ANODIC OXIDATION OF A METAL AND AN AMALGAM UNDER CONSTANT CURRENT UNDER CONDITIONS OF SPHERICAL DIFFUSION WITH COMPLEX FORMATION

M. S. Zakharov, V. I. Bakanov, V. V. Pnev

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

The purpose of the present work is to derive equations establishing the dependence of the electrode potential on time in anodic voltammetry with constant current during the electrooxidation of a metal and an amalgam with complex formation under conditions of spherical diffusion. On this question we found only one work in the literature [1], in which an analogous problem was solved for a plane metal electrode.

Let us first consider the electrooxidation of a metal at constant current on a spherical electrode with complex formation. In this case the following process will occur:

\[ M+pA^{x-}\rightleftarrows MA_p^{(px-z)}+ze, \]

where \(M\) is the metal; \(A\) is the ligand; \(p\) is the composition of the complex formed; \(z\) is the positive charge of the metal ion; \(x\) is the negative charge of the ligand.

The dependence \(\varphi-t\) will be obtained under the following conditions: 1) the \(\Psi_1\)-potential is absent; 2) a complex of only one composition is formed; 3) self-dissolution of the metal is absent; 4) before electro-dissolution the complex is absent from the solution; 5) between the components participating in complex formation a state of equilibrium is reached; 6) the electrode surface is equally accessible. For reversible electrode processes, the electrode potential in the case under consideration will be determined by the Nernst equation.

\[ \varphi=\varphi^0+\frac{RT}{zF}\ln\frac{f_{M^{z+}}C_{M^{z+}}(r_0,t)}{f_M C_M(r_0,t)} . \tag{1} \]

Usually one assumes \(a_M(r_0,t)=1\) \((a_M=f_M C_M)\). The concentration of metal ions at the electrode surface will be determined by the instability constant of the following process:

\[ M^{z+}+pA^{x-}\rightleftarrows MA_p^{(px-z)} . \tag{2} \]

Thus, the simple metal ions formed at the electrode surface during electro-dissolution will interact with ligand ions with formation of a complex. As a result, the electrode potential will be determined by the concentration, at the electrode surface, of the ligand ions and the complex. Let us first consider the case when concentration polarization occurs for the ligand ions. The supply of ligand ions

to the electrode is effected by semi-infinite spherical diffusion, and the ligand concentration at the electrode surface will be determined by the expression (in this expression \(z\) is replaced by \(z/p\))

\[ C_{A^{x-}}(r_0,t)=C^0_{A^{x-}}+\frac{i_0 r_0 p}{zFD}\left(1-\exp \vartheta \operatorname{erfc}\sqrt{\vartheta}\right), \tag{3} \]

where \(C^0_{A^{x-}}\) is the ligand concentration in the bulk of the solution.

The equation for the transition time for spherical diffusion, under the condition \(C_{A^{x-}}(r_0,\tau)=\), has the form [2]:

\[ \frac{C_{A^{x-}}}{\lambda r_0} = 1-\exp\left(\frac{D\tau}{r_0^2}\right) \operatorname{erfc}\left(\frac{D\tau}{r_0^2}\right)^{1/2}, \tag{4} \]

where

\[ \lambda=\frac{i_0 r_0}{zFD}. \]

Let us estimate under what conditions the sphericity of the electrode will influence the electrode process. For this purpose we expand the expression in parentheses in equation (3) in a series for small values of the argument. We obtain

\[ C_{A^{x-}}(r_0 t)=C^0_{A^{x-}}+\frac{i_0 r_0 p}{zFD} \left(\frac{2\vartheta}{z^{1/2}}-\vartheta^2+\ldots\right). \tag{5} \]

With an error of 1% or less, at \(\vartheta \geq 1\cdot 10^{-4}\) the series may be restricted to the first term. In this case the equation for the concentration at the surface of a spherical electrode becomes the equation for the concentration at the surface of a plane electrode. Consequently, at \(\vartheta \geq 1\cdot 10^{-4}\) the sphericity of the electrode will affect the electrode process.

From expressions (3) and (4) one can obtain the following expression for the ligand concentration at the electrode surface:

\[ C_{A^{x-}}(r_0,t)=C^0_{A^{x-}}\cdot P_1, \tag{6} \]

where

\[ P_1=1- \frac{1-\exp\vartheta\,\operatorname{erfc}\sqrt{\vartheta}} {1-\exp\vartheta'\,\operatorname{erfc}\sqrt{\vartheta'}}, \tag{7} \]

The concentration of the complex at the electrode surface will be expressed by the equation

\[ C_{MA_p^{(px-z)}}(r_0,t)=\frac{C^0_{A^{x-}}}{p}\cdot P_2, \tag{8} \]

where

\[ P_2= \frac{1-\exp\vartheta\,\operatorname{erfc}\sqrt{\vartheta}} {1-\exp\vartheta'\,\operatorname{erfc}\sqrt{\vartheta'}}, \tag{9} \]

The concentration of the ion \(M^{z+}\) at the electrode surface is determined from the expression for the instability constant \((K_{\mathrm{n}})\) (equation 2) of the complex [1] and equations (6, 8):

\[ C_{M^{z+}}(r_0,t)= \frac{f_{MA_p^{(px-z)}}\,K_{\mathrm{n}}} {p f^p_{A^{x-}}\,(C^0_{A^{x-}})^{p-1}} \cdot \frac{P_2}{P_1}, \tag{10} \]

where \(f_{MA_p^{(px-z)}}\) and \(f_{A^{x-}}\) are the activity coefficients of the complex and the ligand. In deriving equation (10), the diffusion coefficients of the complex and of the ligand were taken to be equal.

If the concentration of the ligand in the solution is large (many times exceeding the complex-formation concentration), then with large-

to sufficient accuracy during anodic polarization the ligand concentration at the electrode surface may be taken equal to its concentration in the bulk. In this case the concentration of the ion \(M^{z+}\) at the electrode surface will be determined by the expression

\[ C_{M^{z+}}(r_0,t)= \frac{f_{MA_p^{(px-z)-}}\cdot K_{\mathrm{н}}} {p f_{A^{x-}}^{p}\cdot \left(C^{0}_{A^{x-}}\right)^{p-1}} . \tag{11} \]

By combining equations (1) and (10), or (1) and (11), equations are obtained for the dependence of the electrode potential on time:

\[ \varphi=\varphi^{\circ}+\frac{RT}{zF}\ln \frac{f_{MA_p^{(px-z)-}}\cdot K_{\mathrm{н}}} {p f_{A^{x-}}^{p}\cdot \left(C^{0}_{A^{x-}}\right)^{p-1}} \cdot \frac{P_2}{P_1}, \tag{12} \]

\[ \varphi=\varphi^{\circ}+\frac{RT}{zF}\ln \frac{f_{MA_p^{(px-z)-}}\cdot K_{\mathrm{н}}} {p f_{A^{x-}}^{p}\cdot \left(C^{0}_{A^{x-}}\right)^{p-1}} . \tag{13} \]

Using equations (12) or (13), one can determine the composition \((P)\) of the complex. For this purpose, when using equation (12), it is necessary to obtain anodic chronopotentiograms with the same transition time at two ligand concentrations. The same transition time at different ligand concentrations is achieved by changing the current density at the electrode. When using equation (13), it is sufficient simply to obtain two chronopotentiograms at two ligand concentrations. From two chronopotentiograms at two ligand concentrations, the following expression can be written:

\[ \Delta\varphi=\varphi_2-\varphi_1 =\frac{RT}{zF}\ln \left( \frac{C^{0,2}_{A^{x-}}}{C^{0,1}_{A^{x-}}} \right)^{p-1}, \tag{14} \]

where \(\varphi_1\) and \(\varphi_2\) are the electrode potentials at two ligand concentrations (and at the same \(\vartheta\) when equation (12) is used). From expression (14) an equation is obtained for determining the composition \((P)\) of the complex

\[ P= \frac{zF\Delta\varphi} {2{,}3RT\lg\left[ \frac{C^{0,2}_{A^{x-}}}{C^{0,1}_{A^{x-}}} \right]} +1 . \tag{15} \]

Knowing \(P\) and \(\varphi^{\circ}\), and measuring \(\varphi\) at any \(t\), from equations (12) or (13) one can calculate the instability constant \((K_{\mathrm{н}})\) of the complex.

It should be borne in mind that all these conclusions are valid under the conditions indicated at the beginning of the paper.

Now let us consider the process of reversible anodic oxidation of an amalgam on a mercury drop electrode (spherical diffusion) with formation of a complex. In this case the following process will occur:

\[ M(\mathrm{Hg})+pA^{x-}\rightleftharpoons MA_p^{(px-z)-}+\mathrm{Hg}+ze . \]

The potential of the amalgam electrode is expressed by the equation

\[ \varphi=\varphi^{\circ}+\frac{RT}{zF}\ln \frac{C_{M^{z+}}(r_0,t)} {C_{M(\mathrm{Hg})}(r_0,t)}, \tag{16} \]

where \(C_{M(\mathrm{Hg})}(r_0,t)\) is the concentration of the amalgam at the electrode surface.

The concentration \(C_{Mz+}(r_0,t)\) for the case in which the transition time for the ligand is less than for the metal atoms dissolved in mercury is determined by equation (10). The concentration of the metal at the surface of the amalgam electrode is determined by the following relations [3]:

for any \(\vartheta\)

\[ C_{M(\mathrm{Hg})}(r_0,t)=C^0_{M(\mathrm{Hg})}-P_3, \tag{17} \]

where

\[ P_3=\frac{i_0 r_0}{z F D_R} \left[ 3\vartheta+0.2-2\sum_{n=1}^{\infty} \frac{\exp(-\mu_n^2\vartheta)}{\mu_n^2} \right]; \]

for \(\vartheta \ge 0.16\)

\[ C_{M(\mathrm{Hg})}(r_0,t)=C^0_{M(\mathrm{Hg})}-P_4, \tag{18} \]

where

\[ P_4=\frac{i_0 r_0}{z F D_R}(3\vartheta+0.2). \]

By combining equations (1, 11, 17 or 18), expressions are obtained for the dependence of the electrode potential on time for the anodic oxidation of an amalgam with formation of a complex under conditions of spherical diffusion:

for any \(\vartheta\)

\[ \varphi=\varphi_0+\frac{RT}{zF}\ln \frac{ f_{MA_p}^{(px-z)}\cdot K_{\mathrm{H}}\cdot P_2 }{ p f_{A^{x-}}^{p}\cdot \left(C^0_{A^{x-}}\right)^{p-1}\cdot \left[C^0_{M(\mathrm{Hg})}-P_3\right]\cdot P_1 }, \tag{19} \]

for \(\vartheta \ge 0.16\)

\[ \varphi=\varphi_0+\frac{RT}{zF}\ln \frac{ f_{MA_p}^{(px-z)}\cdot K_{\mathrm{H}}\cdot P_2 }{ p f_{A^{x-}}^{p}\cdot \left(C^0_{A^{x-}}\right)^{p-1} \left[C^0_{M(\mathrm{Hg})}-P_4\right]\cdot P_1 }. \tag{20} \]

In the absence of concentration polarization for the ligand ions, the concentration \(C_{Mz+}(r_0,t)\) is determined by relation (11). The concentration of metal atoms at the electrode surface in this case will be described by the following equations:

for any \(\vartheta'\)

\[ C_{M(\mathrm{Hg})}(r_0,t)=C^0_{M(\mathrm{Hg})}\cdot P_5, \tag{21} \]

where

\[ P_5=1- \frac{ 3\vartheta+0.2-2\displaystyle\sum_{n=1}^{\infty} \frac{\exp(-\mu_n^2\vartheta)}{\mu_n^2} }{ 3\vartheta+0.2-2\displaystyle\sum_{n=1}^{\infty} \frac{\exp(-\mu_n^2\vartheta')}{\mu_n^2} } \tag{22} \]

for any \(\vartheta'\)

\[ C_{M(\mathrm{Hg})}(r_0,t)=C^0_{M(\mathrm{Hg})}\cdot P_6, \]

where

\[ P_6=1-\frac{3\vartheta+0.2}{3\vartheta'+0.2}. \]

By combining equations (1, 11, 21 and 22), we obtain the equations of the chronopotentiogram in the anodic oxidation of an amalgam with formation of ...

complex under conditions of spherical diffusion in the presence of concentration polarization for the metal atoms in the amalgam and in the absence of such for the ligand ions:

for any $\vartheta'$

\[ \varphi=\varphi_0+\frac{RT}{zF}\ln \frac{f_{MA_p}^{(px-z)-}\cdot K_{\mathrm{n}}} {p f_{Ax-}^{p}\cdot \left(C_{Ax-}^{0}\right)^{p-1}\cdot \left[C_{M(\mathrm{Hg})}^{0}\cdot P_i\right]}, \tag{23} \]

for $\vartheta' \geqslant 0.16$

\[ \varphi=\varphi_0+\frac{RT}{zF}\ln \frac{f_{MA_p}^{(px-z)-}\cdot K_{\mathrm{n}}} {p f_{Ax-}^{p}\cdot \left(C_{Ax-}^{0}\right)^{p-1}\cdot \left[C_{M(\mathrm{Hg})}^{0}\cdot P_i\right]} . \tag{24} \]

At two ligand concentrations at identical $\vartheta'$, from equations (23) or (24) one can obtain equation (15), i.e., the composition of the complex can be determined using an amalgam stationary electrode. The phenomenon of complex formation according to the model formulated above can be established by comparing the calculated curve with the experimental chronopotentiogram.

Consideration of irreversible electrode processes is of no interest, since in this case the reaction products, and consequently also the phenomenon of complex formation, do not affect the electrode potential.

Conclusions

1. Equations have been obtained for the dependence of the electrode potential on time for the cases of anodic oxidation of a metal and an amalgam at constant current with formation of a complex under conditions of spherical diffusion at a stationary electrode.

2. Methods are proposed for determining the composition and the instability constant of the complex.

LITERATURE

1. P. Delahay, C. C. Mattax, T. Berzins, J. Am. chem. Soc., 76, 5319 (1954).
2. G. Mamantov, P. Delahay, J. Am. chem. Soc., 76, 5323 (1954).
3. M. S. Zakharov, V. V. Pnev, V. I. Bakanov, Electrochemistry (in press).