Quantum Numbers, Atomic Orbitals, and Electron Configurations

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Quantum Numbers and Atomic Orbitals

Introduction to Quantum Numbers

Quantum numbers are fundamental to understanding the arrangement of electrons in atoms. They arise from the solutions to the Schrödinger equation, which describes the behavior of electrons as wave functions (ψ). Each electron in an atom is described by a unique set of four quantum numbers, which together specify its energy, location, and spin.

Wave function (ψ): A mathematical function describing the probability of finding an electron in a particular region of space, known as an atomic orbital.
Atomic orbital: A region in space with a high probability of finding an electron.
Energy changes: Occur when electrons move between orbitals, often accompanied by absorption or emission of photons.

The Four Quantum Numbers

Principal Quantum Number (n): Specifies the energy level and size of the orbital. Possible values: n = 1, 2, 3, ...
- Defines the shell (level) of the electron.
- All orbitals with the same n are in the same shell.
- The number of orbitals in a shell:
- Example: For hydrogen, n=1 is the ground state; n=2 is an excited state.
Angular Momentum Quantum Number (l): Specifies the shape of the orbital. Possible values: l = 0, 1, ..., n-1.
- Defines subshells (sublevels) within a shell.
- Letter codes: l = 0 (s), 1 (p), 2 (d), 3 (f), 4 (g), 5 (h), ...
- Example: n=2, l=1 is the 2p subshell.
- Energy increases with l: s < p < d < f
Magnetic Quantum Number (ml): Specifies the orientation of the orbital. Possible values: ml = -l, ..., 0, ..., +l.
- Determines the number of orbitals in a subshell: 2l+1
- Example: s (l=0) has 1 orbital; p (l=1) has 3 orbitals; d (l=2) has 5 orbitals.
Spin Quantum Number (ms): Specifies the spin orientation of the electron. Possible values: ms = +½ or -½.
- Each orbital can hold a maximum of two electrons with opposite spins.

Pauli Exclusion Principle

Pauli Exclusion Principle: No two electrons in the same atom can have identical values for all four quantum numbers. This means:

Each orbital can hold a maximum of two electrons, and they must have opposite spins.

Magnetism and Electron Spin

Diamagnetic substances: All electrons are paired; not attracted to magnets.
Paramagnetic substances: Contain unpaired electrons; weakly attracted to magnets.

Table of Allowed Quantum Numbers

The following table summarizes the allowed quantum numbers, the number of orbitals, and the maximum number of electrons for each subshell:

n	l	ml	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, -2, -1, 0, +1, +2, +3	7	4f	14

Additional info: Table entries for n=3,4,5, etc., have been logically completed based on quantum number rules.

Writing Electron Configurations

Electron Configuration Principles

The electron configuration of an atom describes the distribution of electrons among its orbitals. Electrons fill orbitals in order of increasing energy, following the Aufbau principle ("building-up").

Order of filling: Orbitals are filled in the order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, ...
Periodic table shortcut: The order can be determined by reading diagonally across the periodic table blocks.
Notation: Each occupied orbital is written with a superscript indicating the number of electrons (e.g., 1s2).
Orbital diagrams: Use boxes or circles for orbitals and arrows for electrons (up or down for spin).
Hund's Rule: Electrons occupy degenerate (equal energy) orbitals singly before pairing up.
Ground state: All electrons are in the lowest possible energy levels.
Excited state: One or more electrons occupy higher energy orbitals after absorbing energy.

Properties of Monatomic Ions

Valence and Core Electrons

The valence shell contains the most energetic electrons (highest n), which are involved in chemical bonding. Core electrons are inner electrons that do not usually participate in bonding.

Elements with similar outer shell configurations have similar chemical properties.
Atoms tend to achieve a filled outer shell (noble gas configuration) for stability.

Formation of Ions by Groups

Group I (Alkali metals): Lose one s electron to form +1 cations.
- Li: 1s22s1 → Li+: 1s2
- Na: 1s22s22p63s1 → Na+: 1s22s22p6
- K: 1s22s22p63s23p64s1 → K+: 1s22s22p63s23p6
Group II (Alkaline earth metals): Lose two s electrons to form +2 cations.
- Be: 1s22s2 → Be2+: 1s2
- Mg: 1s22s22p63s2 → Mg2+: 1s22s22p6
Group IIIA metals: Lose three electrons to form +3 cations.
- Al: 1s22s22p63s23p1 → Al3+: 1s22s22p6
Group IV and V metals: Can lose p electrons or both s and p electrons to form cations with different charges (pseudo-noble gas configurations).
- Sn: [Kr] 4d105s25p2 → Sn2+: [Kr] 4d105s2; Sn4+: [Kr] 4d10
- Pb: [Xe] 4f145d106s26p2 → Pb2+: [Xe] 4f145d106s2; Pb4+: [Xe] 4f145d10
- Bi: [Xe] 4f145d106s26p3 → Bi3+: [Xe] 4f145d106s2; Bi5+: [Xe] 4f145d10
Group IV-VII nonmetals: Gain electrons to fill their valence shell (8 electrons).
- C: 1s22s22p2 → C4–: 1s22s22p6
- N: 1s22s22p3 → N3–: 1s22s22p6
- O: 1s22s22p4 → O2–: 1s22s22p6
- F: 1s22s22p5 → F–: 1s22s22p6
Group VIII (Noble gases): Already have a full outer shell; do not form ions.
- Ne: 1s22s22p6
- Ar: 1s22s22p63s23p6
Transition metals (B-group): Usually form +2 cations by losing s electrons, but can also lose d electrons to form other charges.
- Fe: 1s22s22p63s23p63d64s2 → Fe2+: 1s22s22p63s23p63d6; Fe3+: 1s22s22p63s23p63d5

Isoelectronic Species

Ions that have the same electron configuration as a noble gas are said to be isoelectronic with that noble gas. This configuration is especially stable.

Summary Table: Electron Configurations of Selected Ions

Element	Electron Configuration	Ion	Ion Electron Configuration	Isoelectronic With
Na	1s22s22p63s1	Na+	1s22s22p6	Ne
Mg	1s22s22p63s2	Mg2+	1s22s22p6	Ne
Cl	1s22s22p63s23p5	Cl–	1s22s22p63s23p6	Ar
Fe	1s22s22p63s23p63d64s2	Fe2+	1s22s22p63s23p63d6	—

Additional info: Table includes common ions and their isoelectronic noble gases for reference.