Results · §5 · From displacement to molecular scale

Page 559

How far a particle moves in a second

Putting §3's diffusion coefficient into §4's law gives $\lambda_x = \sqrt{t}\sqrt{\frac{RT}{N}\frac{1}{3\pi kP}}$ , and with $N = 6\cdot10^{23}$ , water at 17 °C and particles 0.001 mm across, the paper finds 0.8 micron in one second and about 6 microns in one minute.

As printed

In § 3 haben wir für den Diffusionskoeffizienten $D$ eines in einer Flüssigkeit in Form von kleinen Kugeln vom Radius $P$ suspendierten Stoffes den Wert gefunden: $D = \frac{R T}{N} \frac{1}{6 \pi k P}.$ Diffusion coefficientGas constantAbsolute temperatureAvogadro's numberDynamic viscosityParticle radiusExplain this equation Ferner fanden wir in § 4 für den Mittelwert der Verschiebungen der Teilchen in Richtung der $X$ -Achse in der Zeit $t$ : $\lambda_x = \sqrt{2 D t}.$ Coordinate RMS displacementDiffusion coefficientObservation intervalExplain this equation Durch Eliminieren von $D$ erhalten wir: $\lambda_x = \sqrt{t} \cdot \sqrt{\frac{R T}{N} \frac{1}{3 \pi k P}}.$ Coordinate RMS displacementObservation intervalGas constantAbsolute temperatureAvogadro's numberDynamic viscosityParticle radiusExplain this equation Diese Gleichung läßt erkennen, wie $\lambda_x$ von $T$ , $k$ und $P$ abhängen muß.
Wir wollen berechnen, wie groß $\lambda_x$ für eine Sekunde ist, wenn $N$ gemäß den Resultaten der kinetischen Gastheorie $6 \cdot 10^{23}$ gesetzt wird; es sei als Flüssigkeit Wasser von 17° C. gewählt ( $k = 1{,}35 \cdot 10^{-2}$ ) und der Teilchendurchmesser sei 0,001 mm. Man erhält: $\lambda_x = 8 \cdot 10^{-5} \, \text{cm} = 0{,}8 \, \text{Mikron}.$ Coordinate RMS displacementExplain this equation Die mittlere Verschiebung in 1 Min. wäre also ca. 6 Mikron.

Text on page 559 of the German source, § 5, paragraph 1 · Text on page 559 of the German source, § 5, paragraph 2

In modern notation

\lambda_x = \sqrt{2\,D\,t}

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

Shown in modern letters; Einstein's are in §4 of the German source face.

Premise: The numbers rest on a value of $N$ taken from the kinetic theory of gases. The calculation shows what size of motion to look for; it measures nothing.
Approximation: The printed values are rounded, and the second is marked "ca.", about.
Conditional: The paper writes "wäre", would be: it states what the theory predicts and reports no observation.

Decoder

$3\pi kP$: Half of Stokes's $6\pi kP$ : the 2 in $\sqrt{2Dt}$ cancels a factor 2 of the drag.
$k = 1{,}35\cdot10^{-2}$: The viscosity of water at 17 °C, printed as a bare number in centimetre-gram-second units (poise). In SI units it is $1.35\times10^{-3}$ Pa·s.
0,001 mm: The particles' diameter. The radius $P$ in the formula is half of it.
$N = 6\cdot10^{23}$: The number of molecules in a gram-molecule as the kinetic theory of gases gave it, taken here as an input.
Mikron: The micron, a thousandth of a millimeter; today the micrometer.

Try it

BM-01: The laboratory opens on the paper's case, water at 17 °C and a particle of radius 0.5 micron: read $\lambda_x$ after one second and after sixty, and compare them with the printed 0.8 and 6 microns.

BM-01: Take the printed 0.001 mm as the radius instead of the diameter: how far does $\lambda_x$ move from 0.8 micron? Choose the preset “The radius probe” there.

BM-01: Double the viscosity: does $\lambda_x$ halve? Choose the preset “The viscosity comparison (common random numbers)” 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

Where this stops: A synthetic run generated from an assumed N is not independent evidence for N. Without the radius, the diffusivity fixes only the product aN. Measurement noise, drift, exposure and finite sampling require separate treatment. Read the argument

Common wrong turns

Doubling the viscosity halves the typical displacement.

Argument: A worked example of the combined law
Model: Exact within the model; the inputs are the paper's round values
Evidence: A prediction of the size of the motion, not a measurement
History: Introduced in this paper

Page 560

Determining N from the displacement

Turned around, the same relation gives $N = \frac{t}{\lambda_x^2}\cdot\frac{RT}{3\pi kP}$ , so a measured displacement, with the temperature, the viscosity and the particles' radius, would determine the number of molecules in a gram-molecule; the paper ends by hoping that a researcher will soon decide the question it has raised.

As printed

Umgekehrt läßt sich die gefundene Beziehung zur Bestimmung von $N$ benutzen. Man erhält: $N = \frac{t}{\lambda_x^2} \cdot \frac{R T}{3 \pi k P}.$ Estimated Avogadro numberObservation intervalCoordinate RMS displacementGas constantAbsolute temperatureDynamic viscosityParticle radiusExplain this equation
Möge es bald einem Forscher gelingen, die hier aufgeworfene, für die Theorie der Wärme wichtige Frage zu entscheiden!

Text on page 560 of the German source, § 5, paragraph 3 · Text on page 560 of the German source, § 5, paragraph 4

In modern notation

N = \frac{R\,T\,t}{3\,\pi\,\eta\,a\,\left\langle x^{2}\right\rangle}

Shown in modern letters; Einstein's are in §5 of the German source face.

Conditional: The paper proposes this determination and reports no measurement. It closes by hoping that a researcher will soon decide the question it has raised.

Decoder

$N$: Here the number to be found, an estimate of Avogadro's constant $N_A$ from observed displacements, and no longer an input as in the previous example.
$\frac{t}{\lambda_x^2}$: What the observations supply: the time divided by the mean square displacement along one axis, which is $\frac{1}{2D}$ .
$\frac{RT}{3\pi kP}$: What the inference needs besides the displacement: the temperature, the viscosity and the radius, each known by other means.
"Möge es bald einem Forscher gelingen": May a researcher soon succeed: the paper leaves the test to observation and reports none.

Try it

BM-07: Leave the radius undeclared: which values of $N$ fit the same displacements? Choose the preset “Identifiability family, radius undeclared” there.

BM-07: Load Perrin's 1909 observations, later evidence the paper could not use: what value of $N$ do they give, and how sure is it? Choose the preset “Perrin 1909 historical set” there.

Common wrong turns

Argument: The previous result turned around, proposed as a measurement
Model: Exact within the model, given independent values of the temperature, the viscosity and the radius
Evidence: A proposed test, not evidence
History: Proposed in this paper; carried out later, among others by Perrin in 1909

Reading preferences

§5 · From displacement to molecular scale

The paper's results

How far a particle moves in a second

As printed

In modern notation

Decoder

Try it

Common wrong turns

Determining N from the displacement

As printed

In modern notation

Decoder

Try it

Common wrong turns

§5 · From displacement to molecular scale

What would let us count molecules?