ELECTRICAL THEORY OF STRONG ELECTROLYTE SOLUTIONS1

B. N. Finkelstein, Leningrad.

CHAPTER TWO.

ELECTRICAL CONDUCTIVITY OF STRONG ELECTROLYTE SOLUTIONS.

§ 1. Qualitative Theory.

According to modern views, the carriers of electric current in solutions are ions moving under the action of an external electric field with velocity \(v\), determined by the relation

\[ z \varepsilon X = r v, \]

where \(z\) denotes the valence of the ion, \(\varepsilon\) the charge of the electron, \(X\) the intensity of the external field, and \(r\) the coefficient of proportionality. The right-hand side of the written equation expresses the force of resistance encountered by an ion moving with velocity \(v\); consequently, the quantity \(r\) is analogous to the coefficient of friction.

Let us suppose that each \(cm^3\) of solution contains \(n\) electrolyte particles; such a concentration corresponds to

\[ \eta = \frac{n}{\mathfrak{N}} \]

gram-molecules of electrolyte per \(1\ cm^3\) of solution (\(\mathfrak{N}\) is Avogadro’s number). Within the solution choose a volume bounded by the faces of a cube with edges of \(1\ cm\), and situated in such a way that any two of its parallel faces are perpendicular to the direction of the external electric field and, consequently, to the direction in which the current passes through the solution. If the potential difference on these two faces is equal to \(X\), then the current strength \(F\) passing through the volume we have chosen will be expressed—according to Ohm’s law—by the following formula:

\[ F = \sigma X, \]

where \(\sigma\) is the specific electrical conductivity of the solution.

On the other hand, the current intensity \(F\) may be defined as the amount of electricity passing per unit time through some cross-section of our cube perpendicular to the direction of the external field.

Adopting for the time being the point of view of the classical theory, let us denote by \(a\) the degree of dissociation of our solution; thus, we assume that out of the total number \(n\) of electrolyte molecules contained in \(1\ \mathrm{cm}^3\) of solution, \(an\) molecules have decomposed into ions, the total number of ions formed being equal to \(an\nu = an \sum_{i=1}^{s} \nu_i\) (\(\nu_i\) denotes the number of ions of the \(i\)-th kind formed upon dissociation of one electrolyte molecule). In this case the current intensity \(F\) will be equal to

\[ F = an\varepsilon \sum_{i=1}^{s} \nu_i |z_i| v_i^{(0)}, \]

where \(v_i^{(0)}\) denotes the velocity of an ion of the \(i\)-th kind in the steady motion.

Comparison of the last two formulas leads to the following expression for the specific electrical conductivity of the solution:

\[ \sigma = an\varepsilon \sum_{i=1}^{s} \nu_i |z_i| \frac{v_i^{(0)}}{X}. \]

Calling the mobility of an ion of the \(i\)-th kind, i.e. the velocity imparted to it by a potential difference equal to one volt, the quantity \(u_i^{(0)} = \dfrac{v_i^{(0)}}{X}\), we finally obtain

\[ \sigma = an\varepsilon \sum_{i=1}^{s} \nu_i |z_i| u_i^{(0)} . \tag{36} \]

In practice one usually uses the quantity of molecular electrical conductivity

\[ \lambda = \frac{\sigma}{n} \]

or gram-molecular electrical conductivity

\[ \Lambda = \mathfrak{N}\lambda = \frac{\mathfrak{N}}{n}\sigma = \frac{\sigma}{\eta}. \]

The change in molecular (or molar) electrical conductivity that occurs upon dilution of a solution is explained by the classical theory by the existence of a dependence between the degree of dissociation and the concentration of the electrolyte (Ostwald’s dilution law).

The electrical theory of solutions, based on the hypothesis of complete dissociation of strong electrolytes (\(a = 1\)), leads to the conclusion that the mobilities of ions depend on concentration,

conditioned by electrostatic forces of interaction between ions.

We shall preface the exposition of the quantitative aspect of this theory with several remarks of a qualitative character, intended to elucidate as clearly as possible the physical essence of the phenomena under consideration.

The considerations set forth in the first chapter led us to the conclusion that around each ion present in a solution there forms a shell (“ionic atmosphere”) of other ions; we saw that on the average (in time) negative ions predominate in the neighborhood of each positive ion—and conversely; such a distribution, in the case of ions participating only in thermal motion, has the property of radial symmetry with respect to the ion under consideration. The picture changes when an ion begins to move under the action of an external electric field. It is true that in this case too the forces of interaction between ions tend at each given instant of time to create a “static” distribution in the ionic atmosphere, but since the formation of the latter does not occur instantaneously, but over a certain interval of time (the relaxation time, “Relaxationszeit”), the static distribution does not have time to establish itself around the moving ion. Nevertheless, around a moving ion there is likewise formed an “ionic atmosphere” stationary with respect to it, i.e., one in which the distribution may be regarded as established; but the distribution of potential around the moving ion will no longer possess the radial symmetry of the “static” case (corresponding to the absence of an external field). It is easy to see that the asymmetry amounts to the following: in the direction of motion of the ion, in front of it, there will be created an excess (as compared with the “static” case) of ions of the same sign, and behind it—of the opposite sign. Indeed, owing to the finiteness of the “relaxation time,” the electric forces do not have time to build up the “static” distribution in front of the moving ion and to destroy it behind it. The indicated distribution of electricity in the “ionic atmosphere” surrounding the moving ion causes the appearance of an additional force directed—independently of the sign of the ion—opposite to its motion. We note that the action of this force, which causes a decrease in the mobilities of the ions and increases, as is easy to see, with increasing concentration, is, from the point of view of Arrhenius’s theory of electrolytic dissociation, equivalent to a decrease in the degree of dissociation.

The effect considered here by no means exhausts the phenomena arising in a solution when an electric current passes through it. The point is that under the action of an external electric field all the ions present in the solution are set in motion: the positive ones in one direction, the negative ones in the opposite one—

opposite. We have seen that each ion is surrounded (on the average) predominantly by ions of the opposite sign, which move toward it, carrying the liquid with them; thus the ion under consideration moves not in a quiescent medium, but in a certain flow directed toward it. The additional frictional force arising thereby may be regarded as a consequence of electrophoresis, which appears in the solution when an electric current passes through the latter.

§ 2. Quantitative theory.

The calculations, on which—owing to their considerable complexity—we shall not dwell here, show that the force acting on an ion of the \(j\)-th kind and caused by the finite time of formation of the “ionic atmosphere” around the ion under consideration is equal to

\[ -\frac{e^{2}}{6}\frac{\omega_{j}}{D}z_{j}^{2}\varkappa, \tag{37} \]

where

\[ \omega_{j}=\frac{v_{j}}{kT}\frac{\sum n_i z_i^2 \rho_i}{\sum n_i z_i^2}; \]

\(\rho_i\)—the coefficient of friction characterizing the given particle;
\(v_i\)—the velocity of its motion; the quantity

\[ \varkappa=\sqrt{\frac{4\pi e^{2}}{DkT}\sum n_i z_i^2}. \]

Let us note that the indicated formula is obtained only in the case where, in calculating the distribution of ions in a solution subjected to the action of an external electric field, in the first approximation one neglects the effect due to the motion of the liquid medium. It follows from formula (37) that the force acting on the ion is directed opposite to the motion of the latter (independently of its sign), is proportional to the first power of the velocity, and, consequently, is equivalent to an ordinary frictional force. In Fig. 3 are shown the lines of force of the electric field established around a moving ion. The field pattern in this case is such as if the lines of force lagged behind the moving ion owing to retardation.

Fig. 3.

Compare this additional force with the ordinary friction \(\rho_1 v\) experienced by an ion moving with velocity \(v\). The ratio \(\alpha\) of the additional force to the latter quantity is equal to:

\[ \alpha = \frac{z_1^2 \varepsilon^2 \varkappa}{6 D k T}. \]

For an aqueous solution containing \(\gamma\) moles of a uni-univalent electrolyte per liter of solution, at \(t = 0^\circ\mathrm{C}\) we have:

\[ \alpha = 0.382 \sqrt{\gamma}. \]

Thus, even at sufficiently small concentrations, the additional friction is comparable with the ordinary resistance of the medium to the motion of the ion.

Turning now to the calculation of the electrophoretic force, let us recall that if ions are regarded as undeformable spheres of some finite radius \(b\), and the solvent is treated as a viscous incompressible liquid characterized by the coefficient of internal friction \(\eta\), then, according to the well-known Stokes formula, an ion moving with velocity \(v\) encounters a resistance equal to \(6\pi \eta b v\).

A characteristic feature of the case of interest to us is the existence, in each volume element of the liquid around the moving ion, of a space charge with density

\[ - \frac{D \varkappa^2}{4\pi}\psi_0, \tag{38} \]

where

\[ \psi_0 = \frac{z_j \varepsilon}{D}\frac{e^{-\varkappa r}}{r}, \]

owing to which the external electric field \(E\) creates in the liquid in which the ion moves a volume force

\[ \delta = - \frac{D E}{4\pi}\varkappa^2 \psi_0. \]

The presence of this volume force is entirely disregarded by classical hydrodynamics in the derivation of the Stokes formula. In calculating the electrophoretic effect, the electrical theory of solutions therefore sets itself the task of clarifying the changes which must be introduced into Stokes’ formula on account of the circumstance indicated above.

Let us note that the density of electricity is calculated from the formula for the “static” case

\[ \psi_0 = \frac{z_j \varepsilon}{D}\frac{e^{-\varkappa r}}{r}, \]

B. N. FINKELSTEIN

although in reality, as indicated above, the field around the moving ion is determined by a more complicated formula. However, the approximation thereby admitted proves sufficient for the limiting case of strongly diluted solutions that interests us.

Without dwelling on the integration of the hydrodynamic equations1, we shall give only the final result. The electrophoretic force, added to the Stokes friction \(6\pi\eta v b_i\), acts on the ion in the direction of the external field and is equal in magnitude to \(\varepsilon z_i X \chi b_i\), where \(X\) denotes the field strength.

Let us now proceed to the determination of the mobility of an ion of the \(j\)-th kind. As is known, in the steady motion of an ion the sum of all forces acting on it is equal to zero:

\[ X\varepsilon z_j-\frac{1}{6}\frac{\rho v_j \varepsilon^2}{DkT}\chi z_j^2-6\pi\eta v b_j-z_j\varepsilon\chi b_j X=0, \]

where

\[ \rho=\frac{\sum n_i z_i^2 \rho_i}{\sum n_i z_i^2} \]

is the “mean” coefficient of friction; whence

\[ v_j=\varepsilon z_j X \frac{1-\chi b_j}{6\pi\eta b_j+\frac{1}{6}\frac{\rho v_j \varepsilon^2}{DkT}\chi z_j^2}. \]

Further, assuming that the quantity \(\chi b_j\) is small in comparison with unity, we obtain for the mobility:

\[ u_j=\frac{v_j}{X}= \frac{\varepsilon z_j}{6\pi\eta b_j} \left[ 1-\left( \frac{\rho}{6\pi\eta b_j}\frac{\varepsilon^2 z_j^2}{DkT}+b_j \right)\chi \right]. \]

Let us consider the case in which the solution contains only one electrolyte, each molecule of which dissociates into \(\nu_i\) \((i=1,2,\ldots,s)\) ions of the \(i\)-th kind with charges \(z_i\varepsilon\). The molecular conductivity \(\lambda\) of such a solution is expressed by the formula

\[ \lambda=\varepsilon\sum_1^s \nu_i |z_i| u_i . \tag{39} \]

Bearing in mind that in the limiting case of infinite dilution the additional forces caused by the asymmetric structure of the ionic atmosphere around the moving ion and by the electrophoresis in the surrounding medium vanish, for the conductivity \(\lambda_0\) of an infinitely diluted solution we obtain

\[ \lambda_0=\varepsilon\sum_1^s \nu_i |z_i| u_i^{(0)}, \tag{40} \]

where

\[ u_i^{(0)}=\frac{\varepsilon |z_i|}{6\pi\eta b_i}=\frac{\varepsilon |z_i|}{\rho_i}. \]

As the conductivity coefficient \(f_\lambda\), taking into account the influence of the electrical forces of interaction between ions on the mobility of the latter, N. Bjerrum proposed calling the quantity

\[ f_\lambda=\frac{\lambda}{\lambda_0}=\frac{\Lambda}{\Lambda_0}, \]

which, as experience shows, is always less than unity. For the percentage deviation of \(\lambda\) from \(\lambda_0\) we obtain

\[ 1-f_\lambda=\frac{\lambda_0-\lambda}{\lambda_0} =\frac{\Lambda_0-\Lambda}{\Lambda_0} =\sqrt{\frac{4\pi e^2\nu n}{DkT}} \left[ \frac{\varepsilon^2}{6DkT}w_1+bw_2 \right], \tag{41} \]

where

\[ w_1= \frac{\sum \nu_i z_i^2 \rho_i}{\sqrt{\nu\sum \nu_i z_i^2}} \cdot \frac{\sum \dfrac{\nu_i z_i^4}{\rho_i^2}}{\sum \dfrac{\nu_i z_i^2}{\rho_i}}; \]

\[ w_2=\sqrt{\frac{\sum \nu_i z_i^2}{\nu}}; \qquad b= \frac{\sum \dfrac{\nu_i z_i^2}{\rho_i}b_i}{\sum \dfrac{\nu_i z_i^2}{\rho_i}}; \qquad \nu=\sum \nu_i,\quad \rho_i=6\pi\eta b_i. \]

Let us consider the coefficients \(w_1\) and \(w_2\), which characterize the individual features of the ions. In the following table are given the values of \(w_2\) for various types of electrolytes.

Table 3.

The coefficient \(w_1\), as is easily seen, is a function of the ratio of the mobilities of the ions in infinitely dilute solutions. Indeed, by definition of the quantities \(\rho\), we have:

a) for one–one-valent electrolytes of the KCl type

\[ w_1= \frac{\rho_1+\rho_2}{2} \frac{\dfrac{1}{\rho_1^2}+\dfrac{1}{\rho_2^2}} {\dfrac{1}{\rho_1}+\dfrac{1}{\rho_2}} = \frac{1}{2}\left(\frac{\rho_1}{\rho_2}+\frac{\rho_2}{\rho_1}\right) \]

or, taking into account that

\[ \rho_i=\frac{|z_i|e}{u_i^{(0)}}, \]

we obtain

\[ w_1=\frac12\left(\frac{u_1^{(0)}}{u_2^{(0)}}+\frac{u_2^{(0)}}{u_1^{(0)}}\right); \]

if we introduce the ionic transport numbers

\[ \vartheta_1=\frac{u_1^{(0)}}{u_1^{(0)}+u_2^{(0)}};\qquad \vartheta_2=\frac{u_2^{(0)}}{u_1^{(0)}+u_2^{(0)}};\qquad \vartheta_1+\vartheta_2=1, \]

then finally

\[ w_1=\frac12\left(\frac{\vartheta_1}{\vartheta_2}+\frac{\vartheta_2}{\vartheta_1}\right); \]

b) for two–univalent salts of the type \(K_2SO_4\)

\[ w_1=\frac{\frac12\vartheta_1+2\vartheta_2}{3(2\vartheta_1+\vartheta_2)} \left(8\frac{\vartheta_1}{\vartheta_2}+\frac{\vartheta_2}{\vartheta_1}\right) \]

(the index 1 refers to the divalent ion, index 2—to the univalent ion);

c) finally, for two–divalent electrolytes of the type \(MgSO_4\)

\[ w_1=4\left(\frac{\vartheta_1}{\vartheta_2}+\frac{\vartheta_2}{\vartheta_1}\right). \]

§ 3. Comparison of Theory with Experiment.

a) Aqueous solutions.

From consideration of formula (41) it is easy to see that the Debye theory leads to a result coinciding with the empirical law

\[ 1-f_i\sim\sqrt{n} \]

(\(\sim\) is the sign of proportionality),

established by Kohlrausch on the basis of his classical investigations of the electrical conductivity of very dilute aqueous solutions of strong electrolytes.

Debye and Hückel also carried out a thorough quantitative test of their theory. For this purpose they used the very accurate measurements of the electrical conductivity of aqueous solutions belonging to Kohlrausch. Since all these measurements refer to the temperature \(t=18^\circ C\), substituting into formula (41) the numerical values of the constants corresponding to this temperature, we obtain

\[ 1-f_i=\left[0.278\,w_1+0.233\cdot10^8\,b w_2\right]\sqrt{\gamma}, \tag{41} \]

whence

\[ \overline{\Lambda}=\overline{\Lambda}_0-a\sqrt[3]{\gamma}, \tag{42} \]

where

\[ \overline{\Lambda}=\frac{\Lambda}{9\cdot 10^{11}};\qquad a=\Lambda_0\left[0.278\,u_1+0.233\cdot 10^8\,l\,u_2\right]; \]

\(\Lambda\) and \(\overline{\Lambda}\) are the molar (gram-molecular) conductivities, expressed in practical units.

It was indicated above that formula (41) is valid only for the most dilute solutions; therefore, when comparing with experiment, Debye and Hückel, as the next approximation, use the formula

\[ \overline{\Lambda}=\overline{\Lambda}_0-a\sqrt[3]{\gamma}+\beta(\gamma). \]

Taking from Kohlrausch’s data the values of \(\Lambda\) for five different concentrations

\[ \gamma=0.0001;\quad 0.0002;\quad 0.0005;\quad 0.001;\quad 0.008, \]

they, for each electrolyte separately, calculated \(\Lambda_0\), \(a\), and \(\beta\) by the method of least squares. Having thus determined from experimental data the quantities \(\overline{\Lambda}_0\), one can, from the transport numbers of at least any two ions, calculate the mobilities of all the remaining ions, if only it is assumed that, in the case of infinitely dilute solutions, the mobility of a given ion does not depend on the nature of the other ion, i.e., in other words, does not depend on the electrolyte of which the ion under consideration is a constituent. The hypothesis stated here of the “independence of ion mobilities” was confirmed by Kohlrausch on the basis of his numerous observations. Let us explain what has been said by an example: knowing the transport numbers for the ions K and Cl, one can first calculate the mobilities of the ions (simple and complex) forming the following electrolytes:

\[ \mathrm{KF},\ \mathrm{KBr},\ \mathrm{KI},\ \mathrm{KJO}_3,\ \mathrm{KClO}_3,\ \mathrm{KNO}_3,\ \mathrm{KCNS} \]

\[ \mathrm{LiCl},\ \mathrm{NaCl},\ \mathrm{CsCl}\quad \text{etc.} \]

Knowing the corresponding quantities \(\overline{\Lambda}_0\) and taking hence the mobility of some ion (for example, Br), common to some other electrolytes (for example, bromides), one can calculate the mobilities of the remaining ions. Indeed, by definition, in the case under consideration of monovalent electrolytes,

\[ \overline{\Lambda}_0=\frac{Ne}{9\cdot 10^7}\left(u_1^{(0)}+u_2^{(0)}\right)=L_1+L_2,\qquad \text{where } L_i=\frac{Ne}{9\cdot 10^7}u_i^{(0)}. \]

Next, taking for the ratio of the transference numbers of the K and Cl ions the value

\[ \delta=\frac{0.497}{0.503}, \]

we may write

\[ \overline{L}_{\mathrm{K}}=\overline{\Lambda}_{\mathrm{KCl}}\frac{\delta}{\delta+1}. \]

Let us now compute \(\overline{L}_{J0_3}\) (a quantity proportional to the mobility:

\[ \overline{L}=321.18u^{(0)}): \]

\[ \overline{\Lambda}_{\mathrm{KJ0_3}}=98.41=\overline{L}_{\mathrm{K}}+\overline{L}_{J0_3}; \quad \overline{L}_{\mathrm{K}}=64.58; \]

therefore, \(\overline{L}_{J0_3}=33.83\).

Debye gives somewhat different values: 64.61 and 33.87, owing to the fact that, having at his disposal data for eighteen electrolytes, he also treated these calculations by the method of least squares. The mobilities thus obtained serve for computing the coefficient \(w_1\).

Next, representing (42) in the following form

\[ \frac{\overline{\Lambda}-\overline{\Lambda}_{0}}{\overline{\Lambda}_{0}}=1-f_{\lambda}\frac{a}{\Lambda_{0}}\sqrt{\gamma} \]

and comparing with (41), we obtain

\[ 0.233b\cdot 10^8=\frac{a}{\Lambda_{0}}=0.278w_1 \]

(for uni-univalent electrolytes \(w_2=1\)).

The “mean” values of the ionic radii calculated from this have, in general, the correct order of magnitude (Table 4). Such a result must be considered entirely satisfactory for the theory under consideration, which is the very first approximation to reality, especially since the ionic radii were introduced in calculating the electrophoretic action, and the latter was carried out with the aid of the ordinary “macroscopic” equations of hydrodynamics.

b) Non-aqueous solutions.

A quantitative verification of the theory as applied to non-aqueous solutions encountered an obstacle in the absence of sufficiently accurate experimental data relating to the region of very small concentrations. The recently published work of Frazer and Hartley1, who measured the electrical conductivity of dilute non-aqueous solutions with an accuracy not inferior to Kohlrausch’s for aqueous solutions, fills this annoying gap. The work published by them con—

consists in determining the electrical conductivity of solutions of uni-univalent salts in methyl alcohol; the measurements cover the concentration interval from 0.0001 normal to 0.002 normal. Frazer and Hartley showed that their measurements are expressed very accurately by a formula of type (42). Calculating the value of \(b \cdot 10^8\) by the method indicated above, they obtained here also, in general, the correct order of magnitude (Table 5). The authors note that here, as in aqueous solutions, the mobilities of the ions of alkali metals increase on passing to atoms with higher ordinal numbers.

Table 4.

Table 5.

1. J. E. Frazer and Harold Hartley. Proc. Roy. Soc. London (A) 109, 351, 1925. ↩↩↩