On Gaseous Self-Diffusion in Long Capillary Tubes

A calculation is made of the rate of diffusion of "tagged" molecules in a pure gas at uniform pressure in a long capillary tube of half-length L and radius a. At pressures for which the mean free path λa, the result in the limit L→∞ reduces to that already obtained by M. Knudsen, the diffusion coefficient D being given by 2av̄3, where v̄ is the mean molecular speed. For a capillary of finite length the diffusion coefficient is, to first order in a/L, smaller than this by a factor 1-3a/4L. In the opposite limit of high pressures, for which λa, the result reduces to the elementary kinetic theory expression for the self diffusion coefficient, D=λv̄3. One of the most significant features of the result is that in a long tube the diffusion coefficient drops very rapidly with increasing pressure from its initial value for λL. Thus the initial slope of D as a function of pressure is given by dDd(aλ)-1/2v̄a lnL/a. It is shown that these results account for the anomalous low pressure minima observed by several investigators who have measured the specific flow GΔp through long capillary tubes as a function of mean pressure p̄. The failure to observe such minima with porous media, for which effectively L≈a in each pore, is also explained by these results. The formulae obtained here represent a rigorous solution to the long capillary diffusion problem, valid at all pressures and subject only to the limitations of the mean free path type of treatment.