Osmotic Pressure and Density Fluctuations in Concentrated Emulsions

(Jean Perrin et René Constantin, Comptes Rendus t. 158, p. 1341, 1914).

The above-mentioned works concern the dependence of the osmotic pressure of emulsions on the concentration of granules. Perrin draws an analogy between a dilute emulsion and a liquid whose molecules are correspondingly enlarged. This analogy allows him to apply the Van der Waals equation. In a vertical layer of emulsion the osmotic pressure is

\[ P = P_0 + \Pi \frac{\omega}{s}, \]

where \(P_0\) is the pressure for height \(H = 0\). On the other hand, according to the Van der Waals equation:

\[ P = \frac{RT}{V} + \frac{bRT - a}{V^2}, \]

where \(R\) is the gas constant, \(T\) the absolute temperature, \(V\) the volume of the emulsion containing \(N\) granules. Two equations of state give:

\[ P=P_0+H\frac{\omega}{s}=\frac{RT}{N}\cdot n+\frac{bRT-a}{N^2}\cdot n^2, \]

where \(n\) is the number of granules per unit volume, for a given level \(H\). This relation gives the dependence of \(P\) on \(n\) (concentration).

If, however, one writes two such equations for the numbers \(n\) and \(n_0\) and subtracts them from one another, one obtains:

\[ \frac{\omega}{s}\cdot\frac{H}{n-n_0}=\frac{RT}{N}+\frac{bRT-a}{N^2}\left(n+n_0\right). \]

Plotting along the \(X\)-axis \(\left(n+n_0\right)\), and along the \(Y\)-axis \(\dfrac{H}{n-n_0}\), one obtains a straight line, whose ordinate at the origin of coordinates gives \(\dot N\). Thus, we have a new method for obtaining Avogadro’s constant. The value obtained is \(60.10^{22}\). Once the dependence of the osmotic pressure on the concentration \(\dfrac{dP}{dn}\) (Perrin calls this the osmotic compressibility of the emulsion—la compressibilité osmotique de l’emulsion) has been established, one can test the validity of the relation given by Smoluchowski for fluctuations of density.

\[ \delta=\frac{n-n_0}{n_0}=\frac{1}{\sqrt{-\pi\,\frac{N\eta v_0\,dP}{2RT\,dv_0}}}\quad \text{(I),} \]

where \(v_0\) is the specific volume, \(P\) the osmotic pressure, and \(N\) Avogadro’s number. The relation for \(\delta\) may also be given in another form:

\[ \delta=\frac{n-n_0}{n_0}=\frac{2n_0^{\,n_0}\cdot l^{-n_0}}{n_0!}\sqrt{\frac{\beta}{\beta_0}}\quad \text{(II),} \]

where \(\beta\) is the observed coefficient of compression, and \(\beta_0\) the coefficient of compression under the assumption that Clapeyron’s equation is applicable (the case of an ideal gas). This dependence was verified experimentally for colloidal solutions by Svedberg (Zeitschr. f. phys. Chem. 73, 547, 1910), and for emulsions by Il’in (Zeitschr. f. phys. Chem. LXXXIII B., 5 Heft). But since both solutions and emulsions were taken only weakly concentrated, \(\dfrac{\beta}{\beta_0}\) proved to be equal to unity. René Constantin takes considerably higher concentrations (up to \(\dfrac{1}{15}\)), and as a consequence Clapeyron’s equation is already inapplicable. The values for \(\delta\), calculated by formula (I), give complete agreement with experiment.

The possibility of applying to emulsions not only the Boyle–Mariotte–Gay-Lussac equation (Clapeyron), but also the Van der Waals equation, which gives a considerably more detailed model of molecular phenomena, compels one to arrive at certain interesting conclusions.

There is no doubt that the phenomenon of Brownian motion and the conditions of motion of granules in emulsions do not in fact correspond to the molecular-kinetic model. The possibility of applying the kinetic theory of gases (liquids) to emulsions (to solutions) is based solely on the statistical character of the phenomenon. It seems to me that this circumstance should be emphasized not only with respect to emulsions, but also with respect to “gases.” Very often the molecular-kinetic model is ascribed a much greater real significance than corresponds to reality. One must always remember that the real kinetic theory of gases is only a model, a working hypothesis, and that the laws of kinetic theory are experimentally confirmed proves nothing. Indeed, in Brownian motion, which clearly does not correspond to the molecular-kinetic model, not only Clapeyron’s equation is confirmed, but even Van der Waals’s equation?! And it is clear why. If between the quantities \(x, y, z, \ldots\) there exists the relation:

\[ f(x, y, z, \ldots) = 0 \quad \text{(I),} \]

then this still does not mean that \(x, y, z, \ldots\) are the real coordinates of the phenomenon. \(X, y, z, \ldots\) may be arbitrary functions of the real coordinates \(u, v, t, \ldots\)

\[ x = \mu (u, v, t, \ldots) \]

\[ y = \psi (u, v, t, \ldots) \quad \text{(II)} \]

\[ z = \chi (u, v, t, \ldots) \]

If the relation (II) exists, then the relation

\[ f[\mu (u, v, t, \ldots), \psi (u, v, t, \ldots), \chi (u, v, t, \ldots)] = 0 \quad \text{(III)} \]

will also hold, and so will equation I.

B. V. Ilyin.