Avogadro's Hypothesis

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Avogadro's number is huge!

A mole of dollar bills would make a stack almost 7,000 light years high.
A mole of mini marshmallows would cover the earth to a depth of about 1 kilometer.
A mole of standard men would weigh about 7 earth masses.
A mole of food calories could feed every person on earth for about 111 million years.
Counting a mole of anything at a rate of one object per second would take about 19 quadrillion years.

By 1808, Joseph Gay-Lussac (1778-1850) established that when different gases combine chemically, the volumes of the combining gases are in the ratio of simple integers. Lorenzo Romano Amadeo Carlo Avogadro (1776-1856) took this observation to imply that the number of molecules in a given volume of gas at a given temperature is the same for all gases. As he was unable to prove it, this statement became known as Avogadro's Hypothesis and the number he sought to calculate as Avogadro's Number. To determine Avogadro's Number, it is first necessary to determine the characteristic size and mass of a molecule. Johann Josef Loschmidt (1821-1895) devised a method to determine the diameter of a gas molecule from the density of the liquefied gas and and the mean free path of a molecule in the gaseous phase. As a result, Avogadro's Number is called Loschmidt's Number in German speaking countries.

Source of confusion concerning "Gay-Lussac's Law"

… Gay-Lussac to the conclusion [1808] that "gases combine in very simple proportions" and that "the apparent contraction in volume which they experience on combination has also a simple relation to the volume of the gases, or at least to one of them." This relationship is known as the law of combining volumes and as Gay-Lussac's law.

Gay-Lussac pointed out that the simple regularities of his law were realized only for gases, for which matter behaves in the simplest and most universal manner. It is worth noting that the English chemist John Dalton's laws of definite and multiple proportions in chemical composition (formulated in the same period) referred to combining weights and not to volumes. This basic difference in approach led each scientist to be skeptical of the other's results, and they did not reach mutual understanding. (The Italian physicist Amedeo Avogadro showed how the results of Dalton and Gay-Lussac could be reconciled.) or so says the Encyclopedia Britannica

Lorenzo Romano Amadeo Carlo Avogadro (1776-1856). This quote is too long.

M. Gay-Lussac has shown in an interesting Memoir 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. 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 is always the same for equal volumes, or always proportional to the volumes. 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 so simple as those which the facts just detailed compel us to acknowledge between the volume and the number of molecules. 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 a greater or smaller quantity around them, without 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. But in our present ignorance of the manner in which this attraction for the molecules for caloric is exerted, there is nothing to decide us a 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.

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. 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. 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.

Lorenzo Romano Amadeo Carlo Avogadro. "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: 58-76 (1811).

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- 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. Lorenzo Romano Amadeo Carlo Avogadro.Journal de physique, 73: 58-76 (1811).
- On the Size of the Air Molecules. Johann Loschmidt. Proceedings of the Academy of Science of Vienna 52 (1865): 395-413.

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