Determination of Transport Numbers by the Tubandt Method in Some Materials

A. A. Kabanov, V. V. Karsakov

(Presented by the scientific seminar of the Department of Radiation Chemistry)

1. Introduction

In works [1, 2] we showed the expediency of determining the contribution of individual types of charge carriers to the total electrical conductivity in elucidating the mechanism of solid-phase reactions, introduced certain methodological changes into the Tubandt method, and presented experimental results for a number of ionic compounds. In the present work, with the aim of further improving the methodology and expanding the measurement limits, we have assessed the experimental capabilities of the Tubandt method with respect to specimen resistance, developed a circuit for a current stabilizer for the optimal limits of the method, and determined transport numbers in four preparations.

2. Optimal Limits of the Tubandt Method with Respect to Specimen Resistance

Doubts concerning the nature of electrical conductivity (predominance of electrons or ions) in solid ionic compounds usually arise for materials having, at room temperature, a resistance of approximately \(10^5\ \Omega\cdot\text{cm}\) and higher. Therefore, an estimate of the optimal upper limit of the Tubandt method with respect to specimen resistance is of primary practical importance. From the laws of electrolysis [3], by means of simple transformations, one can obtain the relation

\[ R=\frac{0.0374\cdot U\tau A}{Zm}, \tag{1} \]

where: \(R\) is the resistance of the specimen, \(\Omega\);
\(U\) is the voltage across the specimen, V;
\(\tau\) is the electrolysis time, hours;
\(A\) is the atomic weight of the ion dominating in the electrical conductivity;
\(Z\) is the valence of the ion;
\(m\) is the mass of the substance transferred to the electrode.

In the simplest case the specimen consists of three identical pressed tablets [1, 2].

Preliminary experiments showed that the most convenient specimens in experiments are tablets 20 mm in diameter and 2 mm thick, taking into account strength, good contact, saving of material, and elimination of edge effects associated with the use of high voltages. From an analysis of known experiments on transport numbers [2, 4], the optimal field strength should be taken as \(1\ \text{kV}/\text{cm}\).

Increasing the voltage raises the probability of breakdown of the specimens, their sintering, and emission of electrons from the electrode material and, apparently, promotes dendrite formation in materials with cationic conductivity. The optimal electrolysis time should be taken as 20 hours. With increasing time, the danger of sintering of the specimens increases and mass evaporation from the surface of the specimen increases. The minimum change in weight that can be reliably recorded with an analytical balance is \(10^{-4}\) g. Substituting the optimal electrolysis parameters into formula (1) and taking into account that, when the transport number of the dominant ion deviates from unity, the resistance limit decreases, we obtain:

\[ R_{\max}=4.5\cdot 10^{7}\frac{A\cdot t}{Z}. \tag{2} \]

Here \(t\) is the transport number of the ion dominant in electrical conduction, and \(R_{\max}\) is the resistance of the specimen. Cationic and cation–electronic electrical conduction are the most common cases of electrical conduction for solid ionic compounds [4]. Taking this into account, the magnitude of the optimal resistance at different cation transport numbers (\(t_+\)) can be explained using cesium salts as an example.

\[ \begin{aligned} \text{For } t_+ &= 1 && R_{\max}=6\cdot 10^{9}\ \Omega,\\ \text{For } t_+ &= 0.01 && R_{\max}=6\cdot 10^{7}\ \Omega. \end{aligned} \]

By the Tubandt method, in the best case it is possible to determine the ion transport number with an accuracy of up to \(1\%\) (\(t_+ \ge 0.01\)) [4].

Thus, when measuring transport numbers by the Tubandt method, a current stabilizer is required for

\[ I_{\min}=10^{-6}\ \text{A}. \]

3. Experimental procedure and apparatus

The measurement scheme is shown in Fig. 1. Specimen 1 consisted of three pellets I—II—III, 20 mm in diameter, with the same thickness of about 2–3 mm.

Fig. 1. Diagram of the apparatus for measuring transport numbers

The pellets were prepared by pressing at room temperature from 4/DA-grade reagents thoroughly dried over \(P_2O_5\), under a pressure of \(10\ \text{t}/\text{cm}^2\). Before electrolysis, the weight of the pellets was stabilized at \(60\text{–}80^\circ\text{C}\). The specimen was clamped between platinum plates 2

using a spring and brass electrodes. The cell frame was made of Teflon. The cell with the specimen was placed in thermostat 3. Next to the cell there was a fourth tablet, analogous to the first three (between glass plates), for monitoring the mass evaporating from the surface of the specimen during electrolysis. An accurate estimate of mass evaporation was made from the change in the weight of the middle (II) tablet, whose weight should not change according to the laws of electrolysis. To measure the charge passed through the specimen, current stabilizer 4 was used. The charge was determined by integrating the current over time. The current stabilizer was chosen to be of the compensation type, since such stabilizers have the best characteristics [5–7].

The circuit of the stabilizer developed by us is shown in Fig. 2. The voltage drop produced by the output current (the electrolysis current) across resistance \(R_1\) and amplified by the intermediate-frequency amplifier was compared with the voltage of a second-class standard cell \((E_{\text{op}})\). The difference signal modulates a rectangular voltage, which is amplified and fed to high-voltage rectifier 5. The rectified voltage supplies the load (specimen 1). The current-stabilization circuit was first trained for a long time in order to eliminate error due to aging of the circuit elements. To start the current stabilizer, a pulse generator 6 of the G-5-15 type was used. The current flowing in the circuit was recorded by an electronic multipoint recorder 9 from the voltage drop across specimen resistance \(R\), which was selected with allowance for the resistance of specimen 1 and the sensitivity of the recorder (\(\sim 10\) mV). The temperature of the specimen was determined by a copper-constantan thermocouple 7, placed in the brass electrode and calibrated with millivoltmeter 8 of the M82 type. A second thermocouple 10 served for continuous monitoring of the temperature with recording on the tape of recorder 9. As voltage sources required for the operation of the current stabilizer, standard circuits of the UIPP-1 and VS-12 types were used.

The stabilizer used provided, in the experiments, a current-stabilization accuracy on the order of \(10^{-3} \div 10^{-6}\), and no worse than \(\pm 2\%\), with load-resistance fluctuations within one order of magnitude. Trial experiments with \(KMnO_4\) specimens 6 mm thick at \(50^\circ C\) for 15–20 h showed a current-stabilization accuracy on the order of \(10^{-6}\) and \(\pm 1\%\). The transport numbers of cations, anions, and electrons were estimated from the change in the weight of the electrode tablets together with the electrodes. In doing so it was taken into account that simultaneous charge transport by cations, electrons, and anions at a field strength below \(10^4\) V/cm is practically impossible [8], and that the sum of the transport numbers of cations \((t_+)\), anions \((t_-)\), and electrons \((t_e)\) is equal to unity. For example, the cation transport number is calculated as

\[ t_+ = \frac{Q_+}{Q} = \frac{m_2}{I \tau k}, \tag{3} \]

where

\(Q_+\) — charge carried by the cations;
\(Q\) — total charge that passed through the specimen;
\(m_2\) — increase in the weight of the cathode and the adjoining tablet;
\(k\) — electrochemical equivalent of the cation.

The measurement procedure for transport numbers is described in more detail by us in [2], where, together with new data for \(KMnO_4\), results are also given for CuO and CuCl with known transport numbers. The total measurement error depended mainly on the accuracy of determining the charge and mass and on average amounted to \(\pm 4\%\).

4. Results and Discussion

The transport number was determined for 7 preparations. The results are presented in the table, with the main parameters of the electrolysis indicated.

Table

$T$ — electrolysis temperature;
$3R_0$ — initial resistance of the specimen composed of three identical pellets;
$3R_{\mathrm{k}}$ — final resistance of the specimen;
$m_{\mathrm{av}}$ — averaged change in the weight of the near-electrode pellets with electrodes;
$m_{\mathrm{ev}}$ — loss of mass due to evaporation from the surface of the specimens (estimated from the middle pellet). For all materials, the value of $m_{\mathrm{ev}}$ differed only insignificantly from the corresponding value for the control pellet.

The results obtained for KMnO$_4$ agree well with the data obtained by us earlier [2] in the interval 100–180°C by a somewhat different method (different electrodes, thin pellets, another current stabilizer). Potassium, rubidium, and cesium permanganates have a negative thermoelectric emf [1], which, taking into account the values of the transport numbers, indicates electronic conduction. All three permanganates differ practically not at all in the magnitude of their electrical conductivity and activation energy in the interval 80–160°C [1]. From this it may be concluded that conduction in them occurs by the movement of electrons over the anion sublattice. A similar conclusion was made by the authors of [9] also for the electronic stage of the thermal decomposition of KMnO$_4$. Potassium manganate, like MnO$_2$, is a solid product of the decomposition of KMnO$_4$. Thus, taking into account the electrophysical properties of MnO$_2$ [10], it may be concluded that both potassium permanganate itself and the solid products of its decomposition are semiconductors (within the measurement error).

We also attempted to determine transport numbers for potassium nitrate, ammonium perchlorate, and silver sulfite, pressed at 3–5 t/cm$^2$. For the first two preparations, the experiments were carried out at 200°C. For KNO$_3$ the duration of electrolysis was 7 hours, $3R_0 = 2.8$ MΩ, $3R_{\mathrm{k}} = 0.5$ MΩ, $m_{\mathrm{ev}} = 0.0003$ g. Stabilization of the weight of KNO$_3$ and ammonium perchlorate was carried out at 100°C. The pellets making up the specimen, at a current of 1.5 mA, stuck together during electrolysis. From the blackening of the cathode pellet one can only draw a qualitative conclusion about the presence of considerable cationic conduction. In specimens of perchlora-

ammonium, composed of tablets 30 mm in diameter, \(3R_0=(1—5)10^8\) ohm, the transport numbers could not be estimated even qualitatively because of the considerable sublimation of the preparation, which is difficult to assess. For silver sulfite, experiments were conducted at \(120^\circ\text{C}\) with a current of the order of \(10^{-3}\) A. As a rule, after 1—2 hours a short circuit was observed, which may be associated only with dendrites of silver atoms and is a qualitative indication of significant cationic conductivity. This may also be confirmed by the fact that no thermoe.m.f. could be detected in \(\mathrm{Ag}_2\mathrm{SO}_3\) in the range \(40—130^\circ\text{C}\).

Conclusions

1. The Tubandt method can be successfully applied not only to binary salts [4], but also to more complex compounds.

2. Potassium, rubidium, and cesium permanganates, as well as potassium manganate, should be regarded as semiconductors.

3. Potassium nitrate and silver sulfite evidently possess cationic conductivity.

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