Atomic Radius

Chlorine's atomic radius is 102 pm. It is also 175 pm. Both figures are correct, both are published, and they differ by nearly three quarters of the smaller one.

The reason is not sloppiness. It is that an atom has no surface. The electron cloud thins out with distance and never quite reaches zero, so there is no point at which the atom stops and empty space begins. Every atomic radius you will ever see is therefore an operational quantity — derived from the distance between two atoms in some specific situation, then halved — and it depends on which situation was chosen.

Once that is clear, the trends make sense and the contradictions in reference tables stop being alarming.

Four radii for one chlorine atom

The four definitions in common use measure genuinely different things.

Covalent radius. Take two chlorine atoms bonded together in Cl₂, measure the distance between their nuclei, and halve it. This is the value used for non-metals in most tables. For chlorine it gives about 102 pm.

Metallic radius. Take a metal crystal, measure the distance between neighbouring nuclei, and halve that. It applies only to metals, and gives sodium about 186 pm.

Van der Waals radius. Take two atoms that are touching but not bonded — chlorine atoms in adjacent molecules in solid chlorine — and halve the distance. It measures how close an atom lets another approach without bonding. Chlorine's is about 175 pm.

Ionic radius. Derived from spacings in ionic crystals, with an unavoidable complication: a measurement gives the sum of a cation and an anion radius, and splitting that sum between them requires an assumption. Different assumptions produce different sets, which is why Shannon's and Pauling's ionic radii disagree by tens of picometres for the same ion. Sizes of ions relative to their parent atoms are dealt with under ions, cations and anions.

The covalent and van der Waals values are the pair most often confused, and the gap between them is large because a bonded pair is pulled together by shared electrons while a non-bonded pair is merely resting against each other's clouds.

Noble gases are the awkward case that exposes all of this. They form almost no bonds, so their tabulated radii are usually van der Waals values. Put those in the same column as the covalent radii of their neighbours and argon appears larger than chlorine, reversing the period trend. That reversal is an artefact of mixing two definitions, not a fact about argon.

Halving across period 2, doubling down group 1

With one consistent set of covalent radii — the figures below are from the widely used 2008 compilation by Cordero and co-workers — the trends are unambiguous.

Across period 2, in picometres: lithium 128, beryllium 96, boron 84, carbon 76, nitrogen 71, oxygen 66, fluorine 57. The atom shrinks by more than half while gaining six electrons.

That sounds backwards until you recall what is happening to the nuclear pull. Each step adds a proton and an electron, but the electron joins the same shell, where it screens the others poorly. The effective nuclear charge rises steeply, and the whole cloud is drawn inward. More electrons, smaller atom.

Down group 1: lithium 128, sodium 166, potassium 203, rubidium 220, cesium 244. Here the effective nuclear charge is very nearly constant from sodium downward, and the increase comes entirely from each element's outermost electrons occupying a shell of higher principal quantum number, further from the nucleus.

Two mechanisms, working on different variables, giving opposite signs. Almost every apparent complication in atomic size is one of these two competing with the other.

The transition metals that barely change size

Run across the first transition series and the shrinkage largely stops. Scandium is 170, titanium 160, vanadium 153, chromium 139, iron 132, cobalt 126, nickel 124 — and then copper goes back up to 132, and zinc is 122.

Ten protons are added across that stretch, which across a main-group period would halve the atom. It does not happen because the added electrons go into the 3d subshell, which lies inside the 4s electrons that determine the atom's size. Inner electrons screen well, so the added nuclear charge is largely cancelled before the outer electrons feel it.

The result is a row of metals with remarkably similar atomic sizes, which is precisely why the transition metals alloy with one another so readily: substituting one for another in a lattice barely disturbs it. Steel, brass, bronze and every stainless alloy depend on that interchangeability.

Hafnium is the size of zirconium, and it matters

The most consequential size anomaly on the periodic table sits between periods 5 and 6.

Ordinarily, an element is larger than the one directly above it. Zirconium's covalent radius is 175 pm. Hafnium, one full period below, is also 175 pm — and by metallic radius the two differ by about one picometre, with hafnium marginally the smaller.

The cause is the lanthanide contraction. Between lanthanum and lutetium, fourteen electrons are added to the 4f subshell. The f orbitals are unusually poor at screening — they are diffuse and keep little density near the nucleus — so effective nuclear charge creeps up across the whole lanthanide series and the atoms shrink steadily, from 207 pm at lanthanum to 187 pm at lutetium. By the time the sixth period resumes at hafnium, that accumulated contraction has cancelled the size increase that an extra shell should have delivered.

Two consequences follow, and both are more than curiosities.

Zirconium and hafnium are chemically almost indistinguishable, and separating them is one of the harder problems in industrial chemistry. It has to be solved, because hafnium absorbs neutrons strongly and zirconium barely at all, so nuclear reactor cladding needs zirconium with the hafnium taken out — and hafnium always occurs with zirconium in nature.

The elements after hafnium are extraordinarily dense. Tungsten, rhenium, osmium, iridium, platinum and gold pack atoms no larger than their period-5 counterparts while carrying nearly twice the atomic mass.

The densest element was decided by a lattice spacing, not a balance

The contraction also produced the two densest elements known, and the long argument over which of them wins turned on exactly the distinction this page is about.

Weighing cannot settle it. Voids and traces of the neighbouring platinum metals in a real ingot only ever push a measured density downward, and the two candidates are a tenth of a per cent apart, so determinations made that way scattered and disagreed about the order for the better part of two centuries. Working from the lattice removes both defects: diffraction gives the unit cell, the number of atoms in it is known, and the density falls out arithmetically. On those figures osmium leads iridium by about 0.03 g/cm³ — still a tenth of a per cent, but now around twice the quoted uncertainty, which is why the ranking has stopped moving. The osmium page carries the numbers and the two conditions under which the answer changes.

The transferable point is the one this page opened with. A quantity taken off a crystal lattice and the same quantity inferred from a bulk sample are not the same measurement, and when they disagree it is usually the lattice that is telling the truth.

Mistakes that come from mixing tables

Most errors in radius questions are not reasoning errors.

• Comparing values from different sources. Two tables using different definitions can differ by 40% on the same element. Take every value in a comparison from one set.
• Comparing a metallic radius with a covalent one. Many general-purpose tables silently do exactly this, listing metallic radii for metals and covalent radii for non-metals in a single row. Sodium at 186 next to chlorine at 102 exaggerates a real trend with two incompatible measurements.
• Expecting the d-block to shrink like a main-group period. It does not, for the reasons above, and a question about copper versus nickel is testing whether you know that.
• Assuming atomic and ionic rankings agree. They frequently invert, because forming an ion can remove or add a whole shell.

The habit worth building is to ask, before answering any question about size, which radius the question means. If it does not say — and many do not — state the definition you are using. That sentence is often worth a mark on its own, and it is the difference between knowing a trend and understanding what is being trended.