and these equations imply that when diffusion and other motions cease, the fluids satisfy the separate conditions of equilibrium dp1/dx − X1ρ1 = 0. The assumption made in the following account is that terms such as Du1/Dt may be neglected in the cases considered.
A further property based on experience is that the motions set up in a mixture by diffusion are very slow compared with those set up by mechanical actions, such as differences of pressure. Thus, if two gases at equal temperature and pressure be allowed to mix by diffusion, the heavier gas being below the lighter, the process will take a long time; on the other hand, if two gases, or parts of the same gas, at different pressures be connected, equalization of pressure will take place almost immediately. It follows from this property that the forces required to overcome the “inertia” of the fluids in the motions due to diffusion are quite imperceptible. At any stage of the process, therefore, any one of the diffusing fluids may be regarded as in equilibrium under the action of its own partial pressure, the external forces to which it is subjected and the resistance to diffusion of the other fluids.
5. Slow Diffusion of two Gases. Relation between the Coefficients of Resistance and of Diffusion.—We now suppose the diffusing substances to be two gases which obey Boyle’s law, and that diffusion takes place in a closed cylinder or tube of unit sectional area at constant temperature, the surfaces of equal density being perpendicular to the axis of the cylinder, so that the direction of diffusion is along the length of the cylinder, and we suppose no external forces, such as gravity, to act on the system.
The densities of the gases are denoted by ρ1, ρ2, their velocities of diffusion by u1, u2, and if their partial pressures are p1, p2, we have by Boyle’s law p1 = k1ρ1, p2 = k2ρ2, where k1,k2 are constants for the two gases, the temperature being constant. The axis of the cylinder is taken as the axis of x.
From the considerations of the preceding section, the effects of inertia of the diffusing gases may be neglected, and at any instant of the process either of the gases is to be treated as kept in equilibrium by its partial pressure and the resistance to diffusion produced by the other gas. Calling this resistance per unit volume R, and putting R = Cρ1ρ2(u1 − u2), where C is the coefficient of resistance, the equations of equilibrium give
| dp1 | + Cρ1ρ2(u1 − u2) = 0, and | dp2 | + Cρ1ρ2(u2 − u1) = 0 (1) |
| dx | dx |
These involve
| dp1 | + | dp2 | = 0 or p1 + p2 = P (2) |
| dx | dx |
where P is the total pressure of the mixture, and is everywhere constant, consistently with the conditions of mechanical equilibrium.
Now dp1/dx is the pressure-gradient of the first gas, and is, by Boyle’s law, equal to k1 times the corresponding density-gradient. Again ρ1u1 is the mass of gas flowing across any section per unit time, and k1ρ1u1 or p1u1 can be regarded as representing the flux of partial pressure produced by the motion of the gas. Since the total pressure is everywhere constant, and the ends of the cylinder are supposed fixed, the fluxes of partial pressure due to the two gases are equal and opposite, so that