generally do not participate in bonding.

• Elements with similar properties have similar valence shell configurations.

• Atoms tend to achieve a filled valence shell for stability (octet rule).

Formation of Cations and Anions

Monatomic ions are formed by the loss or gain of electrons, resulting in cations (positive) or anions (negative). The electron configuration of the ion often matches that of the nearest noble gas (isoelectronic).

• Group I (Alkali metals): Lose one s electron to form +1 cations.

• Group II (Alkaline earth metals): Lose two s electrons to form +2 cations.

• Group III metals: Lose three electrons to form +3 cations.

• Group IV and V metals: Can lose p electrons or both s and p electrons, forming cations with pseudo-noble gas configurations.

• Group IV-VII nonmetals: Gain electrons to fill their valence shell, forming anions.

• Group VIII (Noble gases): Already have a full valence shell and do not form ions.

• Transition metals: Usually lose s electrons to form +2 cations, but can also lose d electrons for other charges.

Examples of Electron Configurations for Ions

Element	Electron Configuration	Ion	Ion Configuration
Li	1s22s1	Li+	1s2
Na	1s22s22p63s1	Na+	1s22s22p6
K	1s22s22p63s23p64s1	K+	1s22s22p63s23p6
Be	1s22s2	Be2+	1s2
Mg	1s22s22p63s2	Mg2+	1s22s22p6
Al	1s22s22p63s23p1	Al3+	1s22s22p6
Sn	[Kr] 4d105s25p2	Sn2+	[Kr] 4d105s2
Sn	[Kr] 4d105s25p2	Sn4+	[Kr] 4d10
Pb	[Xe] 4f145d106s26p2	Pb2+	[Xe] 4f145d106s2
Pb	[Xe] 4f145d106s26p2	Pb4+	[Xe] 4f145d10
Bi	[Xe] 4f145d106s26p3	Bi3+	[Xe] 4f145d106s2
Bi	[Xe] 4f145d106s26p3	Bi5+	[Xe] 4f145d10
C	1s22s22p2	C4–	1s22s22p6
N	1s22s22p3	N3–	1s22s22p6
O	1s22s22p4	O2–	1s22s22p6
F	1s22s22p5	F–	1s22s22p6
Ne	1s22s22p6	Ne	1s22s22p6
Ar	1s22s22p63s23p6	Ar	1s22s22p63s23p6
Fe	1s22s22p63s23p63d64s2	Fe2+	1s22s22p63s23p63d6
Fe	1s22s22p63s23p63d64s2	Fe3+	1s22s22p63s23p63d5

Additional info: Table entries for ions have been logically completed based on typical electron loss/gain patterns.