find this constancy when we proceed to calculate the pathvolume for a series of gases from the foregoing numbers.

It therefore follows that we cannot hold ourselves justified in taking the calculated values of s as representing the actual diameters of the gaseous molecules. The reason that we may not do so is obvious. There can be no doubt that the molecules are not spheres in shape; for, as we concluded in § 112 from a large number of facts, they are more probably, without exception, flattish discs of very small thickness. We therefore cannot arrive at true values of their diameter and volume by looking on them as spheres. At most we may expect (§ 113) to obtain from this calculation an estimate of the volume of a larger sphere which the flat disc describes when it rotates, and the mean diameter of a molecule must be less than the diameter of this sphere. We may not, therefore, take the values calculated for s as giving the true size of the molecules, but may see in them only a superior limit which the size of the molecules does not attain.

From these considerations we can conclude only that the gaseous molecules are smaller than a sphere whose diameter is one-millionth of a millimetre. But we may add as very probable that the size of the gaseous molecules will in no way appear to be vanishingly small when compared with that small sphere. This is justified on many other grounds, which we have still to mention.

117. Calculation of the Size of Molecules from the Deviations from Boyle's Law

The above calculated numbers obtain a remarkably good confirmation from the values which we obtain for the same magnitude by a different mode of calculation first given by van der Waals.1 In the theory explained in Chapter IV. of this book, by which van der Waals sought to explain the deviations of actual gases from the Boyle-Gay Lussac law, the grounds of these deviations were found partly in

1 Over de continuïteit van den gas- en vloeistof-toestand,' Leiden 1873, transl. by Phys. Soc. London, 1890, Chap. VI. p. 384. Abstracted in Beibl. to Pogg. Ann. 1877, i. p. 10.

the cohesion of the gases and partly in the space occupied by their molecules. On the basis of this theory the values of two constants a and b, the latter of which represents a measure of the size of the molecules, could be calculated from Regnault's observations on the compressibility of gases and on their expansibility under the action of heat.

This magnitude b is directly connected with the coefficient of condensation v described in the last paragraph, and to recognise this more clearly we will seek with van der Waals to push Clausius' theory of the molecular free path a step further. The correction, which is calculated in fuller detail in § 34" of the Mathematical Appendices, results from regard being paid to the fact that a particle cannot pass over paths between other particles which are equal to the distances apart of these other particles, or, more strictly, of their centroids; for the paths cannot be greater than the length left free between the spheres of action of the particles. For this reason the estimated molecular free path, L = λ πε2,

has to be diminished by an amount which depends on the radius s of the sphere of action. This correction attains its greatest value when the collision is direct and central, in which case the paths of both colliding molecules are together shortened by the radius s. On the average its value is smaller, and equal to

(√2/3) s,

so that the free path would, strictly speaking, be represented by the formula

From this we see that the so-called elemental cube X3, in which a single molecule is contained, is diminished in the corrected formula by

that is, by four times the volume of the molecular sphere. From this remark we at once obtain the meaning of the constant b which comes into van der Waals's theory, since on this theory a similar correction was introduced by putting

the smaller volume vb for the larger volume v Nλ3 of the whole mass consisting of N molecules. The number to be subtracted has thus the meaning

that is, it is equal to four times the volume actually occupied by the whole of the molecules that are contained in unit volume.

Putting in this equation the value of the free path given by the former formula 1/2 TS2NL,

which may without hesitation be here employed without correction, we obtain

bL =

(√2/3)s,

so that we can calculate the molecular diameters from the known values of b and L. In this calculation we have still one precaution to take; for b and L are both dependent on the pressure, b being proportional to the number N and therefore to the pressure, and L being inversely proportional to these magnitudes. The values given in §§ 79 and 116 for the free path have reference to the pressure of one atmosphere, while the values of b calculated by van der Waals and others from his theory presuppose, at least for the greatest part, the pressure of 1 metre of mercury; to compensate for this difference we must multiply the formula for the calculation of the radius of the sphere of action by the ratio of the pressures, and thus put

where b is referred to the pressure of 1 m. of mercury, and p denotes the pressure in metres of mercury for which the value of L, which is employed, holds good.

From the observations made by Regnault and Cailletet on the deviations of gases from Boyle's law, van der Waals1 has calculated the following values:

1 Continuïteit, &c. Chap. VIII. b. §§ 41, 42, pp. 67-9. Phys. Soc. Transl. pp. 400-2. F. Roth, Wied. Ann. 1880, xi. p. 25.

On combining with these the values of the free paths given in §§ 78 and 79, as obtained from Graham's experiments on transpiration, we obtain for the molecular diameters the values:

These numbers are markedly smaller than those given before, but they are of the same order of magnitude, and therefore we may see in them a confirmation of the correctness of the theoretical views from which we have started.

We should obtain a better agreement if we replaced the numerical factor of the formula, 3/2 = 2.12, by a greater value. Our determination of this factor really rests on a not entirely safe footing, and it has not always, therefore, come out the same.1 We might object that the correction, which the value of the free path needs on account of the space occupied by the spheres of action, must not be applied quite in the same way as that which we have to make to Boyle's law for the same reason. The two corrections, therefore, b and 47s3N, need not be equal to each other, but may still differ by a numerical factor; and this factor is obtained by Clausius and G. Jäger3 from the consideration that molecules which are near each other cannot be struck by another colliding particle at every point of their surface if they really occupy space; there is therefore a diminution of their surface to be taken into account in the calculation, and this consideration leads to the formula

In the first edition of this book another smaller value, 1.5, was taken; this results from assuming as strictly valid the calculation first developed in § 34* of the Mathematical Appendices. There are grounds of probability in its favour which depend on the phenomena described in § 118 (Heilborn, Exner's Repert. 1891, xxvii. p. 369; Sydney Young, 1898, Chem. News, lxxviii. p. 200), but the larger value seems to me to be theoretically better established.

2 Mech. Wärmetheorie, 1889-91, iii. pp. 57, 213; Wied. Ann. 1880, x. p. 102.

3 Wien. Sitzungsber. 1896, cv. Abth. 2, p. 97.

from which we get for the determination of s the formula S= (24/5√/2) bL.

The factor, according to these theories, becomes 24/5/2=3.39, and attains therefore a value by which a satisfactory agreement is established.

118. Calculation of the Size of the Molecules from the Dielectric Capacity

Stefan, in his memoir on the theory of the diffusion of gases, drew attention to a simple relation in which the values of the mean free paths of the gaseous molecules stand to the refractivities of the gases. He remarked that the refractive index n of a gas is the smaller the greater the free path L of its molecules, the simple law, indeed, that the product

has a nearly constant value, holding for many gases, especially for those whose properties have been investigated with the greatest exactness.

Now Maxwell's electromagnetic theory of light requires the refractive index of a substance to be equal to the square root of its dielectric capacity; and this law has been shown by Boltzmann's' experiments on seven gases to be very exactly correct. Therefore also the dielectric capacity of a gas must stand in as simple a relation to the mean free path of its molecules as its refractive power.

The surprising fact that a simple connection exists between two such different magnitudes as the dielectric capacity and the molecular free path finds its explanation in an assumption regarding the molecular qualities of dielectric bodies made by Faraday and by Mossotti. The molecules of such substances are assumed to be good conductors of electricity, while the interspaces between them are taken to be insulating. According to this assumption the dielectric polarisation must depend on the size and

Wien. Sitzungsber. 1872, lxv. Abth. 2, p. 341. Compare also Rubenson, Oefv. Kgl. Vetensk.-Akad. Forhandl. Stockholm, 1884, xli. No. 10, p. 3. 2 Pogg. Ann. 1875, clv. p. 421.