Page:Encyclopædia Britannica, Ninth Edition, v. 16.djvu/646

Rh 618 MOLECULE The term &quot; diffusion &quot; has sometimes been applied to the process by which a gas passes through a porous diaphragm. This, however, is now generally denominated transpiration. It has also been applied to the expansion of a gas into vacuum, as on the removal of a diaphragm separating the gas from an exhausted receiver. This is now generally denominated free expansion. We shall understand, as is now usual, by the term diffusion the process by which, when two or more gases are mixed throughout any space in different proportions at different points, but so that if all molecules were of the same gas the whole would be in equilibrium, the different gases pass through each other and tend to equalize the proportions at all points in the space. Suppose, for instance, a tube containing a mixture of two gases, A and B, at constant temperature and constant pressure of the com bined gases throughout the tube and subject to no forces, but the density of gas A increasing and that of B diminishing from one end of the tube to the other. Let the axis of the tube be taken for the axis of x. If N a be the number of molecules of gas A, and N b the number of molecules of gas B in unit volume, we have, owing to the constant pressure and temperature at all points of the tube, N a + N b = A T, a constant. But at a given instant N a and N b at any point are severally functions of x. It will be found that under these circumstances more molecules of gas A pass through any sec tion of the tube, which may be in the plane of yz, in one direction, say from left to right, than in the opposite direction. On the other hand, more molecules of gas B pass from right to left than from left to right. And this will go on till the mixture becomes uniform throughout the tube. The investigation of the rate at which the unequal distribution tends to equalize itself in this simple case that is, the excess of the number of molecules of gas A which cross a section of the tube from left to right over the number crossing in the same time from right to left is the problem of diffusion. We give the results obtained by 0. E. Meyer as follows: if the molecules of the two gases had the same mass and dimensions (to put an ideal case), then the excess of molecules of either gas passing through the section in one direction that is, the stream velocity would be~n -y- 3 ul, where I denotes the mean free path for a molecule having velocity u, and ul is the average value of that function for all molecules of the gas. When we come to deal with two gases, the molecules of one not being of the same size and dimensions with those of the other, we shall find that, in the absence of any common velocity of the two gases at the plane of yz, more, or fewer, molecules of gas A would cross the plane per unit of time from left to right than of gas B from right to left, because, assuming constant pressure and tem perature of the mixture at every point in the tube, the number of molecules of the two gases combined must be the same at every point that is, N a + N b =N, where Nis constant. Heuce dN a _ _dN b dx dx Now the excess of molecules of gas A coming from left to right per unit of time is - ul a, and similarly the excess of mole- o dx cules of gas B crossing from right to left per unit of time is T cfa* ulb if we now dating 1 &quot; 811 b y suffixes a and 6 quantities relating to the two gases respectively. Here l a and l b are mean free paths for velocity u of the two kinds of molecules through the mixed gases, and ul a is not generally equal to ul b. Hence the total number of molecules crossing the plane from left to right ex- 1 dN - ceeds the number coming from right to left by -5- ~r~ (ul a - ul b ). Meyer here assumes that the combined gases have a common velocity - y -j- (ul a - ul b ], and that such common velocity will not affect the relative motion of the molecules. On that hypo thesis the rate of diffusion can be calculated as follows. The pro portion of the stream of the combined gases which consists of mole cules of gas A is _ N a 1 dN a, } N^+N b Tdx~ M.--U 1 Hence the total surplus number of molecules of gas A passing through unit area of the plane per unit of time is dN a dx The expression dN a dx is denned to be the &quot;coefficient of diffusion&quot; of gas A into gas B. It is evidently the same as that of gas B into gas A. The Relation of the Coefficient of Diffusion to Density. It can be shown that l u , the mean free path for a molecule having velo city to, is for any single gas inversely proportional to the density, and for any mixture of gases inversely proportional to X, the ag gregate volume occupied by matter in unit space. Hence, in the expression -- ul -T-, ul is inversely proportional to the density, or to X, as the case may be. Now the rate of diffusion on this theory depends upon ul. 3 dx Hence, given the absolute increase of density of a gas per unit of length, that is, given -r-^, the rate of diffusion ought to vary in versely as the density of the combined gases. On the other hand, given the proportional increase of the density, or -^ -T-, the rate of diffusion ought to be independent of the density, because in that case dN -T varies directly, and ul inversely, as N. The analytical result, dN that at given temperatures, and given the absolute value of, the rate of diffusion is inversely proportional to the density of the gases agrees with the experimental results obtained by Loschmidt for carbonic acid gas and air, carbonic acid gas and hydrogen, hydrogen and oxygen. 1 Relation of the Coefficient of Diffusion to Temperature. The coefficient of diffusion varies directly as the square root of the abso lute temperature, for /h u/h Jo V IT I _^t = i e + and or, if u V/i y, Hence where ^ denotes a certain function, and 1 _ fh This analytical result also agrees fairly with Loschmidt s experi ments above referred to. FRICTION OR VISCOSITY OF GASES. Suppose two layers of gas separated by an imaginary plane, similar in all respects except that the molecules of one have a small common momentum in a certain direction parallel to the plane. We may take the imaginary plane for that of yz, and the average direction of motion of the molecules on one side of the plane, e.g., the left-hand side, for the axis of y, the molecules on the right-hand side of the plane having no average momentum. Then the mole cules crossing from the left to the right side carry with them an average momentum in the direction y, and so tend to impress the right-hand stratum of gas with that mo mentum. On the other hand, the molecules of the right- hand stratum crossing the plane into the left-hand one have, relatively to the molecules in the latter, an average momentum in the opposite direction, and therefore tend to diminish the average momentum of the left-hand stratum. 1 Sitzungsberichte, 1870, Bd. Ixi. S. 380. 2 gee Meyer s Kin. Theorie, p. 295.