Quantum Numbers n, l, m, s — Allowed Values and What Each Controls
An address has a pin code, a locality, a street and a house number, each part narrowing things down until one house is left. The four quantum numbers do the same job for an electron in an atom, picking out one electron and no other. Seen as an address rather than four unrelated symbols, the topic becomes bookkeeping — and bookkeeping is easy to get right.
The allowed values — the whole rule set
Every legal set obeys those four lines; every illegal set breaks one. Note the chain: l depends on n, ml depends on l, and only ms is free.
| Quantum number | Name | Allowed values | What it controls |
|---|---|---|---|
| n | Principal | 1, 2, 3, … (positive whole numbers, never 0) | Shell — size of the orbital and, in a hydrogen-like atom, the energy |
| l | Azimuthal / angular momentum | 0, 1, 2, … up to n − 1 | Subshell — the shape of the orbital (s, p, d, f) |
| ml | Magnetic | every whole number from −l to +l, including 0 | Orientation of the orbital in space |
| ms | Spin | +½ or −½ only | Which of the two electrons in that orbital you mean |
The l values and their letters
| l | Letter | ml values | Orbitals in the subshell (2l + 1) | Maximum electrons |
|---|---|---|---|---|
| 0 | s | 0 | 1 | 2 |
| 1 | p | −1, 0, +1 | 3 | 6 |
| 2 | d | −2, −1, 0, +1, +2 | 5 | 10 |
| 3 | f | −3 … +3 | 7 | 14 |
Counting a whole shell
| n | Subshells present | Orbitals | Check against n² | Max electrons (2n²) |
|---|---|---|---|---|
| 1 | 1s | 1 | 1² ✓ | 2 |
| 2 | 2s, 2p | 1 + 3 = 4 | 2² ✓ | 8 |
| 3 | 3s, 3p, 3d | 1 + 3 + 5 = 9 | 3² ✓ | 18 |
| 4 | 4s, 4p, 4d, 4f | 1 + 3 + 5 + 7 = 16 | 4² ✓ | 32 |
Rebuild that table from memory and you can answer almost any "how many electrons can have …" question without a formula.
Worked example 1 — is this set allowed?
Question: Which of these sets are permitted?
(a) n = 2, l = 2, ml = 0, ms = +½
(b) n = 3, l = 2, ml = −3, ms = −½
(c) n = 4, l = 3, ml = +2, ms = +½
(d) n = 0, l = 0, ml = 0, ms = +½
(a) Not allowed. l must be at most n − 1 = 1; there is no 2d subshell.
(b) Not allowed. With l = 2, ml may only be −2 to +2.
(c) Allowed. n = 4 so l may be 0–3 ✓; l = 3 so ml may be −3 to +3, and +2 is inside ✓; ms = +½ ✓. An electron in a 4f orbital.
(d) Not allowed. n starts at 1; there is no shell zero.
Method: check the chain in order — n, then l against n, then ml against l, then ms. Stop at the first failure.
Worked example 2 — how many electrons fit a description?
Question (i): How many electrons can have n = 4 and l = 1?
That is the 4p subshell. Orbitals = 2l + 1 = 3, each holding 2 electrons.
Answer: 3 × 2 = 6 electrons.
Question (ii): How many can have n = 3?
2n² = 2 × 3² = 18 electrons.
Question (iii): How many can have n = 3 and ms = +½?
Half of those 18, since each of the 9 orbitals holds one electron of each spin.
Answer: 9 electrons.
Question (iv): How many can have n = 3, l = 2 and ml = 0?
Three quantum numbers fixed names one specific orbital, and an orbital holds 2 electrons — one of each spin. Whenever a question fixes n, l and ml, the answer is 2.
Worked example 3 — quantum numbers of the last electron
Question: Give a valid set of quantum numbers for the last electron of chlorine (Z = 17).
Step 1 — write the configuration: 1s² 2s² 2p⁶ 3s² 3p⁵
Step 2 — identify where the last electron sits: in the 3p subshell.
Step 3 — read off: n = 3 (the shell number), l = 1 (p means l = 1).
Step 4 — spin: 3p⁵ means three orbitals take one electron each first (Hund's rule), then
pairing begins. The fifth electron is the paired one — conventionally ms = −½.
A valid answer: n = 3, l = 1, ml = 0, ms = −½
Be careful with ml. Textbooks differ on whether the p orbitals are filled in the order −1, 0, +1 or +1, 0, −1. That is purely a labelling convention — the three p orbitals are identical in energy in a free atom, so no physical fact decides it, and different books give ml = 0 or −1 here. What is not negotiable is n = 3, l = 1 and that the electron is paired. Follow your own textbook's convention and say which you used.
Worked example 4 — counting nodes
Nodes are places where the probability of finding the electron is zero. The counting rules follow straight from n and l:
Question: Count the nodes in a 3s, a 3p and a 3d orbital.
3s: n = 3, l = 0 → angular = 0, radial = 3 − 0 − 1 = 2, total = 2
3p: n = 3, l = 1 → angular = 1, radial = 3 − 1 − 1 = 1, total = 2
3d: n = 3, l = 2 → angular = 2, radial = 3 − 2 − 1 = 0, total = 2
Every n = 3 orbital has exactly 2 nodes in total, as n − 1 = 2 requires; what changes is the split between angular and radial. That check — angular + radial must equal n − 1 — catches an arithmetic slip instantly.
Two rules that decide where electrons go
- Pauli exclusion principle: no two electrons in an atom share all four quantum numbers. This is why an orbital holds only two — n, l and ml are already fixed, so only the two ms values remain.
- Hund's rule: within a subshell, electrons occupy separate orbitals with parallel spins before any orbital takes a second electron.
Common mistakes that cost marks
- Letting l equal n. The maximum is n − 1. There is no 1p, no 2d and no 3f.
- Writing n = 0. The principal quantum number begins at 1.
- Forgetting that ml = 0 is one of the values. l = 1 gives three values (−1, 0, +1), not two.
- Using 2n² for a subshell. 2n² is the whole shell; a subshell holds 2(2l + 1).
- Assuming energy depends only on n. True for one-electron species only; in a multi-electron atom it depends on both n and l.
- Confusing total nodes with radial nodes. Radial nodes are n − l − 1; the total is n − 1.
Where this appears in exams
| Exam | Typical use |
|---|---|
| CBSE/ICSE Class 11 | Structure of atom — allowed sets, orbital counts, electron configurations |
| JEE / NEET | "Which set is not possible" and "how many electrons have …" questions |
| IIT-JAM / CUET-PG | Node counting, orbital degeneracy, hydrogen-like energy levels |
| GATE / CSIR-NET | Angular momentum values, term symbols, spectroscopic selection rules |
Check a set in one click. The quantum numbers tool tests whether (n, l, ml, ms) is allowed, names the subshell, and reports the orbital count, electron capacity and node counts — so a whole page of practice questions can be checked quickly.
Open the Quantum Numbers Calculator →Structure of the atom is the chapter everything else in Class 11 and 12 chemistry rests on — bonding, periodicity and coordination chemistry all build on it. ABC Chemistry teaches it in the Class 11–12 batches at the Gurugram centre and in online classes across India: abcchemistry.in.