Results · §1 · Osmotic pressure from suspended particles

Pages 550, 551, 553

The osmotic pressure of suspended particles

By the molecular-kinetic theory of heat, $n$ suspended particles held in a volume $V^*$ press on the wall that holds them with the osmotic pressure $p = \frac{RT}{V^*}\frac{n}{N}$ , as the same number of dissolved molecules would, provided the particles are far enough apart; classical thermodynamics expects no such force.

As printed

Nach dieser Theorie unterscheidet sich ein gelöstes Molekül von einem suspendierten Körper lediglich durch die Größe, und man sieht nicht ein, warum einer Anzahl suspendierter Körper nicht derselbe osmotische Druck entsprechen sollte, wie der nämlichen Anzahl gelöster Moleküle. Man wird anzunehmen haben, daß die suspendierten Körper infolge der Molekularbewegung der Flüssigkeit eine wenn auch sehr langsame ungeordnete Bewegung in der Flüssigkeit ausführen; werden sie durch die Wand verhindert, das Volumen $V^*$ zu verlassen, so werden sie auf die Wand Kräfte ausüben, ebenso wie gelöste Moleküle. Sind also $n$ suspendierte Körper im Volumen $V^*$ , also $n/V^* = \nu$ in der Volumeneinheit vorhanden, und sind benachbarte unter ihnen genügend weit voneinander entfernt, so wird ihnen ein osmotischer Druck $p$ entsprechen von der Größe: $p = \frac{R T}{V^*} \frac{n}{N} = \frac{R T}{N} \cdot \nu,$ Osmotic pressureGas constantAbsolute temperatureVolumeParticle countAvogadro's numberNumber densityExplain this equation wobei $N$ die Anzahl der in einem Gramm-Molekül enthaltenen wirklichen Moleküle bedeutet.
Durch diese Betrachtung ist gezeigt, daß die Existenz des osmotischen Druckes eine Konsequenz der molekularkinetischen Theorie der Wärme ist, und daß nach dieser Theorie gelöste Moleküle und suspendierte Körper von gleicher Anzahl sich in bezug auf osmotischen Druck bei großer Verdünnung vollkommen gleich verhalten.

Text on pages 550–551 of the German source, § 1, paragraph 3 · Text on page 553 of the German source, § 2, paragraph 8

A comparison, not a premise: Classical thermodynamics, as the paper reports it, expects no force on the wall, because the free energy seems not to depend on where the wall and the bodies are. The pressure is the molecular-kinetic theory's prediction, and the paper sets the two expectations side by side.

Decoder

$p$: The osmotic pressure on the wall that keeps the particles in $V^*$ ; modern texts write $\Pi$ .
$n$ , $V^*$: The number of suspended particles, and the part of the liquid's volume they are confined to.
$\nu$: The number of particles per unit volume, $n/V^*$ . In this paper $\nu$ is a number density, not a frequency; modern texts write $n$ .
$N$: The number of actual molecules in one gram-molecule, today Avogadro's constant $N_A$ .
$R/N$: The gas constant per molecule, today Boltzmann's constant $k_B$ . The paper writes the ratio and has no single letter for it.
"genügend weit voneinander entfernt": Far enough apart from one another: the law is for dilute suspensions, in which the particles do not act on one another.

Try it

BM-02: Switch between the molecular-kinetic mode and the classical expectation: what does each say the partition feels, and does the particles' size change it at the same number per volume?

BM-03: Count the positions 1000 particles can take in two volumes: does the free energy give the same pressure as the partition? Choose the preset “1000 particles (the osmotic-partition default)” there.

Where this stops: Gravity is set aside, and so are the energy and entropy of the surfaces between bodies and liquid (capillary forces), on the assumption that the moves considered do not change those surfaces. §1 states the molecular-kinetic expectation; §2 derives it from the theory. Neither section says which view nature follows; that is for observation. The law is for great dilution. Crowded bodies, or bodies that act on one another, depart from it. Read the argument

Where this stops: The footnote to the heading says this section, and Einstein's earlier papers on the foundations of thermodynamics, are not needed to understand the paper's results. The passage explains the section for the reader who wants to know how the law is obtained. The argument finds only how the free energy depends on V*; it computes nothing else about the integral B. Independence, homogeneity and the absence of forces are assumptions. With particles that act on one another, or crowded ones, J would depend on their positions and the pressure would depart from the dilute law. Read the argument

Argument: §1 states a consequence of the molecular-kinetic theory, and §2 derives it from that theory
Model: Exact for independent particles at great dilution; an idealization beyond it
Evidence: A prediction, not a measurement
History: The law for dissolved substances was known; its extension to suspended bodies is introduced in this paper

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§1 · Osmotic pressure from suspended particles