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

Vol. 164
1967

THEORY OF ANODIC AMALGAM VOLTAMMETRY WITH AN ARBITRARY FORM OF VARIATION OF THE POTENTIAL OR CURRENT OF A SPHERICAL ELECTRODE. THE ELECTRODE PROCESS IS COMPLICATED BY A SUBSEQUENT CHEMICAL REACTION

M. S. ZAKHAROV, V. I. BAKANOV

(Presented by the scientific seminar of the ChTF)

In the present work equations will be obtained for \(\varphi - t\) (AAV with constant current), \(i - \varphi\) (AAV with linearly varying potential), and \(i - t\) (AAV with constant electrode potential) for an electrode process complicated by a subsequent first-order chemical reaction.

Let the following process occur at the electrode:

\[ R \xrightarrow{ze} O + N \underset{k_2}{\overset{k_1}{\rightleftarrows}} Y, \tag{1} \]

where \(R\) is a metal dissolved in the mercury electrode; \(O\) is an ion formed directly in the process of electro-dissolution of the amalgam and then, in the bulk of the solution, converting into ion \(Y\); \(Y\) is not reduced in the potential interval in which \(R\) is oxidized; \(k_1\) and \(k_2\) are the rate constants of the forward and reverse reactions.

To find the equations of the curves indicated above, it is necessary to know the expressions for the concentrations \(O\) and \(R\) at the electrode surface, \(C_R(1,\Theta)\) and \(C_O(1,\Theta)\). The course of the reaction \(O \rightleftarrows Y\) does not affect the distribution of the concentration of \(R\) at the electrode surface.

In this connection, the expressions for \(C_R(1,\Theta)\) at constant current or for any form of variation of the electrode potential may be taken from work [1].

At constant current

\[ C_R(1,\Theta)=C_R^0-\lambda_R\left[3\Theta_R+0.2-2\sum_{n=1}^{\infty}\frac{\exp(-\mu_n^2\Theta_R)}{\mu_n^2}\right], \tag{2} \]

for any form of variation of the electrode potential

\[ C_R(r_0,t)=C_R^0-\frac{1}{r_0}\int_{0}^{t}q(\zeta) \left\{3+2\sum_{n=1}^{\infty}\exp\left[-\mu_n^2\frac{D_R}{r_0^2}(t-\zeta)\right]\right\}d\zeta, \tag{3} \]

where \(C_R^0\) is the concentration of metal atoms in mercury (\(z\)-atom/cm\(^3\)) at \(t=0\); \(\Theta=\dfrac{Dt}{r_0^2}\); \(D\) is the diffusion coefficient, cm\(^2\)/sec; \(r_0\) is the radius of the spherical—

...of the mercury electrode, cm; \(\lambda=\dfrac{i_0 r_0}{zFD}\); \(i_0\) is the current density, A/cm\(^2\);

\(\mu_n\) are the positive roots of the transcendental equation

\[ \mu_n=\operatorname{tg}\mu_n . \]

To obtain the expression \(C_0(1,\Theta)\), it is necessary to solve two equations of Fick’s second law:

\[ \frac{\partial \Psi(Y,\Theta)}{\partial \Theta} = \frac{\partial^2 \Psi(Y,\Theta)}{\partial Y^2} + \frac{\Gamma}{Y}\frac{\partial \Psi(Y,\Theta)}{\partial Y}, \tag{4} \]

\[ \frac{\partial \Phi(Y,\Theta)}{\partial \Theta} = \frac{\partial^2 \Phi(Y,\Theta)}{\partial Y^2} + \frac{\Gamma}{Y}\frac{\partial \Phi(Y,\Theta)}{\partial Y} - \chi \Phi(Y,\Theta) \tag{5} \]

under the following initial and boundary conditions:

\[ \Theta=0,\qquad \Psi(Y,0)=0,\qquad \Phi(Y,0)=0; \tag{6} \]

\[ \Theta>0,\qquad \kappa\frac{\partial \Psi(1,\Theta)}{\partial Y} = -\frac{\partial \Phi(1,\Theta)}{\partial Y}; \tag{7} \]

\[ \frac{\partial \Psi(1,\Theta)}{\partial Y} - \frac{\partial \Phi(1,\Theta)}{\partial Y} = -\lambda(1+K); \tag{8} \]

\[ \Psi=C_0(Y,\Theta)+C_{\Gamma}(Y,\Theta); \tag{9} \]

\[ \Phi=C_{\Gamma}(Y,\Theta)-\frac{\kappa_1}{\kappa_2}C_0(Y,\Theta), \tag{10} \]

where

\[ Y=\frac{r}{r_0}; \]

\[ \Gamma=2\gamma+1 \]

is the electrode-shape coefficient;

\[ \gamma=\frac{1}{2} \]

for a spherical electrode;

\(C_0(Y,\Theta)\) and \(C_{\Gamma}(Y,\Theta)\) are the concentrations of \(O\) and \(\Gamma\);

\[ K=\frac{\kappa_1}{\kappa_2} \]

is the equilibrium constant of the reaction

\[ O \rightleftarrows \Gamma; \]

\[ K_2=K_1 C_N;\qquad \chi=\frac{(\kappa_1+\kappa_2)r_0^2}{D}. \]

Solving the stated boundary-value problem by the Laplace-transform method, we obtain the following equation for the concentration of \(O\) at the surface of a spherical electrode for an electrode process complicated by a reversible subsequent chemical reaction:

\[ C_0(1,\Theta) = \frac{\lambda}{1+K}\left(1-\exp\Theta\,\operatorname{erfc}\sqrt{\Theta}\right) + \frac{K\lambda}{(1+K)(1-\chi)} \times \]

\[ \times \left[ 1-\chi^{1/2}\operatorname{erf}\sqrt{\chi\Theta} -\exp\Theta\,\operatorname{erfc}\sqrt{\Theta}\exp(-\chi\Theta) \right]. \tag{12} \]

For \(K=0\), equation (12) becomes the expression for the distribution of concentrations at the electrode surface in the absence of kinetic complications.

We shall consider quasireversible and reversible processes, since in the case of completely irreversible processes the occurrence of the subsequent reaction \(O \rightleftarrows \Gamma\) has no effect on the \(\varphi-t\) or \(i-\varphi\) curves. The occurrence of this reaction also does not affect the magnitude of the trans-

…transition time in (AAV with constant current), both in irreversible and reversible processes.

Substituting into the Nernst equation the values \(C_s(1,\Theta)\) and \(C_R(1,\Theta)\) from equations (12), (2), we obtain the dependence of the electrode potential on time in AAV with constant current at \(\Theta' \ge 0.14\) [3]:

\[ \varphi(\Theta)=\varphi_{1/2}-\frac{RT}{zF}\ln\frac{\Theta'-\Theta}{\Theta} +\frac{RT}{zF}\ln\Phi(\Theta), \tag{13} \]

where \(\Theta'=\dfrac{D\tau}{r_0^2}\) is the dimensionless transition time;

\(\tau\) is the transition time, sec;

\[ \Phi(\Theta)= \frac{1-\exp\Theta\,\operatorname{erfc}\sqrt{\Theta}}{3(1+K)\Theta} -\frac{K}{3(1+K)(\chi-1)\Theta}\times \]

\[ \times\left[1-\sqrt{\chi}\operatorname{erf}\sqrt{\chi\Theta} -\exp(\Theta-\chi\Theta)\operatorname{erfc}(\sqrt{\Theta})\right]. \tag{14} \]

At \(\Theta=\dfrac{\Theta'}{2}\), equation (13) is reduced to the form

\[ \varphi_{(\Theta'/2)}=\varphi_{1/2}+\frac{RT}{zF}\ln\Phi_{(\Theta'/2)}. \tag{15} \]

\(\Phi_{(\Theta'/2)}\) is expressed by equation (14), only in the latter \(\Theta\) is replaced by \(\Theta'/2\).

Using equation (15), from the experimental values \(\varphi_{(\Theta'/2)}\), \(\varphi_{1/2}\), \(\Theta'\) and the known \(K\) and \(D\), one can determine by fitting the rate constants \((\kappa_1\) and \(\kappa_2)\) of the chemical reaction.

At \(\Theta'\ge 0.14\) and \(\chi\ge 100\) (fast chemical reactions), the following expression for the electrode potential can be obtained:

\[ \varphi_{(\Theta'/2)}=\varphi_{1/2}+\frac{RT}{tF}\ln\Phi'_{(\Theta'/2)}. \tag{16} \]

where

\[ \Phi'_{(\Theta'/2)}= \frac{1-\exp(\Theta'/2)\operatorname{erfc}\sqrt{\Theta'/2}} {3(1+K)\Theta'/2} +\frac{K}{3\sqrt{\chi}\Theta(1+K)}. \tag{17} \]

Using equation (16), from the experimental data for \(\varphi_{(\Theta'/2)}\), \(\varphi_{1/2}\), \(\Theta'\), \(K\), \(D\), it is easy to calculate \(\kappa_1\) and \(\kappa_2\) of a fast chemical reaction.

Let us consider some limiting cases that are of particular interest.

1. In the case of low current densities (large value of \(\Theta'\)), we represent the error functions of equation (14) in the form of a conditionally convergent series [4, p. 95], restricting ourselves to the first two terms of the series at \(\dfrac{\chi\Theta'}{2}>9\), with an error of less than \(1\%\), and \(\left(\dfrac{\pi\Theta'}{2}\right)^{-1/2}\ll 1\); we obtain the following expression for \(\Phi_{(\Theta'/2)}\):

\[ \Phi''_{(\Theta'/2)}= \frac{2K}{3(1+K)\Theta'} \left[\frac{1}{K}+\frac{1}{1+\sqrt{\chi}}\right]. \tag{18} \]

In the case of a fast subsequent reaction \((\chi\gg 1)\), expression (18) is reduced to the form:

\[ \Phi'''_{(\Theta'/2)}= \frac{2}{3(1+K)\Theta'}. \tag{19} \]

From the Nernst equation, upon substituting expression (19) into it, the equilibrium constant of a fast chemical reaction can be calculated from the experimental values \(\varphi_{1/2}\), \(\varphi_{(\Theta'/2)}\), \(\Theta'\).

1. At small values of \(\Theta\), the expression for \(C_R(1,\Theta)\) passes into the expression for semi-infinite diffusion. The problems of semi-infinite spherical diffusion with a subsequent chemical reaction are considered in [5].

2. For \(\kappa_1 \gg \kappa_2\), the chemical reaction may be regarded as irreversible:

\[ R \xrightarrow{ze} O + N \xrightarrow{\kappa'} \Gamma . \tag{20} \]

As \(\kappa \to \infty\), from equation (12) one obtains an expression for determining the concentration at the surface of a spherical electrode for an electrode process complicated by a subsequent irreversible first-order chemical reaction:

\[ C_0(1,\Theta)=\frac{\lambda}{1-\chi_1} \left[1-\sqrt{\chi_1}\operatorname{erf}(\chi_1\Theta)^{1/2} -\exp(\Theta-\chi_1\Theta)\operatorname{erfc}\Theta^{1/2}\right], \tag{21} \]

where

\[ \chi_1=\frac{K_1 r_1^2}{D}. \]

For \(\Theta \geq 0.14\), expression (14) in the case under consideration is reduced to the form:

\[ \Phi^{\mathrm{IV}}_{(\Theta')}= \frac{1}{3(1-\chi_1)\Theta} \left[1-\chi_1^{1/2}\operatorname{erf}(\chi_1\Theta)^{1/2} -\exp(\Theta-\chi_1\Theta)\operatorname{erfc}\Theta^{1/2}\right]. \tag{22} \]

In the case of fast chemical reactions \((\chi_1\Theta)^{1/2} \geq 3\), from expression (22), with an error of less than 1%, we obtain

\[ \Phi^{\mathrm{V}}_{(\Theta')}= \frac{2}{3(1+\chi_1^{1/2})\Theta}. \tag{23} \]

The equation of the \(\varphi\)–\(t\) curve of the reversible process of electrooxidation of the amalgam, complicated by an irreversible subsequent chemical reaction, at \(\Theta=\Theta'/2\) will have the form:

\[ \varphi_{(\Theta'/2)}=\varphi_{1/2}+\frac{RT}{zF} \ln \Phi_{(\Theta'/2)}\left[\Phi^{\mathrm{V}}_{(\Theta'/2)}\right]. \tag{24} \]

Using this equation, taking into account expression (22), from the experimental data \(\varphi_{(\Theta'/2)}\), \(\varphi_{1/2}\), \(r_0\), \(D\), \(\Theta'\), by the method of selection one can calculate the rate constant of any irreversible chemical reaction following the electrode process.

For fast irreversible chemical reactions at \(\Theta=\Theta'/2\), the following expression for the rate constant is obtained:

\[ \lg \chi_1=\lg \frac{2}{3\Theta'} -\frac{(\varphi_{\Theta'/2}-\varphi_{1/2})zF}{2.3RT}. \tag{25} \]

It should be noted that at large \(\Theta'\) (low current densities), at \(\Theta=\Theta'/2\), for any irreversible chemical reactions (proceeding at high and low rates) one can obtain

\[ \Phi_{(\Theta'/2)}= \frac{1}{\dfrac{3}{2}\Theta'(\sqrt{\chi_1}+1)}. \tag{26} \]

Applying Duhamel’s theorem, expressions are obtained for the concentrations of the oxidized form of the element at the electrode surface for any form of variation of the current (potential) of the electrode. For the electrode

process complicated by a subsequent reversible chemical reaction, we shall have

\[ 1)\quad C_0(1,\Theta)=-\frac{r_0}{D_0(1+K)}I_1, \]

where

\[ I_1=\int_0^\Theta q(\xi)\left[\frac{1}{\sqrt{\pi(\Theta-\xi)}}-\exp(\Theta-\xi)\operatorname{erfc}\sqrt{\Theta-\xi}\right]\times \]

\[ \times\{1+K\exp[-\chi(\Theta-\xi)]\}\,d\xi; \tag{27} \]

2) an irreversible chemical reaction

\[ C_0(1,\Theta)=\frac{r_0}{D_0}I_2, \]

\[ I_2=\int_0^\Theta q(\xi)\exp[-\chi_1(\Theta-\xi)] \left[\frac{1}{\sqrt{\pi(\Theta-\xi)}}-\exp(\Theta-\xi)\operatorname{erfc}(\Theta-\xi)^{1/2}\right]\,d\xi. \tag{28} \]

The expression for \(C_R(1,\Theta)\) for an arbitrary form of the electrode current (potential) is given in [1]:

\[ C_R(1,\Theta)=C_R^0-\frac{r_0}{D_R}I_3, \]

\[ I_3=\int_0^\Theta q(\xi)\left\{3+2\sum_{n=1}^{\infty}\exp[-\mu_n^2(\Theta-\xi)]\right\}\,d\xi, \tag{29} \]

\(\xi\) is an auxiliary variable.

The expression for the current density for a first-order electrode reaction has the form:

\[ i(t)=zF\kappa_s\left[ e^{-\frac{\alpha zF}{RT}(\varphi-\varphi^0)}C_0(1,\Theta) - e^{\frac{\beta zF}{RT}(\varphi-\varphi^0)}C_R(1,\Theta) \right]. \tag{30} \]

The quantities entering into this equation have generally known meanings.

By combining equations (27, 29, 30), one obtains the equation for the current density for a quasi-reversible electrode process of amalgam oxidation complicated by a subsequent first-order reaction:

\[ i=ae^{y_1}\frac{r_0}{D_0(1+K)}I_1 - be^{y_2}\left(C_R^0-\frac{r_0}{D_R}I_3\right). \tag{31} \]

Substituting into the Nernst equation the values \(C_R(1,\Theta)\) and \(C_0(1,\Theta)\) from equations (27) and (29), we obtain the equation for the current density of the reversible process of electrooxidation of the amalgam, complicated by a subsequent reversible chemical reaction:

\[ \frac{r_0 I_1}{(1+K)(C_R^0D-r_0I_3)} = \exp\left[\frac{zF}{RT}(\varphi_i-\varphi^0)\right]\exp(y), \tag{32} \]

the values of \(a\), \(b\), \(y_1\), \(y_2\), \(y\) are given in [7].

By combining equations (28, 29, 30), one obtains the equation of the \(i-\varphi\) curve for a quasi-reversible electrode process of electro-dissolution of the amalgam, complicated by a subsequent irreversible chemical reaction:

\[ i=ae^{y_1}\frac{r_0}{D_0}I_2 - be^{y_2}\left(C_R^0-\frac{r_0}{D_R}I_3\right). \tag{33} \]

For the reversible process of electrodissolution of an amalgam, complicated by a subsequent irreversible chemical reaction, we have

\[ \frac{r_0 I_2}{C_R^0 D_R - r_0 I_3} = \exp\left[\frac{zF}{RT}\left(\varphi_i-\varphi^0\right)\right]\exp y_s . \tag{34} \]

Equations (31, 32, 33, 34), by one of the methods of numerical integration, are reduced to algebraic equations, which can be programmed and then used on electronic computers to calculate the values of \(i(t)\) for constructing the corresponding curves. These questions will be considered in a separate paper.

Conclusions

1. Equations have been obtained for the concentration distribution at the electrode surface in the AAV method under spherical diffusion for electrode processes complicated by a subsequent chemical reaction.

2. Equations have been obtained for the \(\varphi—t\), \(i—\varphi\), and \(i—t\) curves for the case under consideration.

REFERENCES

1. M. S. Zakharov, V. V. Pnev. Electrochemistry (in press).

2. P. Delahay, C. C. Mattax, J. Berzins. J. Am. Chem. Soc. 76, 5319 (1954).

3. M. S. Zakharov, V. I. Bakanov, V. V. Pnev. Electrochemistry (in press).

4. P. Delahay. New Instruments and Methods in Electrochemistry, Foreign Literature Publishing House, Moscow, 1957.

5. N. S. Zakharov, V. I. Bakanov. Electrochemistry (in press).

6. W. T. de Vries, E. van Dallen. J. Electroanal. Chem. 10, 183 (1965).

7. M. S. Zakharov, V. V. Pnev. Theory of anodic amalgam voltammetry with an arbitrary form of variation of the potential and current of a spherical electrode. Electrode reaction complicated by a preceding chemical reaction. (This collection).