SOME METHODS OF DETERMINING AVOGADRO'S NUMBER
Students of chemistry and physics are often startled when they learn that the numher of molecules in a thimbleful of air is so great that if each of the inhabitants of the United States (125,000,000) were to count one molecule each second it would take about 10,000 years to complete the job; or again that if these molecules were placed side by side they would encircle our globe about one hundred times. Statements such as these might well startle a student. But though some students accept these statements readily, others wish to know how such a tremendously large number has been obtained. I t has been thought advisable to present a few of the methods, especially since we have some ways of arriving a t such figures which are easily comprehended by the average student. It will be remembered that Amadeo Avogadro (1776-1856), the great Italian chemist, over one hundred years ago proposed the hypothesis that equal volumes of all gases under like conditions contained equal numbers of molecules. How many molecules were to he found in a lifer or a gram molecular volume was not stated. Some fifty or more years passed before any one felt it would be possible to estimate this numher. Loschmidt (1865) appears to have been the first scientist to estimate the number of molecules in one cubic centimeter of a gas, and in some portions of the European continetit this value is known as Loschmidt's numher. For some'years, however, it has been customary to refer to the numher of molecules in a gram molecular volume as Avogadro's number. Most chemists and physicists agree that the most probable value of Avogadro's number is G.OG2 X loz3 1 or 606,200,000,000,000,000,000,000. It will be our purpose then to present some methods of determining this immense figure. From Atomic and Molecular Dimensions Without doubt the simplest possible way to arrive a t an a@roximate value for Avogadro's number is to determine, if possible, the size of atoms or molecules. Such a piece of data, coupled with the value of the volume
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Since some students may not be familiar with this method of representing very large or very small numbers, a few examples will be given. It seems increasingly evident that the best way to express a large or small number is to write it down as an integer and multiply it by 10 with the proper exponent. Thus 348 = 3.48 X 100 = 3.48 X 1 1 103 and 3480 = 3.48 X 10% In a similar way we have 0.01 = -=- = 1 X lo-' 100 10% and 0.0348 = 3.48 X lo-? Multiplying 3.48 X lo2by 4.0 X lo4we get 13.92 X lo6 0.87 or 1.392 X 107. Dividing 3.48 X loa by 4 X 10'we get - = 0.87 X 10-'or 8.7 X 100
occupied by a gram atom or gram molecule, leads immediately to the desired figure. Several illustrations will now be given. Many investigators have shown that very thin sheets of metal may be prepared. The thinnest gold foil ever made was approximately 5 X lo-' an.thick. Michael Faraday2 claimed to have produced gold foil '/,oo wave-length of sodium light, or five millionths of a millimeter. It may be well to mention, too, that in recent years wires of various metals of diameter as small as 2 X 10-6 cm. may be purchased from various manufacturers. It should be quite evident, then, that atoms are smaller than these figures. Other scientists have further shown that very thin uniform coatings of silver may be electrolytically deposited on, say, copper. The important point, for our purposes, is that the properties of copper are lost and those of silver gained when the film reaches a definitethickness, usually about lo-' cm. Electromotive force measurements and observation of the change in phase of reflected light are the properties usually studied to determine when the film is sufficiently thick. Let us now make a simple calculation assuming a film of silver.5 X 10-8 cm. thick to be composed of only one layer of silver atoms." The atomic weight of silver is very nearly 108, while its density is 8.65, therefore 108 108 grams of silver have a volume of -or 12 cc. Now if the atoms of silver 8.65 4 ; arJ or are asskmed to be spherical in shape, the volume of one atom is ; 0 e 3 4 or 8 X lo-" cc. Since one atom has a volume of i X 3.14 X X
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8 X 10V3c;. and one gram atom occupies 12 cc., the number of atoms 12 in a gram atom is readily seen to be = 1.5 X lCP3. This, i t 8 x must he emphasized, is only an approximate value of Avogadro's number, since the atomic dimensions used above are only approximate. Now there is another method of determining the limiting value of molecular dimensions, which yields values which are quite accurate, and yet the principle of the method is easily understood. From a study of oil flms on water, Lord Rayleigh, Devaux, L a n g m ~ i r ,and ~ others have shown the thinnest oil films to be about 5 X 10V cm. thick. This is done in the following manner. A rectangular trough is saupulously cleaned and then half filled with freshly distilled water. Fine talcum powder is sifted (through a ~e meshed linen cloth) on the surface of the water. Strips of Pogg. Ann.. 110, 168 (1860). U n article by Lanmuir [J.Am. Chem. Sac., 39. 1898 (191711 may be read with profit by students intewted in this subject. The earlier literature is critically reviewed by Langmuir in this paper.
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paper are now used to push all the talc particles to one end of the trough. This procedure is repeated a few times to make sure the surface of the water is very clean. The surface of the water is once more covered with fine talc powder. The appearance of the trough is illustrated diagrammatically in Figure 1. A small weighed amount of one per cent solution of oleic acid (dissolved in benzene) is dropped in the center of the trough; it will be observed immediately that the talc particles are pushed back so that the trough now appears FIGURE 1 as shown in Figure 2. Strips . . .......,,. ;':.,.......;...;.,,.:i. of paper are now used to get v~ ;: .;: :*.;I:;.:, ..;T,, :,,,.'. . : ' . : ' >:-< ?:'X the film of oil in the rectangu- ., .../, : . ,.. . . .__, .:.< ., ..: :::: J ~,y.:,:. ,,..:-:. :, ..... . . .;j . ; .,. $:. lar The shape length seen and breadth in Figure of the 3. .;,:.;::.,:!,:? ;.:;!> ,: $' ; ...... .:: oil film may now be easily& .:?::;; .., ...-.. ;..". ;::t,.:.:; ;3 :..: ..... . . measured with a fair degree of;:;.:?>;:;; ,.. , ,.
