Bernoulli supposed a mass of gas to be confined in a vessel with a movable but air-tight lid, such as the cylinder of an air-pump; this gas can be compressed by increasing the pressure on the lid or piston. If, now, the gas consists of a large number of moving particles, and the pressure exerted by it on the walls of the vessel arises from the impacts of the particles against these walls, then equilibrium results when the resultant action of the impacts on the lid is equal to the pressure applied to it.

If the gas is compressed and the volume diminished, the number of impacts of the now more closely packed particles against the walls increases, and for two reasons: first, there is a larger number of particles in the layer of gas immediately adjoining the walls; and, secondly, as the particles are more crowded together, they collide oftener, and, hurled back by the collision, are oftener flung against the walls. If, by the compression of the gas, the volume is diminished in the ratio 1 : s3, the distance between any pair of particles is diminished in the ratio 1 s; the number of particles, therefore, in the bounding layer, which is in contact with a given area of the walls, is increased in the ratio s2: 1; further, the number of collisions that take place between the molecules in a given time is increased in the ratio s 1; and in this same ratio also is the number of impacts of any particle in the bounding layer against the walls increased. Since, then, the number of impinging particles is increased in the ratio s2: 1, and the number of impacts by each in the ratio s : 1, the number of impacts against a given part of the walls in a given time is increased in the ratio s3: 1, which is the inverse of the ratio in which the volume of the gas is diminished. The pressure, therefore, of a gas varies inversely as its volume.

Boyle's law is thus deduced from the hypothesis of molecular impacts.

7. The Admissibility of the Hypothesis

Since Boyle's law can be deduced also from quite different assumptions, this first consequence of the theory is

с

no proof of the exclusive claims of our kinetic hypothesis; but it allows us to judge for what substances, and under what circumstances, the theory may be considered admissible.

Boyle's law is not obeyed by all substances in the gaseous state. The vapours of liquid bodies do not obey it except within certain limits of pressure and temperature, and then only with moderate approximation. Even the so-called permanent gases do not satisfy the law rigorously and under all circumstances.

This was known to Boyle himself, and the inexactness of the law has been confirmed by Musschenbroek and a whole series of physicists, ancient and modern, such as Despretz, Arago and Dulong, Pouillet, Regnault, Siljeström, Mendelejeff and Kirpitscheff, Amagat, Cailletet.

It would be out of place here to enter fully into the results of the numerous investigations undertaken to test Boyle's law, as this work does not profess to be a complete text-book of the physics of gases, and the more detailed textbooks contain full information. A few examples are here sufficient, which show how far the real gases depart from Boyle's law under ordinary circumstances, that is, at mean temperatures and under moderate pressures.

If this law were strictly exact, the product of the pressure p into the volume v of a mass of gas would be a magnitude which would not alter in value when the pressure took the value P and the volume the corresponding value V; we should, therefore, have

The

if, as we assume, the temperature did not alter. following table contains for a series of gases the values of this ratio which Regnault2 found on increasing the pressure from p=1 atmosphere to P=2 atmospheres.

1 Winkelmann, Handbuch der Physik, i. p. 503; Ostwald, Allgemeine Chemie, 2nd ed. i. p. 139; &c.

2 Mém. de l'Acad. de Paris, xxi. 1847, p. 329; xxvi. 1862, p. 260.

This table shows that Boyle's law does not hold exactly for any gas, but that for the gases named it holds with sufficient approximation to be considered for most purposes an exact law of nature. The vapours of liquids, indeed, depart from the law more widely than gases; but from a theoretical calculation to which Clausius' has subjected Regnault's observations on saturated steam, it appears that for this vapour from 0° to 100° the values of the magnitudes, which according to the law should be constant, vary by not more than 5 per cent. at the most. Similar relations were found by Herwig for the unsaturated vapours of alcohol, chlorcform, and carbon disulphide.

2

Accordingly, therefore, the departures of even the vapours of liquid bodies from the law inferred from theory are sufficiently unimportant to be provisionally neglected when we attempt to investigate the other properties of these bodies on the basis of this theory. Strictly speaking, our further conclusions will only be rigorously true for such media as obey Boyle's law exactly; these, however, exist only in the imagination, and are therefore called ideal (or perfect) gases. But, in the main, our considerations are also true for real gases, and the variations between the theory and observation will be of no greater importance than that of the differences in the foregoing numbers.

8. Defects of the Hypothesis

We shall, however, not rest content with this approximation, but try to get nearer the truth by inquiring into the

Pogg. Ann. lxxix. 1850, p. 513, Clausius, Mechanische Wärmetheorie,

2nd and 3rd ed. i. p. 151; transl. Phil. Mag. [4] ii. 1851, p. 102.

possible causes of these variations. If Bernoulli's theory gives, as its necessary consequence, a law that is only approximately exact, the hypothesis underlying the theory cannot be quite true in every respect, but must be defective, even if only slightly.

In the assumptions with which we started there are two different points which cannot be directly proved, and are therefore open to doubt. The first is the assumption that gases are made up of molecules of very small dimensions, and the second is the assumption that in gases there is no cohesion. Neither of these is exactly true, and therefore neither can be admitted except as an approximation to the truth; and in their inexactness lies ample ground for the departures from Boyle's law.

In the first place, if the dimensions of the molecules are not indefinitely small, the calculation which led to the law is not exact. For it is only if the space actually occupied by the molecules is absolutely negligible in respect of the volume which contains them that we may justifiably conclude that the frequency of collision is increased in the ratio s : 1 by a diminution of the volume in the ratio 1: s3. If this condition is not fulfilled there is less actual distance between the molecules, which, therefore, collide the oftener with each other and in the same ratio impinge the oftener against the walls of the vessel-in other words, the pressure is greater than according to the former calculation; and as this increment in the pressure is the more considerable the less the volume, the pressure must increase at a greater rate than the volume diminishes. The denominator PV of the ratio considered in § 7, wherein P denotes the higher pressure, is on this account greater than the numerator pv, so that the ratio pv/PV has, as actually happens with hydrogen, a value less than 1.

A deviation in the reverse direction occurs when the second hypothesis is sensibly in fault and the gas has marked cohesion. For such a property will tend to lessen the volume, which will, therefore, on this ground diminish more rapidly than the pressure increases; PV will thus be smaller than pv and the ratio pv/PV greater than 1, as is

the case with all the gases in the table of §7, except hydrogen.

Probably both influences occur in nature, and the numbers in Regnault's table seem to show that in the case of most gases the influence of cohesion is predominant so long as the pressure lies within certain limits. But when higher pressures are employed all gases exhibit, according to the observations of Natterer,' Amagat,' and Cailletet, the same behaviour as under lower pressure is noted with hydrogen. The product PV increases with the pressure P, because on account of the dimensions of the molecules the volume V cannot diminish so much as the law requires.

These considerations, which we shall repeatedly have again to take up and extend, show that the departures from the strict law can also be explained by the theory. Since the probability of the theory is, therefore, in no respect prejudiced by the inexactness of Boyle's law, we are entitled to draw further inferences and conclusions, first of all, from the simple theory, and to reserve their correction for later chapters.

9. Increase of Pressure by Heat

Bernoulli also saw that his theory accounted not only for Boyle's law, but also for the observed increase in the pressure of a gas to which heat is communicated. According to the laws of thermodynamics heat is energy; an increase of heat, therefore, consists in an augmentation of the speed of the molecular motion, and this increase of speed entails increase both in the number of impacts of the molecules of the gas against the vessel and also in the strength of these impacts. For a double reason,

therefore, the resultant action of the impacts in a given

Pogg. Ann. lxii. 1844, p. 132, xciv. 1855, p. 436.

Comptes Rendus, lxxxviii. 1879, p. 336; Ann. Chim. [5] xix. 1880, p. 345, [6] xxix. 1893, p. 68.

p. 267.

C. R. lxx. 1870, p. 1131, lxxxviii. 1879, p. 61; Journ. de Phys. viii. 1879,

See Chap. IV. §§ 40-51.

C