Electrons, energy levels and atomic orbitals

Three nested ideas, from biggest to smallest:

• Shell (principal quantum number, $n$) — the main energy level. $n = 1, 2, 3, \dots$ Higher $n$ means higher energy and further from the nucleus.
• Sub-shell — within a shell, electrons occupy s, p, d (and f) sub-shells of slightly different energy.
• Orbital — a region of space that can hold a maximum of 2 electrons, which must have opposite spins.

Ground state means every electron is in the lowest available energy level (the normal, unexcited arrangement).

Orbitals and capacities per sub-shell:

Sub-shell	Number of orbitals	Max electrons
s	1	2
p	3	6
d	5	10

(So shell $n$ contains $n^2$ orbitals and up to $2n^2$ electrons — shell 1 holds 2, shell 2 holds 8, shell 3 holds 18.)

Sub-shells fill from lowest energy upwards. The order you must know is: $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p$

The key surprise is that 4s is lower in energy than 3d, so the 4s sub-shell fills before 3d. This is why potassium and calcium fill 4s (giving $\dots 4s^1$ and $\dots 4s^2$) before the 3d block begins at scandium.

Why this ordering? Sub-shell energies depend on the balance between attraction to the nucleus and inter-electron repulsion / shielding. After 3p is full, the 4s sub-shell happens to be slightly lower in energy than 3d, so electrons enter 4s first.

Full notation lists every sub-shell: e.g. iron ($Z = 26$) is $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^6\,4s^2$

Shorthand notation replaces the inner electrons with the previous noble gas: Fe $= [\text{Ar}]\,3d^6\,4s^2$.

Two exceptions you must memorise (stability of half-filled and fully-filled d sub-shells):

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

Configurations of ions — remove 4s first. Although 4s fills before 3d, once 3d is occupied it drops below 4s, so when an atom is ionised the 4s electrons leave first.

• Fe $= [\text{Ar}]\,3d^6\,4s^2$
• Fe$^{2+} = [\text{Ar}]\,3d^6$ (lost both 4s electrons)
• Fe$^{3+} = [\text{Ar}]\,3d^5$ (then one 3d electron)

For p-block ions, simply remove from (cations) or add to (anions) the outer p sub-shell: Cl ($[\text{Ne}]3s^23p^5$) → Cl$^-$ ($[\text{Ne}]3s^23p^6$); Ca → Ca$^{2+}$ ($[\text{Ar}]$).

Scope. Only the elements hydrogen to krypton are assessed, and always in the ground state.

In the electrons-in-boxes notation each box is one orbital and each arrow is one electron; arrows point up or down to show opposite spin.

Two rules:

• Pauli exclusion principle — a maximum of 2 electrons per orbital, and they must have opposite spins (↑↓).
• Hund's rule — within a sub-shell, electrons occupy orbitals singly first (all spins parallel) before any pairing, because this minimises repulsion between electrons.

Worked — nitrogen ($1s^2\,2s^2\,2p^3$). The three 2p electrons go into the three separate 2p orbitals singly:

$2p:\;[\uparrow\,][\uparrow\,][\uparrow\,]\qquad 2s:\;[\uparrow\downarrow]$

Worked — oxygen ($2p^4$): three orbitals, so one must pair: $2p:\;[\uparrow\downarrow][\uparrow\,][\uparrow\,]$.

Free radical. A free radical is a species with one or more unpaired electrons (e.g. $\text{Cl}\bullet$, $\text{CH}_3\bullet$). They are very reactive and feature in free-radical substitution (Topic 14).

You must be able to describe and sketch these shapes.

• s orbital — spherical, centred on the nucleus. Same shape for 1s, 2s, 3s (just larger for higher $n$).
• p orbital — dumbbell-shaped (two lobes either side of the nucleus). There are three p orbitals at right angles, one along each axis: $p_x$, $p_y$, $p_z$.