STUDY OF THE KINETICS OF THE PROCESSES OF FORMATION AND DISSOCIATION OF INTERMETALLIC COMPOUNDS IN MERCURY (IMC) BY THE APN METHOD AT A LINEARLY VARYING POTENTIAL

B. F. Nazarov, N. P. Mikheeva, V. S. Baev

(Presented by the scientific seminar of the Department of Physical and Colloid Chemistry)

Previously, the APN method at a linearly varying potential, on the basis of the parametric theory, was used to study IMC in mercury mainly with the aim of determining their phase nature in mercury and calculating thermodynamic constants [1].

The aim of the present work is to study the kinetics of the processes of formation and dissociation of IMC by the APN method at a linearly varying potential. For this purpose, the task was set of obtaining the equation of the voltammetric curve for the process of electro-dissolution of an amalgam from a mercury spherical electrode, complicated by the dissociation of an IMC dissolved in mercury, according to the equation

\[ AB_n \underset{k_2}{\stackrel{k_1}{\rightleftarrows}} A+nB \tag{1} \]

(\(k_1\) and \(k_2\) are the rate constants of the processes of dissociation and formation of the IMC), so that the kinetic constants of the IMC can be determined from the parameters of the anodic peak. In the case of dissociation of an IMC by reaction (1), under the condition that one of the components, for example \(B\), is taken in excess, the process of electro-dissolution of metals from a mercury spherical electrode is described by a system of two partial differential equations with the following initial and boundary conditions:

\[ t=0,\quad C_1=\mathrm{const},\quad C_2=\mathrm{const},\quad C_{AB}=\mathrm{const}; \]

\[ t>0,\quad \left.\frac{\partial C(r,t)}{\partial r}\right|_{r=0}=0, \]

\[ \left.\frac{\partial C(r,t)}{\partial r}\right|_{r=0} = - k_1 C_{AB}+k_2 C_1 C_2; \]

\[ C_1=C_1^0 e^{\frac{zF}{RT}(\varphi_0-\varphi)},\quad C_1^0=\mathrm{const}; \]

\[ D\left.\frac{\partial C(r,t)}{\partial r}\right|_{r=R} = D_R\left.\frac{\partial C_1}{\partial r}\right|_{r=R} + D_0\left.\frac{\partial C_{AB}}{\partial r}\right|_{r=R}. \tag{2} \]

\(C_1\), \(C_2\), and \(C_{AB}\) are the concentrations of metals \(A\), \(B\), and the IMC \(AB\) in the amalgam (we take \(n=1\)).

A necessary condition in the present problem is the constancy of the concentration of ions of one of the metals in the solution \((C_1^0=\mathrm{const})\), which is realized at an electrolysis potential close to the half-wave potential. As shown in [2], for \(\varphi_{\mathrm{e}}<\varphi_{1/2}\), in solving the stated problem the Nernst equation is used without additional boundary conditions; this made it possible, in solving the problem by the operational method, to obtain a simple expression for the anodic current of electro-dissolution of an amalgam from a spherical electrode, having the following form

\[ \frac{iR}{zFsDC_1} = 2k_2C_2 \sum_{1}^{\infty} \left| \frac{\sigma}{k_1-\sigma} \cdot \frac{\left(e^{-k_1t}-e^{-\mu_n^2\gamma\sigma t}\right)} {\mu_n^2\gamma\sigma-k_1} - \left( \frac{1}{k_1-\sigma}+\frac{1}{k_2C_2} \right) \cdot \frac{e^{-\sigma t}-e^{-\mu_n^2\gamma\sigma t}} {\mu_n^2\gamma-1} \right|, \tag{3} \]

where

\[ \gamma=\frac{R_0T}{zF}\frac{D}{R^2\omega}, \qquad \sigma=\frac{zF}{RT}\omega, \qquad \mu_n=n\pi, \tag{4} \]

\(R\) is the radius of the spherical electrode, \(\omega\) is the rate of change of the potential; the remaining notation is standard.

Equation (3) was calculated on an electronic digital computer for various values of \(\sigma\), \(\gamma\), \(k_1\), and \(k_2\). From the curves in the coordinates \(iR/zFsDC_1\)—\(\sigma t\), it was noted that the rate constant of formation \((k_2)\) affects

Fig. 1. Dependence of the half-peak width on \(k_1\).
1 — \(\gamma=0.05;\ \sigma=0.5;\)
2 — \(\gamma=0.05;\ \sigma=0.25;\)
3 — \(\gamma=0.038;\ \sigma=0.262;\)
4 — \(\gamma=0.038;\ \sigma=0.5\)

Fig. 2. Theoretical dependence of the height of the anodic peak of electro-dissolution of an amalgam on the rate constant of IMC formation for different \(k_1\). All curves were calculated for \(\sigma=0.25\); the solid lines were constructed for \(\gamma=0.05\), \(k_1\) for curves 1—5 increases from left to right and is equal, respectively, to 0.01; 0.05; 0.1; 0.5; 1. For curves a, b, c, \(k_1\) is equal, respectively, to 0.01; 0.05; 0.1; \(k_1\) for curve 2 is equal to 0.03.

only the height of the anodic peak, whereas the dissociation rate constant \((k_1)\) changes the peak potential and the half-peak width. From an analysis of the obtained...

methods are proposed for calculating the kinetic parameters of IMCs in mercury. Figure 1 presents the theoretical dependence of the half-peak width on the rate constant of IMC dissociation for different values of \(\gamma\) and \(\sigma\). From this graph, using the experimental half-peak width, converted into \(\sigma \cdot t\) according to equation (4), the dissociation rate constant is determined.

Figure 2 shows the theoretical dependence of the magnitude of the anodic peak of electro-dissolution of the amalgam on the formation rate constant in logarithmic coordinates; from it the product \(k_2 \cdot C_2\) is determined for a known \(k_1\); \(C_2\) is the free concentration of the second metal in the amalgam, and the other quantities \((z, F, S, D, C_1)\) are known.

The dissociation rate constant can be determined from the shift of the peak potential from the theoretical dependence of the peak potential on \(\lg k_1\) for different \(k_2 \cdot C_2\), shown in Fig. 3.

Experimental part

To verify the relations obtained, the Cd—Au—Hg system was chosen, in which, according to [3, 4], the IMC CdAu is formed and, at low concentrations of the metals, it is soluble in mercury [1]. The experimental verification consisted of the following: electrolysis of a Cd solution was carried out at an indeterminate-current potential \(\varphi_{\mathrm{e}}\) \(\left(\varphi_{\mathrm{e}} \approx \varphi_0\right.\) in the equation

Fig. 3. Theoretical dependence of the peak potential on \(\lg k_1\), constructed for \(\sigma=0.25\); solid curves 1, 2, 3 correspond to the values \(\lg k_2 C_2 = 2.0;\ 1.3;\ 0.0;\ \gamma = 0.05.\) Curves 4, 5, 6 correspond to \(\lg k_2 \cdot C_2 = 1.3;\ 0.0\) and \(1.0;\ \gamma = 0.1\)

Fig. 4. Anodic peaks of electro-dissolution of Cd amalgam: 1—on a mercury spherical electrode, \(C_{\mathrm{Cd}} = 3.6 \cdot 10^{-4}\) g-at/l; \(\varphi_{\mathrm{e}} = 0.64\) V (rel. sat. c. e.); 2—on Au amalgam, \(C_{\mathrm{Au}} = 1.05 \cdot 10^{-2}\) g-at/l

Nernst equation

\[ \varphi = \varphi_0 + b \ln \frac{C_u}{C_a}. \]

At the chosen potential, the height of the anodic Cd peak does not depend on the electrolysis time (under our conditions already after one minute), which indicates that equilibrium between the amalgam and the solution has been reached. Then, under the same conditions, electrolysis of the Cd solution was carried out on Au amalgam obtained by electrolysis of a solution containing \(\mathrm{Au}^{3+}\) ions in a separate beaker. When the anodic polarogram was recorded, a noticeable increase in the height of the anodic Cd peak and a change in its parameters (half-peak width, peak potential) were observed. Figure 4 shows the anodic Cd peaks recorded on mercury and on gold amalgam. The increase in the height of the anodic Cd peak on Au amalgam occurs because Cd,

present in mercury, had reacted with gold with the formation of an intermetallic compound, but, owing to the imposed potential, a new portion of Cd enters the mercury; and the higher the Au concentration, the more Cd enters the mercury. During electro-dissolution of the amalgam an increased (total) anodic peak is obtained, corresponding to the discharge of free Cd and of Cd obtained as a result of dissociation of the intermetallic compound.

\[ \mathrm{CdAu} \rightleftarrows \mathrm{Cd} + \mathrm{Au}. \]

Thus, from the area of the anodic peak obtained, the concentration of the intermetallic compound is calculated, equal to \(C_{\text{i.m.c.}} = C_{\text{tot.}} - C_{\mathrm{Cd}}\), where \(C_{\text{tot.}}\) is the concentration calculated from the area of the total peak, and \(C_{\mathrm{Cd}}\) is the concentration of Cd in the amalgam, calculated from the anodic Cd peak in the absence of gold. In the present case \(C_{\text{i.m.c.}}\) is equated to the concentration of Cd and Au atoms obtained as a result of dissociation of the intermetallic compound, because the intermetallic compound formed has the composition CdAu \((n = 1)\). To calculate the dissociation constant by the formula

\[ K=\frac{[\mathrm{Cd}][\mathrm{Au}]}{[\mathrm{CdAu}]} \]

it is necessary to determine the concentration of free Au in mercury, equal to \(C_{\mathrm{Au}} = C_{\mathrm{Au}}^{\circ} - C_{\text{i.m.c.}}\), where \(C_{\mathrm{Au}}^{\circ}\) is the initial concentration of Au in mercury, determined from the magnitude of the electrolysis current during preparation of the gold amalgam.

The kinetic constants \(k_1\) and \(k_2\) for the CdAu intermetallic compound are determined from the peak half-width, from the magnitude of the peak-potential shift, and from the peak height in the plots presented in Figs. 1–3.

Table

\(k_1\), \(k_2\), and \(K\) for the intermetallic compound in the Cd—Au—Hg system

The table gives the values of \(k_1\) (calculated by two methods), \(k_2\), and \(K\). Since data on the study of kinetic parameters in the Cd—Au—Hg system are almost absent from the literature, the applicability of the proposed method may be judged from the satisfactory agreement of the values of the dissociation constant with those found by other methods.

REFERENCES

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2. V. A. Nemov. Candidate’s dissertation, TPI, Tomsk, 1972.
3. N. Hautmann, K. Schölzel, Z. Phys. Chem., 9, 106, 1956.
4. L. F. Kozin, G. F. Cherkasova, M. I. Erdenbaeva, Izv. AN KazSSR, Ser. Khim., No. 3, 42, 1964.