in which A, B, n are constants. Similarly Battelli1 put

so that he had one more constant at his command. G. Jäger,2 on the contrary, added to a the factor

These formulæ exhibit much better agreement with experiment than the simpler formula of van der Waals, as is to be expected from the greater number of disposable constants. Yet, as Korteweg3 has remarked, the formula of Clausius deviates from the observed behaviour of gases in many other regards more largely than that of van der Waals. For our theory the more complicated formulæ are less valuable than the original simpler one on account of the difficulty of their interpretation.

Amagat has completed van der Waals's equation by giving it the form

{p + Av ̃3(ve) } { v − b + B (v — b)"}

=

R(1 + ad), which contains five constants, A, B, e, b, n. This formula has proved itself good in a comparison with the observed behaviour of hydrogen.

Boltzmann and Mache 5 assume the formula

48. The Cohesion of Gases

If it can appear scarcely doubtful that the defects of the theory, even after the corrections just applied, depend on the cohesion having been insufficiently treated, there may yet arise doubts as to the mode in which a strict theory

1 Memorie di Torino [2] xliv. 1893, p. 27.

2 Wiener Ber. ci. 1892, p. 1675.

3 Wied. Ann. xii. 1881, p. 143.

Comptes rendus, cxviii. 1894, p. 566.

[He has also found (Comptes

rendus, cxxviii. 1899, p. 538) that the behaviour of CO, in a very wide range of pressure and temperature is well represented by a formula of the type

P+

v − {a + m(v − b) + c(v − b)-'}(1 + að)

kvn - a + √ {(v — B)2 + d2}

5 Wiener Sitzungsanzeiger, 1899, p. 87.

would have to take the influence of this force into account, and as to the change in our views regarding molecules and their motion that are demanded by reason of the cohesion. There are, in fact, two essentially different explanations of the deviations which gases and vapours exhibit from the theoretical laws, and yet both explanations arise at bottom from the same origin.

In order to understand this double possibility we need only remember that the foundation of our theory contains also several unproved and unprovable assumptions, of which only two come here into account. The first is the assumption that gases consist of molecules of invariable mass, the second is the hypothesis that they move in straight lines.

Of these two hypotheses the latter at first sight seems the more doubtful; for in any case it is only true with the limitation or exception that at the moment of a collision the motion, till then along a straight line, must experience a sudden change of direction. But the former, too, as we shall see, is not above doubt, and an inexactness in this hypothesis might just as well cause the deviations as an error in the second of the two hypotheses.

We can frame for ourselves no idea of the cohesion of gases that is essentially different from that of liquids; we imagine, therefore, forces which act attractively from particle to particle in the direction of the line joining them, and whose strength falls off very quickly as the distance increases, so that at a finite or measurable distance the force is infinitely small. It would be as difficult to oppose this customary supposition regarding the nature of cohesion as to contest the essential part of the kinetic theory of gases if we were to ascribe to the forces of cohesion, for a distance of the attracting particles from each other equal to the mean distance apart of those which were the nearest neighbours, a sensible value which comes somehow into consideration. We shall therefore have to assume that attractive forces of any importance are active between two particles only when they actually collide or just graze each other in their paths.

If we assume this idea of the action of the force of

cohesion, we do not on this account need to drop the hypothesis of rectilinear motion, in which the most essential and characteristic peculiarity of Bernoulli's theory consists. It only becomes necessary to modify the hypothesis so that the changes of direction from one straight path to another are not caused suddenly by a collision, but gradually by forces which act continuously, even if they very quickly come into and go out of play. The paths then do not form sharp-angled zigzag lines, but the passage from one straight line to another is brought about by lines that are sharply but continuously curved so that the corners seem to be rounded off. In the next paragraph we shall have to discuss, at least in its general character, the influence which this modification of the hypothesis introduces into the calculation of the pressure.

Still, an effect of quite a different kind is conceivable with the same hypothesis as to the nature of cohesion.' If we assume, with regard to this force, that it acts only in the proportionately rare moments of an actual or very nearly occurring collision between two molecules, the fact established by Joule and Lord Kelvin, that the intensity of the cohesion in gases is very small, will be simply explained by the force acting only during the short time of the collision and being in abeyance during the much longer interval between successive collisions. There would therefore be no contradiction with the observations mentioned if we assume that in the short periods of collision the force acts with very considerable intensity.

But if this assumption is admissible there is nothing inconsistent in the hypothesis that the attractive forces of cohesion might be able, at least now and then under favourable circumstances, to bind together two colliding particles so fast that they traverse the next stretch of their path together as a double molecule. By this a state of equilibrium would be produced in the gas in which, among the

In his memoir on 'Temperature and the Measure of Temperature' (Pogg. Ann. Erg.-Bd. vi. 1874, p. 275) Recknagel also considers effects of two kindsattraction of molecule on molecule and actions within molecules. The latter are perhaps to be interpreted in the manner explained later.

molecules of the same kind, there would be always some of greater mass. The number of the latter would depend on the frequency of the favourable cases of collision, and therefore, also, chiefly on the number of collisions that occur, so that in a denser gas wherein the molecules collide more frequently there will be also more molecules of greater mass.

49. Playfair and Wanklyn's Explanation of the Anomalies

The hypothesis last expounded forms the basis of the explanation of the anomalies regarding vapour-densities that has been given by Playfair and Wanklyn,' an explanation which embraces all the other deviations of actual gases from the theoretical laws.

According to what has already been said regarding the hypothesis, it is only necessary to remember the mechanical definition of temperature, given before in § 14, to see at once the possibility of the explanation of all anomalies. According to § 29, the mean value of the kinetic energy of a molecule, even when of different kinds, forms the measure of temperature. Consequently, in a gas whose molecules are either wholly or in part bound together to form larger aggregates, the kinetic energy contained in unit volume is less than before the aggregation; or, more simply expressed, the pressure is lowered by the aggregation of molecules if the rise of temperature resulting therefrom is compensated. Hence a gas whose molecules may combine together may be more easily and strongly compressed than an ideal gas whose molecules are unalterable massive points. A deviation from Boyle's law will therefore arise in the direction shown by most of the gases included in the table of § 7, with the single exception of hydrogen, the behaviour of which has been already described elsewhere; and determinations of vapour-density will therefore give higher values the higher the pressure during the measurement, as was found, for instance, by Alex. Naumann to be the case for acetic acid.

1 Trans. Roy. Soc. Edin. xxii. pt. 3, 1861, p. 441; Ann. Chem. Pharm. cxxii. 1862, p. 247.

The fact that the vapour-density is smaller at higher temperatures and larger at lower temperatures is explained in the same way; for since the heat-motion loosens the bond between the molecules the molecules will be lighter, and therefore the gas or vapour will be specifically lighter at the higher than at the lower temperatures.

The behaviour of the thermal expansion-coefficients is also directly explained. At lower temperatures not only do the gaseous molecules separate more widely from each other on the addition of heat, but they also split up and require greater space for their greater number. At higher temperatures, at which all the molecules have been already split up, heating brings about merely an increase of speed as in ideal gases; all vapours and condensable gases must therefore at high temperatures attain the same thermal coefficients of expansion as the so-called permanent gases, while at lower temperatures they expand more largely.

If this explanation of the deviations is really true, a conclusion already drawn by Regnault' from his observations on the compressibility of gases must be unconditionally considered as correct. If molecules that are bound together are more and more separated by rise of the temperature, there must be a temperature at which all move singly and no further separation is possible; at this temperature the ground in question of the anomalies would fail, so that the only cause of an anomaly that would remain is the circumstance that the dimensions of the molecules in comparison with their distances apart need not be vanishingly small, a cause therefore which, as in hydrogen, would entail a deviation in the opposite direction. According to this theory, therefore, as Regnault has already conjectured, every gas must at a sufficiently high temperature exert a greater pressure than would be expected by Boyle's law, and a less pressure at lower temperatures, so that for every gas there will be a certain temperature at which it strictly obeys this law. Hydrogen would at very low temperatures behave just like the others.

Mém. de l'Acad. de Paris, xxi. 1847, p. 404.