Quantum Mechanical Model of Atoms
High-Yield Summary
- Bohr's fixed orbits contradict the Heisenberg uncertainty principle (position and momentum can't both be known precisely) and only work for hydrogen — superseded by the wave-mechanical model.
- Electrons occupy orbitals (probability regions), not fixed orbits.
- Four quantum numbers fully describe an electron: n (energy level/size, shell holds 2n²), l (shape/subshell: s, p, d, f; 0 to n−1), ml (orientation, −l to +l; orbital count = 2l+1), ms (spin, +1/2 or −1/2).
- Shells (n) contain subshells (l), which contain orbitals (each holding up to 2 electrons with opposite spins).
- Filling rules: Aufbau's principle (lowest energy first), Pauli exclusion principle (max 2 electrons/orbital, opposite spins), Hund's rule (singly fill degenerate orbitals before pairing).
- Electron configuration filling order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p... Chromium ([Ar]4s¹3d⁵) and copper ([Ar]4s¹3d¹⁰) are Aufbau exceptions, favoring a half- or fully-filled 3d subshell.
The Four Quantum Numbers
| Quantum Number | Describes / Possible Values |
|---|---|
| Principal (n) | Energy level and orbital size; any positive integer (1, 2, 3, ...); shell capacity = 2n² |
| Angular momentum (l) | Orbital shape/subshell; integers 0 to (n−1): 0=s, 1=p, 2=d, 3=f |
| Magnetic (ml) | Orbital orientation in space; integers −l to +l including 0; orbital count = 2l+1 |
| Spin (ms) | Electron spin; only +1/2 or −1/2 |
Key Terms
- Orbital (vs. orbit)
- A probability region where an electron is likely to be found — not a fixed circular path. The core distinction between the quantum mechanical and Bohr models.
- Heisenberg uncertainty principle
- Position and momentum of an electron cannot both be known with perfect accuracy simultaneously.
- Shells / subshells / orbitals
- Nested hierarchy: shells (defined by n) contain subshells (defined by l), which contain individual orbitals.
- Degenerate orbitals
- Orbitals of equal energy within the same subshell (e.g., the three 2p orbitals).
Writing an Electron Configuration
- 1Start at hydrogen and add electrons one at a time, following the subshell filling order.
- 2Apply Aufbau's principle: fill from lowest to highest energy — 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p...
- 3Apply the Pauli exclusion principle: no orbital holds more than 2 electrons, and paired electrons have opposite spins.
- 4Apply Hund's rule: give every orbital in a subshell one electron before any orbital gets a second (paired) electron.
Common MCAT Trap
- Chromium and copper break the predicted Aufbau order — [Ar]4s¹3d⁵ and [Ar]4s¹3d¹⁰, not 4s²3d⁴ / 4s²3d⁹ — because a half- or fully-filled d subshell is more stable.
- 'Orbital' ≠ 'orbit' — Bohr's model uses fixed orbits; the quantum mechanical model uses probability-based orbitals. Conflating them is a common MCAT trap.
- The periodic table itself encodes the filling order (lowest s = 1s, lowest p = 2p, lowest d = 3d, lowest f = 4f) — useful as a fast check on a written configuration.
Quick Recall
What does the angular momentum quantum number (l) describe, and what are its possible values?
How many orbitals are in a subshell with l = 2 (d subshell)?
What is the actual electron configuration of chromium (Z = 24), and why does it deviate from the predicted one?
Which rule requires that degenerate orbitals each get one electron before any is paired?