General Chemistry I · Electronic Structure of Atoms
Quantum Numbers
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In 30 seconds
The solutions to the Schrödinger equation describe electrons with orbitals, each labeled by a set of quantum numbers — the electron's "address." Three quantum numbers (n, l, mₗ) specify an orbital, and a fourth (mₛ) specifies the electron's spin within that orbital. Every electron in an atom has a unique set of four quantum numbers (the Pauli exclusion principle), and the allowed values of these numbers dictate the structure of the periodic table.
Why this matters
Quantum numbers are the grammar of electron structure. They explain why the periodic table has its shape: the first period has 2 elements (one s orbital), periods 2–3 have 8 (s + p), periods 4–5 have 18 (s + p + d), and the lanthanides/actinides add 14 (f). They justify the block structure (s-, p-, d-, f-blocks), the maximum of two electrons per orbital (Pauli), and the magnetic properties of atoms (unpaired spins). Every electron configuration that follows is just an ordered list of these addresses.
The college version
Key Ideas
- Principal quantum number (n): shell and size/energy. n = 1, 2, 3, … Larger n → larger orbital, higher energy.
- Angular momentum (azimuthal) quantum number (l): subshell and shape. l = 0, 1, 2, …, (n − 1). Values 0, 1, 2, 3 correspond to s, p, d, f.
- Magnetic quantum number (mₗ): orientation of the orbital in space. mₗ = −l, …, 0, …, +l (integer steps). There are (2l + 1) values.
- Spin quantum number (mₛ): electron spin. mₛ = +½ or −½ only.
- Orbital counts: s = 1 orbital, p = 3, d = 5, f = 7.
- Shell capacity: a shell n has n² orbitals and holds up to 2n² electrons.
Equations and Variables
| Symbol | Name | Allowed values | Meaning |
|---|---|---|---|
| n | Principal | 1, 2, 3, … | Shell, size, energy |
| l | Angular momentum | 0, 1, …, n−1 | Subshell, shape |
| mₗ | Magnetic | −l … 0 … +l | Orbital orientation |
| mₛ | Spin | +½, −½ | Electron spin |
- Orbitals in a subshell = 2l + 1
- Orbitals in a shell = n²
- Electrons in a shell = 2n²
How It Works
The four quantum numbers are hierarchical, like an address going from city → street → house → which side of the bed.
- n picks the shell (the "city"). For n = 3, we're in the third shell.
- l picks the subshell (the "street"). l can run from 0 to n − 1. For n = 3, l = 0, 1, 2 → the 3s, 3p, and 3d subshells.
- mₗ picks the specific orbital (the "house"). For l = 2 (d subshell), mₗ = −2, −1, 0, +1, +2 → five d orbitals, each pointing in a different direction.
- mₛ picks the spin (the "side of the bed"). Each orbital can hold at most two electrons, one spin-up (+½) and one spin-down (−½).
Putting it together: the shell n = 3 contains one 3s (l=0), three 3p (l=1), and five 3d (l=2) orbitals — nine orbitals total (n² = 9), holding up to 18 electrons (2n² = 18).
Letter code: l = 0 → s, l = 1 → p, l = 2 → d, l = 3 → f. The letters come from early spectroscopic descriptions (sharp, principal, diffuse, fundamental).
Worked Example
For n = 3, list all allowed values of l and, for l = 2, all values of mₗ.
Step 1 — l values: l = 0, 1, 2 (i.e., 0 to n − 1 = 2).
Step 2 — For l = 2, mₗ = −2, −1, 0, +1, +2 (five values).
So the n = 3 shell has subshells 3s, 3p, 3d, and the 3d subshell has five orbitals.
Second example — counting. How many orbitals and electrons are in the n = 4 shell?
Orbitals = n² = 4² = 16. Electrons = 2n² = 2 × 16 = 32.
Third example — identify the subshell. Which subshell has n = 5 and l = 3?
l = 3 → f. So it is the 5f subshell, with 2l + 1 = 7 orbitals holding up to 14 electrons.
Common Confusions
- l's upper limit. l goes only up to n − 1. For n = 1, only l = 0 exists (no 1p orbital). A common trap is writing "1p" — impossible.
- mₗ range. mₗ runs from −l to +l, not from −n to +n. For a p orbital (l = 1), mₗ is −1, 0, +1 only.
- s vs. mₛ. "s" (subshell letter for l = 0) and "mₛ" (spin) are unrelated — don't confuse the letter s with spin.
- 2n² vs. n². n² is the number of orbitals; 2n² is the number of electrons (two per orbital). Know which is asked.
- "Orbital" vs. "subshell" vs. "shell." Shell (n) contains subshells (l); each subshell contains orbitals (mₗ); each orbital holds ≤ 2 electrons.
Eli explains
The same idea, in plain words
Explain it like I’m 10
Every electron in an atom has a home address, and it's written in four parts — like a letter to your friend: city, street, house number, and which bunk in the room.
- n is the city (how far out from the nucleus — bigger number, farther out).
- l is the street (what kind of neighborhood: s, p, d, or f — these are different shapes).
- mₗ is the exact house on the street (which way the orbital is pointing).
- mₛ is the bunk bed (spin up or spin down — each bed fits only two electrons, one on top, one below).
And there's a rule: no two electrons can have the exact same full address. Just like the post office can't deliver two letters to "same city, same street, same house, same bunk," no two electrons can share all four numbers. That's why electrons spread out into different orbitals instead of piling up in one place.
Key takeaways
- n = 1, 2, 3, … ; l = 0 … n−1; mₗ = −l … +l; mₛ = ±½.
- l = 0(s), 1(p), 2(d), 3(f).
- s/p/d/f hold 1/3/5/7 orbitals and 2/6/10/14 electrons.
- A shell n has n² orbitals and 2n² electrons.
- No two electrons share all four quantum numbers (Pauli exclusion principle).
- mₛ is always +½ or −½, never any other value.
- What are the possible values of l when n = 2? Name the subshells.
- How many orbitals are in a d subshell, and how many electrons can it hold?
- What are the allowed values of mₛ?
- Give the four quantum numbers for an electron in a 3p orbital (any valid set).
- Answers: (1) l = 0, 1 → 2s and 2p; (2) five orbitals, ten electrons; (3) +½ and −½; (4) e.g., n = 3, l = 1, mₗ = −1, mₛ = +½ (mₗ can be −1, 0, or +1).
Study tools & related lessonsYou’ll learn to · Related
You’ll learn to
- Name the four quantum numbers and the property each describes.
- State the allowed values for n, l, mₗ, and mₛ and their relationships.
- Recognize s, p, d, and f subshells and count the orbitals in each.
- Determine the number of orbitals and electrons in a shell or subshell.
Sources & references
- OpenStax, *Chemistry 2e*, §6.3 Development of Quantum Theory (quantum numbers).
- OpenStax, *Chemistry 2e*, §6.4 Electronic Structure of Atoms.
- IUPAC "Gold Book" — orbital and quantum number definitions.
This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.
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