Quantum Numbers and Electron Configurations — Detailed Notes

Principal Quantum Number (n)

Purpose: describes the size of the orbital and its energy; used in the Schrödinger equation to describe the probability of finding an electron around the nucleus.
n is a positive integer (whole number): $n \,=\, 1,2,3,\ldots$ and $n \in \mathbb{Z}^+$ .
Practical limit in the real universe: we only observe up to seven shells (n ≤ 7) in known elements; theoretically can go to infinity.
n labels the electron shell (row of the periodic table) and sets the boundary for allowed values of the next quantum number (l).

Angular Momentum Quantum Number (l)

Purpose: describes the shape of the orbital (subshell). Defines the subshell type (s, p, d, f, …).
l is an integer that depends on n: $l \in {0,1,\dots, n-1}$ .
Each value of l corresponds to an orbital type (subshell):
- $l=0 \rightarrow s\, type\, orbital$ (spherical, 1 orbital per subshell)
- $l=1 \rightarrow p\, type\, orbital$ (dumbbell shape, 3 orbitals per subshell)
- $l=2 \rightarrow d\, type\, orbital$ (cloverleaf, 5 orbitals per subshell)
- $l=3 \rightarrow f\, type\, orbital$ (flower-like, 7 orbitals per subshell)
The numerical value represents the number of nodal planes on the orbital: for s, 0 nodal planes; for p, 1; for d, 2; for f, 3. In general, $N_{nodal} = l$ .
Relationship to n: l is restricted by n (lower than n), i.e., higher-n shells can host more complex subshells.

Magnetic Quantum Number (m_l)

Purpose: describes the orientation of the orbital in 3D space.
For a given l, m_l can take integer values from $-l) to +l) in steps of 1:$
- If l=1 (p subshell): $ml \in {-1, 0, +1}$ (3 orbitals: px, py, pz)
- If l=2 (d subshell): $m_l \in {-2, -1, 0, +1, +2}$ (5 orbitals)
- If l=3 (f subshell): $m_l \in {-3,-2,-1,0,+1,+2,+3}$ (7 orbitals)
Note: you don’t need to memorize which specific ml corresponds to each orientation; just that there are (2l+1) possible ml values, giving that many orbitals in the subshell.

Electron Spin Quantum Number (m_s)

Purpose: describes the intrinsic spin orientation of the electron, contributing to its magnetic moment.
Allowed values: $m_s \in {+\tfrac{1}{2}, -\tfrac{1}{2}}$ (two possible spin states per electron).
Spin and mass remark (from the lecture): this spin value is tied to mass-bearing vs massless particles in the simplified discussion:
- Half-integer spins (e.g., $\pm \tfrac{1}{2}$ ) are associated with massive particles (in the teaching context, e.g., electrons).
- Whole-integer spins (e.g., 0, ±1, ±2, …) are discussed as massless or non-mass-carrying in the lecture context (e.g., photons, gravitons, Higgs boson).
Each orbital can hold up to two electrons with opposite spins (Pauli exclusion principle).

Summary of quantum numbers and how they constrain orbitals

Quantum numbers describe the 3D shapes and energies of orbitals that arise from the Schrödinger wave equation ($\phi$):
- Shape and size depend on n and l; orientation on ml; spin on ms.
n fixes the size of the orbital; higher n means larger orbitals (onion-like layering around the nucleus).
l fixes the orbital type (s, p, d, f) and the number of nodal planes (equal to l).
m_l fixes the orientation in space (degenerate orbitals within a subshell share the same energy before considering other interactions).
m_s fixes the spin state; together with other quantum numbers, determines the electron’s address within an orbital.

How many orbitals and electrons per shell (quick references)

Number of orbitals in a given shell (principal quantum number) is: $N_{orbitals} = n^2$ .
Number of electrons that can occupy a given shell: $N_{electrons} = 2n^2$ .
For a given subshell with angular momentum l, the number of orbitals is: $N_{orbitals}(l) = 2l+1$ (e.g., s:1, p:3, d:5, f:7).

Four types of quantum-number questions you may see

1) Name all subshells within a given n (e.g., $n=4$ ).

Use the rule $l \in {0,1,2,3}$ for $n=4$ , giving subshells 4s, 4p, 4d, 4f.
2) How many orbitals are there in a given n (e.g., $n=4$ ).
Count by summing 2l+1 for l=0,1,2,3: $1+3+5+7 = 16$ orbitals.
General fast method: number of orbitals in shell $N_{orbitals} = n^2$ .
3) Given a set of quantum numbers, determine if it’s a valid set.
Rules: $n \in \mathbb{Z}^+, n\ge 1$ ; $l \in {0,1,\dots,n-1}$ ; $ml \in {-l,\dots,+l}$ ; $ms \in {+\tfrac{1}{2}, -\tfrac{1}{2}}$ .
Check consistency among the four numbers; any violation makes the set invalid.
4) Given a set of quantum numbers, identify the corresponding orbital address or identify invalid elements.
Example checks include whether $ml$ aligns with the chosen $l$ , whether $l$ is allowed for the given $n$ , and whether $ms$ is one of the two allowed values.

Putting quantum numbers into context: orbital shapes and orientations

S orbitals (l=0): spherical; 1 orbital per subshell; no nodal planes.
P orbitals (l=1): dumbbell shapes; 3 orbitals per p subshell; 1 nodal plane per orbital; orientations along x, y, z (px, py, p_z).
D orbitals (l=2): cloverleaf shapes (four lobes) plus a donut-oriented one; 5 orbitals per d subshell; 2 nodal planes.
F orbitals (l=3): flower-like shapes; 7 orbitals per f subshell; 3 nodal planes.
Orientation and degeneracy: s types are directionally symmetric; p, d, f subshells have multiple orientation possibilities leading to degenerate orbitals within the same n and l values.

Electron configurations and the key rules

Electron configuration lists all occupied subshells and how many electrons are in them for a given atom/ion.
Three fundamental rules:
- Aufbau principle: electrons fill the lowest-energy orbitals first before higher-energy orbitals; analogy: liquids fill the lowest space first (like water in a cup).
- Hund's rule: when filling degenerate orbitals (same n and l), assign one electron to each orbital before pairing electrons in any orbital. This minimizes repulsion and stabilizes the atom.
- Pauli exclusion principle: no two electrons can have the same set of quantum numbers; at most two electrons per orbital with opposite spins.

Visual and mnemonic methods for orbital filling order

Visual/diagonal filling pattern (as taught): follow a diagonal path through the (n, l) diagram: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p → 7s … (stop at reasonable levels for introductory chem).
Alternative method: build a triangular diagram of n and l values, then draw diagonal lines to indicate the order of filling. This is often memorized by a recursive triangle or by the diagonal rule.
Another teaching approach uses the periodic table as a quick guide to identify the core and valence electrons (condensed configurations) by using noble-gas cores.

Using the periodic table to read electron configurations

Four blocks in the periodic table corresponding to subshells:
- s-block: first two columns; highest-energy electrons are in an s-type subshell. The row number equals the principal quantum number (n).
- p-block: last six columns; highest-energy electrons are in a p-type subshell; row number equals n.
- d-block (transition metals): center; highest-energy electrons are in a d-type subshell; n for the highest-energy d electrons is one less than the row number (i.e., largest n is row, but the d subshell is n-1).
- f-block: bottom two rows; highest-energy electrons are in an f-type subshell; f subshell principal number is two less than the actual row number.
Example: Strontium in the fifth row has highest-energy electrons in the 5s subshell (row number equals n).
Example: Iron (Fe) in the fourth row: highest-energy electrons are in the 3d subshell (one less than row number 4) and 4s, etc.
Example: Europium (Eu) has a complex fill with f-block involvement; its highest-energy electrons involve 4f and 6s as described.
Important note: there are no known elements that require g or higher subshells in introductory chemistry; g, h, and higher subshells remain theoretically possible but are not populated by observed elements (as of the course content).

Condensed electron configurations and core notation

Core electrons: electrons in shells lower in energy than the valence shell; they are often represented by a noble-gas core.
Valence electrons: electrons in the outermost shell (highest n) that participate in bonding and chemistry.
Examples:
- Beryllium (Be, Z=4): neutral; electron count = 4; on the second row (n=2). Configuration by subshells: $1s^2\,2s^2$ ; condensed form: "core" He (1s^2) + $2s^2$ → [He] 2s^2.
- Phosphorus (P, Z=15): neutral; 15 electrons. Fill order: $1s^2\,2s^2\,2p^6\,3s^2\,3p^3$ . Condensed form: [Ne] 3s^2 3p^3. In the 2p subshell, there are three degenerate orbitals; Hund's rule yields three unpaired electrons in 2p before pairing.
- Nickel (Ni, Z=28): condensed form using Ar core. In practice, Ni is described as [Ar] 3d^8 4s^2 (consistent with the lecture's explanation that four s and three d subshells are involved; the four s subshell is filled, then the 3d subshell begins filling to reach 8 electrons in 3d and 2 in 4s).
- Rhenium (Re, Z=75): condensed form with Xe core: [Xe] 4f^{14} 5d^5 6s^2. Note: the 4f subshell fills before the 5d, and the 6s fills as well; the “f-block before d-block” rule applies in this region.

Exceptions to the simple filling pattern

Chromium (Cr, Z=24) and Copper (Cu, Z=29) illustrate notable deviations from the simple Aufbau sequence due to electron-electron interactions and stabilization from half-filled or fully-filled subshells.
- Chromium: expected configuration by simple rules would be [Ar] 4s^2 3d^4, but the actual preferred configuration is [Ar] 4s^1 3d^5, giving two payoffs: one half-filled 4s and one half-filled 3d subshell, which stabilizes the atom.
- Copper: similar stabilization leads to [Ar] 3d^{10} 4s^1 (rather than 4s^2 3d^9).
The lecturer emphasizes teaching the “wrong way” first (the naive pattern) and then showing the correct pattern via payoffs for filled and half-filled subshells.

Practical takeaways for exam-ready mastery

Always verify that the principal number n allows the angular momentum l, and that ml and ms are consistent with l and n.
Use the rules in order when evaluating a set of quantum numbers:
- n must be a positive integer: $n \in {1,2,3,\dots}$
- l must satisfy: $l \in {0,1,\dots,n-1}$
- ml must satisfy: $ml \in {-l,-l+1,\dots,+l}$
- ms must be: $ms \in {+\tfrac{1}{2}, -\tfrac{1}{2}}$
Remember: orbitals per shell grow as $N{orbitals}=n^2$ and orbitals per subshell follow $N{orbitals}(l)=2l+1$ .
The four-block view of the periodic table helps determine the highest-energy electron’s subshell: s-block (rows correspond to n), p-block (same n as row), d-block (one less than row), f-block (two less than row).
Condensed configurations via noble-gas cores simplify writing electron configurations for heavier elements.
The diagonal filling order (and its variants) is the practical method taught for quickly determining the order of orbital filling on exams.
In chemistry, we primarily focus on the valence shell electrons, but the core electrons are essential for understanding shielding and overall energy picture; cores are often represented by noble gases in condensed notation.
Be aware of the shapes, degeneracy, and orientation of orbitals: s (1, spherical), p (3 orientations), d (5 from l=2 with 2 nodal planes), f (7 from l=3 with 3 nodal planes).
The spin state has two values per electron; overall, two electrons per orbital with opposite spins.
Real-world relevance: electron configurations explain periodic trends, magnetic properties, spectroscopy, and chemical reactivity; the organization of the periodic table reflects the filling order and orbital types.
Philosophical/ethical implications (brief): understanding quantum numbers and electron configurations underpins modern chemistry and materials science, enabling advances in energy, medicine, and technology while requiring responsible use of such knowledge.

Quick reference formulas and facts (latex-ready)

Principal number domain: $n \in \mathbb{Z}^+, \; n \ge 1$
Angular momentum domain: $l \in {0,1,\dots,n-1}$
Magnetic quantum number domain: $m_l \in {-l,-l+1,\dots,+l}$
Spin quantum number domain: $m_s \in {+\tfrac{1}{2}, -\tfrac{1}{2}}$
N orbitals per shell: $N_{orbitals} = n^2$
N electrons per shell: $N_{electrons} = 2n^2$
Orbitals per subshell: $N_{orbitals}(l) = 2l+1$
Nodal planes: $N_{nodal} = l$
Subshell types (l values): $l=0\rightarrow s, \; l=1\rightarrow p, \; l=2\rightarrow d, \; l=3\rightarrow f$
Typical filling order (partial list): 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p