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. The Schrödinger equation ($H\psi = E\psi$) yields wave functions ($\psi$), which represent the probability of finding an electron in a specific region of space. These wave functions are called atomic orbitals.

• Atomic orbital: A region in space with a high probability of finding an electron.

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

Quantum Numbers

Each electron in an atom is described by four quantum numbers, which define its energy, location, and spin.

• 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. $l = 0, \ldots, n-1$

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

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

Pauli Exclusion Principle: No two electrons in the same atom can have identical values for all four quantum numbers. Thus, each orbital can hold a maximum of two electrons with opposite spins.

Subshells and Orbitals

The angular momentum quantum number ($l$) divides shells into subshells, each with a characteristic shape and energy. Subshells are denoted by letters:

• s ($l=0$)

• p ($l=1$)

• d ($l=2$)

• f ($l=3$)

• g ($l=4$), h ($l=5$), etc. (rarely encountered)

The number of orbitals in a subshell is $2l+1$. For example:

• s subshell: 1 orbital

• p subshell: 3 orbitals

• d subshell: 5 orbitals

• f subshell: 7 orbitals

Electron Spin and Magnetic Properties

• Electrons have intrinsic spin, creating a magnetic field.

• Paired electrons (opposite spins) result in diamagnetic substances (not attracted to magnets).

• Unpaired electrons result in paramagnetic substances (weakly attracted to magnets).

Table of Allowed Quantum Numbers

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

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, ..., +3	7	4f	14

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

Writing Electron Configurations

Electron Configuration Notation

The arrangement of electrons in an atom's orbitals is called its electron configuration. Electrons fill orbitals in order of increasing energy, following the Aufbau principle ("building-up").

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

• Electron configuration is written as: [orbital]^[number of electrons], e.g., H: 1s1

Orbital diagrams use boxes or circles for orbitals and arrows for electrons, indicating spin direction. According to Hund's rule, electrons occupy orbitals singly before pairing.

Ground and Excited States

• 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

Electrons in the highest energy shell (largest n) are called valence electrons and are involved in chemical bonding. Inner electrons are core electrons and generally do not participate in bonding.

Formation of Ions and Isoelectronic Species

Atoms gain or lose electrons to achieve a stable, filled valence shell, often becoming isoelectronic with the nearest noble gas.

• 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 (e.g., Al): 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 (8 electrons), forming anions.

• Group VIII (noble gases): Already have a full valence shell; 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

• Li: 1s22s1 → Li+: 1s2

• Na: 1s22s22p63s1 → Na+: 1s22s22p6

• K: 1s22s22p63s23p64s1 → K+: 1s22s22p63s23p6

• Mg: 1s22s22p63s2 → Mg2+: 1s22s22p6

• Al: 1s22s22p63s23p1 → Al3+: 1s22s22p6

• 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

• C: 1s22s22p2 → C4–: 1s22s22p6

• N: 1s22s22p3 → N3–: 1s22s22p6

• O: 1s22s22p4 → O2–: 1s22s22p6

• F: 1s22s22p5 → F–: 1s22s22p6

• Ne: 1s22s22p6 (noble gas, no ion formation)

• Ar: 1s22s22p63s23p6 (noble gas, no ion formation)

• Fe: 1s22s22p63s23p63d64s2 → Fe2+: 1s22s22p63s23p63d6; Fe3+: 1s22s22p63s23p63d5

Example: Sodium (Na) loses one electron to form Na+, which has the same electron configuration as neon (Ne), making it isoelectronic with Ne.

Additional info: Electron configurations for ions are written to show how atoms achieve stable, noble gas-like arrangements.