Size is the bottom of the well
Across this rung you have watched bonding pick structure. The bonding-energy curve set whether two atoms bind (guide 1); the three primary bonds set how (guide 2); the secondary bonds glued sheets and chains (guide 3); and directionality decided open versus dense (guide 4). One thread ran quietly through all of it — size. This last guide brings that thread into the open: where an atom's radius comes from, how it changes when the atom turns into an ion, and how relative size together with bond directionality fixes the coordination number, the count of nearest neighbours out of which every structure is assembled.
An atom has no hard edge — its electron cloud just fades out — so 'size' has to be defined operationally. The clean recipe comes straight from guide 1: push two identical atoms together and they settle at the equilibrium spacing, the bottom of the well on the bonding-energy curve. The atomic radius is simply half of that touching distance. For copper the atoms rest about 2.56 angstrom centre-to-centre, so we quote a copper radius near 1.28 angstrom. Two trends across the periodic table follow from the atom itself: radius grows DOWN a group (each row adds a whole new outer shell) and shrinks ACROSS a period left to right (more protons pull the same shell in tighter, a rising effective nuclear charge).
Ions resize: cations shrink, anions swell
The moment an atom becomes an ion, its size jumps. When an atom gives electrons away to become a cation it often loses its entire outermost shell, and the protons that remain pull the smaller cloud in tighter, so a cation is markedly SMALLER than its parent atom — a sodium atom is about 1.86 angstrom, but Na+ is only about 1.02 angstrom, nearly halved. When an atom grabs electrons to become an anion, the extra electrons crowd and repel one another while the same number of protons pulls each one less firmly, so an anion is markedly LARGER — chlorine's covalent radius is near 0.99 angstrom, but Cl- swells to about 1.81 angstrom. This is the ionic radius at work.
One clean pattern nails the idea: the isoelectronic series. O2-, F-, Na+, Mg2+, and Al3+ all carry exactly the same 10 electrons, yet their nuclear charge climbs 8, 9, 11, 12, 13 — and their radius falls steadily from about 1.40 angstrom for O2- down to about 0.54 angstrom for Al3+. Same cloud, more pull, smaller ion. The structural consequence is decisive: in almost every ionic crystal the anion is the big sphere and the cation is a small ball that must find somewhere to sit among the anions. That single fact — small cation, big anion — is exactly what the rest of this guide turns into a rule.
Coordination number: how many neighbours touch
The coordination number is just the count of nearest neighbours that actually touch a given atom or ion. It is the single number that turns 'how atoms bond' into 'what the structure looks like'. Recall the two extremes from guide 4. A non-directional bond has no favoured angle, so it wants coordination as HIGH as geometry allows: identical metal spheres reach 12, the maximum, by close packing like oranges in a grocer's pyramid. A directional covalent bond fixes its angles, so it locks coordination LOW: carbon in diamond bonds to just 4 neighbours at 109.5 degrees, leaving a wide-open cage.
For a non-directional ionic bond the story is purely geometric: coordination is set by how many big anions can crowd around the small cation while every one of them still touches it. Common salt is the model case — the large Cl- ions form a face-centred array and each small Na+ drops into an octahedral hole, a gap ringed by exactly six chlorides. So each Na+ touches six Cl- and vice versa, a coordination number of 6. A bigger cation would push its neighbours apart and let more of them fit; a smaller one would rattle loosely and prefer a tighter hole with fewer neighbours. The bond type does not choose 6 here — the relative sizes do.
So two levers act in series. First, bond directionality decides whether size is even allowed to matter: a directional covalent bond overrides geometry and freezes coordination low, but a non-directional ionic or metallic bond hands the decision entirely to size. Second, when size is in charge, relative size sets the coordination number. That second step is quantitative enough to write down as an actual rule — and the next section does exactly that.
The radius-ratio rule: does the small ion fit?
Model the ions as hard spheres and one clean question decides everything: is the cation big enough to touch all the anions crowded around it? If it just touches them, the packing is stable; if it is too small it rattles in the hole and the structure collapses to a smaller hole with fewer neighbours. The threshold is pure trigonometry. To sit snugly in an octahedral hole (6 neighbours) the cation-to-anion radius ratio must be at least the square root of 2, minus 1 — about 0.414. A tetrahedral hole (4 neighbours) needs only 0.225, and a cubic hole (8 neighbours) needs 0.732. These holes are the interstitial sites of the anion packing: a close-packed layer stack offers one octahedral hole and two smaller tetrahedral holes per anion, and the small cation simply moves in.
radius ratio r/R hole shape coordination example
------------------- --------------- ------------ --------------
0.155 - 0.225 triangular 3 (rare, B-O)
0.225 - 0.414 tetrahedral 4 ZnS (~0.40)
0.414 - 0.732 octahedral 6 NaCl (~0.56)
0.732 - 1.000 cubic 8 CsCl (~0.92)
> 1.000 equal-size packing 12 metals
r = cation radius (the small sphere), R = anion radius (the big sphere)Now put real numbers in. For common salt, r/R is 1.02/1.81 = 0.56, which lands in the 0.414-to-0.732 window, so the rule predicts an octahedral hole and coordination 6 — and rock salt is indeed coordination 6, a direct hit. For caesium chloride, the caesium ion is far bigger, about 1.67 angstrom, so r/R is 1.67/1.81 = 0.92, above 0.732, predicting a cubic hole and coordination 8. And CsCl really does put each Cs+ at the centre of a cube of eight Cl-, a coordination number of 8. Same anion, bigger cation, more neighbours — the rule reads the structure straight off the two sizes.
- Look up the ionic radii: r for the small cation and R for the big anion, using values for the coordination you suspect.
- Form the ratio r/R (it should come out less than 1, since the cation is the smaller sphere).
- Read the predicted hole and coordination off the table: 0.414 to 0.732 means an octahedral hole and coordination 6, and so on.
- Compare with the real structure — and expect the prediction to be right only about two times in three. When it disagrees, that disagreement is itself a clue.
Honest limits and Pauling's rules
The radius-ratio rule is a guideline, not a law — it predicts the right coordination only about half to two-thirds of the time. A crisp failure: rubidium chloride has r/R near 0.84, which sits above 0.732 and should give coordination 8, yet RbCl is plain rock salt with coordination 6. Three honest reasons the rule slips. First, ions are not hard spheres — their clouds are soft and squashable (polarizable), so they overlap and share instead of just touching. Second, the radius you plug in itself depends on coordination number (the tabulated Shannon radii list a different value for each), so the calculation is faintly circular. Third, the rule assumes pure ionic bonding and ignores covalency; any directional, shared-electron character bends the answer.
Even so, the rule earns its place — it is the first of Pauling's rules, five guidelines that let you predict the structures of ionic crystals from sizes and charges. The others build on it: bond strengths must balance so the crystal stays neutral (the electrostatic valence rule), and the anion polyhedra around cations prefer to share corners rather than edges or faces, because sharing an edge or face brings the small, highly charged cations dangerously close and destabilises the crystal. Together they capture a real truth: in a non-directional ionic solid, relative size is the dominant lever on coordination and structure, exactly as this rung has argued.