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 of finding an electron at a particular location. These wave functions are called atomic orbitals, and they define regions in space where electrons are likely to be found.

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

• Energy changes: 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 letter codes:

• $l = 0$: s

• $l = 1$: p

• $l = 2$: d

• $l = 3$: f

• $l = 4$: g

• $l = 5$: h

The number of orbitals in a subshell is $2l+1$. For example, the p subshell ($l=1$) has three orbitals.

Electron Spin and Magnetic Properties

• Diamagnetic: Substances with all electrons paired; not attracted to magnets.

• Paramagnetic: Substances with unpaired electrons; 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, -2, -1, 0, +1, +2, +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 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:

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

• Each orbital is written with a superscript indicating the number of electrons (e.g., $\text{H}: 1s^1$)

Electron configurations can also be represented by orbital diagrams, where each orbital is shown as a box and electrons as arrows (up or down for spin).

Hund's Rule and Ground/Excited States

• Hund's Rule: Electrons occupy orbitals of equal energy singly before pairing up, to maximize unpaired electrons.

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

• Excited State: Electrons have absorbed energy and occupy higher energy orbitals.

Properties of Monatomic Ions

Valence and Core Electrons

The electrons in the outermost shell (highest n) are called valence electrons and are most involved in chemical bonding. Inner electrons are called core electrons and generally do not participate in bonding.

Formation of Ions and Isoelectronic Species

Elements form ions by gaining or losing electrons to achieve a stable, filled outer shell (often matching the configuration of a noble gas, called isoelectronic).

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

• Group II (Alkaline Earth Metals): Lose two electrons to form +2 cations.

• Group III (Aluminum Group): Lose three electrons to form +3 cations.

• Group IV and V Metals: Can lose p electrons or both s and p electrons to form 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 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: $1s^2 2s^1$ → Li+: $1s^2$

• Na: $1s^2 2s^2 2p^6 3s^1$ → Na+: $1s^2 2s^2 2p^6$

• K: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^1$ → K+: $1s^2 2s^2 2p^6 3s^2 3p^6$

• Mg: $1s^2 2s^2 2p^6 3s^2$ → Mg2+: $1s^2 2s^2 2p^6$

• Al: $1s^2 2s^2 2p^6 3s^2 3p^1$ → Al3+: $1s^2 2s^2 2p^6$

• C: $1s^2 2s^2 2p^2$ → C4–: $1s^2 2s^2 2p^6$

• N: $1s^2 2s^2 2p^3$ → N3–: $1s^2 2s^2 2p^6$

• O: $1s^2 2s^2 2p^4$ → O2–: $1s^2 2s^2 2p^6$

• F: $1s^2 2s^2 2p^5$ → F–: $1s^2 2s^2 2p^6$

• Fe: $1s^2 2s^2 2p^6 3s^2 3p^6 3d^6 4s^2$ → Fe2+: $1s^2 2s^2 2p^6 3s^2 3p^6 3d^6$; Fe3+: $1s^2 2s^2 2p^6 3s^2 3p^6 3d^5$

Isoelectronic species: Ions with the same electron configuration as a noble gas (e.g., Na+ and Ne).

Summary Table: Electron Configurations of Common Ions

Element	Electron Configuration	Ion	Ion Configuration
Li	1s22s1	Li+	1s2
Na	1s22s22p63s1	Na+	1s22s22p6
K	1s22s22p63s23p64s1	K+	1s22s22p63s23p6
Mg	1s22s22p63s2	Mg2+	1s22s22p6
Al	1s22s22p63s23p1	Al3+	1s22s22p6
C	1s22s22p2	C4–	1s22s22p6
N	1s22s22p3	N3–	1s22s22p6
O	1s22s22p4	O2–	1s22s22p6
F	1s22s22p5	F–	1s22s22p6
Fe	1s22s22p63s23p63d64s2	Fe2+	1s22s22p63s23p63d6
Fe	1s22s22p63s23p63d64s2	Fe3+	1s22s22p63s23p63d5

Additional info: Table entries for ions have been logically completed for clarity and completeness.