= contained in unit volume which is given by Nλ3 1, we find that the molecular path-volume is determined by the expressions +πs2L = λ3/4√/2 = 1/4√/2N, which show that its value is the same for all gases, since, according to Avogadro's law, the number N has the same value for all kinds of gas. 70. Frequency of Collisions From the value of the free path and of the known magnitude of the molecular speed we can without difficulty determine the frequency of the collision of any particle with others, and the time which on an average elapses between two successive collisions. We need only remember that the speed is simply measured by the length of path traversed in the unit of time. If now a particle traverses on an average in unit time the path, which in general is zigzag-shaped, and between two successive collisions passes over the average length L in a straight line, the time required on an average for the particle to move over the length L is T = L/Q. From this interval between successive collisions we obtain the frequency of collision or the number of collisions that a particle undergoes in unit time, viz. 1/T= Q/L. If we put for the speed in these formula the arithmetic mean calculated by Maxwell's theory, we must also put for the free path L the value L = λ3 | πs2√√2, as calculated (§ 68) on the same theory. If, on the contrary, in accordance with Clausius' theory, we assign to all the molecules the same mean energy and the same value G of the speed which corresponds to it, the interval between successive collisions in this case would be T' = L'|G, where L' is the value of the free path as deduced in § 67 on this assumption, viz. and where G = √(3π, 8) as in § 27. Thus between these two values there is the ratio T': T√3: √π = [or very nearly as 43: 44]; the interval between successive collisions is thus somewhat smaller, and the collisionfrequency a little larger on Clausius' theory than on Maxwell's. 71. Relations of the Free Path to the Pressure and Temperature According to the theoretical formula we have found, the value of the molecular free path depends only on the volume 3 of the elemental cube and the area s2 of the central section of the molecular sphere of action; the molecular speed , by which the value of the temperature of the gas is determined, does not, however, occur in the formula. Of these two magnitudes the elemental cube denotes the small volume in which, on the average, each single molecule only is contained. The size of this space is not altered by mere addition of heat, but can only be altered by the volume of the gas becoming greater or less; it is proportional to this volume, and therefore varies inversely as the density, but is independent of the temperature of the gas. If now the size of the sphere of action were not variable with either the pressure or the temperature of the gas, it would follow that the molecular free path cannot depend on the temperature, but only on the density of the gas; and, indeed, must decrease or increase inversely proportionally to the density, and therefore, if the temperature remains constant, inversely proportionally to the pressure, by reason of Boyle's law. The assumption that the sphere of action of a molecule is actually altered neither by pressure nor by heat has much to entice us; for we have become used to consider a molecule as an aggregate of atoms which can certainly be altered by chemical transformations, but not by processes which belong to the narrower region of physics. If this doctrine, which in former days ruled without question, were true, the size of the molecular sphere of action could not be conditioned by the temperature or the pressure of the gas. But this view is contested by abundant observations and especially by the phenomena of dissociation. Numerous instances of gases and vapours can be cited wherein the molecules are composed of more atoms at lower temperatures than at higher. The vapours and many gases deviate at low temperatures from the laws of the ideal state of gas, especially from those of Boyle and Gay-Lussac, as has been already described in Chapter IV., these deviations being such that the gases have too great a density and an expansibility which is much greater than that of ideal gases. These and many other irregularities force us to the conclusion that the molecules of those gases form bigger aggregates of molecules at lower temperatures than at higher. By increment of heat the molecules break up into smaller ones, and therefore the mass of the molecule is decreased by rise of temperature; consequently their extension in space, and therewith the size of their sphere of action, will both become smaller when the temperature rises. From this we should expect that, by reason of dissociation, the molecular free path for vapours and non-perfect gases increases as the temperature rises. Moreover, even for gases which undergo no dissociation of their molecules, it is possible to suppose that a diminution of the molecular sphere of action may occur and demand explanation. We have only to remember that the sphere of action need not denote the space which the molecule itself occupies or claims for itself; but its radius is the least distance to which the centres, or, more generally, the centres of gravity, of two molecules can approach each other during a collision. To assume that this distance is smaller at higher temperatures than at lower entails nothing that is at all contrary to either reason or probability. For the speed of the particles increases with the temperature, and, therefore, also the intensity of the stress during collision, and it is easily conceivable that the more strongly the particles collide, the nearer they approach each other. We might perhaps suppose that the molecules become looser in their joints during rise of temperature-this is a safe assumption at least with compounds-so that one penetrates into another the more easily and deeply the warmer they are. Or we might assume with Stefan 2 that the molecules are surrounded by atmospheres of ether which, like elastic bodies, are compressed the more during a collision the more intense the blow. We could, finally, share Maxwell's view, according to which two molecules that collide are repelled from each other because, when they approach very near together, they act on each other with repulsive forces. On all these different hypotheses the particles must come the nearer together the greater their relative velocity; or, in other words, the sphere of action is the smaller the higher the temperature, and we ought therefore to expect that the molecular free path increases with rising temperature. This has in fact been proved, as will be later shown, by measurements on viscosity and allied phenomena. W. Sutherland has attempted to give an essentially different explanation of these facts. He assumes forces between the gaseous particles, when very near together, which are not repulsive, as Maxwell takes them, but, on the contrary, attractive; his supposition agrees, therefore, the best with the known observations which Joule and Lord Kelvin made on the heat-phenomena of certain gases streaming from a holder. These attractive forces do Pogg. Ann. 1873, cxlviii. p. 233. 2 Wiener Sitzungsber. 1872, lxv. Abth. 2, p. 339. 3 Phil. Trans. 1866, clvi. p. 257; 1867, clvii. p. 51. Phil. Mag. 1868 [4] xxxv. p. 133. Scientific Papers, ii. pp. 11, 29. Phil. Mag. 1893 [5] xxxvi. p. 507. 5 Phil. Trans. 1853, cxliii. p. 357; 1854, cxliv. p. 321; 1860, cl. p. 325; 1862, clii. p. 579. not remain without effect on the molecular free path; for by such forces as cause approach the probability of a collision is increased, and the mean probable value of the molecular free path is therefore diminished. The process by which Sutherland calculates the amount of diminution of the free path is given in § 35*; I prefer here another way of attaining this object without much calculation. Whether the attractions will bring about an encounter of two particles that pass close by each other, or not, depends on the amount of the two kinds of energy, one of which furthers the encounter, while the other hinders it. While the kinetic energy which the particles possess by reason of their speed, as they rush close by each other, opposes a deviation from the rectilinear path, and, therefore, also the probability of an encounter, the potential energy of the attractive forces, on the contrary, has the effect of promoting the encounter. The number of collisions will therefore be the more increased by the molecular energy the greater the amount E of potential energy which comes into activity on the approach of one particle from an infinite distance to entrance into the sphere of action of another; but this increase will be so much the smaller the greater the kinetic energy of the particles. Hence we assume that the number of encounters which a particle undergoes in unit time, by reason of the attractive forces, is increased by a magnitude which is proportional to the given potential energy E, and, on the contrary, is inversely proportional to the mean kinetic energy of the gaseous molecules, and thus inversely proportional to the magnitude mG2, in which m is the molecular weight of the gas and G represents Clausius' mean value of the molecular speed (§ 27). According to a formula of § 70, the number of encounters in unit time without reference to the molecular attraction is 1/T= Q/L = √/2πs2N/X3 ; |