[Reader's Note: This paper is incomplete.]

I. FREEZING POINT DEPRESSION AND RELATED PHENOMENA.

(Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen)

P. Debye and E. Hückel

(Submitted February 27, 1923)

Translated from

*Physikalische Zeitschrift,* Vol. 24, No. 9, 1923, pages 185-206

The present considerations were stimulated by a lecture by E. Bauer on Ghosh's works, held at the Physikalische Gesellschaft. The general viewpoints taken as the basis for the computation of the freezing point depression as well as of the conductivity lead me, among other things, to the limiting law involving the square root of the concentration. I could have reported on this during the winter of 1921 at the "Kolloquium." With the active assistance of my assistant, Dr. E. Hückel, a comprehensive discussion of the results and their collection took place during the winter of 1922.

It is known that the dissociation hypothesis by Arrhenius explains the abnormally large values of osmotic pressure, freezing point depression, etc., observed for solutions of electrolytes, by the existence of free ions and the associated increase in the number of separate particles. The quantitative theory relies on the extension, introduced by van 't Hoff, of the laws for ideal gases to diluted solutions for the computation of their osmotic pressure. Since it is possible to justify this extension on the basis of thermodynamics, there can be no doubt regarding the general validity of these fundamentals.

However, for finite concentrations we obtain smaller values for freezing point lowering, conductivity, etc. than one would expect on first consideration, in the presence of a complete dissociation of the electrolyte into ions. Let P_{k}, for instance, be the osmotic pressure resulting from the classical law by van 't Hoff for complete dissociation, then the actually observed osmotic pressure will be smaller, so that:

P = f_{o}P_{k}

where, according to Bjerrum,^{1} the "osmotic coefficient" f_{o} thus introduced is intended to measure this deviation independent of any theory-and can be observed as a function of concentration, pressure, and temperature. In fact, these observations do not relate directly to the osmotic pressure but to freezing point lowering, and boiling point rise, respectively, which can both be derived on the basis of thermodynamics, and by means of the same osmotic coefficient f_{o}, from their limiting value following from van 't Hoff's law for complete dissociation.

The most evident assumption to explain the presence of the osmotic coefficient is the classical assumption, according to which not all molecules are dissociated into ions, but which assumes an equilibrium between dissociated and undissociated molecules which depend on the over-all concentration, as well as on pressure and temperature. The number of free, separate particles is thus variable, and would have to be made directly proportional to f_{o}. The quantitative theory of this dependence, as far as it relates to the concentration, relies on the mass action law of Guldberg-Waage; the dependence on temperature and pressure of the constant of equilibrium appearing in this law can be determined thermodynamically, according to van 't Hoff. The complete aggregate of dependencies, including the Guldberg-Waage law, can be proved by thermodynamics, as is shown by Planck.

Since the electric conductivity is determined exclusively by the ions, and since, according to the classical theory the number of ions follows immediately from f_{o}, the theory requires the well known relation between the dependence on the concentration of the conductivity on the one hand and of the osmotic pressure on the other hand.

A large group of electrolytes, the strong acids, bases, and their salts, collectively designated as "strong" electrolytes, exhibits definite deviations from the dependencies demanded by the classical theory. It is especially noteworthy that these deviations are the more pronounced the more the solutions are diluted. [A summary presentation of this subject was given by L. Ebert, "Forschungen ueber die Anomalien starker Elektrolyte," *Jahrb. Rad. u. Elektr.,* 18, 134 (1921).] Thus, as was recognized in the course of developments and following the classical theory, it is possible only with a certain degree of approximation to draw a conclusion from f_{o} as to the dependence of the conductivity on the concentration. Moreover the dependence of the osmotic coefficient f_{o} on the concentration is also represented entirely incorrectly. For strongly diluted solutions, f_{o} approaches the value 1; if 1 - f_{o} is plotted as a function of the concentration *c*, classical theory requires for binary electrolytes, such as KCl, that this curve meets the zero point with a finite tangent (determined by the constant of equilibrium, K). In the general case, provided the molecule of the electrolyte splits into n ions, we obtain from the law of mass action for low concentrations:

1 - f_{o} = ((n - 1) / n) times (c^{n - 1} / K)

so that in cases where splitting into more than two ions occurs, the curve in question should present even a higher order of contact with the abscissa. The complex of these dependencies constitutes Ostwald's dilution law.

Actually observations on strong electrolytes show an entirely different behavior. The experimental curve starts from the zero point at a right angle (cf. Figure 2) to the abscissa, independent of the number of ions, n. All proposed, practical interpolation formulas attempt to represent this behavior by assuming 1 - f_{o} to be proportional to a fractional power (smaller than 1, such as 1/2 or 1/3) of the concentration. The same remark holds with regard to the extrapolation of the conductivity to infinite dilutions which, according to Kohlrausch, requires the use of the power 2.

It is clear that under these circumstances the classical theory can not be retained. All experimental material indicates that its fundamental starting point should be abandoned, and that, in particular, an equilibrium calculated on the basis of the mass action law does not correspond to the actual phenomena.

W. Sutherland,^{2} in 1907, intended to build the theory of the electrolytes on the assumption of a complete dissociation. His work contains a number of good ideas. N. Bjerrum^{3} is, however, the first to have arrived at a distinct formulation of the hypothesis. He clearly stated and proved that, for strong electrolytes, no equilibrium at all is noticeable between dissociated and undissociated molecules, and that, rather, convincing evidence exists which shows that such electrolytes are completely dissociated into ions up to high concentrations. Only in considering weak electrolytes, undissociated molecules reappear. Thus the classical explanation as an exclusive basis for the variation of, for instance, the osmotic coefficient, has to be abandoned and the task ensues to search for an effect of the ions, heretofore overlooked, which explains, in the absence of association, a decrease in f_{o} with an increase in concentration.

Recently, under the influence of Bjerrum, the impression gained strength that consideration of the electrostatic forces, exerted by the ions on one another and of considerable importance because of the comparatively enormous size of the elementary electric charge, must supply the desired explanation. Classical theory does not discuss these forces, rather, it treats the ions as entirely independent elements. A new interaction theory has to be analogous in some respects to van der Waals' generalization of the law of ideal gases to the case of real gases. However, it will have to resort to entirely different expedients, since the electrostatic forces between ions decrease only as the square of the distance and thus are essentially different from the intermolecular forces which decline much more rapidly with an increase in distance.

Milner^{4} computed the osmotic coefficient along such lines. His computation can not be objected to as regards its outline, but it leads to mathematical difficulties which are not entirely overcome, and the final result can only be expressed in the form of a graphically determined curve for the relation between 1 - f_{o} and the concentration. From the following it will further emerge that the comparison with experience, carried through by Milner, supposes the admission of his approximations for concentrations which are much too high and for which, in fact, the individual properties of the ions, not taken into account by Milner, already play an important part. In spite of this, it would be unjust to discard Milner's computation in favor of the more recent computations by Ghosh^{5} on the same subject. We shall have to revert, in the following, to the reason why we can not agree to Ghosh's calculations, neither in their application to the conductivity nor in their more straight forward application to the osmotic pressure. We will even have to reject entirely his calculation of the electrostatic energy of an ionized electrolyte, which is the basis for all his further conclusions.

The circumstances to be considered in the computation of the conductivity are very similar to those for the osmotic coefficient. Here also the new interaction theory has to make an attempt at understanding the mutual electrostatic effect of the ions with regard to its influence on their mobility. An earlier attempt was made in this direction by Hertz.^{6} He transcribes the methods of the kinetic theory of gases, and, in fact, finds a mutual interference of the ions. However, the transcription of this classical method, and particularly the use of concepts like that of the free path length of a molecule in a gas for the case of free ions surrounded by the molecules of the solvent, does not seem to be very reliable. The final result obtained by Hertz cannot, in fact, be reconciled with the experimental results.

In this first note, we shall confine ourselves to the "osmotic coefficient f_{o}" and to a similar "activity coefficient f_{a}," used by Bjerrum^{7} and stressed in its significance. Even for such (weak) electrolytes, where a noticeable number of undissociated molecules is present, the equilibrium cannot simply be determined by the Guldberg-Waage formula in its classical form:

c_{1}^{m1} c_{2}^{m2} . . . c_{n}^{mn} = K

(c_{1}, c_{2}, . . . c_{n}, are the concentrations, K the constant of equilibrium). It will be necessary, in view of the mutual electrostatic forces between the ions, to write:

f_{a}K

instead of K, introducing the activity coefficient* f_{a}. This coefficient, just as f_{o}, will depend on the concentration of the ions. Though, according to Bjerrum, a relation to be proved by thermodynamics exists between f_{a} and f_{o}, their dependence on the concentration is different for the two coefficients.

* The activity coefficient f

The detailed treatment of conductivity shall be reserved for a later note. This division seems justified, since the determination of f_{o} and f_{a} requires solely a consideration of reversible processes, whereas the computation of mobilities has to do with essentially irreversible processes for which no direct relation to the fundamental laws of thermodynamics exists.

**II. Fundamentals**

As is well known, it is shown in thermodynamics that the properties of a system are completely known, provided one of the many possible thermodynamic potentials is given as a function of the correctly chosen variables. In view of the form in which the term based on the mutual electric effects will appear we choose the quantity:**

** The potential G differs from Helmholtz' free energy F = U - TS only by the factor -1/T. This difference is not essential at all; we define it as in the text; to have immediate connection with Planck's thermodynamics.

**[The remainer of the paper will appear here in stages.]**

**X. General Remarks**

From the preceding discussion it may be concluded that it is inadmissible from a theoretical as well as from an experimental point of view to consider the electric energy of an ionic solution to be essentially determined by the average mutual distance of the ions. Rather, a quantity which measures the thickness of the ion atmosphere or, to connect with something known better, the thickness of a Helmholtz double-layer proves to be a characteristic length. In view of the fact that this thickness depends on the concentration of the electrolyte, the electric energy of the solution also becomes a function of this quantity. The fact that this thickness is inversely proportional to the square root of the concentration is responsible for the characteristic appearance of the limiting laws for highly diluted solutions. Though we must decline to talk in terms of a lattice structure of the electrolyte in the conventional sense, and though, as shown by the development of the subject, taking this image too literally leads to inadmissible mistakes, it still contains a grain of truth. To make this clear, the following two imaginary experiments are carried out. First, we take an element of space, and consider it placed, repeatedly, at arbitrary positions in the electrolyte. It is clear that, in a binary electrolyte, we shall find therein positive and negative ions with equal frequency. Second, we take the same spatial element, and again place it repeatedly in the electrolyte, now not arbitrarily, but always such that it is, for instance, located at a definite distance (of several angstrom units) from an arbitrarily selected positive ion. Now we shall not find positive and negative charges with equal frequency, the negative charges will prevail in number. In that the oppositely charged ions, on the average, prevail in number in the immediate surroundings of each ion, we can see, correctly, an analogy to the crystal structure of the NaCl type, where each Na ion is immediately surrounded by 6 Cl ions and each Cl ion by 6 Na ions. However, it is to be considered an essential characteristic of the electrolytic solution that the measure for this order is determined by the thermal equilibrium between attracting forces and temperature movement, while it is definitely predetermined for the crystal.

The computations and comparison with experience were carried out by taking the conventional dielectric constant for the surrounding solvent. The success justifies this assumption. Though this procedure is justifiable for low concentrations, it should cause mistakes for higher concentrations. In fact, it follows from dipole theory that for high field intensities, dielectrics must show saturation phenomena similar to the known magnetic saturation. The recent experiments by Herweg^{15} may be taken as an experimental confirmation of this theoretical requirement. Since at a distance of 10¯^{7} cm. from a singly charged ion, a field intensity of approximately 200,000 volt/cm. is to be expected, we should be prepared to observe something of these saturation phenomena. It would, of course, be very interesting if an attempt to separate this effect in its consequences from the observations were successful, the more so that nature puts at our disposal field intensities of a magnitude hardly attainable otherwise with conventional experimental means.

In another respect concentrated solutions should show a special behavior. If many ions are present in the surroundings of each single ion, this can be regarded as a change of the surrounding medium with respect to its electrical properties, an effect which has not been taken into account in the preceding theory. The manner in which this may become effective may be indicated by the following considerations. Let us consider one fixed ion and another mobile ion, oppositely charged, and investigate the amount of work required to remove the mobile ion. This work may be regarded as composed of two parts: (1) the ion will require a certain amount of work for its removal, and (2) we shall gain work by filling the space, previously taken up by the ion, with solvent. Experiments concerning the heat of dilution actually provide an indication of the existence of such conditions. Let us take, for example, a HNO_{3} solution of initially low concentration and dilute it with a large quantity of water (i.e., so much that further dilution would not cause any heat effect), cooling will take place, i.e., work must be done in the sense of the previous considerations to separate the ions from one another. If the initial solution has a higher concentration, then, in the same experiment, heat is generated, i.e., work is obtained, if the surrounding of each ion is freed of a sufficient number of other ions which are replaced by water molecules. In conventional language, it is said that a predominant hydration of the ions occurs, and that this is to be regarded as an exothermic process. Obviously the above considerations intend an explanation of this so called hydration on a purely electric basis. In fact an approximate computation can be carried through which gives theoretically Berthelot's rule, valid in this connection for the dependence of the heat of dilution from the initial concentration, and which makes plausible the order of magnitude of the experimentally determined numerical coefficient of this rule. These considerations have some bearing on the freezing point observations inasmuch as they suggest the possibility of computing why and to what extent the curves found for the percentage deviation **Q** (compare the case of KCl) bend downward for higher concentrations and may even cross the abscissa provided the concentration is high enough. In this instance, the freezing point depression exceeds the one expected from classical theory (also, as may be stated explicitly, if the classical theory is used in its unabbreviated form). Until now, one has been resigned, in such cases, to talk about hydration.

However, before conditions for concentrated solutions can be investigated, it must be shown that the irreversible process of electric conduction in strong electrolytes can also be understood quantitatively from our point of view. We reserve the detailed presentation of this subject for a future article. Here only the basic ideas, which will be discussed more thoroughly in that paper, may be indicated. If an ion moving in a liquid is subjected to the influence of an external field, the surrounding ions will have to move constantly in order to form the ion atmosphere. If we now assume for a moment that a charge is suddenly generated in the electrolyte, an ion atmosphere will have to appear which requires a certain time of relaxation for its formation. Similarly, for a moving ion, the surrounding atmosphere will not attain its equilibrium distribution and thus cannot be computed on the basis of the Boltzmann-Maxwell principle. However, the determination of its charge distribution can be carried through on the basis of an obvious interpretation of the equations for the Brownian movement. It can be estimated qualitatively in which direction this effect, caused by the presence of a finite relaxation time, will be operative. At a point in front of the moving ion (i.e., a point toward which it moves) the electric density of the ion atmosphere must increase with time; it must decrease for a point behind the ion. As a consequence of the relaxation time, the density in front of the ion will be slightly smaller than its value at equilibrium; behind it, however, it will not yet have decreased to its equilibrium value. Consequently, during the movement there always exists a slightly larger electrical density of the ion atmosphere behind the ion than in front of it. Since charge density in the atmosphere and charge of the central ion always carry orposite signs, a force braking the ion movement will occur, independent of its sign, and obviously this force will increase with increasins concentration.

This is one effect which operates in the same sense as a decrease in dissociation calculated on the basis of Ostwald's dilution law. However, still another effect is present which must be taken into consideration. In the vicinity of an ion are predominently ions of the opposite sign, which under the influence of the external field will, of course, move in the opposite direction. These ions will, to a certain degree, drag along the surrounding solvent, thus causing the considered single ion not to move relative to a stationary solvent but relative to a solvent moving in the opposite direction. Since, apparently, this effect increases with increasing concentration, we have a second effect operating in the same sense as a decrease in dissociation. The effect can be calculated quantitatively according to the principles used by Helmholtz for the treatment of electrophoresis.

The common factor of the two effects just mentioned consists, as is shown by the computations, in the fact that both are closely related to the thickness of the ion atmosphere, and that, therefore, the generated forces are proportional to the square root of the concentration of the electrolyte, at least in the limit for very low concentrations. Thus we obtain a law, found by Kohlrausch^{16} according to which for low concentrations the percentage deviation of the molecular conductivity from its limiting value at infinite dilution is proportional to the square root of the concentration. Also the proportionality factor thus finds a molecular interpretation

Anticipating the detailed representation of electrolytic conductivity in prospect for a following artiole, we can state as an over-all result that the view, according to which strong electrolytes are completely dissociated, is entirely supported.

[Reader's Note: bibliography to follow.]