Quantum Numbers - MCAT General Chemistry

Quantum Numbers

9 min read Updated Mar 26, 2026

Every electron in an atom has a unique address described by exactly four quantum numbers. No two electrons in the same atom can share all four - this is the Pauli exclusion principle. Understanding these four numbers is essential because they determine everything about where an electron lives and how it behaves.

1. Principal Quantum Number (n)

The principal quantum number describes the energy level (shell) of the electron.

Values: n = 1, 2, 3, 4, … (any positive integer)
Physical meaning: Determines the average distance from the nucleus and the overall energy. Higher n = higher energy, larger orbital, farther from the nucleus.
Maximum electrons in shell n: 2n²

Shell (n)	Max Electrons (2n²)
1	2
2	8
3	18
4	32

The energy gaps between levels decrease as n increases. The jump from n = 1 to n = 2 is enormous compared to the jump from n = 5 to n = 6.

2. Azimuthal (Angular Momentum) Quantum Number (l)

The azimuthal quantum number describes the shape and subshell within a given energy level.

Values: l = 0, 1, 2, …, (n - 1)
Physical meaning: Determines the shape of the orbital and the subshell name
Number of subshells in shell n: n (equal to the value of n)
Maximum electrons in subshell l: 4l + 2

l value	Subshell letter	Shape	Max electrons
0	s	Sphere	2
1	p	Dumbbell	6
2	d	Cloverleaf	10
3	f	Complex	14

Spectroscopic notation combines n and l: an electron in the n = 3 shell, l = 2 subshell is described as being in the 3d subshell.

3. Magnetic Quantum Number (ml)

The magnetic quantum number specifies the orientation of the orbital in space - which specific orbital within a subshell the electron occupies.

Values: ml = -l, -(l-1), …, 0, …, (l-1), l (integers from -l to +l)
Physical meaning: Determines the spatial orientation of the orbital
Number of orbitals in subshell l: 2l + 1

Subshell	l	ml values	Number of orbitals
s	0	0	1
p	1	-1, 0, +1	3
d	2	-2, -1, 0, +1, +2	5
f	3	-3, -2, -1, 0, +1, +2, +3	7

For the p subshell, the three orbitals correspond to the three axes in space: px, py, and pz. Each orbital can hold two electrons, so the p subshell holds 3 x 2 = 6 electrons total.

4. Spin Quantum Number (ms)

The spin quantum number describes the intrinsic angular momentum of the electron - its “spin.”

Values: ms = + $\frac{1}{2}$ or - $\frac{1}{2}$
Physical meaning: Two electrons in the same orbital must have opposite spins (one “spin up,” one “spin down”)

This is a direct consequence of the Pauli exclusion principle. Since each orbital can hold at most two electrons, and they must differ in at least one quantum number, they must have opposite spins.

Putting It All Together

For any electron, the four quantum numbers (n, l, ml, ms) provide a complete description. Here is the full set for the second principal energy level:

n	l	Subshell	ml	ms	Total electrons
2	0	2s	0	+ $\frac{1}{2}$ , - $\frac{1}{2}$	2
2	1	2p	-1, 0, +1	+ $\frac{1}{2}$ , - $\frac{1}{2}$ each	6
Total					8 = 2(2)²

An electron has quantum numbers n=3, l=2, ml=-1, ms=+

\frac{1}{2}

. What subshell is it in, and is this a valid set of quantum numbers?

Click to reveal answer

The electron is in the 3d subshell, and yes, this is valid. n=3 allows l values 0, 1, 2. l=2 is the d subshell. ml=-1 is within the range -2 to +2. ms=+

\frac{1}{2}

is valid. All four quantum numbers follow the rules.

How many orbitals are in the n=3 shell? How many total electrons can it hold?

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9 orbitals, 18 electrons. The n=3 shell has subshells s (1 orbital), p (3 orbitals), and d (5 orbitals) = 9 orbitals total (or n² = 3² = 9). Each orbital holds 2 electrons, so 9 x 2 = 18 (or 2n² = 2 x 9 = 18).