Elements and Atoms: Chapter 9
Equal Numbers in Equal Volumes: Avogadro

Amedeo Avogadro (1776-1856; see portrait at the Edgar Fahs Smith collection, University of Pennsylvania) was an Italian chemist and physicist best known for today for Avogadro's law and Avogadro's number. Avogadro did not discover or determine Avogadro's number; its determination occured late in the 19th century and early in the 20th and it was named in his honor. [Perrin 1908] It is the number of atoms in 1 gram of hydrogen, or equivalently the number of atoms in one mole: 6.022x1023 .

This chapter includes Avogadro's law, that equal volumes of gas contain equal numbers of particles, presented as a hypothesis. Combined with Dalton's atomic hypothesis (chapter 7), the law explains Gay-Lussac's observations about combining volumes of gases (chapter 8). Avogadro's ideas removed an important obstacle to the determination of atomic weights by allowing molecular formulas to be determined independently of mass measurements, at least for volatile compounds. I should say that his ideas would have removed that obstacle if they were adopted by other chemists; however, they were not widely accepted for about 50 years, until Stanislao Cannizzaro made them the foundation of a logical and coherent system [Cannizzaro 1858].


Essay on a Manner of Determining the Relative Masses of the Elementary Molecules of Bodies, and the Proportions in Which They Enter into These Compounds

Journal de Physique 73, 58-78 (1811), as translated in Alembic Club Reprint No. 4, (Edinburgh, 1890)


M. Gay-Lussac has shown in an interesting Memoir (Mémoires de la Société d'Arcueil, Tome II.) that gases always unite in a very simple proportion by volume, and that when the result of the union is a gas, its volume also is very simply related to those of its components. But the quantitative proportions of substances in compounds seem only to depend on the relative number of molecules which combine, and on the number of composite molecules which result.[1] It must then be admitted that very simple relations also exist between the volumes of gaseous substances and the numbers of simple or compound molecules which form them. The first hypothesis to present itself in this connection, and apparently even the only admissible one, is the supposition that the number of integral molecules in any gases [sic] is always the same for equal volumes, or always proportional to the volumes.[2] Indeed, if we were to suppose that the number of molecules contained in a given volume were different for different gases, it would scarcely be possible to conceive that the law regulating the distance of molecules could give in all cases relations as simple as those which the facts just detailed compel us to acknowledge between the volume and the number of molecules.[3] On the other hand, it is very well conceivable that the molecules of gases being at such a distance that their mutual attraction cannot be exercised, their varying attraction for caloric may be limited to condensing the atmosphere formed by this fluid having any greater extent in the one case than in the other, and, consequently, without the distance between the molecules varying; or, in other words, without the number of molecules contained in a given volume being different. Dalton, it is true, has proposed a hypothesis directly opposed to this, namely that the quantity of caloric is always the same for the molecules of all bodies whatsoever in the gaseous state, and that the greater or less attraction for caloric only results in producing a greater or less condensation of this quantity around the molecules, and thus varying the distance between the molecules themselves.[3a] But in our present ignorance of the manner in which this attraction of the molecules for caloric is exerted, there is nothing to decide us à priori in favour of the one of these hypotheses rather than the other; and we should rather be inclined to adopt a neutral hypothesis, which would make the distance between the molecules and the quantities of caloric vary according to unknown laws, were it not that the hypothesis we have just proposed is based on that simplicity of relation between the volumes of gases on combination, which would appear to be otherwise inexplicable.[4]

Setting out from this hypothesis, it is apparent that we have the means of determining very easily the relative masses of the molecules of substances obtainable in the gaseous state, and the relative number of these molecules in compounds; for the ratios of the masses of the molecules are then the same as those of the densities of the different gases at equal temperature and pressure, and the relative number of molecules in a compound is given at once by the ratio of the volumes of the gases that form it.[5] For example, since the numbers 1.10359 and 0.07321 express the densities of the two gases oxygen and hydrogen compared to that of atmospheric air as unity, and the ratio of the two numbers consequently represents the ratio between the masses of equal volumes of these two gases, it will also represent on our hypothesis the ratio of the masses of their molecules. Thus the mass of the molecule of oxygen will be about 15 times that of the molecule of hydrogen, or, more exactly as 15.074 to 1. In the same way the mass of the molecule of nitrogen will be to that of hydrogen as 0.96913 to 0.07321, that is, as 13, or more exactly 13.238, to 1. On the other hand, since we know that the ratio of the volumes of hydrogen and oxygen in the formation of water is 2 to 1, it follows that water results from the union of each molecule of oxygen with two molecules of hydrogen.[6] Similarly, according to the proportions by volume established by M. Gay-Lussac for the elements of ammonia, nitrous oxide, nitrous gas, and nitric acid, ammonia will result from the union of one molecule of nitrogen with three of hydrogen, nitrous oxide from one molecule of oxygen with two of nitrogen, nitrous gas from one molecule of nitrogen with one of oxygen, and nitric acid from one of nitrogen with two of oxygen.[7]


There is a consideration which appears at first sight to be opposed to the admission of our hypothesis with respect to compound substances. It seems that a molecule composed of two or more elementary molecules should have its mass equal to the sum of the masses of these molecules; and that in particular, if in a compound one molecule of one substance unites with two or more molecules of another substance, the number of compound molecules should remain the same as the number of molecules of the first substance. Accordingly, on our hypothesis, when a gas combines with two or more times its volume of another gas, the resulting compound, if gaseous, must have a volume equal to that of the first of these gases. Now, in general, this is not actually the case.[8] For instance, the volume of water in the gaseous state is, as M. Gay-Lussac has shown, twice as great as the volume of oxygen which enters into it, or, what comes to the same thing, equal to that of the hydrogen instead of being equal to that of the oxygen.[9] But a means of explaining facts of this type in conformity with our hypothesis presents itself naturally enough; we suppose, namely, that the constituent molecules of any simple gas whatever (i.e., the molecules which are at such a distance from each other that they cannot exercise their mutual action) are not formed of a solitary elementary molecule, but are made up of a certain number of these molecules united by attraction to form a single one;[10] and further, that when molecules of another substance unite with the former to form a compound molecule, the integral molecule which should result splits itself into two or more parts (or integral molecules) composed of half, quarter, &c., the number of elementary molecules going to form the constituent molecule of the first substance, combined with half, quarter, &c., the number of constituent molecules of the second substance that ought to enter into combination with one constituent molecule of the first substance (or, what comes to the same thing, combined with a number equal to this last of half-molecules, quarter-molecules, &c., of the second substance); so that the number of integral molecules of the compound becomes double, quadruple, &c., what it would have been if there had been no splitting up, and exactly what is necessary to satisfy the volume of the resulting gas.[11]

Thus, for example, the integral molecule of water will be composed of a half-molecule of oxygen with one molecule, or what is the same thing, two half-molecules of hydrogen.

On reviewing the various compound gases most generally known, I only find examples of duplication of the volume relatively to the volume of that one of the constituents which combines with one or more volumes in the other. We have already seen this for water. In the same way, we know that the volume of ammonia gas is twice that of the nitrogen which enters into it. M. Gay-Lussac has also shown that the volume of nitrous oxide is equal to that of the nitrogen which forms part of it, and consequently is twice that of the oxygen. Finally, nitrous gas, which contains equal volumes of nitrogen and oxygen, has a volume equal to the sum of the two constituent gases, that is to say, double that of each of them.[12] Thus in all these cases there must be a division of the molecule into two; but it is possible that in other cases the division might be into four, eight, &c. The possibility of this division of compound molecules might have been conjectured à priori; for otherwise the integral molecules of bodies composed of several substances with a relatively large number of molecules, would come to have a mass excessive in comparison with the molecules of simple substances. We might therefore imagine that nature had some means of bringing them back to the order of the latter, and the facts have pointed out to us the existence of such means.[13] Besides, there is another consideration which would seem to make us admit in some cases the division in question; for how could one otherwise conceive a real combination between two gaseous substances uniting in equal volumes without condensation, such as takes place in the formation of nitrous gas? Supposing the molecules to remain at such a distance that the mutual attraction of those of each gas could not be exercised, we cannot imagine that a new attraction could take place between the molecules of one gas and those of the other. But on the hypothesis of division of the molecule, it is easy to see that the combination really reduces two different molecules to one, and that there would be contraction by the whole volume of one of the gases if each compound molecule did not split up into two molecules of the same nature. M. Gay-Lussac clearly saw that, according to the facts, the diminution of volume on the combination of gases cannot represent the approximation of their elementary molecules.[14] The division of molecules on combination explains to us how these two things may be made independent of each other.


Dalton, on arbitrary suppositions as to the most likely relative number of molecules in compounds, has endeavoured to fix ratios between the masses of the molecules of simple substances.[15] Our hypothesis, supposing it well-founded, puts us in a position to confirm or rectify his results from precise data, and, above all, to assign the magnitude of compound molecules according to the volumes of the gaseous compounds, which depend partly on the division of molecules entirely unexpected by this physicist.

Thus Dalton supposes that water is formed by the union of hydrogen and oxygen, molecule to molecule.[16] From this, and from the ratio by weight of the two components, it would follow that the mass of the molecule of oxygen would be to that of hydrogen as 71/2 to 1 nearly, or, according to Dalton's evaluation, as 6 to 1. This ratio on our hypothesis is, as we saw, twice as great, namely, as 15 to 1. As for the molecule of water, its mass ought to be roughly expressed by 15+2=17 (taking for unity that of hydrogen), if there were no division of the molecule into two; but on account of this division it is reduced to half, 81/2, or more exactly 8.537, as may also be found by dividing the density of aqueous vapour 0.625 (Gay-Lussac) by the density of hydrogen 0.0732.[17] This mass differs from 7, that assigned to it by Dalton, by the difference in the values for the composition of water; so that in this respect Dalton's result is approximately correct from the combination of two compensating errors,--the error in the mass of the molecule of oxygen, and his neglect of the division of the molecule.[18]

Dalton supposes that in nitrous gas the combination of nitrogen and oxygen is molecule to molecule: we have seen on our hypothesis that this is actually the same. Thus Dalton would have found the same molecular mass for nitrogen as we have, always supposing that of hydrogen to be unity, if he had not set out from a different value for that of oxygen, and if he had taken precisely the same value for the quantities of the elements in nitrous gas by weight. But supposing the molecule of oxygen to be less than half what we find, he has been obliged to make that of nitrogen also equal to less than half the value we have assigned to it, viz., 5 instead of 13.[19] As regards the molecule of nitrous gas itself, his neglect of the division of the molecule again makes his result approach ours; he has made it 6+5=11, whilst according to us it is about , or more exactly , as we also find by dividing 1.03636, the density of nitrous gas according to Gay-Lussac, by 0.07321. Dalton has likewise fixed in the same manner as the facts has given us, the relative number of molecules in nitrous oxide and in nitric acid, and in the first case the same circumstance has rectified his result for the magnitude of the molecule. He makes it 6 + 2x5 = 16, whilst according to our method it should be , a number which is also obtained by dividing 1.52092, Gay-Lussac's value for the density of nitrous oxide, by the density of hydrogen.

In the case of ammonia, Dalton's supposition as to the relative number of molecules in its composition is on our hypothesis entirely at fault. He supposes nitrogen and hydrogen to be united in it molecule to molecule, whereas we have seen that one molecule of nitrogen unites with three molecules of hydrogen. According to him the molecule of ammonia would be 5+1=6; according to us it should be , or more exactly 8.119, as may also be deduced directly from the density of ammonia gas. The division of the molecule, which does not enter into Dalton's calculations, partly corrects in this case also the error which would result from his other suppositions.

All the compounds we have just discussed are produced by the union of one molecule of one of the components with one or more molecules of the other. In nitrous acid we have another compound of two of the substances already spoken of, in which the terms of the ratio between the number of molecules both differ from unity. From Gay-Lussac's experiments (Société d'Arcueil, same volume) it appears that this acid is formed from 1 part by volume of oxygen and 3 of nitrous gas[20], or, what comes to the same thing, of 3 parts of nitrogen and 5 of oxygen; hence it would follow, on our hypothesis, that its molecule should be composed of 3 molecules of nitrogen and 5 of oxygen, leaving the possibility of division out of account. But this mode of combination can be referred to the preceding simpler forms by considering it as the result of the union of 1 molecule of oxygen with 3 of nitrous gas, i.e. with 3 molecules, each composed of a half-molecule of oxygen and a half-molecule of nitrogen, which thus already included the division of some of the molecules of oxygen which enter into that of nitrous acid. Supposing there to be no other division, the mass of this last molecule would be 57.542, that of hydrogen being taken as unity, and the density of nitrous acid gas would be 4.21267, the density of air being taken as unity. But it is probable that there is at least another division into two, and consequently a reduction of the density to half: we must wait until this density has been determined by experiment.[21]



It will have been in general remarked on reading this Memoir that there are many points of agreement between our special results and those of Dalton, although we set out from a general principle, and Dalton has only been guided by considerations of detail.[23] This agreement is an argument in favour of our hypothesis, which is at bottom merely Dalton's system furnished with a new means of precision from the connection we have found between it and the general fact established by M. Gay-Lussac. Dalton's system supposes that compounds are made in general in fixed proportions, and this is what experiment shows with regard to the more stable compounds and those most interesting to the chemist. It would appear that it is only combinations of this sort that can take place amongst gases, on account of the enormous size of the molecules which would result from ratios expressed by larger numbers, in spite of the division of the molecules, which is in all probability confined within narrow limits. We perceive that the close packing of the molecules in solids and liquids, which only leaves between the integral molecules distances of the same order as those between the elementary molecules, can give rise to more complicated ratios, and even to combinations in all proportions; but these compounds will be so to speak of a different type from those with which we have been concerned, and this distinction may serve to reconcile M. Berthollet's ideas as to compounds with the theory of fixed proportions.[24]


[1]Avogadro uses the term molecule and never atom. His molecules are similar conceptually to Dalton's atoms, the smallest piece of matter which retained its identity as a particular substance (chapter 7, note 5). Sometimes he modifies the word, as in composite molecule or (below) elementary molecule. An "elementary molecule" is what a modern chemist would call an atom. A "composite molecule" is what a modern chemist would call a molecule (i.e., two or more atoms bound together).

[2]This is Avogadro's law, presented as a hypothesis. In modern terms, it states that equal volumes of gas contain equal numbers of molecules.

[3]Why might one think that equal volumes of gas contain equal numbers of molecules? For one thing, it is the simplest explanation of Gay-Lussac's observations that gases combine in simple ratios by volume. For if atoms combine in simple ratios by volume (as Dalton believed), and if equal volumes of gas contain equal numbers of molecules, then it would follow that gases would combine in simple ratios by volume.The close relationship between Gay-Lussac's observations and Dalton's ideas was not sufficient, however, to convince Dalton either of Gay-Lussac's observations or of Avogadro's hypothesis. (See chapter 8, note 42.)

In fact, the hypothesis of equal volumes containing equal numbers of molecules had already occured to Dalton, but he rejected it: "At the time I formed the theory of mixed gases, I had a confused idea, as many have, I suppose, at this time, that the particles of elastic fluids are all of the same size; that a given volume of oxygenous gas contains just as many particles as the same volume of hydrogenous; ..." [Dalton 1808, p. 188]. The reasons Dalton rejected the hypothesis had to do with caloric supposed to be associated with gaseous particles.

[3a]Note that Avogadro has already referred to the work of two of his contemporaries. Scientists typically build upon, criticise, and modify the work of their predecessors and contemporaries. In this sense, they do not work alone. Often the interaction among scientists and their ideas is international. Here the Italian (or, more accurately, Piedmontese, as Italy was not yet unified) Avogadro refers to work of the English Dalton and the French Gay-Lussac.

[4]Avogadro accepts the conventional wisdom of his time that molecules in a gas are coated with caloric (heat), which overcomes the forces of cohesion apparent in solids and liquids. Dalton pictures this caloric [Dalton 1808] as a mechanical spacer keeping molecules apart, the distance of separation determined by a substance's affinity to caloric. Avogadro maintains that not enough is known about caloric to decide the question; therefore, Dalton's picture should not be an obstacle to Avogadro's hypothesis.

[5]Avogadro recognized that if his hypothesis was correct, it provided the basis for determining the relative atomic and molecular weights of gases and the formulas of gaseous compounds. If equal volumes of gas contain equal numbers of molecules, then their densities (mass per unit volume) are proportional to their molecular weights (mass per molecule). Ratios of gas densities, then, are equal to ratios of molecular weights. "The relative number of molecules [atoms] in a compound" is nothing more than the compound's formula, a list of the atoms in the compound and their relative numbers.

Already Avogadro has begun to give some evidence for his hypothesis (previous paragraph) and to sketch out its implications.

[6]That is, the formula for water is H2O; compare to Dalton's assumed formula, HO.

[7]That is, the formula for ammonia is NH3; compare to Dalton's assumed formula, NH. "Nitrous gas" is known today as nitric oxide, NO; "nitric acid" is known today as nitrogen dioxide, NO2. Dalton got right the formulas for the three compounds of nitrogen with oxygen.

[8]Gay-Lussac presented some examples of these apparent paradoxes. See notes 39 and 40 of the previous chapter.

[9]For example, Gay-Lussac reported that one part oxygen reacts with two parts hydrogen to give two parts water by volume. Why two parts of water, Avogadro asks: after all, if water is H2O, shouldn't the two particles of hydrogen attach to the one particle of oxygen to make one particle of water?

[10]In modern terminology, Avogadro supposes that any gaseous element ("simple gas") exists not as single atoms ("solitary elementary molecule") but as molecules composed of more than one atom. This hypothesis turned out to be correct, but it was not widely accepted at the time. Indeed, there were some good reasons for skepticism. Some scientists were not convinced of the atomic hypothesis in the first place [Brock & Knight 1967, Knight 1967]; they naturally were not convinced that gaseous elements existed as pairs of atoms bound together. In addition, many scientists who thought the atomic hypothesis did make sense (including Dalton) saw the idea of diatomic gases as nonsense. These scientists believed (correctly) that gases were made up of widely separated particles. It did not make sense that these particles would form pairs and that these pairs would then stay apart from other pairs: if atoms of the same kind were attracted to each other, they would all cluster together to form a solid or liquid; if they repelled each other (as they were believed to do when they were attached to caloric), then no pairs would form.

Thus, the hypothesis of composite molecules for gaseous elements was one which had no direct evidence in its favor, while it had a considerable quantity of scientific opinion against it. The only point in its favor at the time was that it (in conjunction with the hypothesis of equal volumes of gas containing equal numbers of molecules) explained the available data. Thus the notion of composite molecules was a classic case of an ad hoc hypothesis, one which was not intimately connected with the rest of the theory in which it appeared and which was introduced only to make the rest of the theory work.

[11]Many commentators have remarked that Avogadro's writing in this paper is not always clear. Leonard Nash, for example, writes [Nash 1957], "The French text suffers from occasional obscurity, which must have confused Avogadro's contemporaries even as it does us." Perhaps the obscurity arises (as Nash hints) because Avogadro was not writing in his native Italian; perhaps because the ideas were not entirely clear in his mind. Perhaps the lack of clarity in Avogadro's prose contributed to the difficulty his ideas experienced in becoming established.

What Avogadro is suggesting here may best be illustrated by example. Suppose, he says, that hydrogen and oxygen gases consist not of a single atom but of two atoms bound together. Using modern notation, we would say that these elements have formulas of H2 and O2 rather than H and O respectively. Now suppose that when they react in the observed proportions of two parts hydrogen to one part oxygen, they form a larger molecule which then splits up to give two molecules of water as the ultimate products:

2 H2 + O2 --> [H4O2] --> 2 H2O .
Today, we recognize Avogadro's two suppositions or hypotheses as correct: equal volumes of gas contain equal numbers of molecules, and many gaseous elements exist as diatomic molecules. In fact, by the end of the 19th century, when argon was discovered, the idea of a monatomic gas was shocking. (See chapter 14, note 52.) The actual mechanism of water formation does not involve the formation and then splitting of a double molecule of water, H4O2, though.

Avogadro recognizes that he cannot tell if the original molecules of hydrogen and oxygen contained two atoms or four atoms or some other even number of atoms. For example, he recognized that if hydrogen and oxygen had the formulas H4 and O4, the reaction

2 H4 + O4 --> [H8O4] --> 2 H4O2
would still explain the data: hydrogen and oxygen would still combine in a ratio of 2 to 1; water would still have the ratio of two hydrogen to one oxygen; and the volume of water produced would be the same as the volume of hydrogen, not oxygen. Indeed, André-Marie Ampère proposed a hypothesis quite similar to Avogadro's except that it explicitly proposes tetratomic (4-atom) molecules [Ampère 1814]. In the absence of any additional information, however, the notion of diatomic molecules is more attractive than that of tetratomic molecules because it is simpler.

[12]The reactions mentioned here are:

N2 + 3 H2 --> 2 NH3 ,
2 N2 + O2 --> 2 N2O ,
N2 + O2 --> 2 NO .

[13]Avogadro seems to suggest that there is some sort of natural tendency to keep molecules from getting very large or complicated. He would be surprised, then, by the knowledge chemists have built up about macromolecules (molecules consisting of hundreds or thousands of atoms). Both natural and artificial macromolecules abound. DNA (which carries the genetic code of living things) and hemoglobin (a protein which transports oxygen in the blood) are two examples of natural macromolecules of tremendous biological importance. Most modern plastics are examples of synthetic macromolecules.

[14]That is, the diminution of volume in these reactions is much less than can be explained if the only thing going on was the atoms getting close together ("approximation") by attraction.

[15]See chapter 7, note 18.

[16]Here and below, Avogadro means one molecule of each reactant by "molecule to molecule."

[17]Recall that Avogadro proposed that a ratio of gas densities would be equal to a ratio of molecular weights. Here and in the remainder of this section, he compares the molecular weight of water based on gas densities to that based on his improved formulas. This is an attempt to compare different ways of measuring the same quantity. In fact, the two ways of getting at the molecular weight of water are not quite independent; still, comparing the results shows that Avogadro's methods are at least self-consistent.

[18]To summarize Avogadro's conclusions on water and its constituents: take the mass of H2 to be 1; then that of O2 is 15, and that of H2O is 81/2. Contrast this to Dalton's conclusions, which take the mass of H to be 1, that of O as 7, and that of HO as 8. A modern chemist would say that if the mass of H is taken to be 1, the mass of O is 16 and that of H2O is 18. The differences among these three sets of values stem from several sources. First, all use different analyses for the ratio of combining weights of hydrogen and oxygen in water (1:7 for Dalton, 1:7.5 for Avogadro; 1:8 today). These numbers represent the results of experiments which became progressively more accurate. They do not depend on the atomic hypothesis or Avogadro's hypotheses. Avogadro's figures differ from Dalton's, besides the difference due to accuracy of the analysis, because Avogadro recognizes that water has two atoms of hydrogen for every atom of oxygen. Finally, Avogadro's numbers differ from modern numbers, besides the difference due to the accuracy of the analysis, because Avogadro in essence chooses H2 (the hydrogen molecule) as unity, while the modern system's basis is tantamount to taking the hydrogen atom as unity.

[19]Actually, Dalton's assignment of 5 for nitrogen appears to be based on an analysis of ammonia, not of nitrous gas (NO). See chapter 7, note 20 for the uneasy coexistence of data on ammonia and on NO in Dalton's work.

[20]This "nitrous acid" must be nitrous anhydride, N2O3. The reaction actually involves four times as much "nitrous gas" as oxygen, not three as stated:

4 NO + O2 --> 2 N2O3 .
Nitrogen forms several compounds with oxygen, including NO, N2O3, NO2, N2O4, and N2O5, and frequently these gases appear in mixtures. Therefore it is not surprising that Gay-Lussac's report was erroneous.

[21]Avogadro suspects that the product of this reaction divides not just in two, as was the case in the examples he has already given, but in half again. Apparently Avogadro thought that such a large molecule was unlikely to be stable. (See footnote 13 above.)

[22]I have omitted sections IV-VII of this paper. --CJG

[23]Dalton, on the contrary, considered that his atomic hypothesis was general, and that conclusions drawn on the combining ratios of gases were restricted to gases. (Cf. chapter 8, note 40.) I must agree with Dalton, with two provisos: Avogadro's ideas can be extended to volatile compounds which are not gases at room temperature; more important, Avogadro corrects a serious flaw in Dalton's system.

[24]See previous chapter, note 6 regarding the question of definite proportions and variable proportions. Avogadro attempts to make a distinction between situations in which definite proportions apply and those in which variable proportions are permitted; however, it is not the modern distinction between compounds and mixtures (chapter 8, note 40).


See also Carmen J. Giunta, "Using History to Teach Scientific Method: the Role of Errors," Journal of Chemical Education 78, 623-627 (2001)
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