Size, grip and reactivity

Atomic radius, ionisation energy and electronegativity all trend the same way across the table, and all for the same reason: the net pull the nucleus exerts on an outer electron. Three measurements, one quantity underneath.

1870 – 196416 min

By the end you should be able to

Explain effective nuclear charge, and compute one with Slater’s rules
Say why atomic radius, ionisation energy and electronegativity are three readings of one quantity
State what a quoted atomic radius actually measures, and why there is more than one answer
Show how Pauling derived electronegativity from measured bond energies rather than asserting it

The first person to show periodicity as a graph rather than a table was not Mendeleev. Lothar Meyer, 1870. Take an element. Take its atomic weight, divide by its density, and you have the volume occupied by a fixed number of its atoms — what he called the atomic volume. Plot that against atomic weight. What comes out is a sawtooth. The alkali metals sit on the peaks, spectacularly, and nothing else comes close; the curve then falls steeply and climbs again to the next alkali metal. Caesium occupies about seven times the volume per atom that carbon does, despite weighing eleven times as much. Meyer had no electrons, no nucleus, no shells and no quantum mechanics. He had a balance, a density measurement and a piece of graph paper. And the shape of the periodic table came out of them. He is usually a footnote to Mendeleev, and the reason is fair — Mendeleev predicted missing elements and Meyer did not — but the atomic volume curve is the older and more direct piece of evidence that something physical about the atom repeats. This lesson is about what that something is.

Before going further, an awkward question about the quantity Meyer plotted. What is the radius of an atom? The previous lesson answered this in passing and it is worth making explicit: an atom has no edge. The electron density falls away exponentially and reaches zero nowhere. There is no surface, so there is nothing to measure to. What can be measured is the distance between two nuclei, which is a real, sharp, reproducible quantity. Every atomic radius you have ever been given is one of those distances, split between the two atoms by a convention. That is not a defect. A convention applied consistently gives numbers that are consistent with each other, and consistent numbers are all a trend needs. But it does mean that a radius carries a hidden clause — as measured this way — and that different ways give different answers.

The experiment

The structure of rock salt, and the first table of atomic radii

William Henry Bragg and William Lawrence Bragg · 1913; the first radius table in 1920 · Leeds and Cambridge

The question: Crystals are built of atoms in a regular arrangement. Can the actual distances between those atoms be measured, rather than inferred from density and guesswork about packing?
The apparatus: A beam of X-rays directed at a cleaved crystal of sodium chloride, and an ionisation chamber to measure the reflected intensity as the crystal is rotated. Bragg’s law relates the angles at which reflections appear to the spacing between planes of atoms. The X-ray wavelength and the plane spacing are determined together, using the known density and molar mass of the crystal to fix the scale.

Theory predicted

Chemists expected a lattice built of NaCl molecules — discrete pairs, as the formula suggests, arranged in a crystal. Nothing in the chemistry of 1913 suggested otherwise.

They measured

The reflections gave a sodium-to-chlorine distance of about 282 pm, in a lattice with each sodium surrounded by six chlorines and each chlorine by six sodiums. There are no pairs and no molecules. And for the first time, an interatomic distance was a measured number rather than an estimate.

How sure could they be? The distances were good to about 1% almost immediately, and to far better than that within a decade. This is the point at which the size of an atom stops being an order-of-magnitude argument from gas kinetics and becomes a laboratory measurement.

Why it mattered

Given thousands of such distances, you can extract a set of atomic radii that reproduce them additively. W. L. Bragg published the first such table in 1920. Every radius quoted in this lesson descends from that method — and the modern set used in the simulation, Cordero’s of 2008, is the same idea run over some 228,000 crystal structures at once.

Effective nuclear charge: one quantity under three trends

An outer electron does not feel the full charge of the nucleus. The electrons lying between it and the nucleus screen part of that charge, and what is left is the effective nuclear charge, Z_eff. It is the net inward pull, and it is the only thing in this lesson that has to be understood. Now walk across a row of the table, one element at a time. Each step adds one proton. More pull. Each step also adds one electron — but into the same shell, sitting alongside the electron we are watching rather than between it and the nucleus. An electron in the same shell is a poor screen: on average it spends about as much time further out as further in. So the pull wins. Every step across a row raises the net charge on the outermost electron, and it does so all the way from group 1 to group 18. Everything follows from that: The atom shrinks, because the outer shell is pulled in harder. The first ionisation energy rises, because the electron is held on harder. The electronegativity rises, because the atom pulls harder on a pair it is sharing with a neighbour. Going down a group the arithmetic reverses. The outermost electron starts in a new shell, much further out, behind a full extra layer of screening — and distance beats charge. All three trends invert.

Start on period 2 with the Z_eff overlay on. Then press "Down a group" and drag the group slider to 1. Watch which line the blue one follows.

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Atomic radius across the second row, with the effective nuclear charge drawn over it. From lithium to fluorine the nucleus gains six protons and Z_eff rises by 3.90 — the other 2.10 is absorbed by the electrons added alongside. The atom shrinks from 128 pm to 57 pm while gaining six electrons, which is the fact worth sitting with.

Two sentences, and they are the whole of periodic trends. Across a row, the charge wins. Protons are added to the nucleus and electrons to the same shell, and same-shell electrons screen badly. Net pull rises; the atom contracts, holds its electrons harder, and pulls harder on shared ones. Down a column, the shell number wins. Z_eff on the outermost electron barely changes — sodium, potassium, rubidium and caesium all sit at about the same net charge — but that electron is now in the next shell out, and further out means less strongly held. The atom expands, ionises more easily, and pulls more weakly. The thing to notice is that these are not two rules. They are one rule, and which term dominates depends on whether you are adding screening or adding a shell.

Switch between the three properties without touching anything else. Then set the period slider to 4 and watch what the d block does to all three at once.

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The same table, coloured by three different measurements. They are not three trends that happen to agree — they are three instruments pointed at the same quantity, which is why the green Z_eff line tracks all of them. The radius runs the other way from the other two for the obvious reason: a harder pull means a smaller atom but a higher energy to remove an electron.

The ionisation energies are not a smooth climb, and the bumps are the most interesting thing on the graph. Across period 2, two elements go the wrong way: Boron is lower than beryllium — 800.6 against 899.5 kJ/mol, a drop of nearly 100 despite an extra proton. Oxygen is lower than nitrogen — 1313.9 against 1402.3 kJ/mol. Both are real, both reproduce to a fraction of a per cent, and neither is explained by effective nuclear charge. They are explained by the orbitals from the previous lesson. Boron is the first element to put an electron into a 2p orbital. A p orbital has an angular node through the nucleus, so a p electron spends less time close in than a 2s electron does, and it sits behind the filled 2s pair. It is easier to remove than the arithmetic of Z_eff suggests. Oxygen is the first element forced to put a second electron into an already-occupied 2p orbital. Two electrons confined to the same orbital repel each other, and that repulsion pays part of the cost of removing one. This is worth more than a smooth line would be. A trend with two exceptions, each of which has a specific structural cause, is a trend you can test — and the same two dips appear in period 3, at aluminium and sulphur, exactly where the same argument says they should.

And here is the case that shows the argument really is about screening rather than about counting protons. Walk across the transition metals, from calcium to zinc. The nucleus gains ten protons — more than the seven added across the whole of period 2 — and the atoms barely change size. Calcium is 176 pm; zinc is 122 pm. The reason is where the electrons go. Across period 2 they go into the same shell as the electron being watched, and screen it badly. Across the d block they go into the 3d subshell, which lies inside the 4s electrons that are being pulled on. Inner electrons screen almost perfectly. So Z_eff on the outer electron creeps from 2.85 to only 4.35 across ten elements — a rise of 1.50 — against 3.90 across six in period 2. Same rule, opposite behaviour, because the shielding is different. That is the strongest evidence in the lesson that the rule is real: it correctly predicts where the trend should nearly vanish. It also has a consequence you will meet later. The contraction across the f block is what makes hafnium almost exactly the same size as zirconium, one row above it — the lanthanide contraction — and it is why zirconium and hafnium are the hardest pair of elements in the table to separate chemically.

A black and white photograph of Linus Pauling in 1962, in a suit and tie, looking slightly to the side. — Pauling introduced electronegativity in 1932, in the first of the papers that became *The Nature of the Chemical Bond*. He is the only person to have won two unshared Nobel Prizes — chemistry in 1954 for the bond work, peace in 1962 for the campaign against atmospheric nuclear testing, which is what this photograph was taken for.

Which leaves electronegativity, and it is the one that sounds like an opinion. How hard does an atom pull on a shared pair of electrons? You cannot put an atom on a machine and read that off. There is no instrument for it, and no isolated experiment that produces a number. It sounds like the kind of quantity a chemist asserts because it makes the story come out right. Pauling found a way to measure it anyway, in 1932, and how he did it is the reason the scale is worth anything at all.

There is no such thing as “the” atomic radius

Chlorine, in three data books, in picometres: 102 — the covalent radius. Half the Cl–Cl distance in a chlorine molecule. 175 — the van der Waals radius. Half the closest approach of two chlorines that are not bonded, from packing in crystals. 181 — the ionic radius of Cl⁻. Extracted from the Na–Cl distance in rock salt by deciding how to split it between the two ions. Those differ by nearly a factor of two, and none of them is wrong. They are answers to three different questions: how close do bonded chlorines get, how close do unbonded ones get, and how big is a chloride ion. There is a fourth, the metallic radius, which is half the nearest-neighbour distance in the solid metal — and which does not exist for the noble gases at all, because they form no bonds to halve. Switch the simulation’s radius set and six cells go blank. That hole is the honest answer to "what is the radius of argon", and any table that fills it has quietly substituted a different measurement. The ionic radii carry an extra difficulty that is worth naming. Crystallography measures the distance between two nuclei; it does not tell you where one ion stops and the next begins. Landé, Goldschmidt, Pauling and Shannon each divided that distance differently, and their tables disagree by tens of picometres while all reproducing the same measured lattices exactly. The trends survive every set intact. Radii fall across a period and rise down a group in all four. The absolute numbers do not survive, and a radius quoted to three figures without saying which set it came from is the quiet dishonesty of this corner of chemistry.

You might think

Effective nuclear charge stays the same going down a group, so the size increase is caused entirely by the extra shell.

Actually

That is what Slater’s rules say, and Slater’s rules are wrong about it. They assign Z_eff = 2.20 to sodium, potassium, rubidium and caesium alike, because the rules count every electron two shells in or deeper as a perfect screen — worth exactly 1.00 — and no real electron screens perfectly. Fitting actual self-consistent wavefunctions, Clementi and Raimondi got 1.279 for lithium, 2.507 for sodium, 3.495 for potassium and 4.985 for rubidium. The net pull nearly quadruples going down four rows. The atoms get bigger anyway, because the outer electron has moved out by three whole shells and distance beats charge — so the conclusion is right and the reasoning that is usually given for it is not. Slater published the rules in 1930 as a way of fitting one-electron wavefunctions to many-electron atoms; they were never claimed to be more than a fit, and they get the shape of the trend across a period to within a few per cent, which is why they survive in textbooks. Use them for the direction and the rough size of a change, and stop there.

Problem

The net pull on a fluorine electron

Fluorine has 9 protons and the configuration 1s² 2s² 2p⁵. Use Slater’s rules to find the effective nuclear charge felt by one of its 2p electrons. This is the number that has to explain why fluorine is smaller than lithium and holds its outermost electron more than three times as hard.

Nuclear charge: Z = 9
Slater groups: (1s) (2s, 2p) (3s, 3p) (3d) …
Same group: 0.35 each — but 0.30 if the group is 1s
Shell n − 1: 0.85 each
Shell n − 2 or deeper: 1.00 each
Anything further out: 0 — it does not screen at all

The power of an atom in a molecule to attract electrons to itself.
Linus PaulingThe definition of electronegativity, from The Nature of the Chemical Bond, 1939. Note the clause that is usually dropped when it is quoted: *in a molecule*. Electronegativity is not a property an isolated atom has — it is a property of an atom in a bond, which is why it had to be extracted from bond energies and could not be measured on an atom sitting on its own.

1870Lothar Meyer plots atomic volume against atomic weight and gets a sawtooth, with the alkali metals on the peaks.
1913The Braggs determine the structure of rock salt from X-ray reflections. An interatomic distance becomes a measurement.
1920W. L. Bragg publishes the first table of atomic radii, by splitting measured internuclear distances.
1926Goldschmidt compiles ionic radii from crystal data; Pauling produces a rival set in 1927, disagreeing by tens of picometres while fitting the same lattices.
1930Slater publishes his shielding constants — a set of empirical rules for the effective nuclear charge, fitted to spectroscopic data.
1932Pauling defines electronegativity from the excess strength of bonds between unlike atoms, and fixes hydrogen at 2.1.
1934Mulliken proposes an independent scale — the mean of ionisation energy and electron affinity — and it correlates closely with Pauling’s.
1958Allred and Rochow define a third scale directly from effective nuclear charge and covalent radius, tying electronegativity back to Z_eff explicitly.
1963Clementi and Raimondi compute effective nuclear charges from self-consistent wavefunctions, and show how far Slater’s rules are off down a group.
1964Slater publishes empirical atomic radii from crystal structures. The noble gases are left blank, because they form no bonds to measure.

Where this leaves us

The table is now explained twice over. Its shape — the widths 2, 6, 10 and 14, and rows of 2, 8, 8, 18, 18 and 32 — comes from counting the solutions of one equation. Its gradient — everything that varies smoothly along a row and down a column — comes from one quantity, the net pull the nucleus exerts on an outer electron. Size, ionisation energy and electronegativity are three instruments pointed at that one thing, which is why they trend together, and why they break together at boron and at oxygen for reasons about orbitals rather than about charge. That is where Act 5 set out to get to. Periodicity was a fact about measurement in 1869. It is now a fact about the atom. But the last of those three properties has quietly written a cheque the next act has to cash. If electronegativity measures how unequally two atoms share a pair, then a large difference ought to mean something categorically different from a small one — a transfer rather than a sharing, salt rather than methane. And underneath that sits the question nobody has answered yet, which is harder than it sounds. Two neutral atoms carry no net charge and have no reason whatever to attract one another. Nothing in classical physics permits a chemical bond to exist at all.