Why Chromium and Copper Break the Usual Electron-Filling Order

Every electron in an atom has its own address, described by four numbers, and it occupies an orbital with a particular shape. A few simple rules decide the order in which orbitals fill — and explain the surprising configurations of elements such as chromium and copper.

This lesson covers quantum numbers, the shapes of orbitals, and the rules for writing electronic configurations.

**The principal quantum number (n) gives the shell and main energy, the azimuthal quantum number (l) gives the subshell and orbital shape, the magnetic quantum number ($m_l$) gives the orbital's orientation, and the spin quantum number ($m_s$) gives the electron's spin, $+\frac{1}{2}$ or $-\frac{1}{2}$.

The four quantum numbers:

- Principal (n)** — 1, 2, 3 and so on; a shell has $n^2$ orbitals and holds up to $2n^2$ electrons
- Azimuthal (l) — 0 to $n-1$; l = 0, 1, 2 and 3 are the s, p, d and f subshells
- **Magnetic ($m_l$)** — whole numbers from $-l$ to $+l$, giving $2l+1$ orbitals in a subshell
- **Spin ($m_s$)** — $+\frac{1}{2}$ or $-\frac{1}{2}$

Worked example 1. For an electron in a 3d orbital, n = 3 and l = 2, so $m_l$ can be $-2,-1,0,+1$ or $+2$, and $m_s$ is $+\frac{1}{2}$ or $-\frac{1}{2}$.

Worked example 2. The fourth shell (n = 4) has $4^2=16$ orbitals and can hold $2\times 4^2=32$ electrons.

Checking a set. n = 2 with l = 2 is not allowed, because l can be at most $n-1=1$.

An everyday example. A postal address with state, city, street and house number pins down one home, just as four quantum numbers pin down one electron.

The substance. No two electrons in an atom share all four quantum numbers — this is Pauli's exclusion principle, which limits every orbital to two electrons.

An s orbital is spherical, a p orbital is dumb-bell shaped with two lobes on opposite sides of the nucleus, and most d orbitals have four lobes in a cloverleaf pattern, with the number of nodes set by n and l.

s orbitals (l = 0):

- Spherically symmetrical around the nucleus
- One s orbital in each shell, growing larger from 1s to 2s to 3s

p orbitals (l = 1):

- Dumb-bell shaped, with two lobes and a nodal plane passing through the nucleus
- Three p orbitals — $p_x$, $p_y$ and $p_z$ — lie along the three axes and have equal energy

d orbitals (l = 2):

- Five d orbitals: $d_{xy}$, $d_{yz}$, $d_{xz}$, $d_{x^2-y^2}$ and $d_{z^2}$
- Four have cloverleaf shapes, while $d_{z^2}$ has two lobes along the z-axis with a ring around its middle

Nodes:

$$\text{Radial nodes}=n-l-1\text{Angular nodes}=l\text{Total nodes}=n-1$$

Worked example. A 3p orbital has $3-1-1=1$ radial node and 1 angular node, 2 in total. A 4d orbital has $4-2-1=1$ radial node and 2 angular nodes.

An everyday example. The skin of a tabla has still rings and lines where it barely moves when struck — much like the nodes of electron waves in an orbital.

The substance. An orbital's shape shows where an electron is likely to be, not a path it follows — and at a node the chance of finding the electron is zero.

Electrons fill orbitals in order of increasing energy (Aufbau principle), no orbital holds more than two electrons, which must have opposite spins (Pauli's exclusion principle), and orbitals of equal energy are singly occupied before pairing begins (Hund's rule).

The Aufbau principle and the (n + l) rule:

- The orbital with the lower value of (n + l) fills first; if two have the same value, the one with lower n fills first
- Filling order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s

Worked example 1. For 4s, n + l = 4 + 0 = 4; for 3d, n + l = 3 + 2 = 5 — so 4s fills before 3d.

Hund's rule. Nitrogen, $1s^{2}2s^{2}2p^3$, has three unpaired electrons, one in each 2p orbital, rather than a pair and a single electron.

Worked example 2 — iron (Z = 26):

$$\mathrm{Fe}:1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{6}4s^2=[\mathrm{Ar}]3d^{6}4s^2$$

Ions. Electrons leave the outermost shell first, so 4s electrons are lost before 3d electrons:

$$\mathrm{F}{\mathrm{e}}^{2+}:[\mathrm{Ar}]3d^6\mathrm{F}{\mathrm{e}}^{3+}:[\mathrm{Ar}]3d^5$$

Exceptions — extra stability of half-filled and fully filled subshells:

- Chromium (Z = 24): $[\mathrm{Ar}]3d^{5}4s^1$, not $3d^{4}4s^2$
- Copper (Z = 29): $[\mathrm{Ar}]3d^{10}4s^1$, not $3d^{9}4s^2$

An everyday example. Passengers boarding an empty bus usually take separate seats before anyone sits beside a stranger — the same pattern as Hund's rule.

The substance. The ion with a half-filled 3d subshell is the more stable one — $\mathrm{F}{\mathrm{e}}^{3+}$, with $3d^5$, resists change better than $\mathrm{F}{\mathrm{e}}^{2+}$.