Recent Investigations on the Anomalies of Strong Elements.

L. Ebert. Neuere Forschungen über die Anomalien starker Elektrolyte.
Jahrb. d. Radioaktivität und Elektronik 18, 134—196 (1921).

Soon after the introduction into science of the theory of electrolytic dissociation, it was found that a whole series of phenomena could be fitted only with difficulty into the framework of the original theory. Phenomena of this kind—anomalies—prove to be especially clearly expressed precisely in strong electrolytes, i.e., in those bodies in which the characteristic properties of the class of electrolytes are predominantly manifested.

The following anomalies are the most important:

Applying the law of mass action to the dissociation reaction of an electrolyte, we obtain

\[ \frac{c'c''}{c_0}=K \tag{1} \]

(\(c',c''\)—concentrations of ions, \(c_0\)—of neutral molecules, \(K\)—constant). If there is only one electrolyte in the solution, then (1) becomes

\[ \frac{\alpha^2}{1-\alpha}c=K \tag{2} \]

where \(\alpha\)—the degree of dissociation of the electrolyte, and \(c\)—its concentration. If in (2), instead of \(\alpha\), one substitutes the quantity \(\frac{\mu_v}{\mu_\infty}\) (\(\mu_v\)—molar conductivity at dilution \(v\) liters per one mole), equal to \(\alpha\) according to Arrhenius’s theory, then genuinely constant quantities are obtained for a number of weak acids and bases; for strong electrolytes, however, it turns out that, although in some cases the quantity \(K\) does remain constant in a very narrow concentration interval, when broader intervals are considered, generally speaking, an increase of \(K\) is observed with increasing concentration \(c\). In other words, in order that \(K\) should remain constant, one must substitute in (1), instead of \(c'\) and \(c''\), not the quantity \(c\frac{\mu_v}{\mu_\infty}\), but another one, which increases more slowly with concentration. Wegscheider1, who studied a series of electrolytes of different strengths, found that the constancy of \(K\) depends on the magnitude of the total concentration of ions in the solution: as soon as the latter exceeds the limit of 0.03 mole per liter, the value of \(K\) begins to increase.

If another electrolyte with a common ion \(c''\) is added to a sparingly soluble electrolyte, then according to (1) we have:

\[ c'c''=c_0K \tag{3} \]

As \(c''\) increases, the “solubility product” \(c'c''\) and the concentration of undissociated molecules \(c_0\) must remain constant. Experiment shows, however, that in fact the quantity \(c'c''\), with increasing total concentration of ions, slowly grows, while \(c_0\) falls. As a result, as in the first case, \(K\) increases. Still sharper anomalies are observed upon the addition of ions of higher valency.—The electromotive force \(E\) of a concentration chain in which the potential difference between two solutions has in some manner been reduced to zero, according to Nernst must be equal to

\[ E=K \ln \frac{c'_1}{c'_2} \tag{4} \]

where \(K\) is a constant, and \(c'_1\) and \(c'_2\) are the concentrations of the corresponding ion. However, calculating \(c'_1\) and \(c'_2\) from the electrical conductivity of the solution and substituting the values obtained into (4), we obtain, as Bjerrum in particular showed,\(^1\) a value exceeding the measured e.m.f. \(E\); in order that equality (4) should hold, it is necessary, as in the case of equation (1), to substitute into (4) in place of \(c'\) not \(\alpha \dfrac{\mu_v}{\mu_\infty}\), but another quantity, which is the smaller compared with \(\alpha \dfrac{\mu_v}{\mu_\infty}\) the greater the ion concentration. This quantity is called the “activity of the ion”; it can be calculated from the experimentally found values of \(E\).

Remarkable results were obtained for cells of the following type, investigated by Brönsted:\(^2\)

\[ Cd\ (\text{amalg.})\ \bigg|\ \begin{matrix} CdSO_4(c_1)\\ MgSO_4(2-c_1) \end{matrix} \ \bigg|\ \begin{matrix} CdSO_4(c_2)\\ MgSO_4(2-c_2) \end{matrix} \ \bigg|\ Cd\ (\text{amalg.}) \]

The value \(c_1\) varied within the limits \(1/10\)–\(1/320\), the value \(c_2\) within the limits \(1/20\)–\(1/640\). The e.m.f. of such cells, in which the solvent is a concentrated solution of ions, proved equal to \(K \ln \dfrac{c_1}{c_2}\); in other words, the activity of the \(Cd\) ions in this case was proportional simply to the concentration of cadmium ions, independently of the magnitude of the ratio \(\dfrac{\mu_v}{\mu_\infty}\).

Experiment has shown that there are also other quantities characterizing the properties of strong electrolytes which, within certain limits, are proportional to the total concentration of the electrolyte. These properties are called properties of the first kind; they include: color, refractive power, heat of neutralization, and, in a number of cases, the catalytic effect produced by the electrolyte.

One may try to explain the deviations from the gas laws observed in strong electrolytes and the existence of properties of the first kind by introducing into the classical theory all sorts of additional assumptions and qualifications (Drucker,\(^3\) and others); a number of investigators, however, beginning with Bjerrum,\(^2\) have adopted an essentially new point of view, which may be formulated as follows: strong electrolytes in solution are practically completely dissociated, so that the value \(\alpha\) may be taken as equal to unity. The properties of the first kind are those properties which depend only on the concentration of the ions; they may serve as a measure of the latter. The other properties—molar electrical conductivity, osmotic pressure, activity of ions—are determined not only by the number of ions, but also by that electric field which exists between the ions according to Coulomb’s law.

Whereas, according to Arrhenius’ original theory, the change in the properties of an electrolyte with concentration was determined by a change in the value of \(\alpha\), according to the new theory it is determined by the increase in the interaction between the electric charges of the ions. The decisive factors here are the following: 1) the magnitude of the electric charges, i.e. the valency of the ions, 2) the dielectric constant of the solvent, 3) the mean distance between two ions, proportional to

\[ \frac{1}{\sqrt[3]{c_i}}, \]

where \(c_i\) is the total concentration of ions in the solution (Totalionenkonzentration).

\(^1\) Ztschr. f. Elektrochem. 17, 392 (1911).

\(^2\) Medd. fr. K. Vetensk. Nobelinstitut. 5 Nr. 25, (1919); D. Kgl. Danske Vidensk. Selsk. Math.-fys. Medd. III, Nr. 9, 1920.

\(^3\) Ztschr. f. phys. Chem. 96, 381 (1920).

\(^4\) Ztschr. f. angew. Chem. 22, 1265 (1909); Samml. chem. u. chem. techn. Vorträge 21, 1. (1915); Ztschr. f. Elektrochem. 24, 321, (1918); Ztschr. f. anorg. u. allg. Chem. 109, 275, (1920).

In the case of weak electrolytes, whose degree of dissociation is less than unity, according to the new theory one must also take account of interionic forces, as soon as the ion concentration exceeds a certain value.

The principles of the kinetic theory of gases make it possible to calculate the magnitude of the osmotic pressure of a completely dissociated electrolyte, taking into account the action of the field between the ions. A calculation of this kind was first carried out by Milner\(^1\), who showed that the gas laws cannot be applied to electrolytes, since the distribution of ions cannot be random; the most probable distribution is that in which oppositely charged particles are somewhat closer together, while particles with the same charge are farther apart from one another. Proceeding from such considerations, Milner showed that the decrease in the mobility of ions caused by mutual attraction is sufficient to explain the decrease of the osmotic coefficient observed when the electrolyte concentration is increased. Later J. Ch. Ghosh\(^2\), starting from certain ideas about the potential of electric forces, arrived at the following formula for a binary electrolyte, which agrees very well with experimental data:

\[ f_o = 1 - K \sqrt[3]{c} \tag{5} \]

\(f_o\) is the osmotic coefficient, i.e. the ratio of the osmotic pressure to the concentration, divided by the limiting value of this ratio at infinite dilution; \(K\) is a constant quantity; it is inversely proportional to the dielectric constant of the solvent.

In a similar way the new theory considers the question of the conductivity coefficient

\[ f_\mu=\frac{\mu_v}{\mu_0}. \]

According to the classical theory,

\[ \mu_v=\alpha(u+v) \]

where \(u\) and \(v\) are the mobilities of the ions—constant quantities; the change in \(\mu_v\) is determined by the change in \(\alpha\).

According to the new theory \(\alpha=1\), while the mobilities of the ions decrease with increasing concentration under the action of the electric field. R. Hertz\(^3\) was the first to carry out the corresponding calculation quantitatively and arrived at a formula with two empirical constants. R. Lorenz\(^4\) showed that there is very good agreement between Hertz’s formula and experimental data. Analogous calculations were made by Milner\(^5\) and Ghosh\(^6\).

The simple dependence which, according to the classical theory, exists between the quantities \(f_o\) and \(f_\mu\) no longer exists in the new theory.

If \(A\) is the work which must be expended in order to remove one mole of some component from a given system, then the quantity \(a\), determined by the equation

\[ A = RT \ln a + \mathrm{const.}, \]

is called the activity of this component\(^7\). According to the classical view, the activity of a dilute electrolyte is equal to its concentration; according to Bjerrum

\(^1\) Phil. Mag. (6) 23, 551 (1912); 25, 742 (1913).

\(^2\) Journ. Chem. Soc. 123, 449, 627, 707, 790 (1918); Ztschr. f. phys. Chem. 98, 211 (1921).

\(^3\) Ann. d. Phys. (4) 37, 1, (1912).

\(^4\) Ztschr. f. anorg. u. allg. Chem. 111, 55 (1920); 113, 135 (1920); 114, 209 (1920); 116, 45 (1921); 116, 161 (1921).

\(^5\) Phil. Mag. (6) 35, 352 (1918).

\(^6\) Loc. cit. Journ. Chem. Soc. 117, 1390 (1920).

\(^7\) See especially Lewis and Randall, Journ. Amer. Chem. Soc. 43, 1112 (1921).

\(a=f_a c\), where \(f_a\)—the “activity coefficient”—in a dilute solution is a quantity less than unity, depending on the existence of the electric field; \(f_a=1\) only at infinite dilution.

The activity coefficient can be found by a number of methods from experimental data, and also calculated with the aid of a thermodynamic relation from the coefficient \(f_0\). Approximately, the following relation holds:

\[ \ln f_a=-4K\sqrt[3]{c}, \]

where \(K\) is the constant from (5).

Similar activity coefficients may also be introduced for individual ions. Their magnitude in dilute solutions of similarly constituted strong electrolytes is determined primarily by the total concentration of ions in the solution. The introduction of the coefficients \(f_a\) makes it possible to eliminate a whole series of contradictions that were observed when the classical theory was applied to the phenomena of dissociation of electrolytes upon dilution, changes in the degree of dissociation, and the solubility of one electrolyte upon addition of another, etc. For the extension of the theory to more concentrated solutions, it is necessary to take into account also the hydration of the ions.

A. Frumkin.

1. Ztschr. f. phys. Chem. 69, 603 (1909). ↩