In Sublevels and Orbitals you met shells, sublevels and orbitals: 2p, 3d and so on. The quantum theory of the atom gives each of these an address made of numbers. Four quantum numbers describe one electron completely.
Key idea
$n$ (principal: 1, 2, 3, …) sets the shell, its size and most of its energy. $l$ (angular momentum: 0 to $n - 1$) sets the subshell's shape: $l = 0, 1, 2, 3$ are s, p, d, f. $m_l$ (magnetic: $-l$ to $+l$ in whole steps) picks one orbital, its orientation. $m_s$ (spin: $+\tfrac{1}{2}$ or $-\tfrac{1}{2}$) is the electron's spin. No two electrons in an atom share all four (the Pauli exclusion principle).
Why it works
The rules come from solving Schrödinger's equation for the electron in an atom (the physics belongs to a later course; here we use its results). Each rule counts something real. Since $l$ runs up to $n - 1$, shell $n$ has $n$ subshells: shell 1 has only 1s, shell 2 has 2s and 2p. Since $m_l$ runs from $-l$ to $+l$, a subshell has $2l + 1$ orbitals: 1 s, 3 p, 5 d, 7 f. Each orbital holds two electrons of opposite spin. So shell $n$ holds $n^2$ orbitals and $2n^2$ electrons: 2, 8, 18, 32.
| $n$ | $l$ | Subshell | $m_l$ values | Orbitals |
|---|---|---|---|---|
| 1 | 0 | 1s | 0 | 1 |
| 2 | 0, 1 | 2s, 2p | 0; −1, 0, +1 | 1 + 3 = 4 |
| 3 | 0, 1, 2 | 3s, 3p, 3d | …; −2 to +2 for 3d | 1 + 3 + 5 = 9 |
| 4 | 0, 1, 2, 3 | 4s, 4p, 4d, 4f | …; −3 to +3 for 4f | 16 |
Worked example: Allowed or not?
Which of these sets $(n, l, m_l, m_s)$ is not allowed? (a) $(3, 2, -2, +\tfrac{1}{2})$ (b) $(2, 2, 0, -\tfrac{1}{2})$ (c) $(4, 0, 0, +\tfrac{1}{2})$
- (a): $l = 2 \le n - 1 = 2$ and $m_l = -2$ is between $-2$ and $2$. Allowed: a 3d electron.
- (b): $l$ must be at most $n - 1 = 1$, but here $l = 2$. Not allowed: there is no 2d subshell.
- (c): a 4s electron. Allowed.
Answer: (b), because $l$ cannot equal $n$.
Worked example: Counting electrons
How many electrons in an atom can have $n = 4$ and $l = 2$? How many can have $n = 3$?
- $n = 4$, $l = 2$ is the 4d subshell. $m_l$ takes 5 values ($-2$ to $+2$), so 5 orbitals, each with 2 electrons: $10$.
- $n = 3$: $n^2 = 9$ orbitals, so $2n^2 = 18$ electrons.
Answer: 10 and 18.
Common mistake
Letting $l$ equal $n$ (writing a "2d" or "1p" orbital), or giving $m_l$ only positive values. $l$ stops at $n - 1$, and $m_l$ runs through every whole number from $-l$ to $+l$, zero included. Check by counting: a p subshell must come out with 3 orbitals ($-1, 0, +1$) and a d subshell with 5.