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 in a particular 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: Occur when electrons transition between orbitals, often accompanied by absorption or emission of photons.

Example: In a hydrogen atom, the electron in the $n=1$ orbital is in its ground state; if it moves to $n=2$, it is in an excited state.

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 $-\frac{1}{2}$

Additional info: The total number of orbitals for a given $n$ is $n^2$.

Subshells and Orbital Types

The angular momentum quantum number ($l$) divides shells into subshells, each with a characteristic shape and letter designation:

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

• Subshell notation: For example, $n=2$, $l=1$ is the 2p subshell.

• Energy order: Energy increases with $l$ (s < p < d < f).

Orbital Orientation and Electron Spin

• Number of orbitals per subshell: $2l+1$ (e.g., s: 1, p: 3, d: 5, f: 7).

• Electron spin: Each orbital can hold two electrons with opposite spins.

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.

Magnetic Properties

• Diamagnetic: All electrons are paired; not attracted to magnets.

• Paramagnetic: Contains unpaired electrons; 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: Higher shells follow the same pattern, with more subshells and orbitals.

Electron Configurations

Aufbau Principle and Filling Order

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

• Filling order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f

• Periodic table: The order can be deduced from the periodic table layout.

Notation: Orbitals are written with a superscript indicating the number of electrons (e.g., H: 1s1).

Orbital Diagrams and Hund's Rule

Orbital diagrams visually represent electron placement. Each orbital is shown as a box or circle, and electrons as arrows (up or down for 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 electrons in the outermost shell (highest $n$) are called valence electrons and are most involved in chemical bonding. Inner electrons are core electrons and usually do not participate in bonding.

• Valence shell: Outermost shell; determines chemical properties.

• Core electrons: Inner shells; generally inert in reactions.

Formation of Ions and Isoelectronic Species

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

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

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

• Group IIIA metals: Lose three electrons to form +3 ions.

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

• Non-metals (Groups IV-VII): Gain electrons to fill valence shell (8 electrons).

• Noble gases (Group VIII): Already have filled shells; do not form ions.

• Transition metals: Usually lose s electrons first, then d electrons, forming various positive 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
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: Ions with filled shells are more stable and are isoelectronic with noble gases.

----------------------------------------