ELECTRICAL FORCES BETWEEN IONS IN SOLUTIONS1

Niels Bjerrum, Copenhagen.

When Svante Arrhenius advanced his famous ionic theory in 1887, there soon appeared many works developing this theory further. Nevertheless, the development of one part of this theory—the study of the forces acting between ions—began considerably later. For many years the action of electrical forces between ionic charges was neglected. To be sure, the existence of these interionic forces was not doubted, but attention was quite rightly directed more to the proposition, characteristic of the theory, that free ions behave in general like neutral molecules. The small difference between free ions and molecules caused by the presence of interionic forces was usually neglected. In the last ten years the question of interionic forces has been taken up seriously, thanks to which the classical ionic theory has received a new and interesting extension. In what follows we shall consider several such works.

I.

1. The laws of Coulomb and Faraday give the following expression for the force acting between two ions:

\[ K=\frac{E^2}{Dr^2} \tag{1} \]

(\(E\)—charge, \(r\)—distance between the ions, \(D\)—dielectric constant of the solvent).

For large values of \(r\) this formula is undoubtedly justified. But when \(r\) reaches molecular dimensions, i.e. when there are only very few solvent molecules between two ions, or one, or none at all, then the applicability of this formula becomes doubtful. Here we shall not concern ourselves with the theory of the dielectric constant \(D\) and the question of the applicability of its usual

values for small values of \(D\). We shall content ourselves with considering a series of experimental data indicating that the above equation retains its applicability down to the smallest distances, although perhaps not with complete exactness.

1. W. Ostwald (Will. Ostwald) was the first to draw attention to one of the actions of forces between ions. In 1892 he pointed out that the second dissociation constant of dibasic acids is always smaller than the first. He explained this by the fact that the electrical attraction between the hydrogen ion and the negative charge of the monobasic acid anion \((\mathrm{HR}^{-})\) makes it difficult for the second hydrogen atom to split off. In accordance with this conception, Ostwald found that the difference between the two dissociation constants becomes the greater, the closer to one another the two hydrogen atoms are situated in the acid; thus, for example, in oxalic acid it is greater than in glutaric acid.

Ostwald’s qualitative reasoning can be made quantitative\(^1\). Let us consider solutions of a dibasic symmetric acid \(\mathrm{H}_{2}\mathrm{R}\). Around the negative ion \(\mathrm{HR}^{-}\) the concentration of hydrogen ions is greater than around the molecule \(\mathrm{H}_{2}\mathrm{R}\). According to Boltzmann, the first concentration is greater than the second by a factor

\[ e^{\frac{\varphi}{kT}} \]

\[ \text{times} \]

(\(k\) is Boltzmann’s constant, \(T\) is the absolute temperature, \(\varphi\) is the work which must be expended in order to remove the ions \(\mathrm{HR}^{-}\) and \(\mathrm{H}^{+}\) from one another). According to Coulomb’s law (1), this work (in a sufficiently dilute solution) is, as is known, equal to:

\[ \varphi=\frac{E^{2}}{Dr}, \tag{1} \]

where \(r\) is the distance between the charges of the ions. Hence the ratio of the concentrations of hydrogen ions around \(\mathrm{HR}^{-}\) and \(\mathrm{H}_{2}\mathrm{R}\) is equal to

\[ e^{\frac{E^{2}}{kTDr}}. \]

This leads to the following expression for the ratio between the first and second dissociation constants of acids:

\[ \frac{K_{1}}{K_{2}}=4e^{\frac{E^{2}}{kTDr}}. \tag{2} \]

For more exact calculations in this formula one ought to replace \(r\) by the intramolecular distance between the hydrogen atom and the negative charge in the singly charged ion \(\mathrm{HR}^{-}\). The number 4 is a statistical factor, appearing because, first, in the acid

\(^1\) Bjerrum. Zeitschr. f. physik. Chemie 106, 219 (1923).

H₂R has two detachable hydrogen ions, while, in the anion R⁻, there are two positions available for substitution where hydrogen ions can be fixed. The exponential expression represents, in essence, the electrostatic action arising as a result of interionic forces between free charges. For water at 18° \(D = 81\), and equation (2) may be written in the following form:

\[ \log \frac{K_1}{K_2}=\log 4+\frac{3.1}{r} \tag{3} \]

In this formula \(r\) must be expressed in Ångström units
(\(\mathring{A}=10^{-8}\) cm).

If, by this formula, one calculates \(r\) for the normal dibasic acids of the oxalic-acid series, the following values are obtained (see Table 1, 4th column):

TABLE 1

Molecular dimensions of normal dicarboxylic acids of the oxalic-acid series.

From the structure of diamond we know that the distance between two carbon atoms connected by a single bond is approximately \(1.5\ \mathring{A}\). If one adds one more \(\mathring{A}\) for each oxygen atom, then for the various acids one obtains, in the second column, the chain lengths of the molecule measured along the carbon chain. The true length of the molecule is in all probability smaller, owing to its curved or zigzag form. On the basis of his experiments on oil films on water, Langmuir¹ calculated for the molecular length \(1.3\ \mathring{A}\) per carbon atom (see the 3rd column).

With the exception of the lower members of the series (C₂ and C₃), the values of \(r\) calculated from the dissociation constants appear quite acceptable.

¹ Langmuir. Journ. Americ. Chem. Soc. 39, 1348 (1917).

The numbers for oxalic and malonic acids, at first glance, give the impression that here one would have had to take account of a lower (approximately by a factor of two) value of the dielectric constant. However, these deviations can in no case be explained solely by the fact that here the use of the ordinary dielectric constant of water was erroneous. Besides the electrostatic action proper, we must also expect an influence of the atomic chain (as a result of electron displacement). The influence of the introduction of hydroxyl and haloids on the strength of organic acids is, in all probability, explained chiefly by displacement of electrons through the atomic chain. If the effect of substitution in the γ-position (or in an even more distant one) is small, then with substitutions in the α- and β-positions a considerable influence of the atomic chain is observed. By this influence one can explain the values, too low, which are obtained for oxalic and malonic acids.

1. The applicability of formula (3) is excellently confirmed by the example of phenolphthalein. For this dibasic acid, Rosenstein found \(\dfrac{K_1}{K_2}=4\). Since, according to the formula of phenolphthalein, \(r\) must be taken equal to \(8\ \mathring{\mathrm A}\), from equation (3) one could expect

\[ \frac{K_1}{K_2}=10 \quad \text{(approximately).} \tag{4} \]

From this I concluded (loc. cit.) that the doubly charged ion of phenolphthalein is only 40% in the colorless form corresponding to the undissociated acid. If the concentration of this form is taken for the calculation, one obtains \(\dfrac{K_1}{K_2}=10\). The remaining 60% must be present in the form of the transformed red quinoid form. Acree and Birge\(^{1}\), on the basis of colorimetric measurements, suppose that about 44% of the doubly charged ion is present in the form of the transformed colored ion, which confirms my reasoning.

1. In alcohols, whose dielectric constant is smaller than that of water, the difference between the first and second dissociation constants of one and the same acid (with the same value of \(r\)) should, according to the theory, be greater than in water. Thus, for example, in methyl alcohol \((D=35)\) it should be:

\[ \log \frac{K_1}{K_2}=\log 4+\frac{7.2}{r}. \]

From Ebert’s\(^{2}\) determinations of the dissociation constants of tartaric and succinic acids in methyl alcohol, \(r\) may be calculated by formula (4). The values found (see Table 1, 5th column) approximate—

\(^{1}\) Acree and Birge. Journ. Americ. Chemic. Soc. 41, 1031 (1919).
\(^{2}\) L. Ebert. Berichte der Deutsch. Chem. Ges. 58, 175, 1925.

tively the same as in water, but nevertheless they slightly exceed them. This could have been expected, since the action through the carbon chain here must play a relatively smaller role than in an aqueous solution, in view of the strong influence of free charges in an alcoholic solution.

1. In exactly the same way as a negative charge hinders the cleavage of a positively charged hydrogen ion, a positive charge facilitates its cleavage. Therefore positive charges make an acid stronger. Equation (3) may be extended to this effect; however, it must be borne in mind that the statistical factor is not always equal to 4, but must be determined in each individual case. Let us give the following examples.

The first and second hydrolysis constants of the hexaaquochromium ion were determined at \(17^\circ\) as approximately \(1\cdot 10^{-4}\) and \(0.006\cdot 10^{-4}\).\(^{1}\) These constants refer to the cleavage of the hydrogen ion from

\[ \mathrm{Cr(H_2O)_6^{+++}} \quad \text{and from} \quad \mathrm{Cr(H_2O)_5OH^{++}}. \]

Since the statistical factor here probably must be taken as equal to \(\dfrac{12}{1}:\dfrac{10}{2}\), from the ratio of the hydrolysis constants there is obtained a value for \(r\) (the distance between the ionizing hydrogen atoms and the center of gravity of the complex) equal to \(1.7\,\text{\AA}\), which is quite probable.

The hydrolysis constant of the dichlorotetraaquochromium ion at \(25^\circ\) is \(4\cdot 10^{-6}\). Therefore this ion may be regarded as an acid,\(^{2}\) considerably weaker than the hexaaquochromium ion, although it also contains two electronegative chlorine atoms. This unexpected circumstance is readily explained by the fact that it possesses only one positive charge, while the hexaaquochromium ion possesses three.

1. The electrostatic considerations set forth above may also be extended to ampholytes.

The author\(^{3}\) derived the formula:

\[ \frac{K_S\cdot K_B}{K_{H_2O}}=\frac{n}{x(1-x)} \tag{5} \]

Here \(K_S\) and \(K_B\) are the dissociation constants of the acidic and basic groups of the ampholyte, \(x\) is the fraction of undissociated ampholyte present in the form of the amphoteric ion, and \(n\) is a factor expressing the ratio of the dissociation constants of the acids

\[ \mathrm{NH_2\cdot R\cdot COOH} \quad \text{and} \quad \mathrm{NH_3^+\cdot R\cdot COOH}. \]

\(^{1}\) Bjerrum. Zeitsch. für physik. Chem. 73, 724 (1910).

\(^{2}\) Cf. the determination of the acids in Brönsted, Rec. trav. chim. Pays-Bas 42, 718 (1922), or in Zeitschr. phys. Chem. 108, 185 (1924).

\(^{3}\) Bjerrum. Zeitschr. für physik. Chem. 104, 148 (1923).

If, instead of \(n\), we introduce its electrostatic expression and take logarithms, we obtain

\[ \log \frac{K_S \cdot K_B}{K_{H_2O}} = \frac{3,1}{r} - \log x(1-x). \tag{6} \]

The difficulty in verifying this formula lies in the fact that \(x\) is usually unknown.

On the grounds given earlier it may be assumed that for aminobenzoic acids \(x\) lies between 0.1 and 0.9. According to the investigations of Rördam\(^{1}\), in this part, not very accurately determined, for the ortho-acid \(x\) lies near 6, for the para-acids—near 0.4. Recently Eiler found that \(x\) must be very small. But in fact it is hardly less than \(1/4\). If \(x(1-x)\) is put approximately equal to 0.2, then from the known dissociation constants one may calculate the following:

For comparison, the values of \(r\) for phthalic and isophthalic acids, calculated by formula (3), are given.

In view of the unreliability of the numerical material, one may say that the values found for the \(o\)- and \(m\)-aminobenzoic acids are quite acceptable. Conversely, the low value of \(r\) found for the \(p\)-acid should prompt new determinations of the dissociation constants and of \(x\) for this acid. The circumstance that for \(m\)-phthalic acid a higher value was obtained than for \(m\)-aminobenzoic acid is explained by the fact that the ionized carboxyl groups of phthalic acid are similarly charged and therefore repel one another, whereas the ionized amino and carboxyl groups in aminobenzoic acid possess charges of opposite sign and therefore attract one another. In view of this, in the first case the molecule is stretched out, while in the second it is bent.

The preceding consideration of amino acids gives us an example of how electrostatic actions in molecules with unequally bound, ionizing hydrogen atoms may be calculated.

1. The electrostatic method of consideration may also be made the basis of calculations in those cases when the matter concerns po-

\(^{1}\) H. N. K. Rördam. Studies on Acidity. Dissertation. Kopenhagen, 1925.

successive eliminations of other ions, except hydrogen. As an example, let us consider the complex constants of the rhodanides of chromium1. Here the elimination of six rhodan groups occurs in the complex \(\mathrm{CrRh}_6^{------}\), with \(\mathrm{H_2O}\) taking the place of \(\mathrm{Rh}^{-}\), and in the end the hexaaquochromium ion is obtained:

\[ \mathrm{CrRh}_6^{------} \to \mathrm{CrRh}_5\mathrm{aq}^{-----} \to \mathrm{CrRh}_4\mathrm{aq}_2^{----} \to \mathrm{CrRh}_3\mathrm{aq}_3^{---} \to \mathrm{CrRh}_2\mathrm{aq}_4^{--} \to \mathrm{CrRh}\mathrm{aq}_5^{-} \to \mathrm{Cr}\mathrm{aq}_6^{+++}. \]

In the following Table 2, the second column gives the calculated values for the six dissociation constants.

TABLE 2

Dissociation constants of the hexarhodanochromium complex at \(50^\circ\)

Mean value [[unclear: \(0,\ldots\)]]; [[unclear: \(5.1^\circ\)]]

In calculating these values, concentrations were taken into account instead of activities. If they are recalculated, insofar as this is possible at present, into activities—which is equivalent to extrapolating these values to an ion concentration equal to zero—then the numbers of the third column are obtained. For statistical reasons these constants must be treated as the fractions shown in the fourth column. If the corresponding corrections are introduced, the numbers of the fifth column are obtained. If the decrease of these numbers with increasing number of the constant is explained as a consequence of the increasing positive charge of the complex, depending on the elimination of the rhodan ion, then the action of each charge can be measured by the differences given in the sixth column. From their arithmetic mean, equal to 0.6, one can calculate

\(r = 5.5\) Å. This value is too large; however, in order of magnitude it is acceptable. The fact that it is too large, in any case, does not show that for water one must take a value of \(D\) smaller than the usual one, even for the rhodanide ion, which, so to speak, is in contact with the chromium complex.

1. Interionic forces manifest themselves not only under conditions of chemical equilibrium, but also in the rates of chemical reactions.

In 1898 Emil Fischer reported his observation that hydroxyl ions usually produce a stronger saponifying or splitting action on neutral substances than on analogous substances with acidic properties. Thus, for example, dimethyl acetoacetic ester is saponified more rapidly than acidic acetoacetic ester. Fischer adds that van’t Hoff drew his attention to a possible explanation of this phenomenon by the fact that the acidic forms in alkaline solution are present in the form of negative ions, e.g.

\[ \mathrm{CH_3CO^-:CH \cdot COOC_2H_5}. \]

The negative charge of these ions repels the hydroxyl ions and thereby hinders their splitting action.

In 1909 Julius Meyer again advanced this idea. As was indicated earlier, in the esters of dibasic acids the second alkyl is always saponified with greater difficulty than the first. Meyer explained this by the fact that the anion of the half-ester electrically repels hydroxyl ions.

Quantitatively the electrostatic view may be expressed in the following form:

\[ \log \frac{K_1}{K_2} = \log 2 + \frac{3.1}{r} \tag{7} \]

(\(K_1\) and \(K_2\) are the rate constants of saponification of the first and second alkyls).

For the dissociation constants the following equation holds:

\[ \log \frac{K_1}{K_2} = \log 4 + \frac{3.1}{r} \tag{8} \]

According to formulas (7) and (8), the specifically electrostatic effect is the same for the saponification and dissociation constants. The whole difference between these two formulas reduces to the fact that the statistical factor for the saponification constants is equal to 2, and for the dissociation constants—4.

Skrabal, who in recent years has studied the processes of saponification in detail, derived from his experiments the purely empirical conclusion that between the ratio of the dissociation constants of dibasic

between acids and the ratio of the saponification-rate constants of the corresponding complex esters there is a parallelism. He also noted that the ratio of the saponification constants, as the distance between the acid groups in the molecule increases, approaches 21. These observations agree well with formulas (7) and (8).

Unfortunately, Skrabal carried out a large part of his measurements with solutions in which the concentration of ions exceeded 0.1 normal; and since the concentration of ions has a strong influence, especially on the second saponification constant [cationic catalysis of Holmberg], it is difficult to pass with certainty from Skrabal’s determinations to the rate constants at an ion concentration equal to zero. In addition, Skrabal performed a number of his experiments in a mixture of 50% alcohol and 50% water. These circumstances make difficult the quantitative use of the extensive and interesting experimental material obtained by Skrabal.

If one keeps chiefly to determinations at low salt concentrations, then the measurements of Goldschmidt and of Scholz, Jul. Meyer, and Skrabal give, for the ratio of the first and second saponification constants, the following values:

oxalic acid 19,000 (high salt concentration),
malonic acid—about 100,
succinic acid—about 10.

From this, by formula (7), one can calculate the values of \(r\) placed in the last column of Table 1, which agree well with those determined by other methods.

1. In general, one may say that the material presented on the preceding pages gives us the right, with sufficient approximation, to calculate the forces acting between two ions on the basis of Coulomb’s law, taking the ordinary dielectric constant of the solvent, even for ions situated very close to one another.

II.

1. Of special interest is the influence of interionic forces on the osmotic pressure, on the active mass or activity, and on the electrical conductivity of ions. Van Laar, Malmström, Sutherland, Bjerrum, and Kjellin had already drawn attention to the significance of interionic forces in this field. However, only Milner (1913) touched upon the core of this phenomenon, which lies in the grouping of ions

Niels Bjerrum

in solution: oppositely charged ions are, on the average, situated somewhat closer to one another than ions of like charge.

Only Debye and Hückel1 succeeded in 1923 in deriving sufficiently simple, and for dilute solutions entirely exact, formulas expressing the action of interionic forces.

These investigators proceed from the Faraday–Coulomb law, carry out the calculation with spherical ions of diameter \(a\), carrying a charge at the center, and arrive at the following equation for the activity coefficient \(f\) (= activity divided by concentration) of an ion:

\[ -\log f = 0.50\,\frac{z^2\sqrt{\mu}}{1+0.327\sqrt{\mu a}} \tag{9} \]

In this equation the numerical values are taken for water at \(18^\circ\); \(z\) denotes the valence of the ion, and \(\mu=\sum \frac{1}{2}cz^2\) is the quantity introduced into the science of electrolytes by Lewis and Randall—“ionic strength.” At small ionic strength (small ion concentrations) the denominator may be neglected; in this case \(-\log f\), which has a positive value, changes with dilution proportionally to the square root of \(\mu\) and approaches zero; consequently, \(f\) is less than unity and, as the concentration decreases, approaches unity according to the same law. Debye and Hückel derived the corresponding formula for the osmotic pressure of ions and a similar, though more complicated, formula for their electrical conductivity (mobility).

Fig. 1. Dependence between the activity coefficient \(f\) and the ionic strength in a uni-univalent salt \((\mathrm{NO}_3)\mathrm{CSN}\mathrm{H}_3\), \(\mathrm{COI}\)—\([\mathrm{I}_2\mathrm{O}_2]\) \((\mathrm{NO}_2)_2(\mathrm{NH}_3)_2\mathrm{Co}\) in the presence of other salts. The straight line represents the theoretical values according to Debye and Hückel. (From the work of Brönsted and La Mer).

1. The work of Milner and of Debye and Hückel shows that interionic forces do not permit a constant ratio to be assumed between the osmotic action of ions (or their activity) and their concentration. The mobilities of ions likewise are not independent of concentration. On the contrary, we must expect that all these quantities decrease as the ion concentration increases.

The decrease of these quantities with increasing ion concentration, as is known, is in fact observed in all electrolytes. This decrease—

...was at first interpreted as a sign of incomplete dissociation—both for weak and for strong electrolytes. But if the experimentally found decreases are compared with those calculated according to the theory of Debye and Hückel, it turns out that for strong electrolytes complete agreement between the two is observed, and there is no need to explain anything by incomplete dissociation. We shall give several examples of this agreement.

The most accurate method for measuring the activity of ions is, at the present time, the determination of the solubility of salts. In this way Brönsted and La Mer1 found values for the activities that agree excellently with those calculated by the Debye–Hückel formula in dilute solutions up to approximately 0.01 molar concentration. This is seen in Figs. 1–3.

Fig. 2. Dependence between the activity coefficient \(f\) and the ionic strength \(\mu\) of a divalent salt \([Co_2] (NH_3)_4 Co_2 — S_2O_6\) in the presence of other salts. The straight line was calculated by the Debye–Hückel formula. (After Brönsted and La Mer.)

Fig. 3. Dependence between the activity coefficient \(f\) and the ionic strength \(\mu\) for the trivalent salt \([(NH_3)_6 Co]—[(C_2O_4)(NO_2)_2 NH_3 Co]_2\) in the presence of other salts. The straight line was calculated by the Debye–Hückel formula. (After Brönsted and La Mer.)

In the summer of 1925, Rodebush and Hovorka2 made extraordinarily accurate cryoscopic measurements in very dilute aqueous solutions of 0.001–0.01 molar concentration. Their results are shown in Figs. 4 and 5. The solid curves represent the values calculated according to Debye and Hückel. The slight curvature observed in these curves shows that Rodebush and Hovorka took into account the diameter of the ions; otherwise these would be straight lines,

tangent to the drawn curves at a concentration equal to zero. The authors chose such diameters \((a)\) of the ions that the curves passed through the points corresponding to the greatest measured concentration.

The following ion diameters were adopted:

Figs. 4 and 5 show that the osmotic properties of ions can also be understood without introducing the assumption that appreciable quantities of undissociated molecules are present in the salt solutions investigated.

Fig. 4. Dependence of the osmotic coefficient on \(\sqrt{\mu}\) according to Rodebush and Hovorka.

Fig. 5. Dependence of the osmotic coefficient on \(\sqrt{\mu}\) according to Rodebush and Hovorka.

The same conclusion follows from the recently published work of Schreiner and Frivold1 on the depression of the freezing points of lithium chloride in cyclohexanol.

The application of this theory to electrical conductivity is the most difficult of all. But here too the Debye and Hückel theory proved capable of explaining the behavior of strong electrolytes in dilute solutions without resorting to partial ionization.

1. As is known, even earlier2 the conclusion had been reached that the so-called strong electrolytes are practically completely dissociated in solutions. This was led to by investigations of the optical and catalytic properties of electrolyte solutions, and also by the remarkable agreement of the degrees of dissociation for all strong electrolytes calculated according to the classical theory. The degrees of dissociation calculated in this way are determined chiefly by electrical

properties of the system (the charge and concentration of the ions, the dielectric constant of the solvent). The work of Debye and Hückel transformed the hypothesis of the practically complete dissociation of strong electrolytes into an experimentally established fact.

In the present state of our knowledge, we must regard it as an error if the electrical-conductivity coefficient $\frac{\mu}{\mu S}$ of a strong electrolyte is treated as the degree of its dissociation and attempts are made to apply to it the law of mass action.

For weak electrolytes, on the contrary, $\frac{\mu}{\mu S}$ must still be regarded as the degree of dissociation. Here the concentration of ions is so small that the action of the forces between the ions may be neglected and the ions may be considered to have constant mobility.

For electrolytes of medium strength one can (as was shown in 1916 for picric acid in alcohol1) calculate the true degree of dissociation from the electrical-conductivity coefficient, if corrections are introduced into it that take account of the influence of the forces between ions. The stronger the electrolyte, the larger the correction, and the less exact the calculation of the degree of dissociation.

1. Now that we have become acquainted with the influence of the forces between ions on the activity of ions, it is of interest to return to the significance of these forces for the rate of ionic reactions. We have already said how, by means of interionic forces, one can explain why a negative ion acts more weakly on a group situated in a negative ion than on the same group in a neutral molecule or in a positive ion. Let us now consider the influence of the concentration of ions on the rate constant of a chemical reaction.

In a series of very interesting works, Holmberg showed that a whole series of reactions in which hydroxyl ions (or other negative ions) react catalytically with negatively charged ions are accelerated by the addition of salts. Especially strong is the influence of salts with polyvalent cations (cationic catalysis). As Holmberg writes2, this phenomenon can be explained, from the standpoint of the complete dissociation of salts, by the fact that interionic forces increase the rate of the reaction. Holmberg, however, did not develop this idea further.

In an interesting paper, Brönsted3 showed that not only in the cases considered by Holmberg, but also for other ionic reac-

the influence of salt concentration can be quantitatively expressed by the simple formula:

\[ k=k_0\,\frac{f_1 f_2}{f_{12}} . \tag{10} \]

Here \(k\) and \(k_0\) are the rate constants in the salt solution and, respectively, at an infinitely small ion concentration; \(f_1\) and \(f_2\) are the activity coefficients of the reacting ions (molecules); \(f_{12}\) is the activity coefficient of an ion whose charge is equal to the sum of the charges of the reacting ions. With regard to the derivation and interpretation of this important formula there are certain disagreements between Brönsted and the author; however, we agree that this formula correctly expresses the influence of salt concentration on the rate of an ionic reaction.

If in Brönsted’s formula (10) one introduces Debye and Hückel’s expression for the activity coefficient according to formula (9), one obtains:

\[ \lg \frac{k}{k_0} = 0.50\,\frac{2z_1 z_2 \sqrt{\mu}} {1+0.3271\sqrt{\mu a}} \tag{11} \]

(\(z_1\) and \(z_2\) are the numbers of electric charges of the two reacting complexes, taken with their respective signs).

1. Perhaps the most immediate explanation of the influence of salt concentration is given by the following reasoning. Let us consider one of Holmberg’s cases of cationic catalysis—the influence of hydroxide ions on the anion \(\mathrm{CH_2BrCHBrCOO^-}\) of dibromopropionic acid. In the vicinity of this ion the concentration of hydroxide ions, owing to interionic forces, is lower than in the rest of the solution. If it is assumed that the \(\alpha\)-bromine is the place in the molecule where the reaction begins, and that its distance from the negative charge is equal to \(r\) Å, then it follows that the concentration of hydroxide ions near the \(\alpha\)-bromine is

\[ e^{-\frac{E^2}{kTD r}}=10^{-\frac{3.1}{r}} \]

times smaller than in the rest of the solution. When the concentration of ions in the solution is increased, the electric force emanating from the dibromopropionate ion is distributed not only over the hydroxide ions but also over all the ions present. The more ions there are, the smaller the action on each individual ion becomes, and the greater the concentration of hydroxide ions near the dibromopropionate ion. Thus, the action of a salt consists in the fact that the added salt diminishes the electrostatic effect. If one carries out a quantitative calculation, it turns out that for dilute solutions the salt effect does not depend on \(r\), provided \(r\) is not too large.

At high salt concentrations and large values of \(r\), its influence nevertheless becomes noticeable, and the simple formula (11) can be us—

can no longer be used. In all probability, for substances with a high value of \(r\) (i.e., with a large distance between the charge and the reacting group) it will be possible to establish that the ratio between the reaction rate in an ion-poor and in an ion-rich medium approaches the limit \(10^{3.1}\) with increasing ion concentration. In this case it will also be possible to determine in this way the molecular size \(r\).

III.

1. Up to now it has been thought that the new views on strong electrolytes can be opposed by proofs of the presence, in one or another strong electrolyte, of a small number of undissociated molecules. This, however, is incorrect.

One must be prepared in advance to encounter in nature all possible transitions from dissociated electrolytes to 100% electrolytes through intermediate and weak ones—even down to typical nonelectrolytes.

In green dichlorochromium chloride there is undissociated chlorine, which can be demonstrated with silver nitrate, and in concentrated solutions of chloroplatinic acid there are chloro complexes, as indicated by their color.

Of special interest is the degree of dissociation of hydrohalic acids. Proceeding from the catalytic action of hydrogen chloride in alcohol, Schreiner1 determined the dissociation constant of this acid in alcoholic solution as \(k = 10^{-2}\) (approximately), and since the dissociation constant of acids in water is usually \(10^{6}\) times greater than in alcohol, he concluded that in water it is equal to \(10^{4}\). The refractive power of concentrated hydrochloric acid also indicates, in Schreiner’s opinion, incomplete dissociation; by this route he finds for the constant the value \(10^{9}\). A third method for the approximate determination of the dissociation constant of hydrogen chloride was given by Ebert2. Extrapolating the series \(\mathrm{C}_{3}\mathrm{H}_{7}\mathrm{Cl}\), \(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{Cl}\), \(\mathrm{CH}_{3}\mathrm{Cl}\) to \(\mathrm{HCl}\), he calculates the solubility of undissociated \(\mathrm{HCl}\) in water. Since the vapor pressure of \(\mathrm{HCl}\) over normal hydrochloric acid is known, Ebert finds the concentration of undissociated hydrogen chloride in hydrochloric acid and hence—the dissociation constant of hydrochloric acid. By this method \(k = 10^{7}\) is obtained. Ebert points out, however, that this method should give values that are too high because of the nature of the extrapolation.

Thus, by entirely different routes one obtains a result indicating incomplete dissociation of hydrochloric acid;

however, what is at issue is the presence, even in a normal solution, of very small amounts of undissociated hydrogen chloride \((10_x^{-5}—10_x^{-6}\ \mathrm{mol}\) per liter).

According to the conductivity coefficient \(\dfrac{\mu}{\mu_\infty}\), HCl, HBr, HI are equally strong electrolytes; but if measured by Ebert’s method, their dissociation constants are in the ratio \(1:100:250\). These large differences are in agreement with the considerable divergences in acidic properties that Hantsch\(^{1}\) established between these acids in the anhydrous state. If the dissociation constant of hydrogen iodide is 250 times greater than that of hydrogen chloride, this means that the difference in strength between hydrochloric and hydriodic acids is greater than between acetic and monochloroacetic acids.

Whereas formerly the dissociation constants of strong electrolytes calculated from conductivity were not at all specific and were determined chiefly by the valence of the ions, the new dissociation constants are distinguished by their specificity, so characteristic of chemical-equilibrium constants, and vary from electrolyte to electrolyte.

The new views have drawn our attention to the problem of determining the dissociation constants of electrolytes that are almost 100% dissociated. A considerable merit of these views lies in the fact that, thanks to them, we can find a measure of the true strength of these electrolytes.

As a final example of an incompletely dissociated strong electrolyte I shall cite caustic soda. The activity of the hydroxyl ion in solutions of caustic soda is significantly less than in solutions of caustic potash\(^{2}\). If dissociation were complete, the relation should be the reverse, since in other salts the activity of the sodium ion is greater than that of potassium. This is explained by the fact that the sodium ion is hydrated and therefore has a larger volume than the potassium ion. I therefore assume that the low activity of the hydroxyl ion in caustic soda can be explained by its incomplete dissociation. Harned\(^{3}\) drew an analogous conclusion from his measurements.

If one pays attention to the identical structure of \(\mathrm{H_2O}\) and \(\mathrm{OH^-}\), it will appear quite natural that an ion prone to hydration, such as sodium, can also combine with hydroxyl ions. One should even expect that a certain parallelism will be observed between the hydration of cations and the weakness of their bases. In connection

\(^{1}\) Hantsch. Ber. d. Deutsch. Chem. Ges. 58, 612 (1925).

\(^{2}\) As yet unpublished measurements by Unmack (Frl. Unmack). Harned (Journ. Amer. Chem. Soc. 47, 684, 689 (1925)) showed that the mean activity coefficient of caustic soda is less than that of caustic potash.

\(^{3}\) Harned. Zeitschr. für physical. Chemie. 117, 49 (1925).

In this connection it should be mentioned that Kolthoff and Gjaldbæk\(^1\) have proved that magnesium hydroxide hydrate in water is not completely dissociated. Gjaldbæk found for the second dissociation constant of magnesium hydroxide hydrate a value approximately equal to \(10^{-21}\), whereas he considers the first to be infinitely large.

Some as yet unpublished measurements of the electrical conductivity of magnesium methylate and various other magnesium salts in methyl alcohol, carried out by Zechmeister (L. Zechmeister) in the author’s laboratory, showed that magnesium methylate in methyl alcohol is likewise not completely dissociated; it conducts current much worse than magnesium chloride and electrolytes similar to it in methyl alcohol. The magnesium ion thus combines not only with \(H_2O\) and \(OH^{-}\), but also with the \(CH_3O^{-}\) ion, which is close to \(H_2O\).

16. In his generally excellent book The Properties of Electrically Conducting Systems, New York, 1922, Kraus (Ch. A. Kraus) speaks against the newest views on electrolytes chiefly on two grounds: he believes that very large changes in the coefficient of electrical conductivity \(\frac{\mu}{\mu_\infty}\) with concentration, observed even in the strongest electrolytes in solvents with low dielectric constants—for example, in water at high temperature (near the critical point)—cannot be explained by interionic forces. He sees no possibility of finding an explanation for the anomalous increase of the coefficient of electrical conductivity with increasing concentration in solvents with a very small dissociation constant.

In reality, however, with the aid of interionic forces these phenomena too can be explained, if one takes into account the association of ions caused by these forces.

It is clear that at the same concentration oppositely charged ions, as a result of the action of interionic forces, will more often be in immediate proximity to one another than neutral molecules. To estimate the significance of this fact, it is necessary to consider it quantitatively.

An elementary argument shows that the number of neutral molecules, in a given solution located at distances from one another from \(r\) to \(r + dr\), for small values of \(r\) (small in comparison with the mean distance between molecules) is proportional to \(r^2dr\). For oppositely charged ions the corresponding number is proportional to \(r^2dr\, e^{-\frac{E^2}{kTDr}}\),

and for like-charged ions it is proportional to \(r^2dr\, e^{\frac{E^2}{kTDr}}\).

\(^1\) J. M. Kolthoff. Recueil des travaux chim. des Pays-Bas, 42, 969 (1923). J. K. Gjaldbäck. Zeitschr. für anorg. Chem. 144, 283 (1925).

NIELS BJERRUM

In Fig. 6 curves are given showing the frequency of occurrence of pairs: I—uncharged molecules, II—like-charged ions, and III—oppositely charged ions in an aqueous solution at ordinary temperature. The abscissas are the distances between the components of the pair, the ordinates—the frequency of occurrence of pairs with the corresponding distance between the components.

On the curve of ionic pairs consisting of oppositely charged ions (II), at \(3.5\ \text{Å}\) a clear minimum is observed. A similar minimum is also observed in other solvents. It is always found at such a distance between the ions at which the work of separating the ions is equal to \(2kT\), i.e. four times the mean kinetic energy per degree of freedom. Ionic pairs with such a distance occur more rarely than pairs with a greater or smaller distance. If the ions in solution are so small that they can approach one another considerably more closely than this distance, then the frequency of closely associated ionic pairs becomes significant, and this association must be taken into account in order to obtain the correct expression for the influence of interionic forces. In aqueous solutions of sodium and potassium chloride the charges of the ions cannot come so close together. But if the ions are considerably smaller or possess several charges, or if the solvent has a considerably lower dielectric constant than water, then association may take place.

Fig. 6. Relative frequency of occurrence of pairs with distance \(r\ \text{Å}\) between the components. I—uncharged molecules; II—oppositely charged univalent ions in water; III—like-charged univalent ions in water.

Debye and Hückel, in their formulas for more concentrated solutions, do not pay sufficient attention to this association. I have tried to carry out a more exact calculation, in which ions whose distance was less than that corresponding to the minimum on the association curve were not counted as free ions. In this way, it seems, one can indeed go somewhat further; elsewhere I have attempted to develop this idea in more detail1. According to this view one can explain why interionic forces reduce the active mass of ions, their osmotic action, and their electrical conductivity in solvents to small fractions of the values that would correspond to completely free ions. Thus Kraus’s first objection to the newest views on strong electrolytes falls away.

This method of calculation brings the modern picture of an electrolyte solution closer to the classical one, since here the degree of association replaces the former degree of dissociation.

However, it should not be forgotten that the distinction between free and associated ions is rather mathematical in character and does not possess the sharpness characteristic of chemical processes. On the other hand, it is interesting to note that the new picture explains to us why the classical picture of the “degree of dissociation” not only does not completely lose its applicability, but, in those cases where interionic forces cause strong association, even gains by it.

The applicability and usefulness of the model of pairwise associated ions does not increase continuously with the growth of the above-mentioned “associating” external factors (a large number of charges and a high concentration of ions, small ion size and a low dielectric constant). If pairwise association is strong, then the interionic forces necessarily cause further association into higher multi-ion complexes, especially if the solution is already not very dilute. In this way are explained the high degrees of association of strong electrolytes in solvents with low dielectric constants, demonstrated by Walden.

With a clearly expressed pairwise association, the electrical conductivity of the ions is small. In solvents with a very low dielectric constant, at infinitely great dilution, theoretically speaking, the ions, to be sure, will be completely free and unassociated; but at the smallest concentrations accessible to measurement (0.0001–0.01 molar) pairwise association will still be very great, and the electrical conductivity will therefore be small. In concentrated solutions under such circumstances, interionic forces may cause an increase of electrical conductivity with concentration. As the concentration increases, the ionic pairs will come closer together, their mutual interaction will begin to destroy them, triple and higher products of association will begin to appear more often, and owing to this the electrical conductivity will increase. With increasing concentration we approach the state characteristic of molten salts, where the ions are very strongly associated, but not only pairwise; for this reason they are capable of motion and of carrying an electric current.

The schemes I and III given below provide a clear representation of the state of a dilute solution of ions with pairwise association and of the state of a molten salt with high, but not pairwise, association; II depicts an intermediate stage of a comparatively strong solution.

I:
            + −                           + −                          + −

II:
            + −       + −       + −          + −       + −

III:
      + − + − + − + − + − + − + − + − + − + − + −

In the region with predominantly polar association, low electrical conductivity is observed, combined with comparatively considerable osmotic action. If, however, association of a higher order (into multi-ion complexes) takes place, then appreciable electrical conductivity and a small osmotic action may be observed.

1. Summarizing the foregoing, we arrive at the conclusion that the characteristic feature of strong electrolytes is not the freedom of the ions. In a crystal of potassium chloride the ions are bound very firmly. But even in the crystalline state potassium chloride is a typical strong electrolyte. No, what is characteristic of strong electrolytes is that their ions do not combine with one another into chemical molecules with a substantial change in their properties. On the contrary: their ions can come very close to one another without noticeably deforming one another. An ideal strong electrolyte may be called an electrolyte whose ions do not deform one another at all. The theory of such electrolytes has also been developed in the most recent theories and compared with experimental data.

2. In accordance with the conceptions of Kossel and certain other physicist-investigators, one may imagine that there exists a continuous transition from large ions, which deform little and associate poorly, through smaller ions, which associate and deform somewhat more readily, to very small ions that form typical chemical complexes with a strong change in their properties. Thus, there should exist a continuous transition from practically nonassociating KCl through weakly associating KNO$_3$—to complex ions of the type Cr(CN)$_6^{---}$, and further to SO$_4^{--}$, where one may imagine the hexavalent positive sulfur ion bound to four divalent negative oxygen ions. In considering the existing experimental material, it seems to me, however, more probable that there is a more or less sharp distinction between the products of association of weakly deforming ions, on the one hand, and more or less stable chemical complexes, built from strongly deformed or completely disintegrated ions, on the other hand.

It may be that chemical complex formation is connected with the fact that certain electrons, bound in a free ion to one nucleus, become the common property of two atomic nuclei. I consider it most probable that such a transition does not take place continuously, but occurs in such a way that an electron in a definite position suddenly jumps to an entirely new orbit, binding it with both nuclei.

1. Bjerrum. Mat.-Phys. Medd. Kgl. Danske Vid. Selsk. Kopenhagen. 1926. ↩↩↩↩↩↩↩↩↩

2. L. Ebert. Die Naturwissenschaften, 13, 393 (1925). ↩↩↩↩

3. Brönsted, Zeitschr. für physikal. Chemie, 102, 169 (1922). ↩