Here, too, the deviation is the greater the higher the molecular weight and the greater the number of atoms in the molecule.

Winkelmann's diffusion experiments on a series of esters afford a rich material from which we may draw an answer to the question in hand. They give:

As before, the column of numbers marked 'observed' are deduced from Winkelmann's experiments on diffusion, and are taken from Landolt and Börnstein's

Another observation gives 87500.

2 Another observation gives 88400.

tables, while those marked 'calculated' are obtained from the chemical formulæ. There is in general a fairly good agreement between the two columns, which, as in the former cases, is the less the greater the number of atoms combined in the molecule. But differently from what appears in respect to the three other series, the values of the section as calculated from the chemical formulæ are greater than those deduced from the observations, whereas in the other cases it is the observed numbers which mostly are the greater. We shall, therefore, feel inclined to look for the cause of the deviations in the inexactness of the values as deduced from the observations on viscosity or diffusion.

In the foregoing I have taken account of all the values given by Landolt and Börnstein which are referred to the temperature 0° C. I have left out only those values that are given for very much higher temperatures; these cannot be brought into agreement with those calculated from the chemical formulæ, and are mostly much the greater. From this we may conclude that the section of compound molecules is very variable with the temperature, and, as we might expect, increases considerably as the temperature rises.

We can, consequently, expect agreement between theory and observation only when all the numbers are reduced to the same temperature. And so good an agreement is exhibited by the great majority of the values at 0° for gases and vapours that we have to conclude in general that their molecules have a shape that is flat, and not spread out on all sides into space. This view seems to be the most probable, at least for the gaseous state.

113. Molecular Volumes

If the molecules were extended in space on all sides they would behave very nearly as if they were spheres; and no further justification would be needed for looking upon the envelopes, which surround them in such wise

that no other molecule can penetrate with its centroid into them, as real spheres and calling them their spheres of action. But, after the foregoing explanations, we must hesitate to believe in the spherical form of the gaseous molecules and, perhaps too, in the spherical form of their spheres of action.

Against this, however, it may be argued that the flat discs which we call molecules are not at rest, but are conceived as being continually in motion; and since, too, they are continuously turning round, they must exert their actions equally in all directions of space, and we should thereby be justified not only in calling the regions within which their action is sensible their spheres of action, but also in looking upon them as veritable spheres.

But we have to consider that the surface conceived to be constructed about a molecule obtains a somewhat different signification when it is assumed to be spherical. The sphere of action has been enlarged to occupy a greater space, which we may call the molecular volume; for we may very well so term that volume which a molecule at least requires for itself. If the molecule were at rest, this space would be the sphere of action, or that volume into which the forces exerted by the molecule would not allow another to penetrate; but the molecule is in motion, and requires, therefore, a greater space. This will be smallest when the molecule has no forward velocity and executes only rotatory motions; the rotation of the sphere of action then gives rise to the molecular volume, or the space from which the molecular forces strive to drive intruders now this way and now that. The molecular volume is therefore the smallest space required by the molecule in case it is not quite at rest, or, in other words, robbed of its heat.

What I have here called the molecular volume is not essentially different from that which for many reasons has been denoted by this term in theoretical chemistry. As is well known, chemists call the molecular or specific volume the volume measured in cubic centimetres of a mass which in grams is numerically specified by the same number as the molecular weight. It is therein assumed that the sub

stance is in the liquid state, and consequently in a state that is marked by its very slight compressibility. We may therefore assume that the substances in the liquid state have attained nearly the smallest volumes to which they can be compressed. But the specific volume of a molecule in the liquid state is then exactly the same as that other volume which encloses the sphere of action, and which we have also denominated the molecular volume. The single difference that can still exist is due to the choice of the units in which the numbers are expressed; but this difference also comes to nothing if we content ourselves with relative values and do not strive after a knowledge of the molecular volumes in absolute measure.

When molecular volumes were calculated for the liquid state from the molecular weights and the specific gravities, simple relations were found between the calculated values and the chemical composition of the substances. Kopp,1 Schröder, and others were led to propose empirical laws, from which the molecular volume of a liquid compound can be calculated by simple addition of the values of the specific volumes of its components.

Loschmidt and Lothar Meyer3 found similar and just as simple relations when they attempted to estimate the molecular volumes of gases. For this purpose they started from a knowledge of the molecular free paths and of the diametral sections as deduced on the kinetic theory from observations on the diffusion and viscosity of gases. In order to estimate the size of the molecular volume from the section of the sphere of action they neglected the distinction between the sphere of action and the molecular volume, and therefore took the sphere of action as actually spherical.

With this assumption it is very easily possible to compare the volumes of the spheres of action or the molecular volumes

1 Ann. Chem. Pharm. 1855, xcvi. pp. 1, 153, 303; 1856, c. p. 19.

2 Wien. Sitzungsber. 1865, lii. Abth. 2, p. 395.

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

for different gases; for this purpose the theoretical formula for the coefficient of viscosity (§§ 76, 27),

n = mG/4πs2 = √(3π/8) mN/4πs2,

or, more strictly by § 78,

η = 0-30967 m/s2

is of service, in which, as before, m denotes the molecular weight and the mean molecular speed. For two different gases, which we distinguish by the subscripts 1 and 2, we then obtain the ratio

From this formula we see that the ratio of the sizes of the spheres of action of two gases, viz.

V1/V2 = (m/m) 3 (n2/n1) 3,

can be determined from the molecular weights and the coefficients of viscosity.

In order to be able to compare the values of the molecular volume calculated by this formula with those given by Kopp, the molecular volume V, for any normal gas, chosen arbitrarily, with which the others are compared must be put equal to the value found by Kopp. For this purpose Lothar Meyer employed sulphurous acid, because its specific volume seemed to be determined with greater certainty than that of any of the other gases whose viscosity had been accurately measured by Graham. In this way he, and likewise Loschmidt, obtained values for the molecular volume which in many cases agreed really well with those calculated by Kopp from the density of the liquid.

But before I can tabulate the results I must mention a striking circumstance which would be well suited to raise objections against the accuracy of the calculation. Such