The ratio of the ratios

Burn a gram of carbon two ways and you get two different compounds. One holds 1.332 g of oxygen and the other holds 2.664 g — exactly twice as much, and nitrogen does the same trick five times over. A regularity in the ratios themselves is a far stranger thing than a regularity in the ratios.

1803 – 181816 min

By the end you should be able to

State the law of multiple proportions, and say precisely what makes it a claim about ratios of ratios
Explain why a second-order regularity is much harder to explain away than a first-order one
Work out the combining ratios in a series of oxides from analyses by mass
Say where the law stops being useful, and why the early chemists were lucky in their examples

Proust settled that a compound keeps its recipe. You could live with that. Perhaps chemical affinity simply saturates — a metal takes up oxygen the way a sponge takes up water, until it can hold no more, and the point at which it stops is a property of the metal. That explanation costs nothing, commits you to nothing about what matter is made of, and covers everything in the previous lesson. What happens next is not something saturation can produce.

Carbon burns two ways, and by 1800 both were well known. In a good draught, charcoal gives the gas Lavoisier called carbonic acid and we call carbon dioxide. It puts out a flame, it turns limewater milky, it is what your breath is full of. Starve the air supply — a charcoal fire in a closed room, a blast furnace, a lamp burning badly — and you get something else. Carbonic oxide: colourless, faintly smelling of nothing, burning with its own pale blue flame, and lethal in a way that nobody could explain until the chemistry of haemoglobin. It was known as a distinct substance from the 1770s, when Joseph Priestley made it and Carl Wilhelm Scheele studied it. Two gases. The same two elements. Nothing intermediate. Not a range of gases shading from one to the other, which is what a continuous theory of composition would suggest, but two, sharply distinct, each with its own fixed recipe. Which raises a question nobody had thought to ask before Proust made recipes fixed: how are the two recipes related?

The experiment

The two oxides of carbon, weighed against the same gram of carbon

John Dalton, and afterwards everybody · 1803 – 1804, and continuously improved for the next fifteen years · Manchester

The question: Take a fixed mass of carbon and turn it into each of its two oxides in turn. How much oxygen does each take, and is there any relation between the two figures?
The apparatus: A weighed quantity of charcoal, burnt in a measured supply of oxygen — plenty of it for the one gas, deliberately restricted for the other. The product collected and weighed, or the oxygen consumed measured directly over water.

Theory predicted

Under a continuous theory of composition, nothing in particular: the two figures should be whatever they are, with no reason for a relation between them. Under Dalton’s atoms, a ratio of small whole numbers, because the second compound is the first with one more atom of oxygen in it.

They measured

Per gram of carbon: 1.332 g of oxygen for carbonic oxide, 2.664 g for carbonic acid. The second is twice the first — and not approximately. On the modern atomic weights the ratio is 2.0000, and it stays 2.0000 however precisely you measure.

How sure could they be? Dalton’s own analyses were nothing like this good, and it is worth saying so. His figures were out by a few per cent, which is enough to see a ratio of 2 but nowhere near enough to establish that it is exactly 2. What secured the law was Berzelius, who from 1810 spent the better part of a decade re-analysing some two thousand compounds to a fraction of a per cent — accepting the atomic theory while thinking very little of Dalton’s arithmetic.

Why it mattered

This is a regularity in the ratios themselves, not in any single ratio, and nothing about a substance you can divide as finely as you like gives such a thing a reason to exist.

The law of multiple proportions

When two elements form more than one compound, the masses of the second element that combine with a fixed mass of the first stand in a ratio of small whole numbers. Read that twice, because the sentence has a ratio inside a ratio and it is easy to lose one of them. The inner ratio is Proust’s: grams of oxygen per gram of carbon. Every compound has one, and it is fixed. That is the previous lesson. The outer ratio is Dalton’s: take the inner ratios of two different compounds of the same pair of elements, and divide one by the other. That number is a small whole number. This is why the law is so much more than a curiosity about tidy figures. A first-order regularity — every compound has a fixed composition — can be explained by almost anything, including a vague appeal to affinity saturating. A second-order regularity has to be explained by something that relates different compounds to each other. And there is essentially one thing that does that naturally. If matter comes in pieces, and a compound is a fixed grouping of a few of them, then two compounds of the same elements differ by whole pieces. One oxygen, or two. Never one and a half.

The oxygen taken up by one gram of carbon, in each of its two oxides, drawn to scale. The dashed rules are at multiples of the shorter bar.

Loading the proportions…

The second bar does not merely reach past the first. It ends on a rule. That is the whole of the law of multiple proportions as a picture — and once you have seen it, the question is no longer whether the numbers are whole but what could possibly be making them whole.

Now the oxides of nitrogen, which do it five times over. Then switch the series to sulphur, which is the one that stops you misremembering the law.

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Per gram of nitrogen, the oxygen in the five oxides runs 1 : 2 : 3 : 4 : 5. Dalton had three of them by 1803 and could see 1 : 2 : 4. Sulphur is the useful corrective: its two oxides give 2 : 3, not 1 : 2. The law says small whole numbers, and people remember it as whole multiples of the first, which is a stronger claim and a false one.

It is worth being exact about why this is stronger evidence than the previous lesson, because the two laws are often taught as though they were a matched pair. Definite proportions is explainable without atoms. Say affinity saturates. Say each pair of elements has a characteristic saturation point, fixed by whatever forces are involved, and reached whenever the reaction runs to completion. You now have fixed composition, with no commitment about the structure of matter whatsoever. Berthollet would have accepted a version of this; so would most chemists of 1800. Multiple proportions is not. Saturation gives you one number per compound. It gives you no reason at all for the numbers belonging to different compounds of the same elements to stand in a simple relation to each other. Why should the saturation point of the second oxide of carbon be exactly twice the saturation point of the first? There is no answer available in a picture where matter is smooth. And the numbers are small. That is the part carrying the weight. A ratio of 2, or 3, or 3 to 2 — the kind of number you get by counting things on one hand. Something in there is being counted, and there are only a few of it.

The experiment

The oxalates of potash

William Hyde Wollaston, who did not believe in atoms · Philosophical Transactions, 1808 · London

The question: Dalton’s law is drawn from gases, analysed by a man with a theory to promote. Does it hold for salts, analysed by someone who would rather it did not?
The apparatus: Three salts of oxalic acid with potash — the neutral oxalate, the acid salt, and the quadroxalate — each decomposed and the acid content determined.

Theory predicted

Wollaston expected nothing in particular. He was investigating the constitution of super-acid and sub-acid salts, and he was, by temperament and by conviction, unwilling to reason from atoms.

They measured

The quantity of acid combined with a fixed quantity of potash stood in the ratio 1 : 2 : 4. He reported it, remarked that it was striking, and explicitly declined to conclude that matter is atomic.

How sure could they be? Wollaston was arguably the best analyst in England, and his figures were good to about one per cent — enough that 1 : 2 : 4 was not in doubt. Thomas Thomson published a parallel result on the oxalates of strontium in the same volume.

Why it mattered

The law is not an artefact of gas measurements, and it is not an artefact of wanting it to be true. Wollaston is the ideal witness precisely because he refused to draw the conclusion: his data support Dalton and his opinions do not, and it is the data that survived.

The part the story usually rounds off

The law of multiple proportions is a good deal weaker than it is usually made to sound, and the weakness is in the word small. Take nonane and decane, two unremarkable components of petrol: C₉H₂₀ and C₁₀H₂₂. Per gram of carbon, the hydrogen in them stands in the ratio 100 : 99. Those are whole numbers. The law is satisfied. It is also telling you precisely nothing — because any two numbers can be written as a ratio of whole numbers if the whole numbers are allowed to be large enough, and a law with no upper bound on them is not falsifiable. What carries the argument is the smallness, and the early chemists were lucky in their examples. The compounds that could be made pure and analysed in 1808 were nearly all simple ones: two elements, a handful of atoms, ratios you could count on your fingers. Had organic chemistry come first, the law would have been visible in the data and worthless as evidence. It is a good habit to ask, of any law of this kind, what would have to be observed for it to fail. Here the answer is: a ratio like 7 : 5 turning up between two simple binary compounds. Nobody has ever found one — but the reason that is impressive is the simple binary, not the ratio of whole numbers.

You might think

Dalton noticed the whole-number pattern in his analyses and proposed atoms to explain it.

Actually

The order is the other way round, which changes what kind of achievement it was. Dalton came to atoms from a physics problem: why the different gases in the atmosphere do not settle into layers by density, and why water dissolves different gases to such different extents. In 1803 he wrote that he was "nearly persuaded that the circumstance depends upon the weight and number of the ultimate particles of the several gases", and on 6 September of that year his notebook carries the first table of atomic weights anyone had ever drawn up. The law of multiple proportions then falls out of that theory as a prediction, which he checked against the compounds he had. Henry Roscoe and Arthur Harden established this from the notebooks in 1896, and it is not a small correction: Dalton’s own analytical figures were not accurate enough to have established the law inductively. He was not a great experimentalist. He was a man with an idea from the wrong field, which turned out to explain a set of facts in chemistry that nobody had assembled yet.

Problem

Two oxides, one gram of carbon

Analysis gives carbonic oxide as 42.881% carbon by mass, and carbonic acid as 27.292% carbon by mass. How many grams of oxygen combine with one gram of carbon in carbonic acid?

Carbonic oxide: 42.881% carbon, the rest oxygen
Carbonic acid: 27.292% carbon, the rest oxygen

The elements of oxygen may combine with a certain portion of nitrous gas, or with twice that portion, but with no intermediate quantity.
John DaltonFrom his work on the two combinations of nitrous gas with oxygen, autumn 1803 — the same season as the first table of atomic weights in his notebook. It is usually quoted as the earliest clear statement of multiple proportions, and the clause that matters is the last one.

1801Dalton’s law of partial pressures. He is working on the atmosphere, not on compounds.
18036 September: the first table of atomic weights, in a notebook. In October he reads the paper on gases dissolving in water that will carry it.
1804August: Thomas Thomson visits Manchester, and Dalton explains the theory to him over two days.
1805The absorption paper appears in print, with the atomic weight table appended to it almost as an afterthought.
1807Thomson publishes Dalton’s theory in the third edition of his System of Chemistry — a year before Dalton publishes it himself.
1808Wollaston on the oxalates of potash, and Thomson on the oxalates of strontium. Independent confirmation, from a sceptic.
1808A New System of Chemical Philosophy, part one.
1810Berzelius begins the analytical programme that will occupy him for a decade and settle the matter properly.
1896Roscoe and Harden read Dalton’s notebooks and find that the theory came before the law, not after it.

Where this leaves us

Two facts, both of them established by nothing more exotic than a balance: Compounds keep their recipes. Whatever you offer them, however you make them. The recipes are related by small whole numbers. Not within a compound — between compounds of the same two elements. Neither statement mentions the structure of matter. Both are about weighing things. And there is essentially one picture in which both are unremarkable. Suppose the stuff comes in indivisible pieces, each kind with its own weight, and suppose a compound is a fixed grouping of a few of them. Then the recipe is fixed because the grouping is fixed. And the second oxide holds twice the oxygen of the first because it has two pieces where the first had one — which is the only quantity that can differ, when the pieces cannot be cut. That argument is about to be made in print, by a Quaker schoolteacher in Manchester who arrived at it while trying to work out why the atmosphere does not separate into layers.