PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV

Volume 164 　　　　　　　　　　　　　　　　　　　　　　　　　1967

DEPENDENCE OF THE ANODIC PEAK CONSTANT IN THE APS METHOD ON THE RADIUS OF THE MERCURY DROP AND THE RATE OF POTENTIAL CHANGE. II. EXPERIMENTAL VERIFICATION

A. G. STROMBERG, B. F. NAZAROV, V. E. GORODOVYKH

(Submitted by the scientific-methodological seminar of the CTF)

In the preceding article [1] we gave a numerical solution of the theoretical equation obtained in [2] for the anodic peak constant \(K\) in the reverse anodic process of dissolution of a metal from a mercury drop of radius \(r\) (cm) under a linear change of potential (at a rate \(w\) volts/sec), as a function of the parameter \(\varepsilon\):

\[ \varepsilon = \frac{1}{r}\left(\frac{D}{zw}\right)^{1/2}, \tag{1} \]

where

\(D\) — the diffusion coefficient of the metal in mercury, \(\text{cm}^2/\text{sec}\);

\(z\) — the number of electrons participating in the anodic process of dissolution of one metal atom.

The dependence of the relative value \(\xi = \dfrac{K_0}{K}\) of the anodic peak constant on the parameter \(\varepsilon\) (in the interval of calculated values from 0 to 5.0) can be satisfactorily represented by two approximate formulas [1]:

\[ \lg \xi = 0.29\,\varepsilon, \tag{2} \]

\[ \xi - 1 = 1.25\,\varepsilon^2, \tag{3} \]

where \(K_0\) is the value of the anodic peak constant in the Randles–Ševčík equation for semi-infinite linear diffusion, i.e., at \(\varepsilon = 0\).

These approximate formulas (2) and (3) correctly reflect the limiting cases: as \(\varepsilon\) tends to zero (\(r\) or \(w\) tends to infinity), we obtain \(\xi = 1\) and \(K = K_0\); as \(\varepsilon\) tends to infinity (\(r\) or \(w\) tends to zero), we have \(\xi = 0\) and \(K = 0\). From the theoretical equation (1) and formulas (2) and (3) it follows that, for different metals (different \(z\) and \(D\)) and under different experimental conditions (different \(r\) and \(w\)), the dependence of \(\xi\) on \(\varepsilon\) must be the same; i.e., on a plot in the coordinates \(\xi\) and \(\varepsilon\), the experimental points for metals under different experimental conditions must lie on a single curve.

The purpose of the present work is to study the dependence of the anodic peak constant of cadmium and thallium on the radius of the mercury drop and on the rate of potential change, and to compare the experimental data obtained with theoretical calculations.

The electrolytic cell for studying the dependence of the anodic-peak constant on the radius of the mercury drop was described by us earlier [3].

Three series of experiments were carried out:

1. The effect of the radius of the mercury drop on the anodic-peak constant of thallium was studied over the radius range from \(2.23 \cdot 10^{-2}\) to \(11.8 \cdot 10^{-2}\ \text{cm}\) in a solution of composition \(1 \cdot 10^{-3}\ \frac{\text{g-ion}}{L}\ \mathrm{Tl}^{+1} + 1\ \mathrm{M}\ \mathrm{KNO}_3\).

2. The effect of the radius of the mercury drop on the anodic-peak constant of cadmium was studied over the radius range from \(1.67 \cdot 10^{-2}\) to \(9.6 \cdot 10^{-2}\ \text{cm}\) in a solution of composition \(1 \cdot 10^{-3}\ \frac{\text{g-ion}}{L}\ \mathrm{Cd}^{+2} + 0.5\ \mathrm{M}\ \mathrm{Na}_2\mathrm{SO}_4\).

3. The effect of the rate of change of potential on the anodic-peak constant of thallium was studied over the range of rates from \(4.5 \cdot 10^{-3}\) to \(2.0\ \mathrm{V/sec}\) in a solution of composition \(1 \cdot 10^{-3}\ \frac{\text{g-ion}}{L}\ \mathrm{Tl}^{+1} + 1\ \mathrm{M}\ \mathrm{KNO}_3\).

Experimental conditions: duration of electrolysis, 5 min; electrolysis potential, \(-0.7\ \mathrm{V}\) (S.C.E.) in the experiments with thallium and \(-1.0\ \mathrm{V}\) (S.C.E.) in the experiments with cadmium; solution volume, 50 ml; temperature, \(18^\circ\mathrm{C}\); rate of change of potential in the first series of experiments, \(1 \cdot 10^{-2}\ \mathrm{V/sec}\), in the second series of experiments, \(5.8 \cdot 10^{-3}\ \mathrm{V/sec}\); radius of the mercury drop in the third series of experiments, \(r = 3.8 \cdot 10^{-2}\ \text{cm}\).

The volume of the mercury drop was determined by knowing the number of drops (and the weight of one drop) that flowed out of the polarographic capillary and were suspended on a platinum contact; drops with radii from \(1.6 \cdot 10^{-2}\) to \(6 \cdot 10^{-2}\ \text{cm}\) were suspended on a contact of diameter \(2 \cdot 10^{-2}\ \text{cm}\), while drops with radii of \(6 \cdot 10^{-2}\ \text{cm}\) and larger were suspended on a contact of diameter \(5 \cdot 10^{-2}\ \text{cm}\). The radius of the mercury drop was calculated from the volume of mercury on the assumption that the drop was a perfect sphere.

Experiments at rates of change of potential from \(5.4 \cdot 10^{-3}\ \mathrm{V/sec}\) to \(1.6 \cdot 10^{-2}\ \mathrm{V/sec}\) were carried out on a recording polarograph with an OH-101 recorder (Hungary); experiments at rates of \(1.6 \cdot 10^{-2}\) and higher were carried out on an OP-3 oscillographic polarograph.

The anodic-peak constant was calculated from the experimental data using the formula¹

\[ K = \frac{zF}{3}\frac{Ir}{q}, \tag{4} \]

where \(I\) is the depth of the anodic notch, A; \(r\) is the radius of the mercury drop, cm; \(q\) is the quantity of electricity passing through the electrolyzer during anodic dissolution of the metal, coulomb (determined from the area under the notch); \(F\) is the Faraday constant; \(z\) is the number of electrons per atom of metal. In calculating the parameter \(\varepsilon\) from formula (1), the diffusion coefficients of thallium and cadmium in mercury were taken as \(1.03 \cdot 10^{-5}\) and \(1.63 \cdot 10^{-5}\ \text{cm}^2/\text{sec}\), respectively [4]. The constant \(K_0\) was calculated from formula (5)

\[ K_0 = 2.68 \cdot 10^5 z^{3/2} w^{1/2} D^{1/2}. \tag{5} \]

The results of the experiments are presented in the table and in Figs. 1–3. From Fig. 1 it is seen that, in qualitative agreement with the theory, with an increase in

\[ I = KSC, \]

where \(S\) is the surface area of the mercury drop, \(\text{cm}^2\);

\(C\) is the concentration of metal atoms in the drop, \(\frac{\text{g-atom}}{\text{cm}^3}\).

of the parameter \(\varepsilon\) (with decreasing drop radius \(r\) and rate of change of potential \(w\)) the anodic-current constant decreases. However, there is no quantitative agreement between the theoretical dependence and the experimental data. As is seen from Figs. 2 and 3, the experimental points on the graphs

Table

Effect of the radius of the mercury drop and of the rate of change of potential on the anodic-peak constant

\({}^{1}\) Values of \(K_{0}\), calculated from formula (5).

Fig. 1. Dependence of the anodic-current constant of thallium and cadmium on the radius of the mercury drop and the rate of change of potential. Curve 1—tendency; 2—thallium, effect of \(r\) (1st series of experiments, white circles) and cadmium, effect of \(r\) (2nd series of experiments, double circles); 3—thallium, effect of \(w\) (3rd series of experiments, black circles). The experimental conditions are indicated in the text.

Fig. 2. Verification of interpolation formula (2) by experimental data. The numbers of the curves and the designations of the points are the same as in the caption to Fig. 1.

in the coordinates \(\left(\lg \dfrac{1}{\lg \xi}\ \text{and}\ \lg \dfrac{1}{\varepsilon}\right)\) and \(\left(\lg \dfrac{1}{\xi - 1}\ \text{and}\ \lg \dfrac{1}{\varepsilon}\right)\) satisfac-

...are indeed located on straight lines, and the dependences of \(\xi\) on \(\varepsilon\) in different series of experiments can be represented by the following empirical formulas.

In the 1st and 2nd series of experiments,

\[ \lg \xi = 0.25 \varepsilon^{0.8} \tag{6} \]

or

\[ \xi - 1 = 0.785 \varepsilon . \tag{7} \]

In the 3rd series of experiments,

\[ \lg \xi = 0.141 \varepsilon^{0.8} \tag{8} \]

or

\[ \xi - 1 = 0.413 \varepsilon . \tag{9} \]

In addition, the requirement of the theory is not fully satisfied that the experimental points in the coordinates \(\xi\) and \(\varepsilon\) should fall on a single curve irrespective of the nature of the metal and the experimental conditions. The reason for this discrepancy between theory and experimental data is apparently that the theoretical derivation is carried out for conditions that are not fully realized experimentally. This difference in conditions is connected primarily with the fact that diffusion at a mercury drop is not completely symmetrical. Moreover, for the smallest and the largest mercury drops their shape differs strongly from spherical, which is not taken into account in the calculations.

Fig. 3. Verification of the interpolation formula (3) against experimental data. The numbers of the curves and the designations of the points are the same as in the caption to Fig. 1.

Let us note (in addition to what was said at the beginning of the article) that the empirical formulas (7 and 9) satisfactorily reflect the consequences of the theory of anodic teeth. Let us rewrite these formulas, taking expression (1) into account, in the form:

\[ K = K_0 \frac{r w^{1/2}}{r w^{1/2} + A \left(\frac{D}{z}\right)^{1/2}}, \tag{10} \]

where \(A\) is a coefficient independent of \(r\) and \(w\).

For small \(r\) or \(w\), when

\[ r w^{1/2} \ll A \left(\frac{z}{D}\right)^{1/2}, \]

the term \(r w^{1/2}\) in the denominator may be neglected and, taking (5) into account, one may write

\[ K = \frac{2.68 \cdot 10^5}{A} z^{1/2} w\,3q. \tag{11} \]

Thus, the empirical formulas (7) and (9) correctly reflect the conclusion of the theory [2] that at low rates of potential change \(w\)

the depth of the anodic notch becomes proportional to \(w\) (to the first power). For the depth of the anodic notch we obtain from (11)

\[ I = KSC = \frac{2.68 \cdot 10^5}{A} z^{1/2} w 3q, \tag{12} \]

\[ q = vC, \tag{13} \]

(\(v\) is the volume of the mercury drop). Formula (12) correctly reflects the conclusion of the theory [2] that, at small \(r\) and \(w\), the depth of the anodic notch becomes proportional not to the concentration, but to the amount of metal in the drop.

Conclusions

1. Using reversible anodic notches of thallium and cadmium in the APP method as an example, the dependence of the depth of the anodic notch on the radius of the mercury drop and on the rate of potential change was studied over a wide range of drop radii and rates of potential change.

2. The experimental data, in semiquantitative agreement with the theory, show that, with a decrease in the radius of the mercury drop or in the rate of potential change, the anodic-current constant decreases.

3. Possible reasons leading to a certain discrepancy between theory and experiment have been discussed.

LITERATURE

1. V. E. Gorodovykh, A. G. Stromberg, B. F. Nazarov. Dependence of the anodic peak constant on the radius of the mercury drop and the rate of potential change. I. Theory. This collection.

2. V. E. Gorodovykh. Izv. TPI, 128, 3, 1965.

3. B. F. Nazarov, A. G. Stromberg. Electrochemistry, in press.

4. A. G. Stromberg, E. A. Zakharova. Electrochemistry, in press.

1. The anodic-peak constant is the coefficient in the expression for the depth of the anodic notch: ↩