The table’s shape is a capacity rule and an ordering, and neither is a new idea. Both come out of the previous two sections without anything further being assumed.

Lithium, sodium, potassium and rubidium — $Z = 3, 11, 19, 37$ — are soft metals that react violently with water. Fluorine, chlorine and bromine — $Z = 9, 17, 35$ — are corrosive and grab electrons. Helium, neon, argon and krypton — $Z = 2, 10, 18, 36$ — do almost nothing at all.

Chemistry knew these families for decades before it knew why. This section derives them, and it needs only two ideas, both already established:

1. Each electron occupies an orbital with a definite energy (§11.1a).
2. Orbitals fill subject to the Pauli exclusion principle (§10.4).

That is the whole apparatus. Everything below is arithmetic on $2(2l+1)$.

The sequence of levels

Fig. 11.4 gives the order in which orbitals fill. It follows the rule from §11.1a — energy rises with $n$, and for fixed $n$ it rises with $l$ — but with one consequence that matters enormously: the $l$-dependence is strong enough that shells overlap. 4s dips below 3d; 5s below 4d; 6s below 4f and 5d.

The periods come out of the arithmetic

Fill the levels in order and count. The number of elements before a shell shell The orbitals sharing one principal quantum number: K for n = 1, L for n = 2, M for n = 3. A closed shell is compact, tightly bound and chemically inert, which is what makes the noble gases noble. defined in ch. 11 — open in glossary closes is just the sum of the capacities used:

2, 8, 8, 18, 18 — the period lengths of the periodic table, and they were not assumed anywhere. They are $2(2l+1)$ added up in the order the levels come.

Reading the families off the configurations

Where the pattern frays

The book is candid that periodicity “becomes somewhat erratic” from $Z = 19$, and it is precise about why: 3d and 4s are so close that their order depends on $Z$. Fig. 11.4’s own caption admits it — 4s lies below 3d for calcium ($Z = 20$) but above it for scandium ($Z = 21$).

Two of Table 11.1’s thirty configurations depart from naive filling, and both are in this region:

against the $3d^4 4s^2$ and $3d^9 4s^2$ that counting alone would predict. Both promote one 4s electron to reach a half-filled or filled d subshell.

Where this is going

Two ideas — orbitals with definite energies, and Pauli — have produced the shape of the periodic table, the length of every period, and a structural account of why three families of elements behave as they do. Nothing was fitted.

The obvious question is what the second idea is really worth. Remove the Pauli principle and keep everything else: what would atoms be like? §11.3 answers it quantitatively, and the answer is that there would be no chemistry at all.