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