The General Law of Fall of a Small Spherical Body through a Gas, and its Bearing upon the Nature of Molecular Reflection from Surfaces

Abstract

Law of fall of a small spherical body through a gas at any pressure.---(1) Theoretical derivation. When the ratio of free path to radius of droplet, $\frac{l}{a}$, is small, the resistance to motion is due entirely to viscosity and is proportional to $a$, while when $\frac{l}{a}$ is large the resistance is due entirely to the inertia of the molecules hit and is proportional to ${a}^{2}$. The equation: $F=6\ensuremath{\pi}\ensuremath{\eta}\mathrm{av} {[1+\frac{{A}^{\ensuremath{'}}l}{a}]}^{\ensuremath{-}1}$ satisfies both these theoretical conditions. From Kinetic theory, however, it has been shown that ${A}^{\ensuremath{'}}$ is not constant but varies from a lower theoretical limit of.7004 (diffuse reflection) for $\frac{l}{a}$ small, to 1.164 for $\frac{l}{a}$ large. We therefore put ${A}^{\ensuremath{'}}=A+B{\ensuremath{\epsilon}}^{\frac{\ensuremath{-}ca}{l}}$ and write the complete equation: $F=6\ensuremath{\pi}\ensuremath{\eta}\mathrm{av} {[1+\frac{(A+B{\ensuremath{\epsilon}}^{\frac{\ensuremath{-}ca}{l}})l}{a}]}^{\ensuremath{-}1}$. (2) Experimental verification. By the oil drop method, values of ${A}^{\ensuremath{'}}$ have been determined for a wide range of values of $\frac{l}{a}$, from 0.5 to 134, for oil drops in air. These results are found to agree within the experimental error of \ifmmode\pm\else\textpm\fi{}2 per cent or less with the theoretical equation, and give $A=.864$, $B=0.290$, $c=1.25$. A discussion of results obtained with other drops indicates that while $A$ varies with the nature of the gas and even more with the material of the droplet, ($A+B$) is within two or three per cent the same for most sorts of particles which might settle through the atmosphere. (3) Differentiation between diffuse reflection, specular reflection, radial reflection, and condensation and re-evaporation of molecules. Radial reflection is shown to be thermodynamically and dynamically impossible; condensation and re-evaporation to be inconsistent with the observed value of ($\mathrm{A}+\mathrm{B}$). This observed value is satisfied only by a combination of about 1/10 of specular reflection with 9/10 of diffuse reflection, this last being defined as a reemission from each element of surface and in such directions as to satisfy the Maxwell distribution law, of all molecules which impinge upon that element. The agreement adds new evidence for the existence of specular reflection of molecules since 1/10 had also been indicated by the values of $A$ previously obtained for oil-drops in air. (4) Effects of mechanical roughness of the surface. The apparent discrepancies between the author's results and those of Knudsen and Gaede, which seemed to indicate complete diffuse reflection, are explained by the effects of minute mechanical proturberances which are thought to have existed on the insides of these authors' capillary tubes.