V. D. Kuznetsov.

Relationship between Electrical Conductivity and Internal Friction of Electrolytically Conducting Solids

With increasing temperature, the electrical conductivity of electrolytically conducting solids increases. The reason for this may lie: 1) in an increase in dissociation, i.e., in an increase in the number of free ions, and 2) in an increase in the mobility of the ions, or in a decrease in the resistance forces during their motion.

I know of no direct methods for measuring the dissociation of solids, and for the present it is impossible to say what effect temperature has on dissociation; but one may suppose that dissociation increases with increasing temperature.

The resistance forces during the motion of ions must decrease with increasing temperature. This follows from the fact that the resistance forces have an electrical nature, just as do the elastic and plastic forces, which decrease with increasing temperature. When, as a consequence of a rise in temperature, the bonds between the elements of the crystal lattice weaken, then the forces of interaction between the elements of the lattice and the free ions also weaken. The decrease in the resistance force during the motion of an ion must follow the same law as that by which the shear force changes in the stationary regime, i.e., the force characterizing what is called internal friction.

In a first approximation one may consider that the resistance force \(R\) during the motion of an ion is proportional to the viscosity \(\eta\) of the medium and to the velocity \(v\) of the ion:

\[ R = a v \eta, \tag{1} \]

where \(a\) is a certain constant.

But the velocity is:

\[ v = kX, \tag{2} \]

where \(k\) is the mobility of the ion and \(X\) is the intensity of the electric field. Consequently, in the steady motion of the ion, when the resistance force is balanced by the electric force moving the ion, it must be:

\[ R = ak\eta X = eX, \tag{3} \]

where \(e\) is the charge of the ion, or

\[ k\eta = \mathrm{const}, \tag{4} \]

i.e., the mobility of the ion must be inversely proportional to the coefficient of internal friction.

On the basis of what has been said, one may consider that the electrical conductivity \(\sigma\) is proportional to the degree of dissociation \(\gamma\), i.e., to the number of free ions, and inversely proportional to the viscosity of the medium \(\eta\):

\[ \sigma \eta = A\gamma, \tag{5} \]

where \(A\) is a certain constant.

If the solid is a crystal, and if the crystalline lattice does not break up into individual ions with increasing temperature, i.e., if \(\gamma\) does not depend on temperature, then at every temperature it must be:

\[ \sigma \eta = \mathrm{const}. \tag{6} \]

Observing the electrical conductivity at two temperatures, one can determine the relative change in dissociation and the relative change in the mobility of the ions. Indeed, suppose that at temperature \(\theta_1\) the electrical conductivity of the crystal is \(\sigma_1\); then we raise the temperature to \(\theta_2\) and find, correspondingly, the electrical conductivity of the crystal \(\sigma_2\). After this we rapidly lower the temperature to \(\theta_1\) and again determine the electrical conductivity \(\sigma'\), which in the general case, if the dissociation depends on temperature, will differ from \(\sigma_1\). Let the dissociation increase when the temperature is raised. At \(\theta_2\) there will be an increased dissociation, which upon rapid cooling to \(\theta_1\) cannot at once decrease and reach the normal value corresponding to \(\theta_1\). In order for the dissociation to decrease, it is necessary that a portion of the free ions recombine into neutral molecules. This recombination proceeds the more rapidly, the greater the mobility of the ions, which depends on the internal friction of the medium. In passing from the higher temperature \(\theta_2\) to the lower \(\theta_1\), the internal friction at once assumes the value corresponding to the latter temperature. As a result, at \(\theta_1\) the process of recombination is slowed down, the mobility of the ions decreases, and the number of free ions for the first time remains approximately the same as at \(\theta_2\).

From what has been said it follows that the relative change in dissociation from \(\theta_1\) to \(\theta_2\) is equal to:

\[ \frac{\Delta \gamma}{\gamma}=\frac{\sigma'-\sigma_1}{\sigma_1}, \tag{7} \]

and the relative change in the mobility of an ion is equal to:

\[ \frac{\Delta k}{k}=\frac{\sigma_2-\sigma'}{\sigma'}. \tag{8} \]

The experimental material on the question of the dependence of electrical conductivity on temperature is very extensive. Here we shall consider only those data which refer to crystals, molten salts, and those solid bodies which conduct electrolytically. There are still very few experimental data on crystals.

It turns out that in the overwhelming majority of cases the dependence of the electrical conductivity \(\sigma\) on the temperature \(\theta\) can be expressed by the formula

\[ \sigma_{\theta_2}=\sigma_{\theta_1} e^{\alpha(\theta_2-\theta_1)}. \tag{9} \]

or

\[ \sigma_\theta=\sigma_0 e^{\alpha\theta}, \tag{10} \]

where \(\alpha\) is some constant.

Plotting \(\theta\) along the abscissa axis and \(\Log \sigma\) along the ordinate axis, we must obtain straight lines.

In A. Bogorodskii’s book Materials on the Electrochemistry of Inorganic Compounds in the So-called Fire-Liquid State. Part I. Kazan, 1905, a large amount of material is given on the electrical conductivity of molten salts. Bogorodskii expresses the dependence of electrical conductivity on temperature by formulae of the form:

\[ \sigma_\theta=\sigma_0[1+\alpha\theta]. \tag{11} \]

or

\[ \sigma_\theta=\sigma_0[1+\alpha\theta+\beta\theta^2], \tag{12} \]

which are approximate values of formula (10) when it is expanded in a series. But the experimental data cited by him are excellently expressed by formula (10). As examples one may point to \(\mathrm{AgNO_3}\) (from \(272^\circ\) to \(375^\circ\)), \(\mathrm{PbCl_2}\) (\(508^\circ\)—\(664^\circ\)), \(\mathrm{NaCl}\) (\(720^\circ\)—\(780^\circ\)), etc.

The works of Foussereau¹ give numerical values of \(\sigma\) at various \(\theta\) for \(\mathrm{NaNO_3}\) \((52^\circ—356^\circ)\), \(\mathrm{KNO_3}\) \((30^\circ—300^\circ)\), \(\mathrm{KClO_3}\) \((145^\circ—359^\circ)\), and \(\mathrm{ZnCl_2}\) \((59^\circ—310^\circ)\). The results fit formula (10) very well; moreover, for \(\mathrm{NaNO_3}\) and \(\mathrm{KClO_3}\), at the melting temperature the electrical conductivity at once increases, and above the melting point it increases according to the same law, but more slowly than for the solid state.

For some salts which, upon an increase in temperature, pass into other modifications or decompose, the electrical conductivity changes by jumps at the transition points. For each modification, formula (10) is for the most part applicable.

The electrical conductivity of ordinary glass, which, as is known, conducts electrolytically, according to the experiments of Foussereau², likewise obeys formula (10).

G. G. Urazov³ showed that silver sulfide and cuprous sulfide conduct current electrolytically, and that the dependence of \(\sigma\) on \(\theta\) is expressed by formula (10), while the polymorphic transformations of both sulfides are clearly reflected in the course of the electrical-conductivity curves.

G. v. Rautenfeld⁴ found that for rock salt within the limits \(630^\circ—790^\circ\) the same formula is valid.

In recent years, for expressing the dependence of \(\sigma\) on \(\theta\), another formula has begun to be used, which can be derived theoretically from the Van’t Hoff—Planck formula for dilute solutions of electrolytes.

Let us suppose that the substance \((AB)\) partially dissociates into its constituent parts \(A\) and \(B\). Let the system consist of \(n_1, n_2\), and \(n_3\) gram-molecules of the substances \((AB)\), \(A\), and \(B\). The concentrations of these substances are:

\[ \gamma_1=\frac{n_1}{n};\quad \gamma_2=\frac{n_2}{n}=\gamma_3=\frac{n_3}{n}=\gamma, \tag{13} \]

where:

\[ n=n_1+n_2+n_3. \]

The equilibrium constant in this case is equal to:

\[ K(p,T)=\frac{\gamma^2}{\gamma_1} \tag{14} \]

and the Van’t Hoff isochore takes the form:

\[ d\log\gamma=\frac{q}{2R}\cdot\frac{dT}{T^2}, \tag{15} \]

where \(q\) is the heat of dissociation of one gram-molecule of the substance, \(R\) is the gas constant, and \(T\) is the absolute temperature.

Integrating (15), we obtain:

\[ \log\gamma=-\frac{\nu}{T}+C, \tag{16} \]

where:

\[ \nu=\frac{q}{2R}. \]

The constant \(C=\log\gamma_\infty\) at \(T=\infty\); consequently:

\[ \log\gamma=-\frac{\nu}{T}+\log\gamma_\infty \]

or

\[ \frac{\gamma}{\gamma_\infty}=e^{-\frac{\nu}{T}}. \tag{17} \]

Since the electrical conductivity \(\sigma\) is proportional to the concentration of ions, or to the dissociation, then:

\[ \sigma=\sigma_{\infty}e^{-\frac{\nu}{T}}, \tag{18} \]

or

\[ \log \sigma=-\frac{\nu}{T}+\log\sigma_{\infty}, \tag{19} \]

where \(\sigma_{\infty}\) is the electrical conductivity at an infinitely high temperature.

Formulas (18) and (19) are used in practice. But, in my opinion, they are not sufficiently well founded theoretically. In their derivation the change of internal friction with change of temperature was completely disregarded. Formula (18) is valid: 1) when \(\eta\) does not depend on temperature, or 2) when the dependence of \(\eta\) on \(T\) is expressed by:

\[ \eta=\eta_{\infty}e^{\frac{\mu}{T}}. \tag{20} \]

Indeed, if formula (5) is regarded as valid, then, with (20), one obtains:

\[ \sigma=A\frac{\gamma}{\eta} =A\frac{\gamma_{\infty}}{\eta_{\infty}}e^{-(\nu-\mu)\frac{1}{T}} =\sigma_{\infty}e^{-\frac{\rho}{T}}, \tag{21} \]

where \(\rho=\nu-\mu\).

E. Rasch and F. W. Hinrichsen¹ indicated that antimony chloride, in the range \(100^\circ\)—\(200^\circ\), lead, white and green glass in the range \(200^\circ\)—\(350^\circ\), and phosphorus from \(50^\circ\) to \(210^\circ\), satisfy formula (18).

Ankron² investigated 13 varieties of glass made from melts of \(\mathrm{SiO_2}\) with \(\mathrm{Na_2O}\) and \(\mathrm{CaO}\). He found that formula (18) holds for all varieties, and that the values of \(\nu\) and \(\sigma_{\infty}\) depend on the concentration of the molecules \(\mathrm{Na_2O}\) and \(\mathrm{CaO}\) in the glass melt.

A. F. Ioffe³ indicates that the conductivity of Iceland spar, quartz, alum, and copper sulfate is expressed by (18). F. v. Rautenfeld⁴ found that, for Iceland spar in the range from \(232^\circ\) to \(552^\circ\),

\[ \sigma=0.01995\cdot e^{-\frac{5756.5}{T}}. \tag{22} \]

However, the salts \(\mathrm{AgBr}\)⁵, \(\mathrm{KClO_3}\), \(\mathrm{ZnCl_2}\), \(\mathrm{NaNO_3}\), and \(\mathrm{KNO_3}\)⁶, while satisfying (10) excellently, do not at all obey (18).

The results of Schultze’s⁷ experiments with \(\mathrm{ZnCl_2}\) in the range \(225^\circ\)—\(700^\circ\) do not satisfy either (10) or (18).

The validity of formula (5) could be proved only by the simultaneous study of the dependence of \(\sigma\), \(\eta\), and \(\gamma\) on temperature. Unfortunately, there is not a single work in which all three quantities have been studied simultaneously.

A. A. Shaposhnikov¹ showed that for crystalline quartz along the optical axis, within the limits \(0^\circ\)—\(95^\circ\), the mobility of the ions may be expressed as:

\[ k_\theta = k_0 e^{0.0331\theta}. \tag{23} \]

and the dissociation—as:

\[ \gamma_\theta = \gamma_0 e^{0.056\theta}. \tag{24} \]

He drew this conclusion from measurements of \(\sigma\) and \(\sigma'\) (see p. 50).

If one “regards quartz as a special kind of electrolyte, in which, owing to the peculiarities of dissociation, the electrical conductivity outwardly resembles the conductivity of ionized gases,” then for quartz the internal friction along the optical axis, according to my assumption, may be expressed as:

\[ \eta_\theta = \eta_0 e^{-0.033\theta} \tag{25} \]

This could be tested by studying the angular velocity of shear parallel to the optical axis at different \(\theta\).

A. F. Ioffe² finds that from \(17^\circ\) to \(325^\circ\) the electrical conductivity of quartz in different directions can best be expressed by formula (18).

For anhydrous pure salts, formula (6) should hold:

\[ \sigma\eta = \text{const} \]

or, if the resistance is denoted by \(\rho\), the formula:

\[ \frac{\eta}{\rho} = \text{const}. \]

Here we have, for the present, the following data, which speak in favor of these formulas.

1. Foussereau³ determined at different temperatures the resistance \(\rho\) and viscosity \(\eta\) for molten \(\mathrm{NaNO_3}\) and \(\mathrm{KNO_3}\), and obtained the following figures:

1. In 1912 and 1918, studying the torsion of rock salt at different temperatures, I found⁴ that within the limits \(14^\circ\)—\(405^\circ\) the internal friction can be expressed as:

\[ \eta_\theta = \eta_0 e^{-0.024\theta} \tag{26} \]

I assumed that the dissociation of rock salt should not depend on temperature, since otherwise the salt would decompose upon heating, and expected that for it there should be

\[ \sigma=\sigma_0 e^{0.024\theta} \tag{27} \]

Several months after my work appeared in print, volume 72 of Annalen der Physik was received in Tomsk, containing F. v. Rautenfeld’s work on the passage of electricity through crystals (p. 617). For rock salt, in the range from \(630^\circ\) to \(790^\circ\), he found:

\[ \sigma_\theta=0\cdot 0_9\,152\cdot e^{0.0237\theta} \tag{28} \]

Such complete agreement, while perhaps purely accidental, compels one to take a more attentive attitude toward the question under consideration concerning the dependence between \(\sigma\) and \(\eta\), and to turn to the study of the internal friction of solid crystalline salts at different temperatures.

It would be interesting to study the dependence of \(\eta\) on \(\theta\) for anhydrous crystalline salts. In particular, on the basis of Foussereau’s experiments¹ on the electrical conductivity of \(\mathrm{NaNO_3}\) and \(\mathrm{KNO_3}\), one may expect that for \(\mathrm{NaNO_3}\), within the range \(52^\circ\)–\(300^\circ\), there should be

\[ \eta_\theta=\eta_0 e^{-0.072\theta} \tag{29} \]

and for \(\mathrm{KNO_3}\), within the range \(30^\circ\)–\(300^\circ\),

\[ \eta_\theta=\eta_0 e^{-0.060\theta}. \tag{30} \]

1. C. A. 96, 1325 (1881); Ann. de chim. (6) 5, 317 (1885). ↩↩↩↩

2. Elastic and electrical properties of quartz. Izv. Petr. Polit. In. 24 (1915), p. 111. ↩↩↩

3. Ann. de chim. et phys. (6) 5, 317 (1885). ↩↩↩

4. Izv. Tom. Tekhn. In. 43, issue 3, p. 47 (1923). ↩↩↩

5. Kohlrausch. W. A. 17, 642 (1882). ↩

6. Foussereau. l. c. ↩

7. Ztschr. f. anorg. Chem. 20, 333 (1899). ↩