Electron configuration exceptions — Cr, Cu, and why half/fully-filled orbitals are stable

Question

Why do chromium (Cr) and copper (Cu) have unexpected electron configurations? Explain the stability of half-filled and fully-filled d-orbitals.

(NEET 2024 asked Cr configuration; JEE Main tests exceptions in MCQs regularly)

Solution — Step by Step

According to the Aufbau principle, we would expect:

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

But the actual configurations are:

Cr: $[\text{Ar}] 3d^5 4s^1$ (one electron moves from 4s to 3d)
Cu: $[\text{Ar}] 3d^{10} 4s^1$ (one electron moves from 4s to 3d)

When electrons with the same spin occupy different orbitals of the same subshell, they can exchange positions. Each such exchange releases a small amount of energy called exchange energy.

A half-filled d-subshell ( $d^5$ ) has 10 exchange pairs ( $\binom{5}{2}$ ). A $d^4$ configuration has only 6 exchange pairs ( $\binom{4}{2}$ ). More exchange pairs = more stability. So Cr prefers $d^5$ over $d^4$ .

Similarly, a fully-filled $d^{10}$ also has maximum exchange energy for that configuration.

Half-filled ( $d^5$ ) and fully-filled ( $d^{10}$ ) subshells have spherically symmetric electron distribution. Each d-orbital has exactly 1 electron or exactly 2 electrons — no orbital is preferentially more occupied than others.

This symmetry minimises electron-electron repulsion and stabilises the atom.

This exception only works because the energy difference between 3d and 4s is very small for elements in the middle of the first transition series. The extra exchange energy gained by achieving $d^5$ or $d^{10}$ is enough to compensate for promoting one electron from 4s to 3d.

For elements where the gap is large (like Ca), this promotion does not happen.

flowchart TD
    A["Aufbau predicts<br/>Cr: 3d⁴4s²"] --> B{Extra stability from<br/>half-filled d⁵?}
    B -->|"Exchange energy gain > promotion cost"| C["Actual: Cr 3d⁵4s¹<br/>Half-filled d-subshell"]
    D["Aufbau predicts<br/>Cu: 3d⁹4s²"] --> E{Extra stability from<br/>fully-filled d¹⁰?}
    E -->|"Exchange energy gain > promotion cost"| F["Actual: Cu 3d¹⁰4s¹<br/>Fully-filled d-subshell"]
    C --> G["Maximum exchange pairs<br/>Spherical symmetry"]
    F --> G
    style C fill:#90EE90,stroke:#333
    style F fill:#90EE90,stroke:#333

Why This Works

The Aufbau principle is a general guideline, not an absolute law. When the energy gained from exchange interactions and symmetrical distribution exceeds the small energy cost of promoting a 4s electron to 3d, the atom “breaks the rule” because nature always prefers the lowest energy state.

These exceptions are specific to elements near the $d^5$ and $d^{10}$ configurations. Other elements like Mn ( $d^5 s^2$ ) and Zn ( $d^{10} s^2$ ) already have half-filled or fully-filled d-subshells naturally without needing any exception.

Alternative Method

For quick recall: Cr and Cu are the two main exceptions in the first transition series. Mo and Ag (their heavier analogues in period 5) show the same exceptions. The easy mnemonic: “CrCu — the copper-chrome rule” — whenever the Aufbau configuration gives $d^4 s^2$ or $d^9 s^2$ , it becomes $d^5 s^1$ or $d^{10} s^1$ .

Common Mistake

Students extend this exception to ALL elements and assume any element “close” to $d^5$ or $d^{10}$ will show anomalous configuration. This is wrong. Nickel (Ni, Z=28) is $3d^8 4s^2$ , NOT $3d^{10} 4s^0$ . The exception only occurs when one electron promotion is sufficient to achieve the half-filled or fully-filled state. Two-electron promotions are too energetically costly.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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