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

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

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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

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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

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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

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

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Bern, May 1905.Show the German source

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(Received 11 May 1905.)Show the German source