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

§ 1. On the osmotic pressure to be ascribed to the suspended particles.Show the German source

In the partial volume $V^*$ of a liquid of total volume $V$, let $z$ gram-molecules of a non-electrolyte be dissolved.Show the German source If the volume $V^*$ is separated from the pure solvent by a wall that is permeable to the solvent but not to the dissolved substance, then this wall is acted on by the so-called osmotic pressure, which, for sufficiently large values of $V^*/z$, satisfies the equation:Show the German source $$p V^* = R T z.$$$p$Osmotic pressure$V^{\ast }$Volume$R$Gas constant$T$Absolute temperature$z$Solute amountExplain this equationShow the German source

If, on the other hand, instead of the dissolved substance, small suspended bodies are present in the partial volume $V^*$ of the liquid, bodies that likewise cannot pass through the wall permeable to the solvent, then according to the classical theory of thermodynamics one is not to expect — at least when gravity, which does not interest us here, is neglected — that a force acts on the wall; for on the usual view the “free energy” of the system appears to depend not on the position of the wall and of the suspended bodies, but only on the total masses and the qualities of the suspended substance, of the liquid and of the wall, and on pressure and temperature.Show the German source It is true that the energy and entropy of the boundary surfaces (capillary forces) would also come into consideration in calculating the free energy; we can, however, disregard these, since in the changes of position of the wall and of the suspended bodies that are to be considered, changes in the size and nature of the surfaces of contact are supposed not to occur.Show the German source

From the standpoint of the molecular-kinetic theory of heat, however, one arrives at a different view.Show the German source According to this theory a dissolved molecule differs from a suspended body only in size, and one does not see why a number of suspended bodies should not correspond to the same osmotic pressure as the same number of dissolved molecules.Show the German source One will have to assume that the suspended bodies, as a consequence of the molecular motion of the liquid, carry out an irregular motion in the liquid, even if a very slow one; if they are prevented by the wall from leaving the volume $V^*$, they will exert forces on the wall, just as dissolved molecules do.Show the German source If, therefore, $n$ suspended bodies are present in the volume $V^*$, that is, $n/V^* = \nu$ in unit volume, and if neighboring ones among them are sufficiently far apart, then there will correspond to them an osmotic pressure $p$ of magnitude:Show the German source $$p = \frac{R T}{V^*} \frac{n}{N} = \frac{R T}{N} \cdot \nu,$$$p$Osmotic pressure$R$Gas constant$T$Absolute temperature$V^{\ast }$Volume$n$Particle count$N$Avogadro's number$\nu$Number densityExplain this equationShow the German source where $N$ denotes the number of actual molecules contained in one gram-molecule.Show the German source In the next section it is to be shown that the molecular-kinetic theory of heat does in fact lead to this extended view of osmotic pressure.Show the German source

Bern, May 1905.Show the German source

(Received 11 May 1905.)Show the German source