On the Motion of Particles Suspended in Liquids at Rest, as Required by the Molecular-Kinetic Theory of Heat

§ 5. Formula for the mean displacement of suspended particles. A new method for determining the true size of atoms.Show the German source

In § 3 we found, for the diffusion coefficient $D$ of a substance suspended in a liquid in the form of small spheres of radius $P$, the value:Show the German source $$D = \frac{R T}{N} \frac{1}{6 \pi k P}.$$$D$Diffusion coefficient$R$Gas constant$T$Absolute temperature$N$Avogadro's number$k$Dynamic viscosity$P$Particle radiusExplain this equationShow the German source Further, in § 4 we found, for the mean value of the displacements of the particles in the direction of the $X$-axis in the time $t$:Show the German source $$\lambda_x = \sqrt{2 D t}.$$${\lambda }_x$Coordinate RMS displacement$D$Diffusion coefficient$t$Observation intervalExplain this equationShow the German source By eliminating $D$ we obtain:Show the German source $$\lambda_x = \sqrt{t} \cdot \sqrt{\frac{R T}{N} \frac{1}{3 \pi k P}}.$$${\lambda }_x$Coordinate RMS displacement$t$Observation interval$R$Gas constant$T$Absolute temperature$N$Avogadro's number$k$Dynamic viscosity$P$Particle radiusExplain this equationShow the German source This equation shows how $\lambda_x$ must depend on $T$, $k$ and $P$.Show the German source

We shall calculate how large $\lambda_x$ is for one second if $N$ is set equal to $6 \cdot 10^{23}$ in accordance with the results of the kinetic theory of gases; let the liquid chosen be water at 17° C ($k = 1{,}35 \cdot 10^{-2}$), and let the diameter of the particles be 0,001 mm.Show the German source One obtains:Show the German source $$\lambda_x = 8 \cdot 10^{-5} \, \text{cm} = 0{,}8 \, \text{Mikron}.$$${\lambda }_x$Coordinate RMS displacementExplain this equationShow the German source The mean displacement in 1 minute would therefore be about 6 microns.Show the German source

Conversely, the relation found can be used to determine $N$.Show the German source One obtains:Show the German source $$N = \frac{t}{\lambda_x^2} \cdot \frac{R T}{3 \pi k P}.$$$N$Estimated Avogadro number$t$Observation interval${\lambda }_x$Coordinate RMS displacement$R$Gas constant$T$Absolute temperature$k$Dynamic viscosity$P$Particle radiusExplain this equationShow the German source

May a researcher soon succeed in deciding the question raised here, which is important for the theory of heat!Show the German source

Bern, May 1905.Show the German source

(Received 11 May 1905.)Show the German source