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299

CHAPTER X

ON THE DIRECT PROPERTIES OF MOLECULES

109. Section of the Molecules

As the investigations in Part I. of this book made it possible to calculate in absolute measure the speed of the molecular movements, so the phenomena discussed in Part II. enable us to determine also in absolute measure the length of the paths traversed by the molecules. All the elements therefore which are concerned in the motion of the molecules are fully known.

Still, the conclusions which the theory lets us draw respecting the properties of the molecules are not thereby exhausted; and first of all we may seek to determine the extension of the molecules in space.

When we remember that the length of the paths is determined by the probability of a collision, and that this probability depends on the size of the molecules, it becomes at once clear that the knowledge of their molecular free path enables us to form a judgment as to their extension in space. In 1865, directly after the first experimental investigations of the viscosity of air had led to the knowledge of the free path, Loschmidt' made an attempt to determine the sizes of molecules. Later on, in 1867, there appeared two other memoirs with the same aim by my brother Lothar Meyer and Alexander Naumann,3

Wien. Sitzungsber. 1865, lii. Abth. 2, p. 395; Schlömilch's Zeitschr. f. Math. u. Physik, 1865, 10th year, p. 511.

2 Ann. Chem. Pharm. 1867, 5. Suppl.-Bd. p. 129.

Ibid. 1867, 5. Suppl.-Bd. p. 252.

then two in 1870 by Lord Kelvin,' and in 1873 one by Maxwell.2

In the formula found earlier (§ 68) for the free path,

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its value is expressed in terms of the size of the elemental cube whose edge has length λ, and of the diametral section of the sphere of action (§§ 44, 63) whose diameter is s. If we use the relation between the size of this cube and the number of molecules contained in unit volume, which is given by Nλ= 1,

the former formula may be written in the shape

1 = √2πs2NL,

which shows that, if the free path L is known in absolute measure, the magnitude

Q=πs2N = 1/4√/2L,

or the sum of the diametral sections of the spheres of all the molecules contained in unit volume, can be also expressed in absolute measure.

110. Numerical Values

From the values of the molecular free paths as obtained from the observations on viscosity carried out by Graham and by Kundt and Warburg, which are tabulated in § 79, I have calculated the following values of the magnitude Q by the above formula.

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'Silliman's Amer. Journ. 1870, 1. pp. 38, 258; Ann. Chem. Pharm. 1871, clvii. p. 54; Nature, 1883, xxviii. p. 203; Exner's Repert. 1885, xxi. pp. 182, 217.

2 Phil. Mag. 1873 [4] xlvi. p. 453; Ency. Brit. 9 ed. iii. p. 36; Scient. Papers, ii. pp. 361, 445.

The numbers express in square centimetres the sum of the diametral sections for the molecules contained in one cubic centimetre of the gas under the pressure of one atmosphere, or, more correctly, for their molecular spheres of action. What is noticeable in the table at the first glance is the large value of these numbers, which seems to be out of harmony with the assumptions made on the nature of gases. The tabulated number for air, for example, tells us that, if all the molecules contained in a cubic centimetre of air under ordinary pressure could be ranged close together in a plane, they would cover an area of 1·84 square metres with their spheres of action. This large value seems to suggest a rather dense packing of space with the air particles and the assumption that the molecules of bodies cannot be of small size.

It requires, however, but little consideration to see the error in this conclusion. The sum Q of the sections may also attain its value by reason of the largeness of the number N of the molecules, and in this case we should at once be able to conclude that the value of the section of a molecule is really small; for if we consider the number of molecules N to increase by division of the molecules, so that the section of any one molecule becomes correspondingly smaller, yet the sum of the sections will thereby increase. This is easily perceived when we recall the mathematical formulæ for the section and volume of spheres. If, for instance, a sphere is divided into two equal parts, the sum of the sections of the two smaller spheres is greater than the section of the original sphere in the ratio

2(1): 1 = 2 : 1 = 1·26 : 1.

But without mathematical calculation this is easily seen. Suppose we pound a bit of a solid substance to powder, then a larger surface can be strewn with the powder so obtained the finer the powder. We may analogously suppose the molecules of air, which in spite of their light weight can cover so much area as is given above, to form an extremely fine dust, like grains that are very small but of enormous number.

This conception reconciles in the simplest fashion the apparent contradictions between the properties of gases, and explains the striking circumstance that the phenomenathose of viscosity, for instance-remain essentially unaltered even when the gas is very greatly rarefied. For the free path, which a molecule can attain in the fine dust of molecules, remains a small magnitude under all circumstances. It is at once obvious from the numbers given that a cubic centimetre of gas of ordinary density is almost as good as impenetrable by another molecule of gas; but even if this gas has been rarefied to about a three-thousandth part of its normal pressure, and so to a pressure of about mm. of mercury, the number of molecules contained in a cubic centimetre would still yet suffice to thickly cover the six faces of the cube which they fill; this cubical space therefore seems to remain almost as impenetrable as before, and we see that the molecular free path will be still very small even now.

111. Section of Compound Molecules

If we compare the tabulated values of the sums of the sections for different gases, we easily perceive that a simple law holds in several cases. The value 25100 for hydrochloric acid is very nearly equal to 24400, which is the arithmetical mean of the values 9900 and 38800 for hydrogen and chlorine respectively. Nitric oxide, however, does not follow this rule, since its value for Q, viz. 19200, is greater than the mean (18000) of those for nitrogen and oxygen, viz. 18600 and 17400; but in this case the conformity to the law indicated might be hidden by these three numbers differing from each other by scarcely more than the possible errors of their determination. The same law appears in another similar case; thus the difference of the numbers 27000 and 18700 for CO, and CO respectively, viz. 8300, is with tolerable exactness the half of the number 17400 found for O,.

These examples seem to indicate that the section of a molecule is equal to the sum of the sections of the atoms

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