Coefficients of Slip in Gases and the Law of Reflection of Molecules from the Surfaces of Solids and Liquids

Abstract

First order correction term to Stokes Law of fall of droplets.---(1) Empirical. The experiments by which the value of the electronic charge was determined by the droplet method gave consistent results only when this law was modified by the factor ($1+\frac{\mathrm{Al}}{a}$), where $\frac{l}{a}$ is the ratio of mean free path to radius of droplet. (2) Hydrodynamic theory gives as a first approximation ($1+\frac{\ensuremath{\xi}}{a}$), where $\ensuremath{\xi}$ is the coefficient of slip. (3) Kinetic theory gives ($1+\frac{0.7004l}{a}$) in case all the molecules are diffusely reflected from the surface of the droplet, where $l$ is defined by the relation $\ensuremath{\eta}=.3502\ensuremath{\rho}\overline{c}l$. If, however, the fraction reflected diffusely is $f$, the fraction ($1\ensuremath{-}f$) being specularly reflected, then the factor is [$1+0.7004(\frac{2}{f}\ensuremath{-}1)(\frac{l}{a})$]. The theory developed by Cunningham gave the numerical constant as 0.79, but this value is too high since experimental values of $A$ nearly as low as 0.70 have actually been obtained.Coefficient of slip between gases and solids.---(1) Stokes' law method. Since $\ensuremath{\xi}=Al$, $\ensuremath{\xi}$ may be computed directly from $A$. (2) Rotating cylinder method of determining viscosity is also capable of giving values of $\ensuremath{\xi}$ accurate to one per cent. If ${\ensuremath{\theta}}_{0}$ is the limiting deflection for high pressures, and $\ensuremath{\theta}$ the deflection for a low pressure, then $\ensuremath{\xi}$ for that low pressure is $\frac{({\ensuremath{\theta}}_{0}\ensuremath{-}\ensuremath{\theta})}{K}$, where $K$ is a geometrical constant of the apparatus. Values obtained by this method agree closely with those obtained by the first method. (3) Capillary effusion method. If the rate of flow of gas for unit pressure difference is ${T}_{0}$ for high pressures and $T$ for a low pressure, then $\ensuremath{\xi}$ for that low pressure is $\frac{R(T\ensuremath{-}{T}_{0})}{4{T}_{0}}$, where $R$ is the radius. (4) Values of $\frac{\ensuremath{\xi}}{l}$ vary with the surface, for instance from 0.70 for air-mercury, and 0.87 for air-oil and air-glass, to 1.07 for air-fresh shellac. They also vary with the gas, for instance from 0.81 for hydrogen-oil and 0.82 for C${\mathrm{O}}_{2}$-oil to 0.86 for air-oil, and 0.90 for helium-oil.Coefficient of diffuse reflection of gas molecules, determined from the relation $A=(\frac{2\ensuremath{\eta}}{\ensuremath{\rho}\overline{c}})(\frac{2}{f}\ensuremath{-}1)$ gives values of $f$ which vary with the surface from 0.79 for air-fresh shellac and 0.89 for air-glass to 1.00 for air-mercury. The values also vary with the gas from 0.87 for helium-oil and 0.895 for air-oil to 0.92 for C${\mathrm{O}}_{2}$-oil and hydrogen-oil. The values are for 23\ifmmode^\circ\else\textdegree\fi{} C.