Pauli Exclusion & the Periodic Table

Begin with an accounting problem. Hydrogen's lowest orbital, 1s, lies at −13.6 eV, and nature runs downhill: an excited atom sheds light until it can fall no further. So why don't all ninety-two of uranium's electrons crowd into one tiny 1s cloud? If they did, each element would differ from its neighbour only by degree, and the repeating rhythm of chemistry that Dmitri Mendeleev tabulated in 1869, with lithium resembling sodium resembling potassium, would have no reason to exist. The rhythm exists, so something forbids the pile-up.

That something was found by Wolfgang Pauli, then twenty-four, in Hamburg. In the autumn of 1924 he proposed that the electron carries a fourth quantum property beyond the three orbital labels, which he cautiously called a 'two-valuedness not describable classically.' Edmund Stoner had just noticed that the number of electrons in each closed noble-gas shell equals the number of distinct states a single electron can take in that shell. Pauli saw the law hiding in that count and published it in early 1925: no two electrons in an atom can share the same four quantum numbers. Later that year George Uhlenbeck and Samuel Goudsmit identified the two-valued property as spin.

Stated generally, the principle concerns fermions, particles whose spin is a half-integer multiple of ħ, such as electrons, protons, neutrons and quarks: no two identical fermions can occupy the same quantum state. 'Identical' is literal: electrons are truly indistinguishable, so asking which electron is which has no meaning. 'The same quantum state' means the complete description, spatial wavefunction and spin together. Two electrons may share an orbital if their spins differ, or share a spin direction if their orbitals differ, but never both.

Recall the four labels from chapters 16 and 17. The principal quantum number n sets the shell; the orbital quantum number l, running from 0 to n − 1, sets the shape, lettered s, p, d and f for l = 0, 1, 2 and 3; the magnetic quantum number mₗ, with 2l + 1 whole-number values from −l up to l, sets the orientation; and the spin quantum number mₛ takes just two values, plus one-half and minus one-half. An orbital is fixed by n, l and mₗ together, which leaves only the spin free. Because spin has exactly two values, an orbital holds at most two electrons, with opposite spins: the chemist's electron pair.

Now count. A subshell, meaning all orbitals that share n and l, has 2l + 1 orbitals and so holds 2 × (2l + 1) electrons: two for s, six for p, ten for d and fourteen for f. A shell, meaning all orbitals that share n, has n² orbitals and so holds 2n² electrons: 2, 8, 18 and 32 for n = 1 to 4. Stoner and Pauli read these numbers off spectra before Schrödinger's equation existed; they are pure exclusion arithmetic, one electron per distinct state.

Exclusion also explains last chapter's silver atoms. Silver has 47 electrons, yet the Stern-Gerlach magnet split the beam in two, as if each atom were a single spin-one-half particle. Forty-six of those electrons fill complete subshells, where every orbital holds one spin-up and one spin-down electron, so their spins and orbital motions cancel. What remains is a lone 5s electron, with l = 0 and therefore no orbital magnetism. Magnetically, a silver atom is one electron's spin wearing a very heavy coat.

Exclusion says how many electrons each orbital can take, not which orbitals fill first. That is the job of the Aufbau principle, German for 'building up', which Niels Bohr developed in the early 1920s: add electrons one at a time, each into the lowest-energy orbital still free. In hydrogen, 2s and 2p share one energy, apart from tiny relativistic effects; in every other atom, shielding breaks the tie. Inner electrons sit between an outer electron and the nucleus and cancel part of the nuclear charge, so the outer electron feels a smaller effective nuclear charge, written Z_eff.

Orbital shape now matters. An s orbital has nonzero probability density at the nucleus itself, whereas p, d and f orbitals vanish there, more steeply as l grows. An s electron therefore penetrates the shielding core, spends time near the almost bare nucleus, and is bound more tightly than a p electron of the same shell, which in turn beats a d electron. When this splitting outgrows the gap between neighbouring shells, orbitals from different shells interleave, and that first happens in the fourth row.

The pattern is the n + l rule, called the Madelung rule after Erwin Madelung, who stated it in 1936, though Charles Janet and Vladimir Karapetoff had published it by 1930. Subshells fill in order of increasing n + l, and on a tie the smaller n goes first. That gives 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, and onward. Note the step beginners miss: 4s, with n + l = 4, precedes 3d, with n + l = 5. Potassium's nineteenth electron enters 4s, and NIST's spectroscopic tables put its lowest 3d level 2.67 eV above the 4s ground state, above even 5s. Electrons do not fill strictly in order of n.

Friedrich Hund, reading atomic spectra in 1925, settled how electrons share a partly filled subshell: orbitals of equal energy are occupied singly, with parallel spins, before any takes a second electron. This is Hund's rule of maximum multiplicity. Carbon's two 2p electrons sit in different orbitals with the same spin. Nitrogen's three spread across all three p orbitals with their spins aligned, giving the total spin of three-halves seen in its spectrum. Only at oxygen is a 2p pair forced to form.

Why parallel spins? Same-spin electrons need an antisymmetric spatial wavefunction, which vanishes wherever they would meet; this Fermi hole keeps them apart on average. Careful calculations for atoms such as carbon add a twist: the arrangement wins mainly because electrons that screen one another less are drawn closer to the nucleus. Either way, parallel spins give the lower energy. Hund's rule is a rule of thumb about energies, and a few unusual molecules break it; exclusion is a law, and nothing has ever broken it.

The recipes falter in the fourth row. The n + l rule predicts four 3d and two 4s electrons for chromium, element 24, but its ground state has five 3d and one 4s, six unpaired spins in all. Copper, element 29, should have nine 3d and two 4s electrons but has a full ten 3d and a single 4s. Textbooks cite the stability of half-filled and filled d subshells, largely Hund's exchange energy at work; the deeper point is that 3d and 4s are so close in energy here that small effects decide. About twenty elements, all in the d and f blocks, defy the simple rule.

Now the table assembles itself. A row, or period, begins when a new s subshell starts to fill and ends with a noble gas, when the next p subshell is full (in the first row, when 1s is full). In n + l order the rows are 1s; then 2s and 2p; 3s and 3p; 4s, 3d and 4p; 5s, 4d and 5p; 6s, 4f, 5d and 6p; and finally 7s, 5f, 6d and 7p. Add up the capacities and you get rows of 2, 8, 8, 18, 18, 32 and 32 elements, exactly as in the real table. The blocks follow too: two s columns on the left, six p columns on the right, the ten-wide d block of transition metals between them, and the fourteen-wide f block printed underneath.

Columns, or groups, collect elements whose outermost electrons, the valence electrons, are arranged alike. Only they take part in ordinary chemistry; core electrons are bound far too tightly for a reaction's few electron-volts to disturb. Lithium, sodium and potassium each have one s electron outside a closed core; fluorine and chlorine each have seven valence electrons, one short of a closed shell. Mendeleev grouped elements by mass and chemical resemblance; Henry Moseley's X-ray measurements of 1913 and 1914 showed the true ordering number to be the nuclear charge Z; quantum mechanics explained why the pattern repeats.

The noble gases, helium through radon, have completely filled outer subshells, and ionizing them is expensive: helium's 24.59 eV is the largest first ionization energy of any element, and neon's is 21.56 eV. They rarely react. Right after them come the alkali metals, each with one lonely s electron in a new shell. Sodium's outer electron costs only 5.14 eV to remove, while the next one, taken from the closed neon-like core, costs 47.3 eV, more than nine times as much. Sodium gives up exactly one electron, and gives it eagerly.

That is the quantum meaning of 'sodium is reactive'. Its 3s electron sits outside ten core electrons that shield most of the eleven protons, so it feels a net pull of a little under two elementary charges and is easily lost. Hand it to chlorine. Removal costs 5.14 eV, chlorine's electron affinity returns 3.61 eV, a net cost of just 1.53 eV, and the electrical attraction of the two ions at their bond length of 0.236 nm, about 6.1 eV, more than repays it. In water, a similar hand-off drives sodium's violent fizzing. Chemistry is electrons seeking lower energy under the rule of exclusion.

Here is a disturbing fact. A nucleus is roughly ten thousand to a hundred thousand times smaller across than its atom, so almost all of an atom's volume holds nothing but electron probability cloud. Why, then, is a brick hard? A popular answer says atoms are 'full', packed with stuff or sealed by completed shells. Neither survives scrutiny. The cloud is not a substance, and completed shells are not the secret: iron, with an unfilled 3d subshell, is far harder than solid neon, whose shells are all closed and which melts at about 25 kelvin.

The real answer joins exclusion to the uncertainty principle of chapter 8. Squeeze two atoms together and their electron clouds overlap. Bosons could simply crowd into the low-energy states already occupied; electrons cannot, so they are forced into states with more nodes and higher momentum, and confining any electron more tightly raises its momentum further, since Δx·Δp ≥ ħ/2. Kinetic energy therefore climbs steeply as the volume shrinks, and an energy that rises under compression is exactly what a pressure is. Chemists call it Pauli repulsion. It is not a fifth force, but electromagnetism and kinetic energy filtered through exclusion.

Paul Ehrenfest made this argument in 1931, when presenting Pauli with the Lorentz Medal. In 1967 Freeman Dyson and Andrew Lenard proved rigorously that ordinary matter is stable, its binding energy growing only in proportion to the number of particles, and their proof required exclusion. Dyson also showed that if all the charged particles were bosons, the binding energy would grow faster than the number of particles, so bulk matter would keep collapsing and merging two lumps would release enormous energy. Elliott Lieb and Walter Thirring found a far simpler proof in 1975. The stiffness of your chair is a theorem about fermions.

Exclusion also holds up dead stars. When a Sun-like star exhausts its nuclear fuel, its core contracts until it is supported not by heat but by electrons that cannot crowd into occupied states. Ralph Fowler made this argument in December 1926, months after Enrico Fermi and Paul Dirac worked out the statistics of particles that obey exclusion. The result is a white dwarf. Sirius B packs about 1.02 solar masses into a sphere roughly 5,600 km in radius, slightly smaller than Earth, for a mean density near 2.7×10⁹ kilograms per cubic metre. This electron degeneracy pressure persists even at absolute zero, since it comes from exclusion, not heat.

Worked example 3 reveals something alarming: at such densities exclusion drives the fastest electrons to momenta above mₑc, the electron's mass times the speed of light, so they move at about three-quarters of light speed. Relativity then softens their resistance. Slow electrons give a pressure that grows as the five-thirds power of density, ultra-relativistic ones only as the four-thirds power, the same rate at which gravity's demand rises as a star contracts. On a voyage to England in 1930, aged nineteen, Subrahmanyan Chandrasekhar began the calculations that led to a startling conclusion: white dwarfs have a maximum mass, now put at about 1.4 solar masses. Stoner and Wilhelm Anderson found related limits using simpler uniform-density models, and Chandrasekhar shared the 1983 Nobel Prize in Physics.

Beyond that limit, as in the collapsing core of a massive star, electrons are forced onto protons, making neutrons and neutrinos. Neutrons are fermions too, and their degeneracy pressure, aided by the strongly repulsive short-range part of the nuclear force, halts the collapse at a radius of about 12 km for 1.4 solar masses: a mean density around 4×10¹⁷ kilograms per cubic metre, a little above that inside an atomic nucleus. Neutron stars have a maximum mass too. As of 2026 the heaviest well-measured examples are a little over two solar masses, and the theoretical ceiling, probably below three, remains uncertain. Beyond it lies a black hole.