June 28, 1900] 



NATURE 



213 



which may be quantitatively investigated by the same simple 

 mathematical treatment as the "flow " of heat in a bar when 

 the permanent state has been reached, or the " flow " of elec- 

 tricity between any two regions of a conductor maintained at a 

 constant diff^erence of potential. 



By a long series of experiments of this nature it was found 

 that the ditfusivity constant, k, for very dilute COo does not 

 materially depart from the value assigned to it by Loschmidt 

 and others, when experimenting with much higher ratios of 

 mixture, and that the difference is certainly not of sufficient 

 magnitude to be taken into serious account in the study of the 

 natural processes of gaseous exchange in the assimilating organs 

 of plants. 



In the static diffusion of a gas, vapour, or solute, as the case 

 may be, the amount of substance diffusing in a given time, all 

 other conditions being the same, is directly proportional to the 

 sectional area of the column. It is found, however, that if the 

 flow is partially obstructed by interposing at any point in the 

 line of flow a thin septum pierced with a circular aperture, the 

 rate of flow across unit area of the aperture is greater than it 

 would be across an equal area of the unobstructed cross-section 

 of the column at this point. If the margin around the aperture 

 has a width of at least three or four times its diameter, the rate 

 of flow is now found to be directly proportional to the linear 

 dimensions of the aperture and not to its area, so that the 

 velocity of flow through unit area varies inversely as the 

 diameter. 



A large number of experiments on the diffusion of carbon 

 dioxide, water-vapour and sodium chloride in solution are 

 given in support of this proposition. All these show that the 

 rate of diffusion across such a septum, all other conditions being 

 the same, is directly proportional to the diameter of the aperture, 

 and not, as might have been expected, to its area. 



Exactly the same result is obtained when small circular discs 

 of an absorbent, such as a solution of caustic alkali, are sur- 

 rounded by a wide rim and exposed to perfectly still air, the 

 amount of carbon dioxide absorbed under these conditions being 

 proportional to the diameters of the discs. 



If, however, there are any sensible air currents the absorption 

 becomes proportional to the areas. 



These two sets of phenomena may be explained as follows : — 



In the case of the absorbing disc in perfectly still air, the con- 

 vergent streams of carbon dioxide creep through the air towards 

 the absorbing disc, establishing a steady gradient af density, and 

 this creep will be a flux perpendicular to the lines of equal 

 density, which form curved surfaces or "shells" surrounding 

 the disc and terminating in the rim. The state of things is 

 exactly analogous to the electric field in the neighbourhood of a 

 conductor of the same shape and dimensions as the absorbent 

 disc.^ In the case of the gas, the curves or "shells" of equal 

 density are the analogues of the similarly curved surfaces of 

 equipotential above the electrified disc, whilst the converging 

 lines of creep or flux of the gas are the analogues of the lines 

 or tubes of force which bend round into the disc as they 

 approach it. 



If we consider two such absorbent discs of different diameters, 

 the curved surfaces in each system corresponding to a given 

 density will be found at actual distances from the discs which 

 are in the same proportion to each other as are the diameters of 

 the discs. In other words, the gradient of density on which 

 the rate of flow depends will be proportional to the diameters 

 of the discs, which is exactly what is found experimentally. 



This case of an absorbent disc is the exact converse of one 

 which has been theoretically investigated by Stefan, viz. the 

 conditions of evaporation of a liquid from a circular surface. He 

 found that the lines of flux of tlie vapour proceeding from the 

 surface of the liquid must be hyperbolas, whilst the curved 

 surfaces of equal pressure of the vapour must form an orthogonal 

 system of ellipsoids, having their foci, like the hyperbolas, in 

 the bounding edges of the disc. This was a purely mathematical 

 deduction which has never been verified experimentally, but it 

 will be seen that the exactly converse phenomena of diffusion 

 are in complete agreement with it. 



In the other case of a diffusive flow through a circular aper- 

 ture in a diaphragm, the lines of flow, which are convergent as 

 they approach the aperture, bend round their foci situated in the 

 edges of the disc and form a divergent system on the other side. 

 If the chamber into which they pass is a perfectly absorbent one, 



1 The authors are indebted to Dr. Larmor for this suggestion of the 

 electrostatic analogy. 



NO. 1600, VOL. 62] 



and is sufficiently large, there will be formed on the inner side 

 of the diaphragm a system of density shells similar to those out- 

 side, but with the gradient of density centrifugally instead of 

 centripetally arranged. This system of shells is termed nega- 

 tive, and is as effective as the outer positive system in regulating 

 the flow according to the "diameter law," so that this law will 

 still hold good even if the outer air currents are sufficient to 

 sweep away the external positive shells altogether. 



All the known facts of diffusion through circular apertures in 

 a diaphragm are in complete accord with the above explanation, 

 which is fully elaborated in the original paper. 



By diffusing colouring matter through apertures in a septum, 

 under such conditions as to prevent convection currents, the 

 " density shells" have been rendered visible, and it has been 

 shown that their ellipsoidal form is exactly that which is- 

 demanded by the above hypothesis. Moreover, this method 

 gives an experimental demonstration of the more rapid projec- 

 tion of the diffusing particles from the edges of the aperture 

 than from a point nearer its centre, a fact completely in harmony 

 with the deduction of Stefan regarding the evaporation olf 

 liquids under analogous conditions. 



The various cases which present themselves in practice with- 

 regard to the rate of diffusion through single apertures in a 

 diaphragm are then discussed from the above point of view, and 

 simple formulae for the determination of this rate for single and' 

 double systems of density shells are established : (l) for cases 

 where the thickness of the diaphragm is negligible, and (2) for 

 other cases where the apertures become more or less tubular. 

 In a subsequent section of the paper it is shown how closely the 

 observed facts conform to these deductions, and that in static 

 diffusion through apertures in a septum we have a new and 

 accurate method for the determination of the diffusivity con- 

 stants of atmospheric CO.^, of the vapours of liquids, and of 

 substances in a state of solution. 



Since the velocity of the diffusive flow through unit area of an- 

 aperture in a diaphragm varies inversely with the diameter, it 

 might reasonably be expected that a diaphragm could be sO' 

 perforated with a series of very small holes arranged at suitable 

 distances from each other, as to exercise little or no sensible 

 obstruction when it was interposed in a line of diffusive flow, 

 although the aggregate area of the small holes might represent 

 only a small fraction of the total area of the septum. Multi- 

 perforate diaphragms of this kind were found to possess all the 

 remarkable properties which had been anticipated. 



The material used for the septa was very thin celluloid, 

 which was perforated at regular intervals with holes of about 

 0*38 mm. in diameter. Details of a number of experiments 

 with such diaphragms are given, in which it is shown that they 

 may be so arranged as to produce but little obstructive influence 

 on the diffusive flow of a gas when the total area of the aper- 

 tures amounts only to about 10 per cent, of the area of the 

 septum, and that nearly 40 per cent, of the full diffusive 

 flow may be maintained when the number of the apertures is so 

 far reduced as to represent an area of only i 25 per cent, of the 

 full area of the septum. 



The explanation is to be found in the local intensification of 

 the gradient of density in the immediate neighbouihood of the 

 diaphragm, and which does not extend to the column away 

 from the apertures. This disturbance of gradient is brought 

 about by the rapid convergence of the lines of flux, and their 

 divergence on the other side, with the consequent formation 

 of a system of " density shells" over each aperture. A system 

 of perforations of this kind may be compared with a systenv 

 of conductors electrified to a common potential, the density of 

 the diffusing substance above the apertures corresponding to 

 electric potential, and the non-absorbing portions of the 

 diaphragm to a surface formed by lines of electric force. 

 Just as the electric capacity of a plate is not much reduced by 

 cutting most of it away, so also is it possible to block out a 

 large portion of the cross-section of the diffusing column with- 

 out materially altering the general static conditions on which the 

 flow depends. 



The importance of these results in relation to diffusion through 

 porous septa is next considered, diffusion through a thin porous 

 septum being only an extreme case of free diffusion through a 

 multiperforate diaphragm, whose apertures are so far reduced in 

 size as to materially interfere with the mass movement of the 

 diffusing substance. 



A section of the paper is devoted to the application of these 

 new observations to the processes of gaseous and liqjiid diffu- 



