hindered by a partition. The explanation of the extreme slowness of this free diffusion between gases is now the special problem before us.1 Passing over the older observations of Dalton,2 Berthollet, Graham, and others, I choose, as instances, the experiments carried out by Loschmidt on the diffusion of gases. 5 A glass tube, 975 mm. long and of 26 mm. diameter, which can be closed at both ends by flat glass plates and glass stopcocks, was cut into two parts of equal lengths, which could at will be shut off from each other or put into connection by means of a slider made of a thin sheet of steel in which was a hole as large as the inner section of the tube. After the two halves had been filled with different gases under the same pressure, the slider was opened so that the gases began to mix, and after the lapse of a measured time-from half an hour to an hour-the slider was again closed, and the gas in each half of the tube was then analysed in order to determine the degree to which the mixing had progressed during the time of the experiment, and from this to discover the speed of diffusion. 6 A theory developed by Stefan was employed for the reduction of these experiments. This starts with the assumption, made also by Maxwell' and Boltzmann,8 We must not conclude that, because Thos. Graham so judiciously distinguished these differing phenomena by different names, he was the first to employ these names. The term diffusion was used by Dalton (On the Tendency of Elastic Fluids to Diffuse through Each Other,' Mem. Manch. Soc. 1805, new series, i. p. 244) and Priestley (Trans. Amer. Phil. Soc. 1802, v. p. 15; 'Experiments and Observations relating to Various Branches of Natural Philosophy,' Birmingham 1781, iii. sect. 27, § 3, p. 390. 2 In different places; Gilb. Ann. 1807, xxvii. p. 388. 3 Mém. d'Arcueil, 1809, ii. p. 463. Quarterly Journ. of Sc. Lit. and Art, 1829, p. 74; Pogg. Ann. 1829, xvii. p. 375. 5 Experimentaluntersuchungen über die Diffusion von Gasen ohne poröse Scheidewände,' Wien. Sitzungsber. Abth. 2, 1870, lxi. p. 367; lxii. p. 468; continued by Wretschko, lxii. p. 575; and by Benigar, lxii. p. 687. • Wiener Sitzungsber. Abth. 2, 1871, Ixiii. p. 63; 1872, lxv. p. 323. Phil. Mag. [4] 1860, xx. p. 21; 1868, xxxv. p. 199. Scientific Papers, i. p. 392; ii. p. 57. 8 Weitere Studien u.s.w.' Wien. Sitzungsber. Abth. 2, 1872, lxvi. p. 324; 1878, lxxviii. p. 733; 1882, lxxxvi. p. 63; 1883, lxxxviii. p. 835. of the idea of a resistance to diffusion which each gas experiences from the particles of the other gas which meet it. This resistance is taken to be proportional to the densities of both gases and to the difference of their speeds of diffusion. The working out of this idea leads to formulæ which have a great likeness to the equations which come into Fourier's theory of the conduction of heat. This similarity is not only in respect of the mathematical form, but is founded on the nature of the matter. Just as heat spreads in a conducting body, so in diffusion a gas spreads from one region to another. The speed with which heat is transmitted is determined for each substance by a constant which is termed the conductivity; in like manner the speed with which one gas penetrates into another is determined by a magnitude which we might call the diffusivity, but which is more usually termed the coefficient, or constant, of diffusion. The meaning of these two constants is quite analogous. We obtain the strength of the flow of heat by multiplying by the conductivity the difference of the temperatures at two places, which are distant from each other by unit length along the line of flow, i.e. the so-called rate of fall of temperature. We likewise obtain the intensity of the flow of diffusion if we multiply by the coefficient of diffusion the difference of the density of the diffusing gas at two places whose distance apart is equal to unit length. But we can express the meaning of this coefficient also in a somewhat different way by replacing the density by the pressure which, by Boyle's law, is proportional to it. We may then say that the amount of partial pressure of one of the gases transmitted by the diffusion is given by the difference in the values of this partial pressure at two places which are separated by unit length (or, as we may say more shortly, by the rate of fall of the partial pressure), multiplied by the coefficient of diffusion. From this determination of the flow of diffusion it is easy to see that the coefficient, which holds for the diffusion of one gas into any other, must be equal to that upon which the diffusion of the second gas into the first depends. For since no inequality in the total pressure is brought about by diffusion in an experiment wherein the initial pressures of the two diffusing gases are equal to each other, the strengths of the two flows which occur in opposite directions, but otherwise under the same circumstances, must be equal, and, therefore, also the two coefficients must be equal. Loschmidt's observations showed that the value of this coefficient is inversely proportional to the total pressure of the gases, so that, as is almost obvious, the mixing occurs the more rapidly the more rarefied the gases. Loschmidt found further that the coefficient alters with the temperature in accordance with the law, that it increases nearly proportionally to the square of the absolute temperature. Finally it appeared that a regular relation must exist with the molecular weights, which, however, did not succeed in disclosing itself with full clearness. We have to develop the value of this coefficient from the conceptions of the kinetic theory. 94. General Theory of Diffusion The slowness with which two diffusing gases mix together is to be explained on the kinetic hypothesis in a way which is so fully analogous to the conceptions underlying the theory of friction that we may speak of diffusion as a kind of reverse friction of the two gases on each other. The cause of the phenomenon, that the forward flow of one layer is transmitted only very slowly by friction to a distant layer, lies only in the shortness of the molecular path. With all their swiftness of motion, the molecules transmit a part of their own momentum only to those particles that are quite close to them and with which they collide after traversing a very short path. A transference of momentum to a greater distance takes place, therefore, only by the interaction of very many particles in their to-and-fro motions; it is, consequently, carried on by no means in a straight direction, and experiences, therefore, a considerable retardation, which appears to us as a consequence of friction. The proof that the shortness of the molecular paths must cause an equal slowness in the mixing or diffusion of two gases may be deduced in an exactly corresponding way. This is just the point on which Clausius' laid special stress in his celebrated memoir on the free paths of molecules. Into a space already filled with one gas the molecules of a second gas penetrate only slowly, because by their frequent collisions with the molecules of the first gas they are driven hither and thither, so that it is impossible for them, even with their enormous speed, to penetrate into the interior of the other mass in straight lines. The velocity of diffusion therefore depends not only on the speeds with which the molecules of the diffusing gases move, but also no less on the length of the free path which a molecule of one sort traverses among molecules of the other sort. It is thus conditioned by the same elements as the friction of gases, viz. molecular speed and free path. A closer examination of the way in which these twoelements are connected together leads to a knowledge of the value of the coefficient of diffusion and teaches us to see its meaning and its relations to other magnitudes. 95. Theoretical Value of the Coefficient of At each moment during the interval occupied by an experiment the diffusing gases may be in such a state of motion that the equilibrium of the total pressure exerted by the mixture is nowhere disturbed; there is then at every point the same pressure p, and this is the sum of the partial pressures Pi and P2 which each of the two gases would exert by itself, or Similarly, by Avogadro's law, there is the same number of molecules in each unit volume, which we will represent by N = N1+N2, 1 Pogg. Ann. 1858, cv. p. 239; Abhandlungen, Abth. 2, 1867, p. 260; Phil. Mag. 1858 [4] xvii. p. 81. where N, and N, are put for the number of molecules of the first and second sort in the unit volume. But the mixture is not homogeneous; the nearer to one end of the tube, the more in excess are the molecules of one kind, just as those of the other kind are at the opposite end. The pressure and density of one gas diminish along the tube in one direction, while those of the other gas increase in the same direction in equal measure. If the experiment has already lasted some time, the diminution of pressure all along the tube will have become uniform, so that we can represent the pressure and the number of particles of the one gas at the distance x from the junction of the diffusiontubes by formulæ of the form P1 = P+ px, N1 = N + nx, while the same magnitudes for the other gas are P2P Pрx, N2 = N N — nx. Here the magnitude p which determines the increase and decrease of the partial pressures is the same for our problem as what is called in hydraulics the slope of pressure, or the diminution of pressure in unit length along a line of flow. The analogous magnitude n determines the decrease or increase of density along the same length; it will result, in agreement with the views mentioned in § 93, that the strength of the diffusion-flow is proportional to it. If such a uniform distribution of pressure and density has not yet been established along the whole tube, the foregoing simple formulæ can still be employed without error, if we use them for only a very short portion of the tube, and therefore, for instance, if by x we understand a length which is shorter than a molecular free path, as in the following calculation. We have to determine how many particles of each kind cross any section of the tube in a given time in consequence of the inequality of the pressure and density that has been described, or, more correctly, how many more cross in one direction than in the other. If the distribution were uniform, the number of particles which in unit time meet |