V. D. Kuznetsov.

Effect of Temperature on the Electrical Conductivity of Liquid Dielectrics.

(The article contains 1 page of drawings.)

§ 1. Introduction.

Proceeding from the conception of the ionic character of the passage of electricity through liquid dielectrics, we must come to the conclusion that the temperature of the liquid affects its electrical conductivity.

It is difficult to admit that temperature in itself is a strong ionizing agent creating new ions. This is apparently possible only for gases, which can be heated so much that the living force of the thermal motion of gas molecules can, upon collision, split neutral molecules into ions. The influence of temperature on conductivity is chiefly indirect.

With an increase in temperature the coefficient of internal friction \(\eta\) of liquids decreases; consequently the resistance encountered by an ion in its motion through the liquid under the action of electrical forces decreases, and the mobility of the ion increases. All experiments with the slow, steady motion of solid or liquid bodies inside liquids lead to the conclusion that the resistance is directly proportional to the viscosity. Consequently, it may be expected that the mobilities of ions are inversely proportional to \(\eta\).

The expression for the saturation current strength does not include the mobilities of ions; therefore, it would seem, the saturation current should not depend on temperature. At small voltages, when the current obeys Ohm’s law, the current strength at a given voltage is proportional to the mobilities of the ions and must be inversely proportional to the viscosity. With an increase in the temperature of a dielectric liquid, the current strength must increase, and one may expect that the saturation current will occur at a lower potential difference.

If the saturation current increases with increasing temperature, this means that the rate of formation of the number of ions increases. This may occur in the case when temperature itself is an ionizing agent, or when the penetrating power of ionizing rays changes with the change in temperature, or when the process of ionization is facilitated with increasing temperature.

Everything said applies to pure dielectrics. If, however, the dielectrics are not sufficiently pure, if various foreign substances capable of dissociating are present in them, then the dependence of the electrical conductivity on temperature becomes considerably more complicated. The more impurities there are in the dielectric, the more its own independent conductivity is obscured, and the more varied is the influence of temperature.

Studying the influence of temperature on the conductivity of petroleum, I found that the electrical conductivity of petroleum that had first been heated and then cooled returns very slowly to its normal value corresponding to the low temperature, whereas in passing from a low temperature to a higher one the stationary state is established without delay. This phenomenon served as the basis for the following view of the process of the influence of temperature on the conductivity of unpurified dielectrics.

V. D. Kuznetsov. “Effect of Temperature on the Electrical Conductivity of Liquid Dielectrics.”

Fig. 2.

Labels in the apparatus drawing:

• to the pump
• [[unclear: vertical label near the upper inlet]]
• [[unclear: small label near the lower inlet]]

Fig. 1.

Fig. 3.

Castor oil.

Dependence on temperature:

1. viscosity \(\eta\),
2. electrical conductivity \(\lambda\),
3. dissociation \(C\).

Curve labels:

\[ \eta = \psi(t) \]

\[ \lambda = \varphi(t) \]

\[ c = \chi(t) \]

As is known, with increasing temperature the degree of dissociation increases and the process of dissociation itself is accelerated. In addition, the viscosity of a liquid decreases with increasing temperature, and the ions encounter less resistance in their motion. Both of these factors increase the electrical conductivity. If a liquid is cooled after being heated, then its viscosity immediately assumes its normal value, but the dissociation proves to be increased. The more rapidly the liquid is cooled, the more the process of recombination is delayed and the higher is the conductivity in comparison with the normal conductivity corresponding to the lower temperature. If the dissociation is artificially increased in some way, then in order to eliminate it the temperature should be raised, which facilitates ionization. In addition, an elevated temperature promotes purification of the dielectric, by passing an electric current, from dissolved impurities.

Thus the influence of temperature may be very diverse. If a dielectric has normal conductivity, then heating it is accompanied by an increase in conductivity. If, however, the dissociation of substances dissolved in the dielectric has in some way been artificially increased, then heating, by helping the current to remove ions, decreases the conductivity.

From what has been said it is clear that, for electrical purification of liquids, one must if possible raise its temperature higher, pass a current of high voltage, and then carefully remove the electrodes or separate them by partitions and, together with them, the dissolved impurities that have settled on them.

The resistance of liquid dielectrics depends on the strength of the current passing through it. If, when studying the dependence of electrical conductivity on temperature, a strong current is passed through the dielectric, then the influence of temperature is interwoven with the influence of the current, and it is difficult to determine exactly what share of the change in conductivity is due to temperature. Therefore experiments with one and the same dielectric at different temperatures are advantageously carried out by passing a current of weak voltage through it. With such an arrangement the influence of temperature comes to the fore, and the influence of the current may be neglected.

The influence of temperature on the conductivity of poorly purified liquid dielectrics may manifest itself 1) through a change in viscosity and 2) through a change in the degree of dissociation. Apparently, the influence of each factor can be taken into account separately if the investigation is conducted in the following order.

Let us measure the conductivity \(\sigma_1\) at temperature \(t_1\), then raise the temperature to \(t_2\) and measure the conductivity \(\sigma_2\). Next, let us very rapidly cool the liquid to the initial temperature \(t_1\) and again determine the conductivity \(\sigma'_1\), which, generally speaking, will differ from \(\sigma_1\) and \(\sigma_2\). Owing to the lowering of the temperature, and consequently owing to the increase in resistance to the motion of ions, the process of recombination is slowed, and the excess of ions over the normal number corresponding to temperature \(t_1\) disappears the more slowly, the lower this temperature. If the transition from \(t_2\) to \(t_1\) is made rapidly, then one may neglect the number of ions that have had time to recombine, and one may consider that the difference between the values \(\sigma_1\) and \(\sigma'_1\) is due exclusively to the difference in the numbers of ions, i.e. to the difference in dissociation at temperatures \(t_2\) and \(t_1\), since in both measurements the viscosity is one and the same. Thus the ratio

\[ a=\frac{\sigma'_1-\sigma_1}{\sigma_1} \]

characterizes the relative change in the degree of dissociation when the temperature changes from \(t_1\) to \(t_2\), while the ratio

\[ b=\frac{\sigma_2-\sigma'_1}{\sigma_2} \]

characterizes the influence of viscosity in the transition from \(t_2\) to \(t_1\).

§ 2. The Process of Ion Development.

Applying kinetic theory to ions in liquid dielectrics, as was done by Van der Bijl¹, we shall assume that the kinetic energy of the thermal motion of an ion situated among the molecules of a liquid dielectric is equal to the vis viva of a gas molecule at the same temperature. Moving ions, like the molecules of a dissolved substance, produce an osmotic pressure, and Clapeyron’s formula is applicable to them. The vis viva of a gas molecule at normal atmospheric pressure and at a temperature of \(20^\circ\mathrm{C}\) is equal to

\[ E=\frac{mv^2}{2}=5.87\cdot 10^{-14}\ \text{ergs}, \]

and the vis viva of an ion is equal to this same quantity.

Under the action of an ionizer, a molecule of the dielectric splits into an electron and a positively charged residue, or into two parts, of which one has a positive and the other a negative charge. These fragments of the molecule attract neutral molecules to themselves and soon find themselves surrounded by them.

An ion attracts a neutral molecule with a force \(F\), which, according to Langevin², may be expressed as

\[ F=\frac{K-1}{2\pi N}\cdot\frac{e^2}{r^5}, \]

where \(N\) is the number of molecules in \(1\ \mathrm{cm}^3\), \(r\) is the distance between the molecule and the ion, \(K\) is the dielectric constant of the medium, and \(e\) is the elementary charge. The work performed by the electric forces in attracting a molecule from infinity to the surface of the ion is equal to

\[ A=\int_{\rho}^{\infty} F\,dr =\frac{e^2(K-1)}{2\pi N}\int_{\rho}^{\infty}\frac{dr}{r^5}, \]

or

\[ A=\frac{K-1}{8\pi N}\cdot\frac{e^2}{\rho^4}; \]

where \(\rho\) is the radius of the ion.

For simplicity of reasoning, although this is not essential, let us suppose that the ion is at rest, while the molecule approaches it, possessing at the moment of collision the vis viva \(E\). We shall regard the collisions between molecules and between a molecule and an ion as perfectly elastic. After a collision with the ion, the molecule may combine with it and increase its mass, or it may move away from it and remain free as before. If, at the moment of impact, the kinetic energy of the molecule \(E>A\), then no combination will occur, since the molecule, rebounding from the ion, overcomes the work \(A\) of the attractive forces. Conversely, if \(E<A\), the molecule will necessarily combine with the ion. Under the condition \(E=A\), either may occur. Since the kinetic energy of an ion in a state of equilibrium must always be equal to the mean value of the energy of the molecules, then, as neutral molecules join the ion, i.e. as the mass of the ion increases, its velocity decreases and it becomes less mobile. As the volume of the ion grows, owing to the attachment of molecules to it, the work \(A\), inversely proportional to the fourth power of the distance \(r\), decreases rapidly, while the mean value of \(E\) of the molecules remains unchanged. Suppose, for example, that the positive residue of a molecule, after losing its electron, is surroun—

covered by a single layer of neutral molecules, then the radius of this positive ion is 3 times greater than the radius of the original positive center, \(a\); consequently, the work \(A\) is 81 times smaller. If, at first after the formation of the centers of the ions, the condition \(E>A\) holds, under which the attachment of only some of the slowest molecules is possible, then, as the ion grows, the ratio \(E:A\) decreases, approaching unity. At \(E=A\) the growth of the ion ceases and a stationary state is established, in which the number of molecules attaching themselves to the ion becomes equal to the number of molecules separating from it. From the condition

\[ \frac{m v^2}{2}=\frac{K-1}{8\pi N}\cdot \frac{\varepsilon^2}{\rho^4} \]

we obtain the value of the upper limit of the ion radius:

\[ \rho=\sqrt[4]{\frac{K-1}{4\pi N}\cdot \frac{\varepsilon^2}{m v^2}} . \]

With increasing temperature the velocity \(v\) of the molecules increases and, consequently, \(\rho\) decreases.

§ 3. Experimental data and their explanation.

The first rather detailed investigation of the influence of temperature on the electrical conductivity of liquid dielectrics was carried out by Koller¹, who studied petroleum ether, methyl and ethyl alcohols, olive, almond, and vaseline oils, benzene, toluene, xylene, turpentine, etc. The measuring condenser consisted of a crystallizing dish and two horizontal disks, with an area of \(3420\ \mathrm{mm}^2\), separated from one another by pieces of glass. The potential difference varied from \(1/8\) to 1 Daniell. Most of the liquids investigated were indefinite in chemical composition and were not subjected to careful purification; therefore the results can hardly be used for theoretical considerations. The liquids under investigation were heated in a sand bath to a definite temperature, and then their resistance was measured at various lower temperatures. It turned out that the resistance of the liquids under investigation usually decreases with increasing temperature.

In 1908 G. Jaffé² noted that the ionization of petroleum ether under the influence of \(\gamma\)-rays increases considerably with increasing temperature. The current strength, beginning with sufficiently large field intensities (above 500 volts/cm), was equal to:

\[ i=a+cV, \]

where \(a\) and \(c\) are constants, \(V\) is the potential difference. The experiments showed that, when the temperature was raised from \(0^\circ\) to \(40^\circ\mathrm{C}\), the coefficient \(a\) increased from \(1.69\) to \(2.45\), while the coefficient \(c\) decreased from \(1.54\cdot 10^{-4}\) to \(1.11\cdot 10^{-4}\).

This result can be explained from the point of view I have adopted in the following way. In ether there exist normal, easily mobile ions and heavy, slowly moving ions; the former give a current that reaches saturation at a certain potential difference, while the latter give a current obeying Ohm’s law. In § 2 we saw that the development of an ion ceases when the mean kinetic energy \(E\) of the molecules becomes equal to the work \(A\). At \(E<A\), neutral molecules are attracted to the ion and increase its size. Heavy ions are formed from slowly moving molecules, and normal ions from faster ones. At a given temperature a certain ratio is established between the numbers of normal and heavy ions, and, consequently, between the coefficients

$a$ and $c$. With an increase in temperature the energy $E$ increases, whereas the work $A$ does not depend on temperature. Some slow molecules which at low temperature would be attracted to the ion and would become bound to it, at a higher temperature, after collision with the ion, will rebound from it and remain free. The probability of formation of slow ions will decrease, while that of readily mobile ions will increase. As a result the saturation current $a$ will increase, while the current $cV$ will decrease.

Further, Jaffé¹ investigated the dependence of the conductivity of pure hexane on temperature in the range from $0^\circ$ to $40^\circ$. The measuring cylindrical condenser was placed in a thermostat, 40 cm in diameter and 30 cm high, filled with paraffin oil, which partly absorbed the external radioactive radiation. The hexane had been carefully distilled 8 times and, consequently, may be regarded as a very pure preparation. When a potential difference of $+400$ volts was applied, a saturation current was obtained which increased very slightly with increasing temperature; when $-400$ or $-800$ volts was applied, the saturation current decreased somewhat with increasing temperature. However, the observed changes in the current did not exceed the limits of error, and therefore it may be considered that, within the range from $0^\circ$ to $40^\circ$, temperature has no influence. On this basis it may be asserted that pure hexane does not possess electrolytic conductivity and that its electrical conductivity is due exclusively to the presence of normal readily mobile ions.

The influence of temperature on the electrical conductivity of ethyl ether was investigated by Schröder². It turned out that with unheated platinum electrodes, i.e., electrodes containing occluded gases, the current decreased with increasing temperature from $0^\circ$ to $30^\circ\text{C}$. When, however, the electrodes had been freed from gases by strong heating with the aid of an electric current, then, on the contrary, within the range from $10^\circ$ to $34^\circ$ the temperature coefficient of conductivity proved greater than zero. Further, the electrodes were deliberately saturated with hydrogen and oxygen. The influence of temperature proved to be the same as with unheated electrodes. The temperature coefficient had a negative value and was somewhat smaller in absolute magnitude than with unheated electrodes. In Schröder’s experiments, with unheated electrodes and with electrodes deliberately saturated with gases, the current increased faster than the voltage. This could occur, apparently, in the case when the field itself liberates occluded substances from the electrodes. In my opinion, the results of Schröder’s experiments can be explained in the following way. It is known that the occlusion of gases decreases with increasing temperature. The electrodes are freed from foreign substances, and the conductivity of dielectrics is determined by them chiefly. With increasing temperature the quantity of occluded substances decreases, the field liberates them in smaller quantity, and the conductivity decreases. With well-heated electrodes, in a purified dielectric there are present chiefly normal ions, the number of which increases with rising temperature at the expense of slow ions, and the current increases.

The electrolytic saturation current in solutions of lead oleate in hexane was first observed by Jaffé³. He measured the temperature coefficient, which is a characteristic of the dependence of the rate of ion formation on temperature. The measuring condenser consisted of two flat vertical disks 40 mm in diameter, separated by an interval of 2 mm. It turned out that the temperature coefficient from $0^\circ$ to room temperature is less than zero; then it increases strongly as the temperature is raised to $40^\circ$,

Consequently, at room temperature its change has a minimal influence on the rate of ion formation. The negative temperature coefficient is so unusual a phenomenon that the author himself recommends treating the results obtained with great caution and not accepting them without further verification.

Ch. T. Bialobrzeski¹ measured the spontaneous and ionization currents through ozokerite (mountain wax) of English origin, with a melting point of 75°, as the temperature was varied from 20° to 100° C. A flat capacitor was placed in a porcelain cylindrical cuvette. On the bottom of the cuvette there stood, on three legs, a brass nickel-plated disk, while the upper plate consisted of a hollow brass cylinder with an aluminum bottom 0.01 mm thick, surrounded by a guard ring. Ionization was produced by a preparation of 1 mgr. of radium bromide, enclosed in a platinum cup with a mica bottom. The dielectric layer was acted upon chiefly by β and γ rays and only partly by α rays. The investigations were carried out with distances between the electrodes of 1 and 10 mm. The current strength was determined by observing the rate of deflection of the needle of a quadrant electrometer. The electrometer was insulated three minutes after the capacitor had been connected to the storage-battery bank. With such an arrangement of the observations, the limiting value of the conductivity was not reached; this conductivity always decreases strongly in the first minutes after the beginning of the current, and the results obtained by the author, in my opinion, have a somewhat arbitrary character. The point is that the decrease of the current depends on its initial value: the stronger the current, the more rapidly it decreases. If all the obtained results were to be reduced to limiting conductivities, the reduction would have to be made many times larger for strong currents than for weak ones, and in any case the reduction coefficient would not be the same. In the solid state the dielectric exhibited a small and quite stable conductivity; in the molten state, however, its spontaneous conductivity was sometimes so considerable that it masked the action of the ionizing rays.

It turned out that, with increasing temperature, the spontaneous and ionization currents at first increased slowly, but at approximately about 75°, i.e. near the melting point, the increase occurred so rapidly that over an interval of 10° the currents increased 8-fold. With a further rise in temperature the increase in the currents again became slower, but not to the same extent as for the solid state. Within the indicated temperature limits the spontaneous current increased 27.5-fold, and the ionization current 15.2-fold. The ratio of the ionization current to the spontaneous current fluctuated from 11 to 18.

On the basis of the results obtained, the author finds it possible to draw the following conclusions. The change in both currents is due exclusively to the fact that the mobilities of the ions increase when the body passes from one state into another, or when its temperature changes. It does not appear possible to explain the increase in the currents by saying that, with increasing temperature, the number of free ions in the dielectric increases, since all investigations of the ionization of matter indicate that the latter is a molecular phenomenon, and the number of free ions cannot undergo large changes when a body passes from one state into another if this transition is not accompanied by a considerable change in density. If it is assumed that the number of ions arising under the action of radium rays does not depend on temperature or on the state of the substance, and that, upon melting of the ozokerite, there occurs an increase in the number of ions responsible for the spontaneous current, then one should expect that

the ratio of the ionization current to the spontaneous current should decrease with increasing temperature, but experiments do not reveal this. Further, one may suppose that the electricity in both currents is carried by the same ions. This conclusion is supported by the circumstance that an approximate parallelism exists between the ionization and spontaneous currents. It is possible that the spontaneous current is partly due to the presence of heavy, slow ions, but the mobilities of both these and other ions may change equally with temperature.

§ 4. Relation between electrical conductivity and viscosity.

There are very few experimental data on the dependence between the electrical conductivity and the viscosity of dielectric liquids. Only one work, by Koller,¹) bears directly on the question under consideration.

Koller measured, at various temperatures, the electrical resistances and viscosities of several poorly purified liquid dielectrics. To determine the viscosity, the author used the method of flow through capillary tubes. The apparatus (Fig. 1) consisted of a glass tube with two spherical bulbs, ending in a horizontal capillary. The lower bulb and the capillary were immersed in the liquid under investigation. This bulb was larger than the upper one; therefore only the liquid located in the lower part of the apparatus and having the same temperature as the rest of the liquid mass flowed through the capillary. To determine the relative value of the viscosity, the time interval required for the liquid to descend from mark \(a\) to mark \(в\) was observed. The relative viscosity is approximately proportional to this interval. The resistance was determined by a special method, analogous to Bronson’s method.

The experiments showed that, with increasing temperature, the resistance of all the liquids investigated decreases, and faster than the viscosity. The curves of the dependence of the resistance \(R\) and the viscosity \(\eta\) on temperature may be divided into two groups. For some liquids (linseed, castor, mineral olive, and vaseline oils), the decrease in \(R\) occurs somewhat faster than the decrease in \(\eta\), and in a first approximation it may be assumed that \(R\) is proportional to \(\eta\). For other liquids (petroleum ether, benzene, toluene, and xylene), the fall of the curve \(R=\varphi(t)\) is considerably greater than the fall of the curve \(\eta=\psi(t)\).

The work of Ch. T. Białobjeski²), who determined the mobilities of ions in liquid hydrocarbons, may also be referred to the question under consideration. The starting materials for obtaining the objects studied were petroleum ether, ligroin, kerosene, and liquid paraffin. From these substances fractions were separated with boiling points between \(47^\circ—52^\circ\), \(52^\circ—58^\circ\), \(58^\circ—70^\circ\), \(75^\circ—82^\circ\), \(82^\circ—90^\circ\), \(90^\circ—105^\circ\), \(150^\circ—200^\circ\), and \(200^\circ—250^\circ\). Liquid paraffin was not subjected to fractional distillation at all. For four liquids the sums of the mobilities \((k_1+k_2)\) of the ions were calculated:

\[ \begin{array}{c|cccc} \text{fraction:} & 52^\circ—58^\circ & 82^\circ—90^\circ & 200^\circ—250^\circ & \text{liq. paraffin} \\ \hline (k_1+k_2)\ \dfrac{\text{cm.}}{\text{sec.}} \bigg/ \dfrac{\text{volt.}}{\text{cm.}} & 11\cdot 7\cdot 10^{-4} & 3\cdot 7\cdot 10^{-4} & 1\cdot 5\cdot 10^{-4} & 0\cdot 5\cdot 10^{-4} \end{array} \]

From the tables of Landolt—Börnstein one may use three numbers giving the viscosities of approximately the same fractions:

\[ 0\cdot 00288,\quad 0\cdot 00382,\quad 0\cdot 01257. \]

The products of the sums of the mobilities by the viscosities of the liquids are:

\[ 11\cdot 7\cdot 2\cdot 88\cdot 10^{-7}=33\cdot 7\cdot 10^{-7},\quad 3\cdot 7\cdot 3\cdot 82\cdot 10^{-7}=14\cdot 1\cdot 10^{-7}\quad \text{and} \]

\[ 1\cdot 5\cdot 12\cdot 6\cdot 10^{-7}=18\cdot 1\cdot 10^{-4} \]

¹) Koller. Wiener. Sitzungsber. 98, 894 (1889)
²) T. Bialobjeski. Le Radium. 8, 293 (1911)

show that for low-boiling fractions the mobilities of the ions decrease more rapidly than inversely proportional to the viscosity, while on passing to high-boiling fractions they begin to vary approximately inversely proportional to the viscosities.

This, apparently, is all that we have at present concerning the question posed in the title.

Let us try to find the theoretical dependence between the electrical resistance \(R\) and the viscosity \(\eta\).

We represent the ion as an elementary charge surrounded by a group of neutral molecules. Consequently, the motion of an ion in a liquid dielectric may be regarded as the motion of a certain body (liquid or solid).

Stokes theoretically considered the case of the motion of a sphere in a viscous medium under the following conditions: 1) the liquid is unbounded and, when the sphere moves, is at rest at infinity; 2) the velocity of motion \(v\) is so small that its higher powers, beginning with the second, may be neglected; 3) the sphere is solid; and 4) the medium is continuous. The resistance \(W\) encountered by the sphere from the liquid is expressed by:

\[ W = 6\pi r v \eta, \tag{1} \]

where \(r\) is the radius of the sphere.

It may be asserted on the basis of experiments that Stokes’s formula is also applicable to liquid spheres moving in a liquid medium. Theory does not establish a lower limit for the radius of the sphere, but Perrin’s experiments on microscopic particles participating in Brownian motion showed that Stokes’s law is applicable to them.

Ions participate, together with the molecules, in thermal irregular motion, and their velocity changes almost continuously in magnitude and direction; at the same time, however, they all have a constant translational velocity in the direction of the electric field.

It is difficult to say to what extent the laws of motion of a solid or liquid body in a viscous medium are applicable to a moving ion. We shall assume that they are applicable.

On an ion situated in an electric field of strength \(X\) there acts the force

\[ F = eX, \tag{2} \]

where \(e\) is the elementary charge. Under the influence of this force the ion begins to move, encountering along its path the resistance \(W\) of the medium. Some time after the beginning of the action of the force \(F\), the motion of the ion becomes uniform; this means that at every given instant

\[ F = W, \tag{3} \]

Suppose that the ion moves in a dielectric liquid possessing a considerable coefficient of internal friction (castor oil, glycerin, etc.). In this case we may expect the motion to be steady and the resistance \(W\) to be expressed by:

\[ W = \alpha \eta u, \tag{4} \]

where \(\alpha\) is a certain constant, and \(u\) is the velocity of the ion. Introducing the mobility \(k\) of the ion, we may write

\[ W = \alpha \eta kX. \tag{5} \]

From (2), (3), and (5) we have:

\[ eX = \alpha \eta kX, \]

or

\[ \eta k = C = \mathrm{Const}. \tag{6} \]

The current density is expressed by:

\[ j=\varepsilon(n_1 k_1+n_2 k_2)X, \tag{7} \]

where \(n_1\) and \(n_2\) are the numbers of \(+\)- and \(-\)-ions, and \(k_1\) and \(k_2\) are their mobilities. On the basis of (6):

\[ j=\frac{\varepsilon(n_1 C_1+n_2 C_2)X}{\eta}, \tag{8} \]

where \(C_1\) refers to the \(+\)-ion, and \(C_2\) to the \(-\)-ion.

Formula (8) shows that, in the case of steady motion of an ion, the current density through the dielectric, and consequently also its electrical conductivity, is inversely proportional to the coefficient of internal friction.

Passing to the case of unsteady motion of the ion, we may assume that:

\[ W=\beta\eta u^2=\beta\eta k^2 X^2, \]

and, for established motion:

\[ F=\varepsilon X=\beta\eta k^2 X^2, \tag{9} \]

whence:

\[ \eta k^2=\frac{B^2}{X}, \tag{10} \]

where \(B^2\) is a constant.

Substituting \(k\) from (10) into (7), we find:

\[ j=\frac{\varepsilon(n_1 B_1+n_2 B_2)}{\sqrt{\eta}}\cdot \sqrt{X}, \tag{11} \]

where \(B_1\) and \(B_2\) refer to the \(+\)- and \(-\)-ions.

The last formula shows that, for rapid motion of the ions, the current density increases proportionally to \(\sqrt{X}\) and inversely proportionally to \(\sqrt{\eta}\).

As we have seen, the experimental material concerning the dependence of the conductivity of liquid dielectrics on their viscosity is very scant and does not permit any conclusions to be drawn as to the validity of the theoretical considerations presented above. In this field it is necessary to carry out a whole series of new investigations that could shed light on the matter.

Taking into account that in electrolysis we also deal with ions, though only of a somewhat different type than in dielectric liquids, I believe it will be of interest to present the results of several works relating to electrolytes.

One may expect that for viscous electrolytes—for example, for molten salts—the dependence

\[ R=a\eta, \tag{12} \]

holds, where \(R\) is the electrical resistance and \(a\) is a constant; and for slightly viscous electrolytes, the dependence

\[ R=b\sqrt{\eta}. \tag{13} \]

Foussereau¹ found that, for the molten salts \(\mathrm{NaNO_3}\), \(\mathrm{KNO_3}\), and \(\mathrm{NH_4NO_3}\), formula (12) holds approximately within the range between \(162^\circ\) and \(360^\circ\). The same was found by L. Poincaré². For solutions of \(\mathrm{CuSO_4}\), \(\mathrm{Cu(NO_3)_2}\), \(\mathrm{AgNO_3}\), and \(\mathrm{KOH}\) of various concentrations, G. Wiedemann³ found dependence (12).

Grotian\(^{1}\) and Kohlrausch\(^{2}\) investigated the change in the resistance and viscosity of solutions of various acids as a function of temperature. It turned out that the course of the curves \(R=\varphi(t)\) and \(\eta=\psi(t)\) almost coincides for solutions of HCl, H\(_2\)SO\(_4\), and HNO\(_3\), and that for solutions of sulfuric acid the temperature coefficients of viscosity and resistance have maxima at one and the same concentration. Walden\(^{3}\) showed that, for very dilute solutions of tetraethylammonium iodide \(N(C_2H_5)_4J\) in various organic solvents, the product of the molecular electrical conductivity at complete dissociation \(\sigma_\infty\) and the viscosity of the solvent \(\eta_\infty\) is a constant quantity independent of temperature. Later Walden\(^{4}\), studying solutions of high-molecular salts, found: 1) for tetramethylammonium iodide \(N(CH_3)_4J\), \(\sigma_\infty\eta_\infty=0.745\); 2) for tetraamylammonium iodide \(N(C_5H_{11})_4J\), \(\sigma_\infty\eta_\infty=0.557\); 3) for piperidylacetic acid triamylamine \(N(C_5H_{11})_3C_6H_2(NO_3)_3OH\), \(\sigma_\infty\eta_\infty=0.509\); and 4) for piperidinic-acid piperidine \(C_5H_{11}NH\cdot C_6H_2(NO_3)_3OH\), \(\sigma_\infty\eta_\infty=0.632\). These products remained, for one and the same substance, constant in different solvents (acetonitrile, methyl alcohol, propionitrile, acetylacetone, epichlorohydrin, ethyl alcohol, etc.). From Walden’s law it follows that the mobility of each ion is inversely proportional to the viscosity of the solvent, i.e.

\[ K_1=\frac{B_1}{\eta_\infty}, \qquad K_2=\frac{B_2}{\eta_\infty}, \]

where \(B_1\) and \(B_2\) are constants for the given ions.

Grotian\(^{5}\), for aqueous solutions of KCl, NaCl, CaCl\(_2\), MgCl\(_2\), BaCl\(_2\), MgSO\(_4\), and ZnSO\(_4\), found the dependence:

\[ R=b\eta^n, \]

where \(R\) is the electrical resistance and \(n\) is a constant exponent, varying for different solutions within the limits \(0.75\)—\(0.46\).

Johnson\(^{6}\) studied the dependence between \(\sigma\) and \(\eta\) over very wide temperature intervals and showed that the inverse proportionality between these quantities is only approximate. In general,

\[ \sigma\eta^n=\mathrm{Const}, \]

where \(n\) for aqueous solutions varies, for different ions, from \(0.85\) to \(1.05\). For non-aqueous solutions the exponent \(n\), generally speaking, is close to unity.

Of all the works I consider Walden’s work the most important, since it established a direct connection between \(\sigma\) and \(\eta\), whereas in other investigations dissociation also enters, which increases the number of ions and thereby increases the conductivity. In Walden’s investigations the number of ions remained constant at all temperatures, since all the molecules of the dissolved substance were dissociated.

Between electrical conductivity \(\sigma\), viscosity \(\eta\), and dissociation \(C\), one can theoretically, proceeding from the ionic theory of conductivity, establish the following dependence:

\[ \sigma\eta=aC, \tag{14} \]

where \(a\) is a constant quantity, different for different substances. Consequently, from the change in the product \(\sigma\eta\) with change in temperature \(t\), one can establish the dependence between \(C\) and \(t\). If Stokes’ law is applicable to moving ions, then formula (14) gives the most reliable method for determining

\(^{1}\) Grotian. P. A. 157, 130 (1876); 160, 238 (1877); W. A. 8, 529 (1879)
\(^{2}\) Kohlrausch. P. A. 159, 260 (1876)
\(^{3}\) Walden. Zeitschr. f. anorg. Chem. 54, 129 (1906); 55, 207, 381 (1906)
\(^{4}\) Walden. Izv. Akad. Nauk 559, (1915)
\(^{5}\) Grotian. P. A. 157, 130 (1876)
\(^{6}\) Johnson. Amer. Chem. Soc. 31, 1010 (1909)

dissociation both of liquid dielectrics and of solutions. This is especially important because the existing methods for determining dissociation (the method of measuring only electrical conductivity, the cryoscopic method, and the solubility method) are, generally speaking, not very reliable. Walden’s investigations at complete dissociation apparently speak in favor of the applicability of Stokes’ law to ions and, consequently, of the correctness of formula (14).

For dielectrics such as benzene, toluene, xylene, etc., the viscosity changes little with temperature, whereas for various oils the viscosity decreases strongly with increasing temperature. Consequently, if the former dielectrics are well purified, i.e., if they contain chiefly normal, readily mobile ions, the number of which depends little on temperature, then their electrical conductivity is almost independent of \(t\). This occurred in Jaffé’s experiments. If, however, these dielectrics are poorly purified, then their electrical conductivity may increase considerably with increasing \(t\), owing to the increase in the dissociation of dissolved impurities. Koller’s experimental results may be assigned here. In oils, which cannot be purified by removing all impurities from them, the electrical conductivity should increase strongly with increasing \(t\), owing to the decrease in viscosity and partly to the increase in dissociation.

§ 5. Investigation with Castor Oil.

Castor oil seemed to me the most convenient object for studying the relation between electrical conductivity and viscosity, because its coefficient of internal friction changes strongly with temperature and its electrical conductivity does not change with time.

For purifying the oil it could not be subjected to distillation; it was necessary to confine oneself to hot filtration in the presence of metallic sodium, several pieces of which were placed in the filtering funnel.

For measuring the electrical conductivity, a flat condenser with a guard ring was used\(^1\). It was placed in a copper drying oven with electric heating.

For studying the dependence of viscosity on temperature, a special apparatus was constructed, shown at \(1/4\) natural size in Fig. 2. A box of tinned sheet iron in the form of a rectangular parallelepiped, with a square base \(22 \times 22\) cm and height 32 cm, was closed at the top with a lid, and in the bottom had an opening fitted with a stuffing box. In two opposite faces of the box windows were made, covered with glass. The glass panes were pressed by metal frames with rubber gaskets. Through the lid and through the stuffing box in the bottom passed a glass tube 9 mm in diameter, furnished with a bulb and a “capillary,” which was a rather wide tube \((D = 4.5\ \text{mm})\) of length \(L = 93\ \text{mm}\). On the tube, on both sides of the bulb, two marks were made. The upper end of the tube was connected to an evacuating air pump, and the lower end was lowered into a jar with castor oil. Through the lid passed two rods, connected with each other, the lower ends of which were attached to a stirrer. Water was poured into the box, its temperature being determined by thermometer \(t\). A special tube, lowered into the water, served for admitting water vapor, which heated the water.

By evacuating the air, the liquid was raised above the upper mark and held in this position for some time, sufficient for equalizing the temperature of the bath. After this the stopcock was opened, the liquid began to flow out, and the time interval \(T\) between the passage of the meniscus through the upper and lower marks was determined with a stopwatch. As is known, the ratio of the viscosities \(\eta_1\) and \(\eta_2\) at two temperatures \(t_1\) and \(t_2\) is equal to:

\[ \frac{\eta_1}{\eta_2}=\frac{T_1 d_1}{T_2 d_2}, \]

\(^1\) V. D. Kuznetsov. Izv. Tom. Tekhn. In. Present volume, p. 37.

where \(T_1\) and \(T_2\) are the corresponding intervals of time, and \(d_1\) and \(d_2\) are the densities. In view of the fact that the density of castor oil changes comparatively little with temperature (the density changes by approximately \(0.1\%\) when heated by \(1^\circ\) C), it may be assumed that the viscosity is proportional to the interval of time \(T\).

At one and the same temperature the experiment was repeated several times. The results of the observations are given in Table I.

Table I.  7·9·18

When studying the dependence of electrical conductivity on temperature, the temperature inside the drying chamber was maintained constant for 1–2 hours, and only after this was an electric current, from a potential difference of 487 volts (a battery of small accumulators), passed through the castor oil for several minutes; this potential difference was kept constant throughout the observations by means of an additional auxiliary battery. The current was measured with a “Granta” galvanometer by W. G. Pye & Co., Cambridge. The deflections of the galvanometer were observed in both directions. Table II gives the results of the observations.

\(V = +487\) volts  Table II.  8·9·18

In Fig. 3 the dependences \(\eta = \psi(t)\) and \(\sigma = \varphi(t)\) are represented graphically, with the difference of the galvanometer readings taken as the measure of the electrical conductivity \(\sigma\). The dotted curve \(C = \chi(t)\) gives the dependence of the product \(\eta\sigma\) on temperature, i.e., it shows the course of the dissociation \(C\). This latter curve shows that \(C\), with increasing temperature, gradually increases, asymptotically approaching a certain value. Such a result is quite plausible and raises no objections.

Knowing the dependence of \(C\) on \(t\) for castor oil, it is now interesting to study the dependence of electrical conductivity on temperature for some solution of a salt in castor oil and to determine the dependence of the dissociation of this salt on \(t\). I intend to carry out experiments in this direction.

§ 6. Results.

1. The possible cases of the influence of temperature on the electrical conductivity of purified and unpurified liquid dielectrics have been considered.
2. The question of ion formation has been considered.
3. Experimental data obtained by various investigators have been considered, and explanations of them have been given from the adopted point of view.
4. The question of the dependence of electrical conductivity on internal friction has been considered theoretically,
5. Data confirming the theory have been presented.
6. A method has been indicated for establishing the dependence of dissociation on temperature by measuring electrical conductivity and viscosity.
7. Investigations with castor oil on the influence of temperature on electrical conductivity and viscosity have been presented.

[Received 24·11·23].

1. Foussereau. Ann. de chim. et de phys. 5, 359 (1885). ↩↩↩↩↩

2. L. Poincaré. Ann. de chim. et de phys. 21, 315 (1890). ↩↩↩↩

3. G. Wiedemann. P. A. 99, 177, 228 (1856). ↩↩