FORCES NEAR THE SURFACE OF MOLECULES1

Irving Langmuir, Schenectady

In the first period of the development of the kinetic theory, in the nineteenth century, gas molecules were regarded as solid spheres, and their interaction was limited to the moment of collision. In this way it was possible to give a fairly precise quantitative explanation of the phenomena of viscosity, thermal conductivity, and diffusion of gases, of the dependence of their volume on temperature and pressure, and also to explain other phenomena connected with the mean free path of molecules.

Taking into account also attractive forces, which vary as some power of the distance between the centers of the molecules, van der Waals was able to explain in general terms the transition from the gaseous state to the liquid state and to give a quite satisfactory explanation of those deviations from the laws of ideal gases which are observed in all gases at high pressures.

If, however, other properties of matter are also taken into consideration, it becomes necessary to postulate additional properties of molecules and to take more accurately into account the forces acting between them. For example, in order to study the properties of electrolytes it is necessary to consider the electric charges on molecules or ions. In 1912 Debye showed that the electrical properties of many dielectrics can be explained if it is assumed that the center of gravity of the electrons in a molecule does not coincide with the center of gravity—

positive charges of the nuclei. Thus the molecule possesses what is called a dipole moment, which is measured by the product of the electric charge and the distance by which it has been displaced. It is known, for example, that the dipole moment of the water molecule is \(1.8 \times 10^{-18}\) electrostatic units. A moment of this magnitude can be obtained if the distance of the electron from the proton is \(0.37 \times 10^{-8}\) cm. Since in the water molecule there are 10 electrons, it is only necessary that the center of gravity of these electrons be separated by \(0.037 \times 10^{-8}\) cm from the center of gravity of the three nuclei of the molecule. Thus even the largest of the observed values of dipole moments can be explained by displacements of electrons over distances insignificant in comparison with the diameter of the molecule.

Hardy (H. B. Hardy) (1) in 1912 and the author (2) (3) in 1916 pointed out that the force field around many chemical molecules is extremely asymmetric, owing to which orientation of the molecule occurs at the surface of liquids and in adsorbed films on solid bodies. It was also found that the shape of the molecule, determined by its chemical structure, is of very great importance in the question of surface tension. In recent years especially, thanks to the work of Debye and his collaborators, a large body of material has been collected on the question of electric forces in liquids and solutions containing ions and dipoles. Thanks to this it has been possible to remove many of the most serious difficulties of the earlier theories, especially in the field of the study of electrolytes.

At the present time there is no need for a precise distinction between chemical and physical forces. The chemist must recognize that many of the forces with which he has to deal are electrical in their nature, since the interaction of molecules consisting of electrons and positively charged nuclei must naturally be electrical. However, the simple classical theory of electrical forces, based on Coulomb’s law, is completely insufficient for explaining chemical properties.

If charged particles obey only Coulomb’s law, then the minimum of potential energy is obtained when the positive and negative particles coincide. There must exist some force, corresponding to repulsive forces and keeping the particles at some distance from one another.

Repulsive Forces between Molecules

In many later works on the kinetic theory, in which molecules are regarded as hard spheres, it is assumed that the repulsive forces which must counteract the attractive forces act only during collisions, namely at the moment the spheres come into contact (4). No attempt was made to find an explanation for those forces within the molecules which lead to their mutual repulsion. To explain the compressibility of solids, Born and Lande assumed that the repulsive forces between molecules (or ions) vary inversely as some power of the distance between the centers of the molecules. Thus, for the halide salts of the alkali metals (NaCl, KJ) they showed from the compressibility that the repulsive force varies inversely as the tenth power of the distance, whereas the attractive force due to the charges on the ions varies inversely as the square of the distance, in accordance with Coulomb’s law. Born (6) also attempted to derive the inverse tenth-power law for repulsion from the assumption that electrons and ions are arranged as if in cubic symmetry at the corners of a cube. However, he did not succeed in this for two reasons. First, the method of expansion in a series that he used is not legitimate when applied to such short distances as the distances between ions in salts; and, second, Born proceeded from the assumption that all cubic ions are arranged so that their edges are parallel. However, such an orientation would be unstable under the action of the assumed forces, since these forces would tend to orient—

to arrange cubic ions so that the corner of one lies opposite the face of another, and in such a case the repulsive force will be replaced by an attractive force superposed on the force of attraction between ionic charges.

It is possible that the orientation of the ions proposed by Born (although he does not go into the causes of this phenomenon) actually exists in solid crystals; however, it is obvious that it cannot be regarded as the cause of the emergence of repulsive forces, since the disappearance of this orientation upon dissolution of salts has no appreciable effect on the density or compressibility.

The theory developed by Debye apparently gives a better conception of repulsive forces. Starting from the assumption that electrons move in orbits, according to Bohr’s theory, he comes to the conclusion that the electric fields around molecules are pulsating, or partly oscillatory, fields, and that the amplitude of their oscillations increases very rapidly as one approaches the surface of the molecule. Thus, when an electron approaches a molecule, it acquires, in addition to its inherent motion, an oscillatory motion caused by the pulsating field. Debye showed that this oscillatory motion tends to go 180° out of phase with the field producing it and that, owing to the inhomogeneity of the electric field near the atom, the resulting force will be repulsive.

According to this theory, the origin of repulsive forces between molecules is explained by the distortion of the electron orbits of one molecule caused by the motion of the electrons of neighboring molecules in their orbits. Thus, the appearance of a repulsive force is determined not by the entire molecule as a whole, but is explained by the close approach of the electrons of two molecules. It would seem that, on the basis of the foregoing, it would be advisable to regard these forces as surface forces and to express them rather as functions of the distances between the surfaces of molecules than as depending on the distances between their centers. This hypothesis justifies to a certain extent the theory that regarded molecules as hard elastic spheres.

Even if we accept Debye’s picture of the origin of repulsive forces, we shall see that in reality it transfers the origin of these forces into the quantum theory that determines the stable orbits of electrons in the molecule. At present, however, such problems are solved on the basis of wave mechanics, and not by considering electronic orbits. For some simple cases, such as, for example, the interaction between hydrogen molecules, Hynd and London have succeeded in calculating the forces by means of wave mechanics. But apparently much time will still pass before it is possible by this route to obtain more accurate data on repulsive forces in more complex molecules.

According to quantum theory, each electron of an atom is characterized by four quantum numbers.

Not all quantum numbers are possible; they must satisfy certain conditions—the selection rules. In a normal atom or molecule, with which the chemist is concerned, the electrons are arranged in such a way that the total energy is a minimum subject to the quantum conditions.

Apparently, the most important factor for the chemist is that known as the Pauli principle, which establishes that in an atom there cannot be even two electrons defined by the same combination of four quantum numbers. The periodic system of the elements is based on this principle, and it also determines the arrangement of electrons in the molecule. Further development of such methods is the surest path toward understanding the nature of repulsive forces.

For practical purposes it is best to accept that the molecules of organic compounds have the shape indicated by chemical structural formulas and a more or less solid surface, so that even small deformations require large repulsive forces. In many cases the surface of molecules may be regarded as completely rigid.

Attractive Forces between Molecules1

While, for an understanding of repulsive forces, the aid of quantum theory is necessary, to explain the phenomena of attractive forces between molecules the classical laws are quite sufficient, such as, for example, Coulomb’s law. Thus, the Debye–Hückel theory of electrolytes (8) is based on Coulomb’s law, Poisson’s equation, and Boltzmann’s law, which are laws of classical mechanics. In considering the forces acting between molecules, Debye divided molecules into three groups: 1) ionic, 2) polar, and 3) nonpolar. The ionic type is represented by electrolytic ions, such as, for example, \(K^+\), \(NO_3^-\), etc., and by gaseous ions. In such molecules the number of electrons is not equal to the number of positive charges of the atomic nuclei, so that the electric charge of the molecule as a whole is a multiple of the electron charge \(e\).

The polar type includes molecules having a dipole moment, i.e., such uncharged molecules in which the center of gravity of the negative charges does not coincide with the center of gravity of the positive charges. Thus, nonpolar molecules include all uncharged molecules in which the centers of gravity of the negative and positive parts coincide. From Debye’s point of view, such molecules may also be quadrupoles, octupoles, etc.

For example, positive and negative charges arranged alternately at the corners of a square will give a quadrupole; at the corners of a cube—an octupole. In order to form a clear idea of the attractive forces between molecules, it is useful to take into consideration the actual magnitude of the corresponding forces.

Ions

The electric field at a distance \(r\) from a monovalent ion having charge \(e\) is equal to \(\dfrac{e}{\varepsilon r^2}\), where \(\varepsilon\) is the dielectric—

ical constant of the medium. The charge of the electron \(e\) is equivalent to \(1.43 \times 10^{-7}\) volt cm; thus the electric field is equal to \(1.43 \times 10^{-7}\varepsilon r^2\) volts per cm, and the electric potential corresponding to this force is equal to \(1.43 \times 10^{-7}\varepsilon r\).

If, instead of \(r\), we take the value \(3 \times 10^{-8}\) cm, roughly corresponding to the distance between the centers of two adjacent ions, then the electric field will be equal to \(1.6 \times 10^8\) volts per cm, and the potential to 4.8 volts. If, instead of \(\varepsilon\), we take the value 80, corresponding to the dielectric constant of water, then the values obtained will be only one eightieth of those given above. However, the concept of the dielectric constant, when applied to such insignificant distances, cannot be considered fully justified, since it can hardly be assumed that two molecules in contact are in a medium possessing any known electrical properties. The use of the dielectric constant at distances several times greater than the diameter of a molecule, on the contrary, is quite justified.

The application of the Debye–Hückel theory to electrolytes proves that forces at these large distances are of great importance for understanding the properties of ions. According to Coulomb’s law, the force between two ions varies inversely as the square of the distance. In electrolyte solutions, in which the concentration of positive ions is increased around negative ions and vice versa, the force between two given ions, as the distance increases, decreases more rapidly than inversely as the square of the distance.

Polar molecules

At a distance \(r\) from a dipole molecule having an electric moment \(\mu\), the electric potential will be equal to \(\dfrac{\mu \cos \theta}{r^2}\), where \(\theta\) is the angle between the radius vector \(r\) and the axis of the dipole. Thus, at a distance \(r\) along the axis of the dipole the electric field is equal to \(-\dfrac{2\mu}{r^3}\), the force being directed

along the polar axis. At a distance \(r\) in the equatorial plane of the dipole, the electric force is likewise directed parallel to the polar axis, but is equal to \(+\dfrac{2}{r^3}\).

Debye (9) showed that ordinarily, owing to thermal motion, a dipole or quadrupole molecule (in a gas or liquid) changes its orientation so rapidly that the force produced by the dipole must be regarded not as constant, but as a rapidly oscillating force. Under such conditions many effects produced by this force are proportional to \(\overline{E^2}\), or to the mean square of the field strength.

Thus the effective force may vary inversely as the sixth power of the distance from the dipole.

Nonpolar Molecules

If the molecule is a quadrupole, the instantaneous force in any direction will vary inversely as the fourth power of the distance, and thus the effective force will vary inversely as the eighth power of the distance. For more symmetrical molecules the forces will vary inversely as still higher powers of the distance.

Under the influence of the forces considered above, the various types of molecules in liquids behave differently.

Positive ions tend to concentrate around negative ions according to the Debye–Hückel theory (8). Dipole molecules have a tendency to orient themselves in the field surrounding ions. Thermal motion tends to prevent this orientation, so that a dipole molecule can be fully oriented only if it is close to an ion.

A dipole molecule oriented in the field tends to move in the direction of greater field intensity. The change in potential energy is equal to \(mE\), where \(E\) represents the change in field strength, and \(m\) is the effective dipole moment (the mean moment) in the direction of the field.

A field insufficient for complete orientation of dipoles,

gives an effective dipole moment \(m\), which, according to Debye (10), is equal to:

\[ m=\frac{\mu^{2}F}{3kT}, \]

where \(F\) is the electric force tending to produce orientation.

Since the energy that can produce displacement in a dipole is proportional to \(mF\), we see that at large distances dipoles attract one another directly proportionally to \(F^{2}\) and inversely proportionally to the sixth power of the distance. However, if we take a fixed dipole (for example, one attached to a large organic molecule) acting on another dipole at a very short distance, so that the latter is oriented in the field of the former, then the force of attraction will vary inversely proportionally to the fourth power of the distance.

An electric field whose intensity is equal to \(F\), acting on a nonpolar molecule, deforms or polarizes it, so that it acquires a dipole moment

\[ m=\alpha F, \]

where \(\alpha\) denotes the coefficient of polarization, or polarizability.

A liquid consisting of nonpolar molecules of this kind will have a dielectric constant \(\varepsilon\) greater than unity; the “molar polarization” \(P\) of the liquid is calculated from \(\varepsilon\) by means of the equation:

\[ P=\left(\frac{\varepsilon-1}{\varepsilon+2}\right)\frac{M}{\rho} \]

where \(M\) is the molecular weight, and \(\rho\) is the density of the liquid. The relation between \(P\) and \(\alpha\) is given by the equation:

\[ P=\frac{4\pi}{3}N\alpha=2.54\times10^{24}\alpha, \]

where \(N\) is Avogadro’s number, equal to \(6.06\times10^{23}\). The quantity \(P\) has the dimension of volume and is therefore measured in \(\mathrm{cm}^{3}\). For ordinary organic liquids the value of \(P\) is approximately—

approximately equal to 0.3 of the volume of one gram-molecule. Some values of \(P\) and \(\alpha\) are given in Table 1.

Table 1

From the value \(\alpha = 10^{-23}\), an order of magnitude encountered in most organic liquids, it may be calculated that the field required to impart to a nonpolar molecule a dipole moment equal to \(10^{-18}\) is \(F = 30 \times 10_7\) volts per cm. Such a field will exist at a distance

\[ r = \frac{7 \times 10^{-8}}{\sqrt{\varepsilon}} \ \mathrm{cm} \]

from an electron. A field of this strength will be found at a distance

\[ r = \frac{2.7 \times 10^{-8}}{\sqrt{\varepsilon}} \]

along the axis of a dipole.

From the foregoing we see that the actions of the electric fields of molecules may be divided into three groups: 1) segregation, 2) orientation, 3) deformation. An example of segregation is the accumulation of positive ions around negative ions in electrolytes. Dipolar molecules are oriented in the field produced by any other molecule, and in this way are attracted by the molecule producing the field. Nonpolar molecules, under the influence of the fields of other molecules, are deformed or polarized in such a way that they acquire a dipole moment in the direction of the field, and thus are attracted by the molecule producing the field.

For a quantitative estimate of the strength of segregation and orientation produced by molecular fields, Boltzmann’s equation is used:

\[ \frac{n_1}{n_2} = A e^{\frac{\lambda}{kT}}, \]

where \(n_1\) and \(n_2\) represent the relative numbers of molecules in two given positions or orientations, and \(\lambda\) is the work necessary to transfer a molecule from one of these states to the other. The constant \(A\) includes

the ratio of the a priori probabilities of finding the molecules in the two positions or states under consideration. Sometimes these probabilities depend on geometrical factors, but often they require knowledge of the quantum phenomena accompanying a change of state. In this equation \(e\) is the base of the natural system of logarithms, 2.718, \(k\) is Boltzmann’s constant, \(1.38 \times 10^{-16}\) erg per degree; \(T\) is the absolute temperature.

From Boltzmann’s equation we see that, if the constant \(A\) only slightly exceeds unity, then the effect of segregation and orientation usually becomes at all appreciable only from the moment when the energy \(\lambda\) reaches the same order of magnitude as \(kT\). At room temperature the value of \(kT\) is equal to \(4.1 \times 10^{-14}\) erg. This is the energy acquired by an electron in passing through a potential difference of 0.025 volt.

Debye’s theory of the influence of dipolar molecules on the dielectric constant of liquids shows that the effective dipole moment \(m\) (of dipolar molecules having moment \(\mu\)) is equal to \(\frac{1}{2}\mu\) if the field intensity is such that the work done by the field in orienting the molecule is equal to \(2kT\).

The segregation of ions of one sign around an ion of the opposite sign can be observed at distances \(r\) smaller than the distance at which the potential is equal to 0.025 volt, i.e. when \(r\) is less than

\[ \frac{57 \times 10^{-8}}{\varepsilon}\ \text{cm}. \]

In the case when the value of \(\varepsilon\) is small and is only a few units, then, even in the most dilute solutions, in which the distances between ions are more than ten times greater than the diameter of the molecules, the ions will tend to leave the solution and come into contact with one another. This agrees fully with the fact that salt-like substances such as NaCl, whose crystals are held together by forces of the Coulomb type, are practically insoluble in organic liquids with a low dielectric constant. On the other hand, in water and other liquids with a high dielectric constant, even in solutions of medium concentration

the distances between ions are so great that their potentials with respect to one another are less than \(kT\), which is in complete agreement with the fact that, if salts are soluble in these liquids, their solutions behave as electrolytes.

As we have seen, in order to orient a large part of the dipolar molecules in a liquid, a field \(F\) is necessary such that \(\mu F\) is greater than \(2kT\). This means that, for orientation, a field approximately equal to \(2.4 \times 10^7\) volts per cm is required. The work done in introducing an oriented molecule into the field will be of the same order of magnitude, so that a field sufficient for an almost complete orientation of the dipolar moments will also produce appreciable segregation. The field will have a strength \(2.4 \times 10^7\) at a distance \(r\) equal to \(\dfrac{8 \times 10^{-8}}{\sqrt{\varepsilon}}\) cm from a monovalent ion, or

\[ r = \frac{3 \times 10^{-8}}{\sqrt{\varepsilon}} \text{ cm} \]

along the axis of a dipolar molecule with moment \(\mu = 10^{-18}\). In other directions, not coinciding with the axis of the dipolar molecule, the force will be smaller, and therefore the distance at which effective orientation or segregation will occur will be less than that indicated above. From these calculations one may conclude that the interaction between dipolar molecules with a moment equal to \(10^{-18}\) leads to mutual orientation only in the case when they are in contact. If they were separated by even one molecule, so that the distance between the centers would be \(6 \times 10^{-8}\) cm, then even in the case where they continued to be oriented, the force, varying inversely as the fourth power of the distance, would be only one sixteenth. However, at such a large distance the orientation would be far from complete, so that the force would in reality vary inversely as the sixth power of \(r\) and would be only one sixty-fourth of the force that acts between molecules in contact. In order to deform a molecule sufficiently to impart to it a dipole moment \(\mu = 10^{-18}\), the same electric force is necessary (\(3 \times 10^7\) volts per cm) as is needed for orient-

tion of a dipole molecule with the same moment. Thus, when two molecules possessing dipole moments come into contact, the dipole moment of each of them is approximately doubled, owing to mutual deformation.

If one considers the electric field near a dipole to be proportional to \(r^{-3}\), and takes into account that the distance between the center of one molecule and the surface of the neighboring one is only one third of the distance to the surface following the neighboring molecule, it becomes clear that the intensity of the electric field on one side of a dipole molecule is twenty-seven times greater than on the opposite side.

The heat of vaporization of a liquid, referred to one molecule, gives the energy that must be expended in order to separate the molecules of the liquid from one another. The heat of vaporization of pentane, a nonpolar liquid boiling near room temperature (at \(36^\circ \mathrm{C}\)), is about \(40 \times 10^{-14}\) erg per molecule, or approximately \(10 kT\). According to Trouton’s approximate empirical rule, the absolute boiling temperatures of liquids are proportional to their heats of vaporization, so that in general one may assume, with the same degree of approximation, that for all liquids the heats of vaporization per molecule will be approximately equal to \(10 kT\), where \(T\) is the boiling temperature. This may be regarded as a consequence of Boltzmann’s law. However, in the present case the coefficient \(A\) differs considerably from unity. The energy released when ions approach one another to distances close to molecular dimensions considerably exceeds \(10 kT\), at least in those cases where the medium has a low dielectric constant. Thus, we should not be surprised that substances consisting of ions are solids—melting and boiling only at very high temperatures and soluble only in liquids with a high dielectric constant.

We have found that the energy values obtained when dipolar and nonpolar molecules approach one another (having

value of $\mu$, equal to approximately $10^{-8}$, and for $\alpha = 10^{-23}$) at a distance of $3 \times 10^{-8}\ \text{cm}$ are approximately equal to $kT$ ($T = 300^\circ K$). Since each molecule is usually in contact with a dozen others, one may expect that the energy required to separate a molecule from all its neighbors will be of the order of $10kT$. In fact, however, the magnitudes will be larger, since the greatest effect is obtained when the molecule is deformed on the side facing the neighboring molecule ($r = 1/2$ the molecular diameter), where the electric field is approximately 8 times greater than at the center of the molecule.

Thus we have an explanation of the fact that the boiling points of liquids whose molecules have the properties postulated by us will usually be considerably above room temperature. When molecules come into contact, the energy of nonpolar molecules does not differ very greatly from the energy of dipolar ones; in any case, these quantities are of the same order. It thus becomes clear why the boiling points of liquids containing dipolar molecules are only slightly higher than those of nonpolar substances.

Some liquids consisting of dipolar molecules, such as, for example, anhydrous hydrogen chloride, boil at temperatures considerably below room temperature.

An analysis of the electrical forces known to us which cause the interaction of molecules thus inevitably leads to the conclusion that the forces holding the molecules of organic liquids together act almost exclusively between molecules in contact, and that it is possible, without serious error, to neglect entirely all forces acting at larger distances.

One may consider that this conclusion is fully justified in the case of all liquids consisting of nonpolar molecules. It is applicable with a sufficient degree of accuracy to the majority of liquids containing dipolar molecules. But, in general, when liquids contain free ions, the sphere of action of the forces is much greater, and in such cases it is necessary to take into account the Coulomb forces,

acting over large distances. Over many years, in connection with studies of adsorption, surface tension, and the kinetics of heterogeneous reactions, the author explained these phenomena by the interaction of touching molecules. From an empirical point of view, the results fully justified this method. In many respects this conception coincides with traditional chemical conceptions, since the chemist usually regards chemical action as taking place between molecules in contact.

The physicist, however, probably since the time of Newton, has preferred to consider forces that vary as some power of the distance, or to operate with force fields in space. The considerable success achieved by physics in the development of atomic theory, and more recently also in some of its applications to chemical phenomena (as, for example, the properties of electrolytes), has led many chemists and physicists to the conviction that these methods may also contribute to the solution of more complex chemical problems.

It is necessary, however, to emphasize that, in attempting to calculate the mode of interaction of dipolar or nonpolar molecules in liquids, the physicist encounters problems so complex mathematically that simplifying assumptions become necessary. For example, the potential energy of dipolar and nonpolar molecules relative to one another is expressed by means of an infinite series, the successive terms of which contain factors \(1/r\), \(1/r^2\), \(1/r^3\), and so on.

The coefficient of the first term of this series for dipolar molecules is equal to zero; the calculation of the coefficient of the second term presents great difficulties; although it is known that the coefficients of the third and subsequent terms are not equal to zero, these terms may be neglected, since they contain higher powers of \(r\). However, we see that the most important forces in liquids are the forces between molecules in contact, and that even a small increase in distance greatly reduces

these forces. For example, in the case of a force varying in proportion to \(1/r^9\), which Debye finds in nonpolar molecules, the energy released when two molecules approach one another is reduced by half when the distance is increased by 15%. This change in energy affects the exponent in the Boltzmann equation; therefore the most important interactions occur at still smaller changes in the distance \(r\). Hence it is clear that, even in the crudest approximations, it is illegitimate to omit the term containing \(1/r^3\).

Another example of the approximations that are necessary for the mathematical study of the forces acting between dipole molecules is Debye’s theory, which considers the relations between the dielectric constant and chemical association (11). Debye considers the action of dipoles upon one another as the concentration of dipole molecules in a nonpolar solvent increases. In doing so, he starts from the assumption that the molecules are spheres, and derives complicated equations to explain their interactions.

As a result it turns out that the dipole moment of each molecule increases as the concentration increases, since the molecules tend to line up along a common axis. This phenomenon is explained by the fact that, for a given value of \(r\), the force acting along the polar axis is twice as great as the force acting in the equatorial plane. However, analysis of experimental data shows that, in solutions of many dipole substances in nonpolar solvents, the polarization referred to a single molecule decreases as the concentration increases. This result could be explained if the molecules were not spheres but ellipsoids, in which the length of the equatorial diameter does not reach 80% of the polar diameter. However, by introducing such a representation, we encounter excessively great mathematical difficulties.

The strongest objection to the mathematical consideration of the properties of a liquid as a function of forces varying in proportion to powers of the distance is the fact that in this way it becomes impossi-

we must take into account the results that follow from the complex form of molecules, which, according to structural formulas, characterizes organic substances. Conversely, if it is assumed that the forces between molecules act only at the points of contact, then our problem is mathematically simplified to such an extent that it will be possible to take into account those data concerning the form of molecules which are known to us from chemistry. Of course, in doing so we introduce a number of approximations, but the errors we make will be far smaller than those usually admitted by physicists in considering these problems. I now wish to indicate a path by which a conception of surface forces may be developed, leading to a concrete representation and to a quantitative solution of many problems too difficult to solve by power series. The fact that interaction between molecules takes place at the surface of contact does not necessarily entail the conclusion that the nature of the forces at each given point of the surface is wholly characterized by the atom situated beneath this surface. From chemistry it is known, for example, that the properties of an organic molecule are not simply the sum of the effects of all the individual atoms composing the molecule. If one of the hydrogen atoms of the methyl group of acetic acid is replaced by a chlorine atom, then the greater charge of the nucleus of the chlorine atom, in comparison with the hydrogen atom it replaces, will displace the electrons of the carboxyl group in the direction of the chlorine atom. This displacement of the pair of electrons holding the nucleus of the hydrogen atom of the carboxyl group weakens the bond of the hydrogen nucleus and thus facilitates its transfer to a water molecule (in aqueous solutions of acetic acid), leading to the formation of the ion \(\mathrm{OH}_3^+\), i.e. of the hydrogen ion characteristic of acids. Thus we have a clear indication that, as a result of the replacement of a hydrogen atom by a chlorine atom, there occurs a change in the surface forces, which may spread over the entire surface of the molecule. In order to obtain a clear idea of the magnitude of the changes in the surface forces as a result of the effect,

transmitted from atom to atom in the molecule, it is sufficient to consider chlorinated fatty acids. Table 2 gives the values of the dissociation constants \(K\) for some of these acids, taken from the Landolt-Börnstein tables.

Table 2

Dissociation constants of chlorinated fatty acids

In the third column, under the designation \(\ln r\), are given the natural logarithms of the ratios of \(K\) for the given acid to the value for normal acetic acid.

According to Boltzmann’s equation, this quantity must be equal to \(\frac{\lambda}{kT}\), where \(\lambda\) represents the difference in the work required to detach a hydrogen ion from the molecule of the given acid and from the molecule of acetic acid. Since \(kT\) is equivalent to 0.025 volts, we can obtain the value of \(\lambda\) in volts by multiplying the value of \(\ln r\) by 0.025. The last column gives these values.

Comparing the monochlorinated acids with one another, we see that all three acids in which chlorine is in the \(\alpha\)-position give \(\lambda = 0.105\) volts. The two acids containing chlorine in the \(\beta\)-position give \(\lambda = 0.039\) volts. The only gamma-chlorinated acid for which we have data gives \(\lambda = 0.012\) volts. *

Proceeding from theoretical premises, one should expect that any stress transmitted from atom to atom along the hydrocarbon chain will decrease according to an exponential law with increasing distance. The data mentioned above thus lead us to the conclusion that

the electrical polarization caused by the presence of a chlorine atom in a hydrocarbon chain decreases in the ratio \(2.7 : 1\) when passing from one carbon atom to the neighboring one. If we replace the chlorine atom by other radicals, we may obtain electrical forces of different magnitude; however, the ratio \(2.7 : 1\) will remain in force as the force decreases from atom to atom. Thus the first atom transmits to the next a considerable effect, of which, however, very little remains when it is transmitted to the second or third atom.

This result allows us to ascribe definite properties to different parts of the surface of molecules of aliphatic compounds. For example, the force field around the carboxyl group in a fatty acid must be independent of the length of the hydrocarbon chain, if it consists of more than two carbon atoms. This means that the forces near the surface of the carboxyl group will be practically constant for all acids higher than propionic acid and approximately the same for acetic acid, but may be very different for formic acid. In the same way, it is impossible to regard, with any degree of accuracy, the force field around the carboxyl group simply as the result of the superposition of the effects of the hydroxyl and carbonyl groups, since both these groups are in such close contact with one another in the molecule that they cannot fail to influence one another.

The Principle of Independent Surface Action

From the above analysis of the forces acting between molecules of different types, one may conclude that there exists a large group of substances in which the forces acting between two molecules in contact may be regarded as depending chiefly on the nature of the surface of the molecules in contact.

This principle of independent surface action (12) will always be only an approximation to the truth; however, in

in many cases it is applicable with a quite sufficient degree of accuracy and so simplifies the problem of the interaction between molecules that it makes it possible to obtain useful results in a number of complicated problems not amenable to solution by other means. Let us consider some of these problems.

An example of the application of this principle is the theory of adsorption of monomolecular films on solids and liquids (13). According to this theory, the forces holding an adsorbed molecule or atom on a surface depend on the character of the surface of contact between the molecule and the solid body. If a second layer of molecules is formed, then the forces holding a molecule in the second layer will thus be quite different from those that hold molecules in the first layer. The rates of evaporation of molecules from the first and from the second layer will be very different, especially because they depend on the magnitude of the forces according to an exponential law of the Boltzmann type. Let us consider two general cases. If the forces holding the molecules in the first layer are greater than the forces holding the molecules in the second, then over a wide range of pressures of the adsorbed gas the thickness of the film will not exceed the diameter of one molecule. Thus we arrive at the conclusion that some fraction of the surface \(\Theta\) is covered by adsorbed molecules and that the properties of this adsorbed film depend above all on \(\Theta\). If the forces acting between these molecules and the surface lying beneath them are much greater than the force acting between adjacent molecules, then we may conclude that the rate of evaporation of the adsorbed molecules will be proportional to \(\Theta\). This simple assumption leads to an adsorption isotherm which in many cases has been confirmed experimentally. If, however, the forces acting between adjacent adsorbed molecules are not negligibly small, then significant deviations from this simple law will be observed; however, the theory can easily be extended to this case as well.

These forces may prove to be forces of attraction or else

repulsion; the latter—in those cases when the molecules have been transformed into dipoles as a result of adsorption, or when they are present at a very high concentration.

A second important case of (14) may be considered to be the case when the forces holding the second layer of molecules are greater than those holding the first, or, speaking more generally, when the forces acting between neighboring adsorbed molecules are greater than the forces holding each of the adsorbed molecules on the surface lying beneath it. In such cases it is very difficult to obtain any appreciable quantity of molecules in the first adsorption layer, and individual molecules, atoms, or groups of molecules in the first layer act as nuclei around which larger aggregates or crystals may form. An example of this type is the condensation of cadmium or mercury vapors on the cooled surface of glass. This problem can be formulated quantitatively, and the number of nuclei forming per second can be expressed as a function of temperature and pressure. The calculated values agree well with those obtained experimentally. If a definite number of isolated copper atoms evaporates and deposits on a clean surface, each of them will be a nucleus for the formation of cadmium crystals, provided that the surface temperature and the pressure of cadmium vapor are precisely regulated.

Thus, copper atoms can be directly counted by illuminating the surface in a dark field. Experiments carried out several years ago by Harold Mott-Smith showed that this method, in its further development, may be used for the exact counting of atoms, just as Wilson’s method is used for counting ions.

In the case where atoms adsorbed on a metallic surface are electrically charged or acquire large dipole moments, electrical forces appear which may render the principle of independent surface action inapplicable. For example, the presence of adsorbed thorium (15) or cesium (16) colos-

greatly increases the rate of evaporation of electrons from a tungsten surface at high temperatures. However, the increase in the number of evaporating electrons is not even approximately proportional to the amount of thorium or cesium present on the surface. The same is true for the evaporation of positively charged cesium ions from a tungsten surface. In both cases the heat of evaporation of the electrons or ions varies approximately linearly as a function of $\Theta$—the fraction of the surface covered by adsorbed atoms—and thus, according to Boltzmann’s equation, the logarithm of the rate of evaporation of electrons or ions varies linearly as a function of $\Theta$. In the case of the evaporation of neutral atoms—for example, oxygen atoms—from adsorbed films on incandescent tungsten filaments, the rate of evaporation is much closer to proportionality to $\Theta$.

In investigations of the surface tension of organic liquids (2) (17), as well as of the properties of adsorbed films of organic substances on water (3) (4) and of oil films on water, the principle of independent surface action has proved especially fruitful. It directly provides simple arguments in favor of the assumption that the thickness of these films rarely exceeds one molecule; more precisely, the properties of films having a thickness of more than one layer should differ markedly from the properties of those films which do not completely cover the surface even with a single layer of molecules. This theory also gives direct arguments in favor of the assumption that the adsorbed films of many organic substances on water consist of oriented molecules. Thus, the area occupied by molecules of various fatty acids on the surface of water depends on the interaction between the carboxyl group and the water and does not depend on the length of the hydrocarbon chain. This simple result, confirmed experimentally, shows that the molecules are oriented in such a way that the carboxyl group is in contact with the water, while the hydrocarbon tails form above the carboxyl group a layer possessing all the characteristic proper—

surfaces of a liquid hydrocarbon. This theory is also applicable to pure organic liquids. The total surface energies (surface tension extrapolated to absolute zero) of hexane and hexyl alcohol are practically identical. This is easily explained by the orientation of the molecules, which does not allow the hydroxyl groups to come into contact with the surface, so that in both cases the surface is in fact the surface of a pure hydrocarbon.

Studies of the surface tension of aqueous solutions of various aliphatic compounds show that, in concentrated solutions, the surface proves to be covered with a dense monomolecular film consisting of closely adjacent, mutually oriented molecules, as is observed in oil films. However, in sufficiently dilute solutions the amount of substance adsorbed at the surface is insufficient to cover the entire surface with closely adjoining molecules. Under these conditions the hydrocarbon chain lies flat on the surface of the water, and the energy required to transfer any molecule from the surface into the interior increases by a definite amount for each additional \(CH_2\) group of the hydrocarbon chain. This is an excellent example of the application of the principle of independent surface action, since each \(CH_2\) group produces its effect independently of the others.

This theory also proves very useful in studying other properties of matter. Let us consider, for example, gels containing gelatin or soap in concentrations of less than one percent by weight. Their elasticity demonstrates the existence of a continuous network structure consisting of contacting molecules and extending throughout the entire liquid. The relaxation time of these gels serves as a measure of the rate at which the chain-forming molecules separate from one another or “evaporate.”

More than ten years ago, guided by this theory, I carried out several experiments to determine the diameters of the cross sections of fibers that are ele-

by elements of this network structure. For this purpose I prepared on filter paper several diluted gelatin gels and measured the rate at which water passed through them under a definite pressure.

A modification of Stokes’ law, which gives the velocities of fall of small spheres in liquids, enabled me to calculate the force necessary for moving small cylinders of various diameters through a liquid. This law was then checked experimentally by passing water through a column of glass wool with fibers of known size. Applying this law to the case of the passage of water through a gelatin gel, it was possible to calculate the size of the fibers. Only approximate experiments were made, but the results clearly showed that the diameter of the fibers approaches \(10^{-7}\) cm.

I think that it may with full justification, at any rate as a first approximation, be assumed that the fibers of gels obey the same laws as bodies of comparatively large dimensions, such as, for example, the fibers of glass wool.

Einstein showed many years ago that Stokes’ law can be applied, with some approximation, to the study of the mobility of ions in aqueous solutions. I think that a thorough quantitative investigation of the forces required to push water through various gels could yield much valuable information concerning the structure of these gels.

Very useful conceptions of the mechanism of diffusion in liquids and solids, as well as of the viscosity of liquids, can be obtained if one takes into consideration that contacting molecules may be in two states: the first, in which the surfaces are immovably joined to one another, and the second—when they can move quite freely.

The behavior of molecules is thus analogous to the behavior of a gas condensing on a solid body. Gas molecules strike a surface, remain adsorbed for a certain time, and then evaporate again. Applying this conception to molecules in a liquid, we see that

motion of molecules past one another, which gives rise to viscosity or diffusion, depends on the ratio of the durations for which the molecules are in an immobile and a mobile state of contact; these durations can be calculated from an equation of the Boltzmann type, into which enters the difference of energies between the two states.

This theory explains the fact of the frequent agreement of the temperature coefficient of viscosity and diffusion with Boltzmann’s equation. This is apparently accounted for by the fact that the viscosity of the different members of a hydrocarbon series increases in geometric progression with the length of the chain. It is possible that the principle of the independence of surface action will make it possible to develop a simple theory of viscosity that would accurately take into account the form and size of molecules, as well as the different chemical groups entering into their composition.

This principle has also been repeatedly applied to the mechanism of heterogeneous chemical reactions. The study of the interactions of oxygen with hydrogen and of carbon monoxide with oxygen at low pressures in the presence of a heated platinum wire (18) showed that equations based directly on this principle agree excellently with the experimental results over wide ranges of temperature and pressure. Thus, for example, experiments showed that at low temperatures the reaction rate is exactly proportional to the partial pressure of oxygen and inversely proportional to the pressure of carbon monoxide. This behavior can be fully explained if it is assumed that the reaction rate is determined by the rate at which oxygen molecules can reach the openings in the films of adsorbed carbon monoxide that remain when carbon monoxide molecules evaporate, or that are formed when carbon monoxide is displaced by oxygen that has reached these openings; the poisoning action of carbon monoxide is explained by the fact that the possibility for oxygen to reach these openings would be diminished if the carbon monoxide molecules were able to fill the openings before the arrival of oxygen. Apparently, in these experiments, to achieve

quantitative agreement is possible only by applying the principle of independent surface action.

Another illustration of this phenomenon is the interaction of hydrogen with oxygen in the presence of tungsten filaments at temperatures from 1500 to 2500° \(K\). Oxygen forms \(\mathrm{WO}_3\) at a rate proportional to the pressure of oxygen (19). The oxygen atoms in the adsorbed oxygen film do not react with one another or with tungsten (with the formation of \(\mathrm{WO}_3\)), even at the highest temperatures. At 1500° \(K\) the lifetime of oxygen atoms on the surface is measured in years; at 1860° \(K\) it is approximately 25 min., and at 2070° \(K\)—15 sec. The atoms that leave the film at high temperatures are free atoms, not molecules. Thus, even at negligible oxygen pressures the surface of tungsten is practically completely covered by a single layer of oxygen atoms. Oxygen molecules striking this surface condense on it, but evaporate comparatively quickly from the second layer thus formed; however, while they are in the adsorbed state, they move freely over the surface and are able to fill all the holes that form in the first layer. Molecules adsorbed in the second layer have a certain probability of interacting with molecules of the first layer, and also with the tungsten surface (with the formation of \(\mathrm{WO}_3\)); the holes that are formed in this way, owing to the removal of oxygen from the first layer, are quickly filled by molecules passing from the second layer.

Hydrogen molecules at no temperature are capable of reacting directly with oxygen, either in the first or in the second layer (10), but if they reach the tungsten through one of the holes in the first layer, they immediately react with neighboring oxygen atoms.

Thus, when the pressure of oxygen falls to some low critical value, hydrogen displaces all the oxygen from the surface, and this occurs suddenly; the hydrogen molecules which then enter

give onto the film dissociate into atoms, the degree of dissociation depending on the temperature; the atomic hydrogen thus formed passes to the walls of the vessel (the bulb) and reacts with the \( \mathrm{WO_3} \), which had previously been deposited on them.

This theory, which can easily be expressed quantitatively, is apparently in full agreement with the experimental facts; this constitutes a new confirmation of the principle of independent surface action.

Molecules in adsorbed films on solutions of organic substances in water are often in the state of a two-dimensional gas. Oil films occur in the form of solid or liquid films. In the case of so-called expanded films (12), the “heads” of the molecules act as a two-dimensional gas, while the “tails” form a two-dimensional liquid.

Molecules of oils adsorbed on the surface of solids, such as, for example, those that account for certain lubricating properties of oils, or those associated with the flotation of ores, usually show little or no tendency to move over the surface. In other words, they are apparently firmly attached to the surface.

Many equations derived in the quantitative study of the rates of heterogeneous reactions assumed that the “bare spots” on the surface are distributed according to statistical laws. Such a statistical distribution of molecules would be possible only if the molecules possessed some degree of mobility over the surface, since the reaction removes molecules located adjacent to already existing holes. Folmer and others have shown experimentally that adsorbed atoms often possess great mobility even on solid surfaces and thus act as two-dimensional gases of high viscosity.

I believe that the principle of independent surface action will be very useful in the study of many properties of organic substances which, owing to their complexity,

could not until now be investigated quantitatively. In recent years I have made several attempts in this direction.

When a liquid is divided into two parts along a surface whose area is \(1\ \mathrm{cm}^{2}\), two new surfaces are obtained, with a total area of \(2\ \mathrm{cm}^{2}\). The surface tension, or free surface energy, measures the work of formation of new surfaces expended per unit area. The total surface energy \(\gamma\) is equal to the free surface energy extrapolated to absolute zero, and represents the total change of energy per unit area. For all pure hydrocarbons—for example, pentane and nonane—\(\gamma\) is approximately equal to \(48\ \mathrm{erg}\times \mathrm{cm}^{-2}\). The fact that this value is practically independent of the length of the hydrocarbon chain proves that the surface forces are almost uniform over the whole hydrocarbon molecule. When such a hydrocarbon as hexane evaporates, the molecules pass from the interior of the liquid into the vapor. The work required to transfer a drop of hexane from a large volume of liquid into free space above it is equal to \(S\gamma\), where \(S\) is the surface of the drop. Since we assume that molecules have a surface endowed with definite properties, it may be said that the work required to remove one molecule of hexane from the liquid, i.e. the latent heat of evaporation per molecule, will likewise be equal to \(S\gamma\), where \(S\) is now the surface of the molecule, and \(\gamma\) is the surface energy of the molecule per unit surface.

The molecular surface \(S\) for a vapor molecule can be calculated from the molecular volume (molecular weight divided by density), if the surfaces are assumed to be the same as in the case of a “close packing” of spheres. This assumption is fairly accurate for large molecules, but is only a rough approximation for short hydrocarbon chains. The values of \(\gamma\) can be calculated from the known values of the latent heats of evaporation. Practically for all normal hydrocarbons, with the exception of methane, the value is

\[ \gamma = 34 \pm\ \mathrm{erg}\times \mathrm{cm}^{-2}. \]

This quantity is

of the same order of magnitude as the value 48 found from measurements of surface tension. The fact that \(\gamma\) proves to be constant shows that the work expended in transferring molecules from the liquid phase into the vapor phase is strictly proportional to the molecular surface. In other words: the latent heat of vaporization is proportional to the molecular volume to the two-thirds power.

This theory can easily be extended to the case of the heats of vaporization of various aliphatic alcohols. If \(S\) is the total surface of a vapor molecule, then \(aS\) represents the surface of the “head” of the molecule (the hydroxyl group), and \(cS\) the area of the “tail” (the hydrocarbon chain). Let \(\gamma_a\) and \(\gamma_c\) denote the surface energy per unit area of the “head” and “tail,” respectively, while the molecule is in the vapor phase. Then \(Sa\gamma_a\) will be the total energy of the “head,” and \(Sc\gamma_c\) the total energy of the “tail.”

If the alcohol molecules in the liquid phase were arranged at random, i.e., if they did not orient themselves appreciably with respect to one another and did not tend to form clusters, it could easily be shown that the total surface energy at the interface in the liquid between hydroxyl groups and hydrocarbon chains would be equal to \(Sac\gamma_{ac}\), where \(\gamma_{ac}\) is the surface energy per unit interfacial area.

Therefore we shall be able to calculate the latent heat of vaporization from the difference between the energies in the vapor and liquid phases.

\[ \lambda = S(a\gamma_a - ac\gamma_{ac} + c\gamma_c). \]

If we take \(\gamma_a = 193\), \(\gamma_c = 34\), and \(\gamma_{ac} = 34\), then an excellent agreement is obtained between the observed heats of vaporization and the structure of most monohydric alcohols. We have already seen that \(\gamma_c\) has the value 34, equal to that known for the vaporization of pure hydrocarbons. The value \(\gamma_{ac} = 34\) was found as a result of experiments on the vapor pressure of mixtures of alcohol with water. The surface energy at the interface between water and hydrocarb—

is equal to approximately 59, so that the value 34 is of the proper order of magnitude. The surface energy of water is 117, but, naturally, this is the surface energy of the least active part of the water molecule, whereas the energy \(\gamma_a\), equal to 193, corresponds to the most active part of the hydroxyl group, so that this value too is apparently quite reasonable.

In the case of alcohols having very long hydrocarbon chains, it is not difficult to discern yet another effect from the experimental results. In the vapor molecule the hydroxyl group is often inclined to hide (in any case, partly) among the coils of the hydrocarbon tail, so that the surface energy of the vapor molecules is considerably reduced, beginning with chains five or six carbon atoms long. In shorter chains the hydroxyl group is, in all probability, not masked at all. Especially interesting results, in good agreement with the principle of independent surface action, are found for di- and trihydric alcohols. The heat of vaporization depends to a large extent on whether the individual hydroxyl groups can come into contact in the vapor molecule and thereby lower the surface energy.

This theory can easily be extended to the calculation of the partial vapor pressures in binary mixtures. A complete theory, taking into account orientation and segregation, would be too complicated, but in many cases, when the forces acting between molecules are not too large, these effects may be neglected in the first approximation. If the distribution and orientation of the molecules are taken according to the laws of probability, the total surface energy per molecule in a solution of any concentration, expressed as a function of the surface energy at the interfaces \(\gamma_{ac}\) and of the partial surfaces \(a\) and \(c\), then one can calculate the work necessary for the transfer of molecules from the liquid phase into the vapor phase and, thus, applying Boltzmann’s equation, one can calculate the deviations from Raoult’s law.

Hence, for the partial pressure of a liquid in a binary mixture one obtains:

\[ P_A = A P_A e^{\frac{\varphi S_A \beta^2}{kT}}, \]

where \(P_A\) is the partial pressure of substance \(A\), \(P_A\) is the total pressure of the pure component \(A\); the molar concentration of component \(A\) is represented in this equation by \(A\). \(S_A\) is the surface corresponding to one molecule; \(\varphi\) is a constant characteristic of the binary mixture but independent of the concentration of the components, which can be calculated from \(a\), \(c\), \(\gamma_a\), etc. The quantity \(\beta\) may be called the partial surface of component \(B\) in the binary mixture \(A\) and \(B\). It corresponds to the usual conception of molar concentration, but is expressed not by the ratio of the numbers of molecules, but by the ratio of their surfaces.

Thus,

\[ \beta = \frac{B S_B}{A S_A + B S_B}. \]

This equation, with only one arbitrary constant \(\varphi\), apparently agrees in general better with experimental data than the somewhat similar equation with two arbitrary constants derived by van Laar on the basis of thermodynamic considerations.

Smyth (C. P. Smyth) has recently applied this equation in connection with measurements of the vapor pressures of binary mixtures. The agreement is for the most part quite satisfactory, but, as was to be expected, more polar molecules, for example in mixtures of alcohol with water, give considerable deviations. It is possible that, with further development of the theory, these deviations may for the most part be connected with the orientation and association of molecules in the liquid.

Thus, the views set forth by me may be applied, often with a high degree of accuracy, in studying interactions between molecules of organic substances. The energetic relationships based on the conception of surface forces between molecules, together with the Boltzmann equation, often make it possible to draw conclusions concerning the mechanism of various surfac-

phenomena. For example, it has often been supposed that in stretched oil films on water the molecules can remain in an upright position at the surface without touching. Simple energetic considerations of the kind we have used up to now immediately show that this is impossible. In the case of long chains the molecules may be in contact with one another; with short chains they may be separated from one another, but in that case they must lie flat on the surface of the water. The principle of independent surface action gives us a very powerful weapon against impossible hypotheses.

LITERATURE

1) Hardy H. B., Proc. Roy. Soc. (London) 86A, 634 (1912), 88A, 330 (1913).

1. Langmuir I., Chem. Met. Eng. 15, 468 (1916).

2. Langmuir I., J. Am. Chem. Soc. 39, 1848 (1917).

3. Langmuir I. Colloid Chemistry, ed. by Jerome Alexander, 525–46 (1926).

4. Born M. and Landé A. Verhandl. deutsch. physik. Ges. 20, 210 (1918).

5. Born M., Verhand. d. phys. Ges. 20, 230 (1918).

6. Debye P., Physik. Zeitschr., 22, 302 (1921).

7. Debye P., and Hückel E., Physik. Z. 24, 185, 305 (1923).

8. Debye P., Physik. Z. 21, 178 (1920).

9. Debye P., Polar Molecules, pp. 8, 12, 29, 43, N-Y (1929).

10. Debye P., Handbuch der Radiologie 6, 633–43 (1925).

11. Langmuir I., Colloid Symposium Monograph, vol. III, 1925).

12. Langmuir I., J. Am. Chem. Soc. 40, 1361 (1918).

13. Langmuir I., Proc. Nat. Acad. Sc. 3, 141 (1917).

14. Langmuir I., Phys. Rev. 22, 357 (1923).

15. Langmuir I., and Kingdon K. H., Proc. Roy. Soc. (London) 107A, 61 (1925).

16. Harkins W. D., Brown and Davies, J. Am. Chem. Soc. 39, 354 (1917). Harkins, Davies and Clark J. Am. Chem. Soc. 39, 541 (1917); see also the review of works: Harkins Colloid Chemistry (J. Alexander) (1926).

17. Langmuir I., Trans. Faraday Soc. 17, 621 (1921).

18. Langmuir I., J. Am. Chem. Soc. 37, 1148, (1915).

19. Langmuir I., J. Am. Chem. Soc. 38, 2272 (1916).

1. The term “molecule” is used in a broad sense and includes the concepts of atom and ion. ↩↩