Quantum Numbers
Four quantum numbers (n, l, ml, ms) uniquely describe each electron in an atom
The Four Quantum Numbers
1. Principal Quantum Number (n)
n = 1, 2, 3, 4, ...
- • Determines electron shell and energy level
- • Positive integers only
- • Larger n → higher energy, farther from nucleus
- • Corresponds to periods in periodic table
2. Angular Momentum Quantum Number (l)
l = 0, 1, 2, ... (n-1)
- • Determines orbital shape (subshell)
- • l = 0 (s), 1 (p), 2 (d), 3 (f)
- • Values from 0 to (n-1)
- • Each subshell has characteristic shape
3. Magnetic Quantum Number (ml)
ml = -l, ..., 0, ..., +l
- • Determines orbital orientation in space
- • Values from -l to +l (including 0)
- • Total of (2l + 1) orbitals per subshell
- • Example: p has 3 orbitals (px, py, pz)
4. Spin Quantum Number (ms)
ms = +½ or -½
- • Determines electron spin direction
- • Only two values: +½ (↑) or -½ (↓)
- • Pauli Exclusion Principle: no two electrons can have identical quantum numbers
- • Maximum 2 electrons per orbital (opposite spins)
Allowed Values & Relationships
| Shell (n) | Subshell (l) | Designation | ml values | Orbitals | Max e⁻ |
|---|---|---|---|---|---|
| 1 | 0 | 1s | 0 | 1 | 2 |
| 2 | 0 | 2s | 0 | 1 | 2 |
| 1 | 2p | -1, 0, +1 | 3 | 6 | |
| 3 | 0 | 3s | 0 | 1 | 2 |
| 1 | 3p | -1, 0, +1 | 3 | 6 | |
| 2 | 3d | -2, -1, 0, +1, +2 | 5 | 10 |
Worked Examples
Example 1: Valid Quantum Number Set?
Given: n=3, l=2, ml=0, ms=+½
Check n: n=3 ✓ (positive integer)
Check l: l=2, must be 0 to n-1 (0 to 2) ✓
Check ml: ml=0, must be -l to +l (-2 to +2) ✓
Check ms: ms=+½ ✓
Valid set! Represents 3d orbital electron
Example 2: Invalid Quantum Numbers
Given: n=2, l=2, ml=-1, ms=-½
Check l: l=2 but n=2, so l can only be 0 or 1 ✗
Invalid! l cannot equal n
For n=2, maximum l=1 (2p orbital is highest)
Example 3: Electron in 4f Orbital
Determine possible quantum numbers for 4f:
n = 4 (fourth shell)
l = 3 (f orbital)
ml = -3, -2, -1, 0, +1, +2, +3 (7 orbitals)
ms = +½ or -½
Total electrons in 4f: 7 orbitals × 2 electrons = 14 electrons maximum
Common Mistakes
Setting l ≥ n
Remember: l can only be 0 to (n-1), never equal to or greater than n
Wrong ml Range
ml must be between -l and +l; for l=2, ml cannot be 3
Forgetting Pauli Exclusion
No two electrons can have all four identical quantum numbers
Remember Letter Designations
l = 0→s, 1→p, 2→d, 3→f, 4→g, 5→h (sharp, principal, diffuse, fundamental)