Thin Layers on the Surface of Water

A. N. Frumkin.

If a drop of oil is placed on the clean surface of water, the oil spreads over the water, forming a thin film, until the area covered by the oil reaches a certain limit. No further increase of this area occurs; in other words, an oil layer of a certain thickness is in equilibrium with the clean surface of the water. The first observations in this field are due to Agnes Pockels (A. Pockels) [1]. Pockels found that the surface tension of water on which there is a thin layer of oil does not differ noticeably from the tension of a clean surface as long as the thickness of the layer does not exceed a certain limit. With a further increase in the thickness of the oil layer, the tension rapidly falls. Pockels’s experiments were repeated by Lord Rayleigh. Both Pockels and Rayleigh [2] used the following simple apparatus. A long narrow vessel was filled to the brim with water. Across the vessel two strips were laid; they served as barriers separating off the part of the water surface onto which a known quantity of oil was applied. By moving the strips, it was possible to vary the thickness of the oil layer at will. Rayleigh measured the surface tension by Wilhelmy’s method: a glass plate, suspended vertically from a balance, touched the investigated surface with its lower, sharply ground edge.

Fig. 1.

...ness. Under these conditions, obviously, the magnitude of the surface tension can easily be calculated from the apparent weight of the plate. The results of measurements of this kind in the case of castor oil are given in Fig. 1, where the abscissae indicate the thickness of the oil layer in $\mu\mu$, and the ordinates the corresponding tensions. We see that the tension begins to fall noticeably only when the thickness of the layer exceeds $1.3 \times 10^{-7}\ \text{cm}$ (point $S$); the sharp fall continues up to point $T$; with a further increase in the thickness of the layer there is only a slight decrease in the surface tension (curve $TT'$), which Rayleigh explains by the influence of impurities contained in the oil. Rayleigh points out that the thickness of the oil layer at point $S$, in any case, only slightly exceeds the probable value of the diameter of the oil molecules, so that the classical theory of capillarity is wholly inapplicable to these layers. “If one assumes,” says Rayleigh, “that the oil molecules behave like smooth and hard spheres of the kinetic theory of gases, then the forces acting between them should not manifest themselves until the molecules come into contact.” According to this conception, the tension should remain constant up to the point at which a second layer of molecules begins to form. After this it changes rapidly, acquiring a new constant value at the moment when the construction of the second layer of molecules is completed. Thus, according to Rayleigh, at point $S$ the surface is covered by a single layer, and at point $T$ by a double layer of oil molecules.

Fig. 2. Fig. 3. Fig. 4. Fig. 5.

Unusually elementary methods for investigating these layers were developed by Devaux (Devaux) [3]. Devaux filled a photographic cuvette with water. By passing a strip of filter paper $A$ (Fig. 2) over the surface of the water, one can remove all contamination from the surface and drive it into the space $C$. The cuvette then contains a perfectly clean water surface, which Devaux lightly dusts with a fine, degreased talc powder. If now the surface is touched with a wire that has been in contact with oil, the talc powder runs away from it in all directions, and, if only oil has been taken from it,

not too much, a strictly delineated circle free of talc is obtained on the surface, covered with a monomolecular layer of oil. This layer is, obviously, in equilibrium with the clean surface of the water. One may proceed in another way as well. If one blows at point \(D\) on a surface covered with talc, the talc gathers at the very barrier \(A\). If, however, the surface is first contaminated with oil and only then dusted with talc, then, blowing at point \(D\), we find that the talc recedes only as far as a certain boundary \(E\), which separates the clean surface from the surface covered with oil (\(F\)). This boundary can be straightened by means of a second paper strip, moving the latter until the clean surface \(H\) disappears. If the barrier \(B\) is now moved farther, the oil film is compressed, offering a certain resistance to compression. Moving strip \(B\) back, we find that the oil layer follows it only as far as the boundary corresponding to the position shown in Fig. 4, but does not spread farther.

Using a dilute solution of oil in benzene, a drop of which he placed on the surface of water and which, evaporating, left a perfectly definite amount of oil, Devaux was able to determine with sufficient accuracy the thickness of oil layers at their maximum spreading; in the case of triolein he arrived at the value \(1.1 \times 10^{-7}\ \mathrm{cm}\), while the calculated diameter of a triolein molecule (assuming the latter to be spherical in shape) is \(1.13 \times 10^{-7}\ \mathrm{cm}\). By means of benzene solutions Devaux also produced layers of certain solid substances and found that these layers offer resistance to shear, which disappears only when the quantity of substance per unit surface is already insufficient for the formation of a monomolecular layer.

After Devaux, Marcelin (Marcelin) [4] took up the same question. Marcelin placed a droplet of oleic acid on the surface of water covered with a monomolecular layer of the same substance and sprinkled with talc. From the motion of the talc it was evident that the monomolecular layer was thereby reduced by half; in other words, the thickness of the layer in equilibrium with the visible drop stood in relation to the minimum thickness as \(2 : 1\). Marcelin concluded from this that the thicker layer is bimolecular. Devaux objected to Marcelin’s conclusion, pointing out that the ratio under consideration is not exactly equal to 2, but fluctuates within the limits \(1.3\text{–}1.8\). Devaux believes that all these layers must be regarded as monomolecular; only in the more “thick” layers are the molecules arranged more densely, i.e., closer to one another.

The most remarkable investigations in this field are due to Langmuir (Langmuir). Langmuir’s starting point was the following two ideas, at which he arrived while studying the phenomena

adsorption of rarefied gases, catalytic processes, etc.: 1) the forces of adsorption by their nature do not differ in any way from those ordinary forces of chemical affinity which lead to the formation of definite chemical compounds; 2) the radius of the sphere of action of these forces is extremely small (of the order of \(0.6 \times 10^{-8}\) cm); in any case it does not exceed the dimensions of a single molecule. Langmuir represents the mechanism of formation of a thin layer, say, of oleic acid, in the following way. Molecules of oleic acid consist of a carboxyl group \(COOH\) and a long hydrocarbon chain. The carboxyl group is strongly attracted by water molecules, as follows, for example, from the much greater solubility in water of organic acids as compared with the corresponding hydrocarbons. On the contrary, hydrocarbon chains are weakly attracted by water (the solubility decreases as the length of the chain increases); but these long chains must attract one another very strongly. Therefore, when we apply a limited amount of oleic acid to the surface of water, the forces of chemical affinity will be saturated if all the \(COOH\)-groups come into contact with the water, while the hydrocarbon chains remain as close as possible to one another,—in other words, oleic acid must form a layer in which the carboxyls are immersed deep in the water, whereas the chains are arranged vertically and perpendicular to the surface of the water. The outer part of such a layer consists of \(CH_3\)-groups; since there is no doubt that the force of attraction between individual \(COOH\)-groups is greater than the force of attraction between such groups and \(CH_3\)-groups, after the formation of a monomolecular layer no further spreading of oleic acid over the surface will occur: the \(COOH\)-groups contained in a drop of oleic acid attract all the molecules and prevent the formation of a bimolecular layer. In support of his views, Langmuir points to the fact that the formation of thin layers requires the presence in the molecule of active groups possessing considerable affinity for water (the term “hydrophilic groups,” which Langmuir does not use, would be the most appropriate in this case). Thus, pure paraffins, containing no active groups, as Hardy showed, do not spread over the surface of water1. To explain the changes in the thickness of the oleic-acid layer discovered by Marcelin, Langmuir resorts to the following additional assumption. In the molecule of oleic acid, besides the \(COOH\)-group, there is also a double bond. From a comparison of the solubilities of the corresponding saturated and unsaturated compounds (say, ethane and ethylene) it follows that a double bond too possesses a noticeable affinity for water molecules. Langmuir proposes—

…therefore believes that, when there is a sufficiently free surface of water, the double bond also comes into contact with the water molecules, so that part of the oleic-acid molecule floats on the surface; when there is no longer enough free space, the double bonds break away from the water, and contact is maintained only by the carboxyl groups. As we shall see below, later experiments did not confirm this part of Langmuir’s theory.

To verify the basic proposition of Langmuir’s theory concerning the orientation of molecules of an organic substance in the surface layer, precise measurements of the thickness of these layers are obviously necessary. Indeed, by determining this thickness and assuming the monomolecular character of the layer, we can easily calculate the magnitude of the cross-section of a single molecule and compare it with the length of the same molecule in the direction perpendicular to the surface of the water. Langmuir proceeded in this way. Langmuir’s first measurements consisted in determining, by Devaux’s method, the maximum area that can be covered by a certain amount of various organic substances. The results of these measurements are set out in the following table. Dividing the area of coverage corresponding to one gram-molecule by Avogadro’s number, Langmuir obtains the area of the cross-section of a single molecule (I). The square root of this quantity gives an idea of the linear dimensions of this section (II). The length of the molecule is, obviously, equal to the thickness of the layer, which is easily calculated by dividing the molecular volume of the substance by the area covered by one gram-molecule of it (III). Finally, dividing the length of the molecule by the number of carbon atoms in the chain, we obtain the numbers given in column IV.

Comparing the data relating to the three saturated acids, tristearin, and palmitinoxylic cetyl, we see that each group

$$ -\mathrm{C}\begin{matrix} \mathrm{O}\\[-2pt] \mathrm{O} \end{matrix} $$

occupies on the surface of water an area of different [[unclear: text cut off at page edge]]

$23 \times 10^{-16}$ cm, independently of the length of the hydrocarbon chain and of whether this group belongs to an acid or to an ester. From a comparison of the numbers in columns II and III it is seen that the length of these molecules considerably exceeds their width. The length of the tristearin molecule is equal to the length of the stearic-acid molecule; evidently, in tristearin all three carbon chains are arranged parallel to one another and perpendicular to the surface of the water. It may be supposed that in the chains of the acids the atoms are arranged in a zigzag fashion or along a helix:

\[ \begin{array}{cccccc} & CH_2 & & CH_2 & & CH_2 \\ / & & \backslash / & & \backslash / & & \backslash \\ & & CH_2 & & CH_2 & & CH_2 \end{array} \]

This idea is suggested, for example, by the circumstance that the distances between two neighboring atoms, calculated on the assumption of a rectilinear arrangement (column IV), turn out to be smaller than the distances between two neighboring atoms in diamond, which is extremely improbable. The magnitude of the cross section is sufficiently large to allow a zigzag structure of the chain. From the figures relating to unsaturated compounds it is seen that the presence of a double bond increases the cross-sectional area approximately twofold; Marselen showed that the thickness of layers of oleic acid in equilibrium with drops of oleic acid exceeds the minimum value of the thickness in precisely this ratio. Thus, in the latter layers the cross-sectional area of the oleic-acid molecule is equal to the cross-sectional area of molecules of saturated acids, as required by Langmuir’s theory, and there is no basis for regarding these thicker layers as bimolecular.

After these preliminary determinations Langmuir proceeded to the study of the forces that determine the conditions for the formation of thin layers, in other words, to the measurement of the surface-tension value corresponding to layers of different thickness. This part of Langmuir’s work has most recently been repeated by Adam (Adam) [6]. Adam used the same experimental method as Langmuir, but the results of his work, apparently owing to a more perfect apparatus, are somewhat simpler and more definite than Langmuir’s results; therefore we shall present them directly here. Adam’s apparatus is shown in Fig. 6. An elongated brass vessel with flat paraffined edges was carefully cleaned and filled to the brim with water. All impurities present on the surface of the water were driven, by means of the barrier $CD$—a paraffined glass plate—into the left part of the vessel. Onto the surface between $CD$ and the paraffined copper strip $AB$ floating on the water there was applied, from a capillary pipette, a benzene solution of the substance under investigation. In order to prevent penetration of the forming ...

in this case, the thin layer between strip \(AB\) and the edges of the vessel into a separate strip-shaped space; at the ends of the latter there were placed two small tubes, by means of which two jets of air were directed onto the surface. The position of these tubes and the strength of the jet were precisely regulated with the aid of a special device, omitted from the drawing. By moving the barrier \(CD\) to the right, one can reach the point after which a further reduction in the area of the layer begins to cause a lowering of the surface tension, and the strip \(AB\) is repelled to the right. By placing counterweights on the pan of the balance, to whose beam the strip \(AB\) is attached, one can return the strip to its initial position; knowing the weight of these counterweights and the length of the strip \(AB\), it is not difficult to calculate the magnitude of the lowering of the surface tension corresponding to any position of the barrier \(CD\), i.e. to any thickness of the surface layer. This

Fig. 6.

method, devised by Langmuir, is undoubtedly the fundamentally simplest method of measuring surface tension and, with careful work, can apparently give very accurate results.

The data obtained by Adam are presented by him in the form of a series of curves in which the abscissae express the magnitude of the area per molecule in the surface layer \(\times 10^{-16}\ \mathrm{cm}^2\), and the ordinates the corresponding value of the lowering of the surface tension in dynes per cm. The curves obtained by Adam may be assigned to two types, between which a series of transitions is observed. An example of a curve of the first type is given by Fig. 7 (p. 180), which shows the results of measurements on saturated fatty acids on distilled water, where

It turned out that acids with 14, 15, 16, 17, 18, 21, and 22 carbon atoms give, provided only that the temperature is not too high, completely coincident curves (in the case of acids with 12 and 13 atoms the solubility was too great; with cerotic acid it was not possible to obtain films of sufficient stability). We see that, so long as the area covered by one molecule exceeds a certain value, equal to \(21 \cdot 10^{-16}\,\mathrm{cm}^2\), the whole layer offers no resistance whatever to compression; in other words, the lowering of the surface tension is equal to zero. At this value of the area the resistance to compression increases sharply, so that even under the strongest compressions, of \(50\)—\(60\) abs. units, the area covered by one molecule decreases by only a few percent. Taking into account that, for a layer thickness equal to approximately \(3 \cdot 10^{-7}\,\mathrm{cm}\), such a lowering of the surface tension corresponds to a lateral pressure reaching as much as 200 atmospheres, we see that the coefficient of compressibility of these films is a quantity of the same order as the coefficient of compressibility of liquid paraffins. When the compression exceeds a certain limit, the film is destroyed, and the surface contracts at constant pressure (point \(H\)). The values of the ordinate of point \(H\) fluctuate within very wide limits; if there were no supersaturation phenomena, one would have to suppose that point \(H\) corresponds to a layer in equilibrium with the substance of the layer forming a separate phase. In fact, however, the position of point \(H\) is apparently determined by the presence of nuclei around which condensation of this substance can occur. The following phenomenon is often observed in this case: after the area of the film has contracted to a certain limit, the process stops, and further contraction occurs only when the compressing force is increased, so that a series of ledges is obtained on the curve. Such a phenomenon can be explained by assuming that, during the process of contraction, the condensation nuclei may be thrown out of the contracting layer.

The curve of the acids undergoes a remarkable change when the concentration of hydrogen ions in the solution is increased: as soon as the latter exceeds \(10^{-3.5}\), instead of the curve of Fig. 7 we obtain the curve of Fig. 8. The upper part of this curve remains unchanged, but below point \(G\) the compressibility of the surface layer in this case is much greater, and the maximum area of coverage is equal not to 21, but to \(25 \cdot 10^{-16}\,\mathrm{cm}^2\) per molecule.

Besides the acids, Adam investigated a number of other compounds with a long chain, belonging to the esters, alcohols, amides, and nitriles. The curves obtained with these substances at sufficiently low temperatures have the same form as the curve of Fig. 8, i.e. they consist of an almost vertical segment \(GH\) and a more inclined lower part. The continuation of the segment \(GH\) always intersects the axis of abscissae at the very same point, corresponding to the area

of \(21 \cdot 10^{-16}\ \mathrm{cm}^2\) per molecule. Since the position of this point does not depend on the nature of the polar group included in the molecule, Adam assumes that this part of the curve corresponds to that structure of the surface layer in which the hydrocarbon chains come into direct contact; thus the value \(21 \cdot 10^{-16}\ \mathrm{cm}^2\) is nothing other than the cross-sectional area of the hydrocarbon chain itself. The slope of the lower part of the curve and the point at which it intersects the abscissa axis depend on the nature of the polar group; it may therefore be assumed that, upon compression of the layer, contact first occurs between the polar groups, whose cross-section is larger than that of the chain; under stronger compressions the molecules shift relative to one another, the polar groups of neighboring molecules fall into different horizontal planes, and the chains come into direct contact. It is evident that the stronger attraction of water to the polar group should favor the second grouping, which is in complete agreement with the properties of acid films; the first grouping appears only when the attraction of water to the \(COOH\) group is reduced owing to the presence of hydrogen ions. From the slope of the lower part of the curves one can evidently calculate the cross-sectional area of the polar groups. Adam thus obtains numbers that range between \(21.7 \times 10^{-16}\) for the \(CH_2OH\) group (cetyl alcohol) and \(28.7 \times 10^{-16}\) for the \(-CH=CH-COOH\) group (isooleic acid).

Fig. 7.        Fig. 8.

In their mechanical properties these thin layers approximate now liquids, now solids. In the latter case dust particles on the surface lose their mobility and appear as if fastened to quite definite places. Generally speaking, layers usually “solidify” after the hydrocarbon chains come into close contact; however, this is not observed in all cases: thus

cetyl alcohol films always remain liquid. On the other hand, urea derivatives give “solid” layers even in the absence of contact between adjacent hydrocarbon chains. Generally speaking, the question of the conditions for “solidification” cannot yet be regarded as clarified.

Of considerable interest is the question of the shape of the curve at the very smallest values of the lowering of surface tension. Indeed, one should expect that individual molecules of the surface film, overcoming the attraction of neighboring molecules, may escape onto the free surface of the water, just as vapor molecules escape from a three-dimensional liquid; to the vapor pressure of a three-dimensional liquid there must correspond, for these two-dimensional liquids, a definite value of the lowering of surface tension at which the transition of the condensed layer into a more rarefied, “gas-like” one will occur.

Fig. 9.

In other words, the curves in Figs. 7 and 8 should, before reaching the axis of abscissae, pass into a horizontal straight line. In reality, no such sharp transition is observed; as the axis of abscissae is approached, the curve of the lowering of surface tension bends round and gradually goes to zero. This deviation from the theory is probably due to the influence of unavoidable surface impurities, which must have an especially strong effect in this part of the curve. The analogy with evaporation phenomena appears, however, clearly when we pass to those curves which are observed at higher temperatures. In Fig. 9, after Adam, a series of such isotherms is shown for ethyl palmitate. We see that at higher temperatures the curves, before reaching the axis of abscissae, bend and pass into ...

at some distance almost parallel to this axis and only then approach the zero value. At still higher temperatures the horizontal part of the curve disappears, so that the area of coverage, as compression decreases, increases uniformly, beginning from a certain minimum value. The set of these isotherms strikingly resembles the compression isotherms of gases in the critical region, only here the whole phenomenon takes place in a space of two dimensions, so that instead of pressure we have a lowering of surface tension, and instead of molecular volume—the molecular area of coverage. This analogy can be brought out in another way as well. If one measures the areas of coverage at constant compression and variable temperature, then at a rather precisely defined temperature a sharp increase of this area is observed (Fig. 9). Labrouste [7], who was the first to observe this phenomenon, called it melting of the layer; according to what has been said above, however, it is clear that here we are dealing not with melting, but with the “evaporation” of the layer at a definite pressure. It is interesting to compare the “evaporation temperatures” for various substances. It turns out, as was to be expected, that this temperature rises with increasing chain length: for lauric acid \((C_{12})\) it lies below \(0^\circ\), for palmitic acid \((C_{16})\)—at \(28.5^\circ\), for stearic acid \((C_{18})\)—at \(46^\circ\) (at a compression equal to 1.4 absolute units). The introduction of a double bond greatly lowers this temperature, so that in the case of oleic acid, despite the great length of the chain \((C_{18})\), it nevertheless lies below zero, which explains the large area of coverage observed for oleic acid at ordinary temperature.

According to the foregoing, the properties of monomolecular layers above the “evaporation point” are analogous to the properties of gases near their critical temperature; in other words, the deviations from the laws of the ideal gaseous state caused by the presence of intermolecular forces are, in this case, very large. There exist, however, thin layers in which the influence of intermolecular forces proves to be much smaller. These are the layers formed on the surface of water by substances of comparatively small molecular weight, such as, for example, butyric acid or amyl alcohol. Since these substances are soluble in water, it is impossible in this case to study the properties of the surface layer by the methods described; it is necessary to resort to measurement of the surface tension of the corresponding solutions.

As Gibbs showed, any substance, upon the dissolution of which the surface tension of the solvent is lowered, is concentrated in the surface layer; between the magnitude of the lowering of the surface tension \(\gamma\) and the quantity of dissolved substance \(\Gamma\) which accumulates per unit surface, there exists the following simple relation

\[ \Gamma = -\frac{1}{RT}\frac{d\gamma}{d\lg c}, \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (1 \]

where \(c\) is the concentration of the dissolved substance. Thus, knowing the dependence between \(\gamma\) and \(c\), we can find the quantity \(\Gamma\), which corresponds to any value of \(c\), i.e., calculate the thickness of the adsorbed layer. The first calculations of this kind were made by Milner [8]; for a complete analysis of the question we are indebted to Langmuir. Let us see to what results these calculations lead us. Let us turn first to the weakest solutions. In this case experiment shows that the lowering of the surface tension \(\Delta \gamma\) is proportional to the concentration of the dissolved substance

\[ \Delta \gamma = ac, \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (2) \]

where \(a\) is a certain constant. Using the Gibbs formula, we obtain from this:

\[ \Gamma = \frac{a}{RT} c = \beta c \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (3) \]

and

\[ \Delta \gamma = RT\Gamma \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (4) \]

If, instead of the quantity \(\Gamma\), we introduce the quantity \(s = \frac{1}{\Gamma}\), i.e. the area covered by one gram-molecule of the substance in the surface layer, then Eq. (4) becomes

\[ \Delta \gamma s = RT \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (5) \]

Eq. (5) is completely analogous to the Boyle–Mariotte law and to the law of osmotic pressure for dilute solutions; in other words, the molecules of the organic substance located in the surface layer behave, within the range of applicability of Eq. (2), like the molecules of ideal gases. Langmuir showed that the quantity \(\beta\) has a simple physical meaning; namely, if we denote by \(W\) the work that must be expended in order to transfer a gram-molecule of the adsorbed substance from the surface layer into the interior of the solution, and by \(\delta\) the thickness of the surface layer, then

\[ \beta = \delta e^{\frac{W}{RT}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (6) \]

In passing from the preceding member of a homologous series to the following one, the quantity \(W\) must increase by a constant amount equal to the work attributable to one group —\(CH_2\)—. According to equation (6), the quantity \(\beta\) will then increase in a constant ratio; in other words, the surface activity in homologous series must grow in a geometric progression. And indeed, already Traube [9] found that the surface activities in the series of acids, alcohols, and esters increase as the terms of the progression: \(1, 3, 3^2, 3^3\), etc.

On passing to stronger solutions, the magnitude \(\Delta \gamma\) ceases to be proportional to the concentration, and in order to express the dependence between \(\Delta \gamma\) and \(c\) at any concentration it is necessary to resort to more complicated formulas. As Shishkovskii showed \([^{10}]\), in the case of substances with not too large a molecular weight, the experimental data agree well with the following empirical formula

\[ \Delta \gamma = b\, lg\left(\frac{c}{a}+1\right) \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (7) \]

Hence, with the aid of the Gibbs formula, we obtain

\[ \Gamma=\frac{bc}{RT(c+a)} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (8) \]

It follows from (8) that, as \(c\) increases, \(\Gamma\) tends to the limit \(\dfrac{b}{RT}\). Denoting this quantity by \(\Gamma_\infty\), we obtain the following adsorption formula

\[ \Gamma=\frac{\Gamma_\infty c}{c+a} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (9) \]

Langmuir derived this adsorption formula from the usual notions of kinetic equilibrium, assuming that adsorption proceeds in a monomolecular layer and that the forces of interaction between adsorbed molecules may be neglected. Here we cannot dwell either on this derivation by Langmuir or on the experimental verification to which he subjected equation (9) \([^{11}]\); let us note only the following further point. If, by means of equation (9), \(c\) is expressed as a function of \(\Gamma\) and substituted in (7), the following relation is obtained:

\[ \Delta \gamma=-RT\Gamma_\infty\, lg\left(1-\frac{\Gamma}{\Gamma_\infty}\right) =-\frac{RT}{s_\infty}\, lg(1-x), \ . \ . \ . (10) \]

where \(x\) is the concentration of the substance in the surface layer, and \(s_\infty=\dfrac{1}{\Gamma_\infty}\), i.e. the limiting value of the molecular area of coverage. This equation is completely analogous to that which expresses the dependence between osmotic pressure and concentration in the case of “ideal” concentrated solutions, i.e. such concentrated solutions in which the forces of interaction between the dissolved molecules may be neglected.

From the value \(b\) found experimentally it is easy to calculate the value \(s_\infty\) and thus to find the area that falls to one molecule of the adsorbed substance in the saturated layer. For propionic, butyric, valeric, and caproic acids we thus obtain the number \(31 \cdot 10^{-16}\ \mathrm{cm}^2\), very close to that number,

which we obtained for the higher acids from direct observation. The thickness of the saturated layer of an oily acid proves to be equal to \(4.7 \cdot 10^{-8}\ \mathrm{cm}\). In other words, we are dealing here with monomolecular layers, in which the molecules are arranged approximately in the same way as in the films of which we spoke at the beginning of this article; only in this case the influence of intermolecular forces proves to be imperceptible.

In the case of soluble compounds, on the basis of the observed curves \(\Delta \gamma - c\), we could calculate the form of the curves \(\Delta \gamma - c\) that are not accessible to direct observation. In the case of high-molecular compounds we may go by the reverse path and predict the form of the \(\Delta \gamma c\)-curves, starting from the results of the measurements of Langmuir and Adam.

Fig. 10.

With the aid of Gibbs’s formula it is easy to show that the surface tension of water in this case must remain almost unchanged up to a certain concentration, after which there begins a sharp fall of it along an almost logarithmic curve. I have carried out a series of measurements with those acids which are, as it were, a connecting link between completely insoluble higher acids and readily soluble lower ones. These experiments fully confirmed the correctness of this part of the theory, as is evident, for example, from Fig. 10, which gives the results of my observations on lauric acid \((C_{12})\). Capric \((C_{10})\) and caprylic \((C_8)\) acids gave curves that are

intermediate between the curve in Fig. 10 and the generally known curve for the lower homologues. Let us also note that, by introducing into Eq. (10) an additional term analogous to the term \(\frac{a}{v^2}\) in the van der Waals equation, we obtain a formula which, with a suitable choice of the attraction constant, satisfactorily represents any of these curves.

Literature

1) A. Pockels. Nature. 43, 437 (1890).
2) Rayleigh. Phil. Mag. 48, 331 (1899).
3) Devaux. Ann. Rep. Smithsonian Inst. 261 (1913).
4) Marcelin. J. de Phys. 1, 19 (1914).
5) Langmuir. J. Am. Chem. Soc. 38, 2221 (1916); 39, 1868 (1917); Proc. Nat. Acad. Sc. 3, 141 (1917).
6) Adam. Proc. Royal Soc. (A) 99, 336 (1921); 101, 452, 516 (1922); 103, 676 (1923).
7) Labrouste. Ann. de Phys. 14, 164 (1920).
8) Milner. Phil. Mag. (6) 13, 96 (1907).
9) Traube. Lieb. Ann. 265, 41, (1891).
10) Shishkovsky. Zeitschr. f. physik. Ch. 64, 385 (1908).
11) Langmuir. J. Am. Chem. Soc. 40, 1361 (1918).
12) Harkins and Feldman. J. Am. Chem. Soc. 44, 2665 (1922).

1. Harkins (Harkins) [12] has quite recently objected to this part of Langmuir’s reasoning; however, it seems to us, without sufficient grounds (note in proof). ↩