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, and each orbital corresponds to a particular energy level and spatial distribution.

• 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 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 $n^2$.

Letter Codes for Subshells

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

Example: The subshell with $n=2$ and $l=1$ is the 2p subshell.

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.

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

• Paramagnetic: Substances with 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	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: Higher shells follow the same pattern, with more subshells and orbitals.

Electron Configurations

Aufbau Principle and Filling Order

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"). The general filling order is:

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

Additional info: The periodic table can be used to determine the order in which orbitals are filled.

Writing Electron Configurations

• Write the orbitals occupied by electrons, followed by a superscript indicating the number of electrons (e.g., H: 1s1).

• Orbital diagrams use squares or circles for orbitals and arrows for electrons, showing their spins.

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

• 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 generally do not participate in bonding.

• Elements with similar properties have similar valence shell configurations.

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

Formation of Cations and Anions

Monatomic ions are formed by the loss or gain of electrons:

• Group I (alkali metals): Lose one electron to form +1 cations, becoming isoelectronic with the nearest noble gas.

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

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

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

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

• Group VIII (noble gases): Already have full shells; 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
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: Ions are often isoelectronic with noble gases, meaning they have the same electron configuration as a noble gas.