216 DIFFUSION rapid one, its velocity of propagation through the gas being that of sound. Let us now consider two gases in the same vessel, the proportion of the gases being different in different parts of the vessel, but the pressure being everywhere the same. The agitation of the inolcules will still cause more mole cules of the first gas to pass from places where that gas is dense to places where it is rare than in the opposite direction, but since the second gas is dense where the first one is rare, its molecules will be for the most part travelling in the opposite direction. Hence the molecules of the two gases will encounter each other, and every encounter will act as a check to the process of equalization of the density of each gas throughout the mixture. The interdiffusion of two gases in a vessel is there fore a much slower process than that by which the density of a single gas becomes equalized, though it appears from the theory that the final result is the same, and that each gas is distributed through the vessel in pre cisely the same way as if no other gas had been present, and this even when we take into account the effect of gravity. If we apply the ordinary language about fluids to a single gas of the mixture, we may distinguish the forces which act on an element of volume as follows : 1st. Any external force, such as gravity or electricity. 2d. The difference of the pressure of the particular gas on oppo site sides of the element of volume. [The pressure due to other gases is to be considered of no account]. 3d. The resistance arising from the percolation of the gas through the other gases which are moving with different velocity. The resistance due to encounters with the molecules of any other gas is proportional to the velocity of the first gas relative to the second, to the product of their densities, and to a coefficient which depends on the nature of the gases and on the temperature. The equations of motion of one gas of a mixture are therefore of the form where the symbol of operation J prefixed to any quantity denotes ot the time-variation of that quantity at a point which moves along with that medium which is distinguished by the suffix d), or more explicitly In the state of ultimate equilibrium u^ M 2 &c. = 0, and the equation is reduced to which is the ordinary form of the equations of equilibrium of a single fluid. Hence, when the process of diffusion is complete, the density of each gas at any point of the vessel is the same as if no other gas were present. If Vj is the potential of the force which acts on the gas, and if in the equation p l = Jc-^p^ , k 1 is constant, as it is when the temperature is uniform, theu the equation of equilibrium becomes the solution of which is Hence if, as in the case of gravity, V is the same for all gases, but k is different for different gases, the composition of the mixture will be different in different parts of the vessel, the proportion of the heavier gases, for which k is smaller, being greater at the bottom of the vessel than at the top. It would be difficult, how ever, to obtain experimental evidence of this difference of com position except in a vessel more than 100 metres high, and it would be necessary to keep the vessel free from inequalities of tem perature for more than a year, in order to allow the process of diffusion to advance to a state even half-way towards that of ultimate equilibrium. The experiment might, however, be made in a few minutes by placing a tube, say 10 centimetres long, on a whirling npparatus, so that one end shall be close to the axis, while the other is moving at the rate, say, of 50 metres per second. Thus if equal volumes of hydrogen and carbonic acid were used, the proportion of hydrogen to carbonic acid would be about T -? )T greater at the end of the tube nearest the axis. The experimental verification of the result is important, as it establishes a method of effecting the partial separation of gases without the selective action of chemical agents. Let us next consider the case of diffusion in a vertical cylinder. Let wij be the mass of the first gas in a column of unit area extend ing from the bottom of the vessel to the height x, and let v t be the volume which this mass would occupy at unit pressure, then
- ih=t> lf
dm^ dm^ dx dt _dv l 1 dx and the equation of motion becomes 1 dx* dt I dP dx _dv l cPv l _ X dv + ; dv% dv r _ d ?>j di a dt dx dt dx dx* - [ +&c. = ( dx If we add the corresponding equations together for all the gases, we find that the terms in C 12 destroy each other, and that if the medium is not affected with sensible currents the first term of each equation may be neglected. In ordinary experiments we may also neglect the eifect of gravity, so that we get or v 1 + v i =px t where p is the uniform pressure of the mixed medium. Hence dt dt and the equation becomes , dv. 2 ami dt an equation, the form of which is identical with the well-known equation for the conduction of heat. We may write it ^ ! i=D ! dt dc 2 D is called the coefficient of diffusion. It is equal to It therefore varies inversely as the total pressure of the medium, and if the coefficient of resistance, C 12 , is independent of the tem perature, it varies directly as the product k-Jc^ , i.e., as the square of the absolute temperature. It is probable, however, that the effect of temperature is not so great as this would make it. In liquids D probably depends on the proportion of the ingredients of the mixed medium as well as on the temperature. The dimen sions of D are L 2 T~ ! , where L is the unit of length and T the unit of time. The values of the coefficients of diffusion of several pairs of gases have been determined by Loschmidt. 1 They are referred in the following table to the centimetre and the second as units, for the temperature 0C and the pressure of 76 centimetres of mercury. D Carbonic acid and air, , . Carbonic acid and hydrogen, - Oxygen and hydrogen, . . Carbonic acid and oxygen, . Carbonic acid and carbonic oxide, Carbonic acid and marsh gas, Carbonic acid and nitrous oxide, Sulphurous acid and hydrogen, Oxygen and carbonic oxide, Carbonic oxide and hydrogen, Diffusion in Liquids. The nature of the motion of the molecules in liquids is less understood than in gases, but it is easy to see that if there is any irregular displacement among the molecules in a mixed liquid, it must, on the whole, tend to cause each com ponent to pass from places where it forms a large proportion of the mixture to places where it is less abundant. It is also manifest that any relative motion of two constituents of the mixture will be opposed by a resistance arising from 1 Imperial Academy of Vienna, 10th March 1870. 1423 5558 07214 1409 1406 1586 0983 4800 1802
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