On Gaseous Self-Diffusion in Long Capillary Tubes

Abstract

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 $\ensuremath{\lambda}\ensuremath{\gg}a$, the result in the limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$ reduces to that already obtained by M. Knudsen, the diffusion coefficient $D$ being given by $\frac{2a\overline{v}}{3}$, where $\overline{v}$ is the mean molecular speed. For a capillary of finite length the diffusion coefficient is, to first order in $\frac{a}{L}$, smaller than this by a factor $1\ensuremath{-}\frac{3a}{4L}$. In the opposite limit of high pressures, for which $\ensuremath{\lambda}\ensuremath{\ll}a$, the result reduces to the elementary kinetic theory expression for the self diffusion coefficient, $D=\frac{\ensuremath{\lambda}\overline{v}}{3}$. 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 $\ensuremath{\lambda}\ensuremath{\gg}L$. Thus the initial slope of $D$ as a function of pressure is given by $\frac{\mathrm{dD}}{d(\frac{a}{\ensuremath{\lambda}})}\ensuremath{\cong}\ensuremath{-}\frac{1}{2}\overline{v}a \frac{\mathrm{ln}L}{a}$. It is shown that these results account for the anomalous low pressure minima observed by several investigators who have measured the specific flow $\frac{G}{\ensuremath{\Delta}p}$ through long capillary tubes as a function of mean pressure $\overline{p}$. The failure to observe such minima with porous media, for which effectively $L\ensuremath{\approx}a$ 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.