Quantum Numbers
Check whether a set of n, ℓ, mℓ and mₛ is allowed, and see which orbital it names.
holds up to 2 electrons
- ✓n — principalShell 3. It sets the size and most of the energy.
- ✓ℓ — angular momentumd subshell, which is one of the 3 allowed here (0 to 2).
- ✓mℓ — magneticOne of the 5 orbitals in the subshell (−2 to +2).
- ✓mₛ — spin+½ — either value is allowed for any orbital, and the two electrons sharing one orbital must differ here.
The 3d subshell
5 orbitals × 2 electrons = 10 electrons in this subshell.
Four quantum numbers name one electron. n gives the shell, ℓ the subshell shape, mℓ which orbital within it, and mₛ the spin. Each constrains the next: ℓ runs from 0 to n − 1, and mℓ from −ℓ to +ℓ. So n = 3, ℓ = 2 is a 3d orbital.
A nested set of choices
The four numbers are not independent — each one narrows the range of the next, which is why an invalid set is usually invalid for a structural reason rather than an arbitrary one. Choosing n fixes how many subshells exist; choosing ℓ fixes how many orbitals that subshell holds; choosing mℓ picks one of them; and mₛ picks one of the two electrons that orbital can take.
That nesting is where the shape of the periodic table comes from. n = 1 permits only ℓ = 0, so the first shell is a single s orbital holding two electrons — and the first period has two elements. n = 2 permits ℓ = 0 and ℓ = 1, giving 1 + 3 = 4 orbitals and eight electrons, and the second period has eight elements. The 2ℓ + 1 count is the reason for the table’s block widths: 1, 3, 5 and 7 orbitals give the s, p, d and f blocks their 2, 6, 10 and 14 columns.
What each number physically means
n sets the orbital’s size and most of its energy. ℓ sets its shape — spherical for s, two-lobed for p, four-lobed for most d. mℓ sets its orientation in space, which is why the three p orbitals are labelled pₓ, p_y and p_z. And mₛ is intrinsic angular momentum, which has no classical picture at all; the electron is not spinning in any literal sense.
a subshell holds 2ℓ + 1 orbitals; a shell holds 2n² electrons
- 1 Check n is a positive whole number. n = 0 and n = 2.5 are both impossible. There is no upper limit in principle, though known elements reach n = 7.
- 2 Check ℓ against n. ℓ must be 0 to n − 1. With n = 3 that allows 0, 1 and 2 — so 3s, 3p and 3d exist, but 3f does not.
- 3 Check mℓ against ℓ. mℓ runs from −ℓ to +ℓ. With ℓ = 2 that is −2, −1, 0, +1, +2 — five d orbitals.
- 4 Check the spin. mₛ is +½ or −½, with no other option and no dependence on the numbers above it.
- 5 Name the orbital. n = 3 with ℓ = 2 is 3d, and mℓ = 0 picks one particular 3d orbital out of the five.
What each shell allows
Orbitals per subshell is 2ℓ + 1; electrons per shell is 2n².
| n | Allowed ℓ | Subshells | Orbitals | Max electrons |
|---|---|---|---|---|
| 1 | 0 | 1s | 1 | 2 |
| 2 | 0, 1 | 2s, 2p | 4 | 8 |
| 3 | 0, 1, 2 | 3s, 3p, 3d | 9 | 18 |
| 4 | 0, 1, 2, 3 | 4s, 4p, 4d, 4f | 16 | 32 |
| 5 | 0, 1, 2, 3, 4 | 5s, 5p, 5d, 5f, 5g | 25 | 50 |
Pauli, and why four numbers are enough
The Pauli exclusion principle says no two electrons in an atom may share all four numbers. Since mₛ has only two values, that caps an orbital at two electrons — and that single restriction produces the entire structure of the periodic table. It is also why the fourth number is needed at all: n, ℓ and mℓ name an orbital, but two electrons live there, and something has to tell them apart.
Two nuances the rules alone do not capture. Subshells fill in order of n + ℓ rather than n, which is why 4s fills before 3d — and why chromium and copper break the pattern outright, since a half-filled or filled d subshell is more stable than the naive order predicts. And the 5g subshell in the table above is allowed by the rules but unoccupied in every known element, because no atom yet has enough electrons to reach it.
What this page checks is the rules themselves: whether a proposed set of four numbers is internally consistent and which orbital it names. It does not work out an element’s configuration or predict which orbital an electron actually occupies — those depend on filling order and electron–electron repulsion rather than on the four numbers alone.