How the table gets its shape
The shape of the table is the electron configuration. Electrons fill subshells in energy order — s two wide, p six, d ten, f fourteen — and the outline of the table follows from that order and nothing else.
7p66/6 · p-block
Z = 118 Oganesson
Reading that: shell 7, subshell type p, holding 6 of the 6 electrons that a p subshell can take.
- 1s
- 2s
- 2p
- 3s
- 3p
- 4s
- 3d
- 4p
- 5s
- 4d
- 5p
- 6s
- 4f
- 5d
- 6p
- 7s
- 5f
- 6d
- 7p
ShowingBlock
- s-block
- p-block
- d-block
- f-block
The lines mark the seven places the sequence does not simply carry on to the next square: the gaps in the first three periods, and the four detours the footnote creates — barium runs to lanthanum, and ytterbium comes back up to lutetium. Row endings are left undrawn, because moving to the start of the next row is just how a table is read. The rings mark where the rule opens a lower shell than the one it just left: scandium, yttrium, lanthanum, actinium.
Then the rule fails. For twenty elements the published configuration is not the one the rule predicts. Switch on “Show where the rule fails” to recolour the table by that question — the same cells, a different lens.
Three elements, all the way through
Before the animation, one pass by hand. Each element has one more electron than the last; the rule above says which subshell that electron enters; the subshell decides the block, and the block and the count decide the square.
1 H·1 electron→the last one enters 1s→1s1→Group 1, Period 1
1s holds 2 — 1/2. The first row begins.
2 He·2 electrons→the last one enters 1s→1s2→Group 18, Period 1
Group 18 here is the one number on this page the rule did not produce. Helium is s-block by configuration and would sit in group 2; it is placed with the noble gases because it behaves like one. A convention, stated as such.
1s is now full (2/2). This is why the first row has exactly two elements and then stops: s holds two, and there is nothing else in shell 1 to fill.
3 Li·3 electrons→the last one enters 2s→[He] 2s1→Group 1, Period 2
2s holds 2 — 1/2. Shell 1 is finished, so the next electron starts shell 2 — and a new row.
All 118 are placed this way. Nothing below is drawn by hand: every position in the table on this page comes out of that rule and the capacities 2, 6, 10 and 14.
The rule, in full
Electrons fill subshells in order of n + ℓ, lowest first; where two subshells tie, the one with the lower n fills first. n is the shell number and ℓ is 0 for s, 1 for p, 2 for d, 3 for f. That is the entire rule — every sum below is worked out from it, and the order of the table follows.
- 1s1+0=1
- 2s2+0=2
- 2p2+1=3
- 3s3+0=3
- 3p3+1=4
- 4s4+0=4
- 3d3+2=5
- 4p4+1=5
- 5s5+0=5
- 4d4+2=6
- 5p5+1=6
- 6s6+0=6
- 4f4+3=7
- 5d5+2=7
- 6p6+1=7
- 7s7+0=7
- 5f5+3=8
- 6d6+2=8
- 7p7+1=8
Read along and the surprise is 4s before 3d: 4+0 = 4 and 3+2 = 5, so the fourth shell starts before the third one finishes. That single step is why the transition metals sit where they do, and why period 4 is eighteen elements long instead of eight.
The n+ℓ ordering, generally attributed to E. Madelung, Die mathematischen Hilfsmittel des Physikers, 3rd ed. (1936); also known as the Klechkovsky rule.
What if the order were different?
The claim on this page is that the shape follows from the filling order. That is worth testing rather than taking on trust, so here is the other ordering — the one most people assume before they meet the rule: finish shell 1, then shell 2, then shell 3, in order. Switch between them and watch the row lengths.
- 1s1+0=1
- 2s2+0=2
- 2p2+1=3
- 3s3+0=3
- 3p3+1=4
- 4s4+0=4
- 3d3+2=5
- 4p4+1=5
- 5s5+0=5
- 4d4+2=6
- 5p5+1=6
- 6s6+0=6
- …
Row lengths2, 8, 8, 18, 18, 32, 32
2, 8, 8, 18, 18, 32, 32 — the row lengths of the actual periodic table. Nothing here is drawn from a picture of the table; these numbers fall out of the ordering above and the capacities 2, 6, 10 and 14.
The twenty exceptions
Solid boxes are the published configuration. Dashed boxes are what the rule predicts. Where a d or f subshell ends exactly half-filled or filled, you can see it.
24 Cr Chromium
half-filled or filledPublished [Ar] 3d5 4s1
Rule predicts [Ar] 3d4 4s2
29 Cu Copper
half-filled or filledPublished [Ar] 3d10 4s1
Rule predicts [Ar] 3d9 4s2
41 Nb Niobium
Published [Kr] 4d4 5s1
Rule predicts [Kr] 4d3 5s2
42 Mo Molybdenum
half-filled or filledPublished [Kr] 4d5 5s1
Rule predicts [Kr] 4d4 5s2
44 Ru Ruthenium
Published [Kr] 4d7 5s1
Rule predicts [Kr] 4d6 5s2
45 Rh Rhodium
Published [Kr] 4d8 5s1
Rule predicts [Kr] 4d7 5s2
46 Pd Palladium
half-filled or filledPublished [Kr] 4d10
Rule predicts [Kr] 4d8 5s2
47 Ag Silver
half-filled or filledPublished [Kr] 4d10 5s1
Rule predicts [Kr] 4d9 5s2
57 La Lanthanum
Published [Xe] 5d1 6s2
Rule predicts [Xe] 4f1 6s2
58 Ce Cerium
Published [Xe] 4f1 5d1 6s2
Rule predicts [Xe] 4f2 6s2
64 Gd Gadolinium
half-filled or filledPublished [Xe] 4f7 5d1 6s2
Rule predicts [Xe] 4f8 6s2
78 Pt Platinum
Published [Xe] 4f14 5d9 6s1
Rule predicts [Xe] 4f14 5d8 6s2
79 Au Gold
half-filled or filledPublished [Xe] 4f14 5d10 6s1
Rule predicts [Xe] 4f14 5d9 6s2
89 Ac Actinium
Published [Rn] 6d1 7s2
Rule predicts [Rn] 5f1 7s2
90 Th Thorium
Published [Rn] 6d2 7s2
Rule predicts [Rn] 5f2 7s2
91 Pa Protactinium
Published [Rn] 5f2 6d1 7s2
Rule predicts [Rn] 5f3 7s2
92 U Uranium
Published [Rn] 5f3 6d1 7s2
Rule predicts [Rn] 5f4 7s2
93 Np Neptunium
Published [Rn] 5f4 6d1 7s2
Rule predicts [Rn] 5f5 7s2
96 Cm Curium
half-filled or filledPublished [Rn] 5f7 6d1 7s2
Rule predicts [Rn] 5f8 7s2
103 Lr Lawrencium
Published [Rn] 5f14 7s2 7p1
Rule predicts [Rn] 5f14 6d1 7s2
What to look for: in eight of these, the borrowed electron lands a d or f subshell exactly half-filled or exactly full. Chromium takes one early to make 3d5, copper to make 3d10, silver 4d10, gadolinium 4f7. Those configurations are more stable than the rule’s arithmetic predicts.
The other twelve are not explained by that pattern, and this page does not offer an explanation for them. Niobium, ruthenium, rhodium, platinum and the early actinides land nowhere special. A pattern that covers eight of twenty is a useful thing to notice and a poor thing to call a reason — the published values stand on measurement, not on any story told about them.
NIST Atomic Spectra Database ground levels; CRC Handbook 104th ed., "Electron Configuration of Neutral Atoms in the Ground State".