There is no molecule of salt

X-rays show sodium chloride is a lattice with no pairs in it — every sodium has six equidistant chlorines and none of them is its partner. Chemists disbelieved it in print. A second and completely independent line of evidence, the Born–Haber cycle, then turned "how ionic is this bond?" from an assertion into a measurement.

1913 – 193216 min

By the end you should be able to

Explain why the formula NaCl states a ratio and not a molecule
Describe what Bragg’s X-ray structure showed, and why chemists resisted it
Use a Born–Haber cycle to obtain a lattice enthalpy from five measurable quantities
Say how the residual between model and measurement locates a bond on the ionic–covalent scale

1916 produced two theories of the chemical bond, from two people who did not agree. Gilbert Lewis, in Berkeley: a bond is a pair of electrons shared between two atoms, so that both count it towards their eight. Walther Kossel, in Munich, in the same year: a bond is an electron handed over. Sodium has one electron more than neon and chlorine has one fewer than argon, so sodium gives it up, both reach a noble-gas arrangement, and the resulting Na⁺ and Cl⁻ attract each other because opposite charges do. Both men were reasoning from the same evidence — the inert gases, the valences, the periodic table — and both were right, about different substances. That is not a resolution anyone found comfortable. It looked like two rival theories of the same thing, and for a decade the question of which was the real bond was live. What settled it was not an argument. It was a photograph of a crystal.

Underneath that argument sits an assumption nobody had thought to examine. A formula names a molecule. Water is H₂O, so somewhere there is an object made of two hydrogens and an oxygen. Methane is CH₄, so there is an object with one carbon and four hydrogens. This has been automatic since Dalton, it is true a great deal of the time, and it is what a structural formula means. Sodium chloride is written NaCl. So there is an object with one sodium and one chlorine in it. That last sentence is false, and it took X-rays to establish it. Not approximately false, not "the molecules are strongly associated in the solid state" — there is no such object, anywhere in the crystal, at any temperature at which the crystal exists.

The experiment

The structure of rock salt from X-ray diffraction

William Lawrence Bragg, aged 23, working with his father William Henry Bragg · 1913 · Cambridge and Leeds

The question: A crystal reflects X-rays strongly at certain angles and not others. What arrangement of atoms produces the pattern that rock salt produces?
The apparatus: An X-ray tube, a cleaved crystal of rock salt on a rotating table, and an ionisation chamber to measure the reflected intensity as a function of angle. Strong reflection occurs when nλ = 2d sin θ — the elder Bragg’s spectrometer measuring the spacings that the younger Bragg’s equation interprets.

Theory predicted

On the chemistry of the day, a crystal is a regular stack of molecules of NaCl. The diffraction pattern should therefore reveal a repeating unit containing a sodium–chlorine pair, with the two members of a pair closer to each other than to anything else.

They measured

A three-dimensional chess board. Sodium, chlorine, sodium, chlorine, alternating along all three axes of a cube, with the cube edge 5.64 Å. Every sodium has six chlorines around it, all at 2.82 Å — one along each of the three axes, in both directions — and every chlorine has six sodiums the same way. There is no closer pair and no repeating molecular unit at all.

How sure could they be? The spacing follows from the diffraction angles and the X-ray wavelength; the wavelength was pinned down independently from the measured density of the crystal and Avogadro’s number, so the absolute distance is not a free parameter. Modern X-ray work gives the cube edge as 5.6402 Å at room temperature, against Bragg’s 5.62 Å.

Why it mattered

The formula NaCl states a ratio, not a molecule. The one-to-one composition comes from the geometry of the packing — a chess board has as many black squares as white — rather than from any pairing off of atoms. It is the first time a chemical formula was shown to mean something other than what everybody had assumed it meant, and it is the founding result of structural crystallography.

A photographic portrait of William Lawrence Bragg as a young man, in a suit and tie, facing slightly to the left. — Bragg was 23 when he determined the structure of rock salt and 25 when he shared the 1915 Nobel Prize in Physics with his father — still the youngest science laureate. He spent much of the rest of his life being told that the prize had really been his father’s.

Six neighbours, all at the same distance, and none of them is the partner

Pick a sodium ion anywhere inside a crystal of salt. Count what is around it. Six chlorides, every one of them 2.820 Å away. One up, one down, one to each side, one front, one back. They are identical in every respect that can be measured: same distance, same interaction, same everything. Now try to say which of them the sodium is bonded to. There is no answer. Choosing one would be an arbitrary act, and choosing all six would give a formula of NaCl₆ — which is wrong, because each of those chlorides is being shared with five other sodiums in exactly the same way. So the one-to-one ratio is a consequence of the packing. A chess board has equal numbers of black and white squares without any square being paired with another. That is the whole content of the formula. And there is a stronger version of this, which the simulation makes visible: the six neighbours are not even the majority of the story. Count every ion in the crystal, with the correct sign, and the binding comes out about three quarters larger than a lone pair of ions would give. A molecule of salt is not a good first approximation to a crystal of salt. It is a different object.

Start with one sodium ion and one chloride ion alone in the universe, and read the depth of the well. Then press “Count the whole lattice” and watch what the rest of the crystal is worth.

Loading the bond curve…

The ghost curve is a single Na⁺Cl⁻ pair, held at the crystal’s own nearest-neighbour distance so that the Madelung factor is the only thing that changes; the solid curve counts every other ion in the crystal, with the alternating signs, out to infinity. Switching the lattice on deepens the well by about three quarters, and that extra depth is the argument of the lesson made quantitative: the neighbours are not a correction to a bond, they are most of the binding. The inset shows the six equidistant neighbours that the reader is being asked not to pick between.

The factor by which the crystal deepens the well is a pure number that depends on the geometry and nothing else. It is called the Madelung constant, and getting it is harder than it looks. Start from one sodium and count outwards. Six chlorides pull, at distance r. Twelve sodiums push, at r√2. Eight chlorides pull, at r√3. Six sodiums push, at 2r. And so on, with the terms alternating in sign and shrinking only as 1/r while the number of ions in a shell grows as r². That series does not converge. Worse, it is conditionally divergent — you can make it sum to almost any value you like by choosing the order in which you take the terms, which means the naive calculation is not merely difficult but meaningless. Hans Evjen found the fix in 1932: sum over neutral cubes. Take a cube of the crystal centred on your ion and count a site lying on a face as a half, on an edge as a quarter, at a corner as an eighth — so that every partial sum is over a block with no net charge and no net dipole. Do that and it converges immediately.

Poor common salt

Chemists did not take this well, and the objection is worth reading rather than laughing at. In 1927 Bragg wrote in Nature that "in sodium chloride there appear to be no molecules represented by NaCl. The equality in number of sodium and chlorine atoms is arrived at by a chess-board pattern of these atoms; it is a result of geometry and not of a pairing-off of the atoms." Henry Armstrong — Fellow of the Royal Society, a considerable figure, then in his late seventies — replied in the same journal under the heading "Poor Common Salt!". Bragg’s statement, he wrote, "is more than repugnant to common sense. It is absurd to the nth degree, not chemical cricket. Chemistry is neither chess nor geometry, whatever X-ray physics may be." Armstrong was wrong, and he was defending something real. The idea that a formula names a thing had been the organising principle of chemistry since Dalton; it is what makes a reaction equation an account of objects being rearranged rather than a bookkeeping identity. He was also watching physicists arrive with an instrument no chemist could check and pronounce on a substance chemists had studied for two thousand years. The reason it is worth telling is that Armstrong could have been answered, and eventually was, on chemical grounds rather than by appeal to the X-rays. That is the next section.

A single line of evidence, however good, leaves you relying on an instrument. What kills the objection is a second line that shares none of its assumptions. Here is the quantity to aim at: the lattice energy, the energy needed to pull a mole of crystal apart into free gaseous ions, infinitely separated. Nobody can measure that. There is no apparatus that takes a crystal of salt, separates it into a mole of Na⁺ and a mole of Cl⁻ floating freely in the gas phase, and puts a calorimeter round the process. The quantity is defined and inaccessible. But five other quantities are accessible, and Hess’s law of 1840 says that the heat of a reaction does not depend on the route taken. So build a route that goes the long way round and ends up in the same place. Every step measurable, one unknown, one equation.

Problem

A quantity nobody can measure

Use the Born–Haber cycle to find the lattice enthalpy of sodium chloride — the energy required to separate one mole of the crystal into gaseous Na⁺ and Cl⁻ ions. Give it as a positive number, since pulling a crystal apart costs energy.

Enthalpy of formation of NaCl(s): -411.2 kJ/mol
Enthalpy of sublimation of Na(s): +107.3 kJ/mol
First ionisation energy of Na(g): +495.8 kJ/mol
Bond dissociation enthalpy of Cl₂: +242.6 kJ/mol
Electron affinity of Cl(g): -349.0 kJ/mol

Step through the four crystals in the table. Three of them agree with the model to about the same small margin. The fourth does not, and its structure and arithmetic are identical to the others.

Loading the bond curve…

Lithium fluoride, sodium chloride and potassium chloride are all short of the measured lattice enthalpy by about 4% — the same shortfall every time, which is the signature of a model missing one small constant thing rather than getting the chemistry wrong. (It is missing the London dispersion between the ions, mostly.) Silver chloride has the same rock-salt structure and nearly the same nearest-neighbour distance as sodium chloride, and the same model is 14.4% short — 132 kJ/mol of binding that point charges cannot account for. That excess is the covalent character of the bond, and this is how it is measured.

Ionic and covalent are the two ends of one scale

The residual between an ionic model and a thermochemical measurement is not a nuisance. It is the instrument. For a crystal that really is a stack of ions, the model is short by a small and reproducible amount — about 4%, arising from the London dispersion between the ions, which a point-charge model has no term for. That constant shortfall is the calibration. When a crystal comes out short by three times that, something else is holding it together. In silver chloride that something is electron sharing: the silver ion is polarising, its filled d shell is close in energy to the chloride’s electrons, and the bond has genuine covalent character. So the question "is this bond ionic or covalent?" has a numerical answer, and it is not obtained by looking at the two elements’ positions in the periodic table and asserting one. Two other routes reach the same conclusion from different directions. Kazimierz Fajans set out rules in 1923 for when an ionic bond should acquire covalent character — a small, highly charged cation and a large, soft anion — and silver chloride fits them exactly. Linus Pauling in 1932 defined electronegativity from the excess energy of a bond between unlike atoms over the average of the two like-atom bonds, and turned the difference into a percentage of ionic character. Three independent approaches, one continuum.

You might think

Sodium chloride molecules do not exist.

Actually

They do — just not anywhere you have ever met salt. Heat sodium chloride to about 1400 °C and it boils, and the vapour contains genuine diatomic NaCl molecules, 2.361 Å long and bound by 4.26 eV, along with some Na₂Cl₂ dimers. They have been studied by microwave spectroscopy and in molecular beams for decades. The precise claim, and the one this lesson is defending, is that the crystal contains no molecules — and neither does the solution in your kitchen, where the ions are separately surrounded by water and wander independently, which is exactly why salt water conducts electricity and sugar water does not. Getting this right matters more than it looks. "There is no molecule of salt" as an unqualified slogan is the kind of statement that gets quietly corrected by a chemist and then disbelieved wholesale, taking the true and important part down with it.

In sodium chloride there appear to be no molecules represented by NaCl. The equality in number of sodium and chlorine atoms is arrived at by a chess-board pattern of these atoms; it is a result of geometry and not of a pairing-off of the atoms.
William Lawrence BraggNature, 1927 — the sentence that provoked Henry Armstrong’s "Poor Common Salt!" reply in the same journal a few weeks later. Bragg had had the structure since 1913; it took fourteen years for the chemical community to be argued with about what it meant.

1840Hess: the heat of a reaction does not depend on the route. Without this there is no cycle.
1913W. L. Bragg determines the rock-salt structure: a three-dimensional chess board, six neighbours, no molecule.
1916Kossel proposes the ionic bond; Lewis proposes the shared pair. Same year, different bonds.
1918Born and Landé give a lattice energy from point charges and a repulsion going as r⁻ⁿ.
1919Born and Haber, independently, close the thermochemical cycle that measures the same quantity.
1923Fajans’ rules: when a supposedly ionic bond should be expected to acquire covalent character.
1927Armstrong: "absurd to the nth degree, not chemical cricket."
1932Evjen makes the Madelung sum converge; Pauling defines electronegativity and puts a number on ionic character.

Where this leaves us

There are two kinds of bond, and they can be told apart by measurement rather than by assertion. A crystal whose lattice energy a point-charge model reproduces to within a few per cent is held together by charges. One where the model falls far short has something else going on, and the size of the shortfall says how much. And the ionic end of the scale has no molecules in it. The formula of a salt is a ratio the packing imposes, not the composition of an object — a fact that took X-rays to establish and a decade of argument to be accepted. What we still cannot do is say why either bond exists. Kossel’s ions at least have an excuse: they carry opposite charges, and opposite charges attract. That is ordinary electrostatics, and the numbers work. Lewis’s shared pair has no such excuse. Two neutral hydrogen atoms have no charge to attract each other with, no dipole, and nothing in classical physics that says they should do anything at all when brought together. And yet they bind, by nearly five electronvolts, which is an enormous amount of energy for two of the simplest objects in the universe. In the summer of 1927, two postdoctoral fellows in Zurich sat down with an equation eighteen months old and calculated exactly how much.