Electronic configuration of elements (Z = 1 to 30) — exceptions Cr, Cu

1. WHAT is an electronic configuration?

WHY do we care? The configuration decides an element's chemistry — valency, magnetism, color, reactivity. All of the periodic table's patterns come from here.

2. The THREE rules (derive the filling order from first principles)

Rule 1 — Aufbau principle (WHAT + WHY)

WHY does energy set the order? A system is most stable at minimum energy. An electron in a lower-energy orbital is more tightly bound → lower total energy → more stable atom. Nature "chooses" this automatically.

HOW do we know which subshell is lower? Use the (n + l) rule (Madelung rule):

Let's derive the order by computing $(n+l)$:

Reading by increasing $(n+l)$, breaking ties by lower $n$: $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p$

Key surprise: $4s\,(n+l=4)$ fills before $3d\,(n+l=5)$. That single fact explains Cr and Cu below.

Rule 2 — Pauli exclusion principle

So capacities are: $s = 2$, $p = 6$, $d = 10$.

Rule 3 — Hund's rule of maximum multiplicity

WHY? Two electrons in the same orbital repel strongly (same region of space). Spreading them into separate orbitals lowers electron–electron repulsion (exchange energy stabilization). Parallel spins add extra quantum-mechanical stability (exchange).

3. Building Z = 1 to 30 (worked, with WHY at each jump)

I'll use noble-gas shorthand once shells close.

4. The EXCEPTIONS: Cr and Cu (the heart of the note)

Naïve prediction (just following Aufbau blindly):

• Cr (24): $[\text{Ar}]3d^4 4s^2$
• Cu (29): $[\text{Ar}]3d^9 4s^2$

Reality:

• Cr (24): $[\text{Ar}]3d^5 4s^1$
• Cu (29): $[\text{Ar}]3d^{10} 4s^1$

HOW the exchange bonus is counted (derivation)

Exchange energy $\propto$ number of pairs of parallel-spin electrons in a subshell. If a subshell has $k$ parallel electrons, the number of such pairs is: $\binom{k}{2} = \frac{k(k-1)}{2}$

Chromium comparison:

• Configuration A $3d^4$: 4 parallel $d$-electrons → $\binom{4}{2}=6$ exchange pairs (plus the $4s^2$ pair, but $4s$ is a single orbital — no parallel bonus).
• Configuration B $3d^5 4s^1$: 5 parallel $d$-electrons → $\binom{5}{2}=10$ exchange pairs.

Going $A \to B$ gains $10-6 = 4$ extra exchange pairs — a large stabilization that outweighs the small energy cost of moving one electron from $4s$ to slightly-higher $3d$. Hence Cr = $3d^5 4s^1$.

Copper comparison: $3d^{10}$ is a fully complete subshell — maximal symmetry, all orbitals filled — dramatically stable. So the atom prefers $3d^{10}4s^1$ over $3d^9 4s^2$.

5. Reading extra info from a configuration

Flashcards

Connections

• Aufbau principle — the base filling order this note refines.
• Pauli exclusion principle — sets orbital capacities.
• Hund's rule of maximum multiplicity — governs pairing and unpaired-electron counts.
• Quantum numbers (n, l, m, s) — define orbitals and the $(n+l)$ energy trick.
• Shielding and penetration effect — deeper WHY behind $4s$ < $3d$.
• Exchange energy and stability — quantitative basis for Cr/Cu exceptions.
• Magnetic properties of transition metals — uses unpaired-electron counts.
• Periodic trends — configurations explain the s-, p-, d-block structure.

Concept Map

Hinglish (regional understanding)