Results · §3 · The diffusion of small suspended spheres

Pages 555, 556

The diffusion coefficient of suspended spheres

Balancing the drift of spheres of radius $P$ , pulled by a force through a liquid of viscosity $k$ , against their diffusion gives $D = \frac{RT}{N}\frac{1}{6\pi kP}$ : apart from universal constants and the absolute temperature, the diffusion coefficient depends only on the liquid's viscosity and the particles' size, and the force does not appear in it.

As printed

Da dynamisches Gleichgewicht herrschen soll, so muß sein: $\frac{\nu K}{6 \pi k P} - D \frac{\partial \nu}{\partial x} = 0.$ (2)Number densityExternal force per particleDynamic viscosityParticle radiusDiffusion coefficientExplain this equation
Man erhält: $D = \frac{R T}{N} \frac{1}{6 \pi k P}.$ Diffusion coefficientGas constantAbsolute temperatureAvogadro's numberDynamic viscosityParticle radiusExplain this equation Der Diffusionskoeffizient der suspendierten Substanz hängt also außer von universellen Konstanten und der absoluten Temperatur nur vom Reibungskoeffizienten der Flüssigkeit und von der Größe der suspendierten Teilchen ab.

Text on page 555 of the German source, § 3, paragraph 7 · Text on pages 555–556 of the German source, § 3, paragraph 8

In modern notation

D = \frac{R\,T}{6\,\pi\,\eta\,a\,N_A}

D = \frac{k_B\,T}{6\,\pi\,\eta\,a}

Shown in modern letters; Einstein's are in §3 of the German source face.

Conditional: The formula is for spherical particles: the paper gives the drag for spheres of radius $P$ , and particles of another shape meet another drag.

Decoder

$D$: The diffusion coefficient: diffusion carries particles across unit area in unit time at a rate $D$ times the fall of their number per unit volume along $x$ .
$k$: The viscosity of the liquid, which the paper calls its coefficient of friction. It is not Boltzmann's constant; modern texts write $\eta$ .
$P$: The radius of a spherical particle; modern texts write $a$ .
$6\pi kP$: Stokes's drag on a slowly moving sphere: a force $K$ gives the sphere the speed $K/6\pi kP$ . The paper cites Kirchhoff's lectures on mechanics for it.
$K$: A force on each particle, introduced to set up the balance. It cancels, so the result holds without it.
$\frac{RT}{N}$: The gas constant per molecule times the temperature, today $k_B T$ .

Try it

BM-04: In the balance of drift against diffusion, change the applied force: does the diffusion coefficient change? Choose the preset “Einstein equilibrium balance (m = 1, F = 4.047 fN)” there.

Where this stops: Slip at the particle’s surface, its inertia, interactions between particles, unusual liquids, and gases are all left out. Stokes’s law and the osmotic-pressure law are brought in from outside; conservation alone does not give them. Read the argument

Common wrong turns

Doubling the viscosity halves the typical displacement.

Argument: A derivation from the osmotic law and Stokes's law
Model: Exact within the model of dilute, independent spheres obeying Stokes's law
Evidence: A derivation, not evidence
History: Introduced in this paper

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§3 · The diffusion of small suspended spheres

The paper's results