An orbital diagram is a compact way to show how electrons occupy atomic orbitals: each orbital is drawn as a “box,” and each electron is drawn as an arrow (its direction indicates spin). A correct orbital diagram must satisfy the Aufbau filling order, the Pauli exclusion principle, and Hund’s rule.
Core rules used to build an orbital diagram
1) Aufbau principle (energy order): electrons fill lower-energy orbitals before higher-energy orbitals (e.g., 1s before 2s, 2s before 2p, 3p before 4s, 4s before 3d, etc.).
2) Pauli exclusion principle: each orbital holds at most two electrons, and if two electrons share one orbital, they must have opposite spins.
3) Hund’s rule: within a set of degenerate orbitals (same subshell, such as the three 2p orbitals), electrons occupy separate orbitals with parallel spins before pairing.
Orbitals per subshell and electron capacity
For a subshell with angular momentum quantum number \(l\), the number of orbitals is \(2l + 1\), and each orbital holds 2 electrons, so the subshell capacity is: \[ \text{max electrons in subshell} = 2(2l+1). \]
| Subshell | \(l\) | Number of orbitals \(2l+1\) | Max electrons \(2(2l+1)\) |
|---|---|---|---|
| s | 0 | 1 | 2 |
| p | 1 | 3 | 6 |
| d | 2 | 5 | 10 |
| f | 3 | 7 | 14 |
Worked example: orbital diagram for phosphorus (Z = 15)
Step 1: Count electrons
Phosphorus has atomic number \(Z=15\), so a neutral phosphorus atom has \(15\) electrons.
Step 2: Fill orbitals in order (Aufbau) and write the electron configuration
Filling in the usual order gives: \[ 1s^2\;2s^2\;2p^6\;3s^2\;3p^3. \]
Step 3: Convert the configuration into an orbital diagram (Pauli + Hund)
The s subshells have one orbital each (one box), while each p subshell has three orbitals (three boxes). In the 3p subshell, Hund’s rule places the three electrons in three different p orbitals with parallel spins before any pairing occurs.
Determining the number of unpaired electrons
Unpaired electrons are orbitals containing exactly one electron arrow. In the phosphorus orbital diagram, the 3p subshell has three separate boxes each with one electron, so the number of unpaired electrons is \(3\). This implies phosphorus is paramagnetic (it is attracted to a magnetic field) because it has unpaired electrons.
Quick checklist for avoiding common orbital diagram errors
| Common mistake | Correction | Rule involved |
|---|---|---|
| Putting three 3p electrons as one pair plus one single immediately | Place one electron in each of the three 3p orbitals before pairing | Hund’s rule |
| Placing two same-spin electrons in the same orbital | Paired electrons in one orbital must have opposite spins | Pauli exclusion |
| Filling higher orbitals before lower ones (e.g., skipping 2p) | Fill in the standard energy order for subshells | Aufbau |