the diminution of the capillarity with rise of temperature. Mendelejeff has determined the height of this point for several liquids.

These theoretical views are in most excellent agreement with the observations made by Andrews, and also earlier by Cagniard de la Tour and Faraday. Andrews called the temperature of the absolute boiling-point, the critical point, which forms the limit between the vapour state and that of a gas proper. At a temperature below this critical point an elastic fluid can be condensed into a liquid both by pressure and by cooling, that is, by either alone, and thus deserves the name vapour. On the contrary, at a temperature above the critical point there is no pressure high enough of itself to make the fluid become liquid, and both cooling and pressure must be applied together to produce condensation; the fluid is then called a gas.

Conformably to Mendelejeff's theoretical interpretation, therefore, we must consider a gas to be a medium in which the kinetic energy of the molecules is greater than the sum total of the energy of the forces of cohesion which may come into play on condensation to a liquid. In a vapour, on the contrary, the energy of motion does not reach this amount, but it is sufficient to overcome the part of the cohesion-energy which from time to time comes into play in the to-and-fro motions of the vapour molecules.

1

A further investigation of these interesting limiting states between gas, vapour, and liquid that have been touched upon, and especially of the saturated state of vapours, will some day or other presumably form a bridge by which a passage will be found from the kinetic theory of gases to a kinetic theory of liquids-a theory the ideas underlying which have been already expressed by Clausius.

For us, whose aim is the establishment of the laws of gases, the foregoing suffices, on the one hand, to show the necessity of an improvement of our theory, which has otherwise approved itself in so many different respects, and,

[Voigt has elaborated a kinetic theory of vaporisation and of liquids in Gött. Nachr. 1896, p. 341; 1897, pp. 19, 261. See Phys. Soc. Abstracts, iii. 1897, p. 350; Science Abstracts, i. 1898, p. 545.—TR.]

on the other, to point out the means by which the defects that have clung to it so far may be removed. The chief ground of these defects we shall have to seek in our having hitherto taken no account of the cohesion of gases.

43. Rankine's and Recknagel's Modification of Boyle's Law

An attempt to carry the kinetic theory further in this direction, and to find the correction of Boyle's law that is necessitated by cohesion, has been already made by Rankine1 and by Recknagel' with happy results. Recknagel allowed for the influence of the cohesion of the gas, in the calculation of the pressure exerted by it, by assuming at every encounter between two molecules a temporary retardation of their rectilinear motions-what, in fact, might be the simplest way of taking the influence of the curvature of the paths into account. Joule's calculation of the pressure given in § 11 will hereby be so far altered that the number of collisions of a particle in unit time against the wall will be diminished by an amount which increases with the number of collisions made by the particle with other particles. We may, therefore, assume this diminution to be directly proportional to the density of the gas, or inversely proportional to its volume, and thereby obtain for the pressure, the value of which is proportional to this number, not Boyle's law as before, but a more general formula of the shape

where A and B are magnitudes depending on the temperature only; and according to Rankine A is directly and B inversely proportional to the absolute temperature.

This formula agrees well with the experimental results obtained in 1862 by Lord Kelvin and Joule3 on the cooling that accompanies the expansion of gases. Recknagel also finds that the formula represents Regnault's Note in a Memoir by Thomson and Joule, Phil. Trans. cxliv. 1854.

P. 336.

Pogg. Ann. Erg.-Bd. v. 1871, p. 563.

3 Phil. Trans. clii. 1862, p. 588.

observations on the compressibility of carbonic acid with great exactness; since it gives a maximum value for p, this occurring for v = 2B, it suffices also to represent Regnault's determinations of the maximum pressure of the gas as a function of the temperature. It thereby appears -what deserves to be mentioned as especially importantthat the saturated vapour of carbonic acid has the same thermal coefficient of expansion as the permanent gases.

This is, however, not an incontestable proof of the exactness of the hypothesis from which the formula was deduced. For exactly the same formula comes also from the different assumption that in condensable gases a part of the molecules are bound together in pairs, since with this hypothesis we have to take into account a diminution of the number of colliding molecules, which diminution is to be assumed the greater the oftener the molecules collide together, that is, the more particles there are in unit volume.

44. Hirn's and van der Waals's Correction of Boyle's Law

For the same reason a more general theory, which we owe to Hirn1 and to van der Waals,' leads also to the same result, without its appearing necessary to specialise the hypothesis so exactly.

This theory not only considers the force of cohesion, which alone up to the present has been mentioned as a cause of the deviations from Boyle's law, but also takes into account, as a second cause, the circumstance that has been mentioned before (§ 8), viz. that the dimensions of the molecules are of disturbing influence on the exactness of the law. This necessitates a twofold change in the theoretical formula

p = {NmG2,

Théorie Mécanique de la Chaleur, ii. 1864, p. 215; Ann. Chim. Phys. [4] xi. 1867, p. 47.

2 Over de continuïteit van den gas- en vloeistof-toestand,' Academisch Proefschrift, Leiden 1873. Translated into German by F. Roth, Leipsig 1881 [and thence into English for the Physical Society of London]. Abstracted in Pogg. Beibl. i. 1877, p. 10.

if we introduce the number n of molecules in the volume. v instead of the number N contained in unit volume.

In its new shape the formula shows itself as an application of the theorem of the conservation of energy to the system of n molecules, by stating that the energy of motion represented by the term on the left-hand side of the equation finds its equivalent in the volume v being filled at the pressure p. If the medium has internal cohesion, this pressure is not the only equivalent, but there is another pressure arising from the cohesion that must be added to p in the preceding equation.

On the other hand, the volume v of the gas which occurs in the equation must be diminished if the molecules occupy space; for the molecules fly about here and there, not in the whole of the space filled with the gas, but in that part which is left free between them. The number of collisions, therefore, depends only on the extension of this space that is left free, and not on the whole of the volume filled. The intensity, therefore, of the pressure exerted will also be determined only by this smaller volume.

The theoretical formula consequently needs correction on both grounds, and we must put

\nmG2 = $(p + C) (v — b),

where denotes the pressure arising from the cohesion, and b the space by which the volume v has to be diminished.

Of the two new magnitudes introduced, the latter b is made up of the sum of the volumes which are so filled by the n molecules that no other molecule could force its way into any one of them. Possibly the volume filled by a molecule in this sense is actually determined by its own. extension in space; a different assumption, however, is quite possible. Physicists who believe in a luminiferous ether, which is different from ponderable matter, would be disposed to consider the volume of the ether atmospheres condensed about the atoms rather than that of the molecules. The assumption of Clausius, Maxwell, and others is also

permissible, namely, that the forces acting between the molecules drive two encountering molecules away from each other even before the moment of an actual contact. We should then have to consider not the actual space occupied by the molecules, but the sum of larger spaces which surround the molecules; and since we might picture these envelopes as spherical, we might justify the name molecular sphere, which we will retain until in our investigation of the free path (§ 63) we introduce the term sphere of action, used by Clausius, for a sphere with a similar meaning.

One is tempted to take the magnitude b that occurs in the formula simply as the sum of the molecular spheres; but this conclusion could be unhesitatingly pronounced right only if the molecules could be supposed at rest. But as they move about they mutually obstruct each other by their motion in greater proportion than if they were partly at rest; it consequently follows that we shall have to understand by b a multiple of the sum of the molecular spheres. The more exact determination of this we shall leave for Part III. (§117); at present the remark is sufficient that, excepting perhaps the most extreme cases, we have to represent by ba magnitude which, as well as the molecular sphere, is independent of the pressure and volume.

The second magnitude & contained in the corrected formula, viz. the pressure which results from the forces of cohesion, is determined by van der Waals in the same way as Laplace calculated, in his theory of capillarity, a magnitude of similar meaning, which he denoted by K, viz. the pressure against a flat bounding surface. Since each of these pressures, both 6 and K, arises from the mutual actions of attracting and attracted particles, it is proportional to the number of attracting particles on the one hand, and of attracted particles on the other; and it is consequently proportional to the square of the number of particles present, and thus increases in proportion to the square of the density. If we refer all magnitudes varying with the expansion of the gas to the volume, as in the last formula, and not to the density, we have to introduce as a magni