Why Does the Periodic Table Have the Shape It Does?

Hydrogen and helium sit alone at the top. Then come two rows of 8, two of 18 and two of 32, with a strip of elements parked below the main table. That odd outline is a picture of how electrons fill an atom. This post follows our explainer Why Can’t Two Electrons Share a State?, because the rule that sets the shape is the Pauli exclusion principle. It is drawn for our Maria Goeppert Mayer sticker, whose slogan “A full floor is magic” is about the same idea one level down, inside the nucleus, and for the Wolfgang Pauli “One Per State” sticker.

What decides the length of each row?

The row lengths were a puzzle long before anyone could explain them. Wolfgang Pauli recalled in his Nobel lecture that “the series of whole numbers 2, 8, 18, 32... giving the lengths of the periods in the natural system of chemical elements, was zealously discussed in Munich,” including the Swedish physicist Rydberg’s remark that these numbers have the form 2n² (Pauli 1946^[1]). The answer came from quantum mechanics.

Shells, subshells and four quantum numbers

Each electron in an atom is labelled by quantum numbers. NIST’s atomic spectroscopy compendium sets them out: n, the principal quantum number, can be 1, 2, 3 and so on; l, the orbital angular momentum number, runs from 0 to n − 1; and two more numbers give the orientation of the orbit and of the electron’s spin (NIST^[2]). Electrons with the same n form a shell; those with the same n and l form a subshell, written s, p, d, f for l = 0, 1, 2, 3.

The exclusion principle caps each subshell

“The Pauli exclusion principle prohibits atomic states having two electrons with all four quantum numbers the same. Thus the maximum number of equivalent electrons is 2(2 l + 1)” (NIST^[2]). Written compactly, a subshell with orbital number l holds at most

2(2l + 1)

electrons: the factor 2l + 1 counts the allowed orientations, and the 2 counts the two spin directions. That gives 2 for s, 6 for p, 10 for d and 14 for f. Adding the subshells of shell n (l = 0 up to n − 1) gives Rydberg’s number:

2n²

which is 2, 8, 18 and 32 for n = 1, 2, 3, 4. Pauli put the principle in its general form in Hamburg in the spring of 1925, saying an energy level made fully non-degenerate is “already « closed », if it is occupied by a single electron” (Pauli 1946^[1]). He received the 1945 Nobel Prize in Physics “for the discovery of the Exclusion Principle, also called the Pauli Principle” (NobelPrize.org^[3]).

So why are there two rows of 8 and not 8 then 18?

Because subshells do not fill strictly shell by shell. NIST notes that an outer electron’s energy depends strongly on l because of how far it reaches into the inner electrons, called “core penetration”: an s electron dips closer to the nucleus than a p electron of the same shell (NIST^[2]). The measured ground states in NIST’s Atomic Spectra Database show the result (NIST ASD^[4]):

• Argon (element 18) is [Ne] 3s² 3p⁶. Its 3s and 3p subshells are full, but its 3d subshell is empty. NIST notes that only 3s² 3p⁶ 3d¹⁰ is a full shell for n = 3 (NIST^[2]).
• Potassium (19) puts its next electron in 4s, not 3d: [Ar] 4s.
• Only after 4s fills does 3d take its 10 electrons, from scandium to zinc, the first row of transition metals. Gallium (31) then starts 4p.

So each row runs from a new s subshell to the next full p subshell. Row 3 holds 3s + 3p = 2 + 6 = 8. Row 4 holds 4s + 3d + 4p = 2 + 10 + 6 = 18. Rows 6 and 7 also slip in an f subshell, 2 + 14 + 10 + 6 = 32. That is why the numbers come in pairs, 2, 8, 8, 18, 18, 32, 32, and why the 14-wide f block is usually printed as a separate strip to keep the table from getting too wide.

What does the data show?

The clearest test is the ionization energy, the energy needed to remove one electron from a neutral atom. NIST’s values (NIST ASD^[4]) peak at every noble gas and crash at the next element:

• Helium 24.59 eV, then lithium 5.39 eV.
• Neon 21.56 eV, then sodium 5.14 eV.
• Argon 15.76 eV, then potassium 4.34 eV.
• Krypton 14.00 eV, then rubidium 4.18 eV.
• Xenon 12.13 eV, then caesium 3.89 eV.

A full p subshell is hard to break into. The next electron has to start a new, farther shell and is easy to remove. The noble gases fall at atomic numbers 2, 10, 18, 36, 54, 86 and 118, and the gaps between them are exactly the row lengths 2, 8, 8, 18, 18, 32, 32. The seventh row was completed on paper when IUPAC approved the names of elements 113, 115, 117 and 118 on 28 November 2016 (IUPAC^[5]).

Does every element follow the filling rule?

No. The order above is a rule of thumb, and the real ground state is whatever the spectroscopy says has the lowest energy. NIST lists chromium as [Ar] 3d⁵ 4s, not 3d⁴ 4s², and copper as [Ar] 3d¹⁰ 4s, not 3d⁹ 4s² (NIST ASD^[4]). The row lengths still come out right; only the details inside a row shift.

Do nuclei have shells too?

Yes, which is where Maria Goeppert Mayer comes in. Protons and neutrons are also fermions that obey the exclusion principle, and nuclei with full shells are extra stable. Goeppert Mayer and J. Hans D. Jensen shared half of the 1963 Nobel Prize in Physics “for their discoveries concerning nuclear shell structure” (NobelPrize.org^[6]). For what holds those nucleons together at all, see What Holds the Nucleus Together?

FAQ

Why is the first row only 2 elements?

Shell n = 1 has only an s subshell, and 2(2l + 1) = 2 for l = 0. Hydrogen and helium fill it.

Why are noble gases unreactive?

Each ends a row with a full p subshell (helium a full 1s). Their ionization energies are the highest in each row, so they hold on to their electrons.

Why is the f block drawn separately?

An f subshell holds 14 electrons. Printing it inside rows 6 and 7 would make the table 32 columns wide, so it is usually moved below.

Who explained the periodic table’s shape?

Many people. Bohr’s 1922 Göttingen lectures on the periodic system, Stoner’s counting of electron levels, and Pauli’s 1925 exclusion principle together gave the explanation, as Pauli described in his Nobel lecture.

References

1. W. Pauli, Nobel Lecture (13 December 1946), “Exclusion Principle and Quantum Mechanics” (NobelPrize.org PDF). https://www.nobelprize.org/uploads/2018/06/pauli-lecture.pdf
2. W. C. Martin & W. L. Wiese, Atomic Spectroscopy: A Compendium of Basic Ideas, Notation, Data, and Formulas, §3 “Atomic States, Shells, and Configurations” (NIST). https://physics.nist.gov/Pubs/AtSpec/node03.html
3. NobelPrize.org, The Nobel Prize in Physics 1945 (Wolfgang Pauli). https://www.nobelprize.org/prizes/physics/1945/summary/
4. A. Kramida, Yu. Ralchenko, J. Reader & NIST ASD Team, NIST Atomic Spectra Database: Ionization Energies (ground states and ionization energies of atoms and ions). https://physics.nist.gov/PhysRefData/ASD/ionEnergy.html
5. IUPAC, Periodic Table of Elements (including the 2016 validation and naming of elements 113, 115, 117 and 118). https://iupac.org/what-we-do/periodic-table-of-elements/
6. NobelPrize.org, The Nobel Prize in Physics 1963 (Wigner, Goeppert Mayer, Jensen). https://www.nobelprize.org/prizes/physics/1963/summary/

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.