Theory of Anodic Amalgam Voltammetry with a Prescribed Electrode Current or Potential

M. S. Zakharov, V. V. Pnev

(Presented by the scientific-methodological seminar of the KhTF)

The essence of anodic amalgam voltammetry (AAV) consists in the preliminary accumulation of an element in a mercury electrode and the subsequent dissolution of the resulting amalgam at a prescribed electrode current or potential.

The principal questions of the AAV method are the determination of expressions for the dependence of the current on time and for the dependence of the electrode potential on time [1]. These problems can be solved if the equations for the distribution of concentration at the electrode surface are known.

In the present work a general solution will be given for the equation of Fick’s second law under conditions of restricted diffusion for spherical and film mercury electrodes. The problem is solved under the following conditions: 1) there is a large excess of indifferent electrolyte in the solution; 2) supply of the substance occurs only by diffusion; 3) no complicating reactions are present.

The equation of Fick’s second law

\[ \frac{\partial C_R(x,\Theta_R)}{\partial \Theta_R} = \frac{\partial^2 C_R(X,\Theta_R)}{\partial X^2} + \frac{\Gamma}{X}\, \frac{\partial C_R(X,\Theta_R)}{\partial X} \tag{1} \]

was solved under the following initial and boundary conditions:

\[ C_R(X,0)=C_R^{0}, \tag{2} \]

\[ \lim_{X\to 0}\frac{\partial C_R(X,\Theta_R)}{\partial X}=0, \tag{3} \]

\[ \frac{\partial C_R(1,\Theta_R)}{\partial X}=-\lambda_R, \tag{4} \]

where

\[ \Gamma \]

is the electrode-shape coefficient;

\[ \Gamma=0 \]

for a film electrode and

\[ \Gamma=2 \]

for a spherical electrode;

\[ \Theta_R=\frac{D_R t}{r_0^2} \quad \text{or} \quad \Theta_R=\frac{D_R t}{l^2} \]

is dimensionless time;

\[ D_R \]

is the diffusion coefficient of metal atoms in mercury, \(\mathrm{cm^2/sec}\);

\[ t \]

is time, sec;

\[ r_0 \]

is the radius of the spherical electrode, cm;

\[ l \]

is the thickness of the film of a mercury film electrode, cm.

\(X \equiv \dfrac{r}{r_0} \equiv \dfrac{x}{l}\)—the dimensionless coordinate for spherical and film electrodes, respectively;

\(C_R(X;\Theta_R)\)—the concentration of the metal in the amalgam, g-atom/cm\(^3\);

\[ \lambda_R \equiv \frac{i_0 r_0}{n F s D_R} \equiv \frac{i_0 l}{n F s D_R}. \]

The formulated boundary-value problem was solved by the method of Laplace integral transforms. In doing so it was assumed that \(\Gamma = 2\gamma + 1\) [3]. For film and spherical electrodes the value of \(\gamma\) is, respectively, equal to \(-\dfrac{1}{2}\) and \(\dfrac{1}{2}\). The solution in transforms has the form

\[ \overline{C}_R(X,s)-C_R^0 = -\lambda_R (X)^{-\gamma} \frac{I_\gamma\!\left(\sqrt{s}\,X\right)} {s\sqrt{s}\,I_{\gamma+1}\!\left(\sqrt{s}\right)}, \tag{5} \]

where

\(I_\gamma(\sqrt{s}\,X)\) is the modified Bessel function of the first kind of order \(\gamma\). Passing to the original by the expansion theorem for multiple roots [2], one can obtain the equation for the distribution of the metal concentration in the amalgam:

\[ C_R(X,\Theta_R)-C_R^0 = -\lambda_R \left[(2\gamma+2)\Theta_R + \frac{X^2}{2} - \frac{\gamma+1}{\gamma+4} - \sum_{n=1}^{\infty} \frac{(X)^{-\gamma}}{\mu_n^2}\cdot \frac{I_\gamma(\mu_n X)}{I_\gamma(\mu_n)} \exp\!\left(-\mu_n^2\Theta_R\right) \right], \tag{6} \]

where

\(I_\gamma\)—the Bessel function of order \(\gamma\);

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

\[ I_{\gamma+1}(z)=0. \tag{7} \]

From equation (6), in the form of particular solutions, expressions can be obtained for the distribution of the metal concentration in the amalgam for the electrode types under consideration. In doing so it is necessary to take into account the relation of the Bessel functions to trigonometric functions [4]:

\[ \left. \begin{aligned} I_{1/2}(z) &= \sqrt{\frac{2}{\pi z}}\sin z,\\ I_{3/2}(z) &= \sqrt{\frac{2}{\pi z}}\left(\frac{\sin z}{z}-\cos z\right). \end{aligned} \right\} \tag{8} \]

Taking relation (8) into account, the characteristic equation (7) can be reduced to the following expressions:

a) for the film electrode

\[ \sin \mu = 0;\quad \mu_n=n\pi;\quad n=1,2\ldots \]

b) for the spherical electrode

\[ \tg \mu=\mu. \]

The roots of the latter equation are given in [5].

Putting \(X=1\) in equation (7), we obtain the following equations for the concentrations at the surface:

for the film

\[ C_R(1,\Theta_R) = C_R^0 - \lambda_R \left[ \Theta_R+\frac{1}{3} - 2\sum_{n=1}^{\infty} \frac{\exp\!\left(-\mu_n^2\Theta_R\right)}{\mu_n^2} \right], \tag{9} \]

for a sphere

\[ C_R(l,\Theta_R)=C_R^0-\lambda_R\left[3\Theta_R+\frac{1}{5}-2\sum_{n=1}^{\infty}\frac{\exp(-\mu_n^2\Theta_R)}{\mu_n^2}\right]. \tag{10} \]

The series in equations (9, 10) converge rapidly, since \(\mu_n\) are sufficiently large. Therefore, beginning with a certain value of \(\Theta_R\), they may be neglected with a prescribed error. It should be noted that equation (10) for the distribution of the metal in the amalgam during pre-electrolysis at the limiting current for a spherical electrode was obtained in works [6, 7], and equation (9) for a film electrode in [8, 9]. Analogous equations are also given in monograph [2].

The expression for the transition time \(\Theta_R'\) in chronopotentiometry is determined from the condition \(C_R(l,\Theta_R')=0\) (\(\Theta_R'\) is the dimensionless transition time). Since the transition time can be varied by changing the density of the imposed current, one can choose such a \(\tau(\Theta_R')\) that in expressions (9, 10) the series may be neglected with a prescribed error. Calculations show that, with an error not exceeding 1%, the sum in equations (9, 10) may be neglected for \(\Theta_R' \ge 0.14\). In this case the concentration of the metal in the amalgam is related to the transition time and the current density by the simple equation

\[ \Theta_R'=\frac{1}{2\gamma+2}\left[\frac{C_R^0}{\lambda_R}-\frac{1}{(2\gamma+2)(2\gamma+4)}\right], \tag{11} \]

in particular, for a film electrode

\[ \Theta_R'=\frac{C_R^0}{\lambda_R}-\frac{1}{3} \tag{12} \]

and for a spherical electrode

\[ \Theta_R'=\frac{1}{3}\frac{C_R^0}{\lambda_R}-\frac{1}{5}. \tag{13} \]

In the case of the electrooxidation of a complex amalgam consisting of electroactive components, in finding the expression for the transition time one may use the principle proposed in [10]. Using this principle, for an electrode of arbitrary shape one can obtain the following equation for the transition time:

\[ \sum_{m=1}^{m}\tau_m = \frac{1}{4}\frac{y_0^2}{D_{R,m}(\gamma+2)(\gamma+1)} = \frac{1}{2}\sum_{m=1}^{m}\frac{z_m F s y_0 C_{R,m}^0}{(\gamma+1)i_0}. \tag{14} \]

Equation (11) for the transition time of the oxidation process of the \(m\)-th element is reduced to the form:

\[ \tau_m = \frac{y_0^2}{4D_{R,m}(\gamma+2)(\gamma+1)} = \frac{z_m F s y_0 C_R^0}{2(\gamma+1)i_0}, \tag{15} \]

where \(y_0\) is the radius of the mercury sphere or the thickness of the mercury film.

It follows from formula (12) that, in the case of electrooxidation of a complex amalgam at constant current, the transition time for the \(m\)-th component does not depend on the transition time of the \((m-1)\)-th component, whereas under conditions of semi-infinite diffusion the transition time of the reduction process of the subsequent element (a cathodic process is considered) depends on the transition time (concentration) of the previously reduced element.

The series in equation (6) converges slowly for small values of \(\Theta_R\); therefore, for small \(\Theta_R\), it is more convenient to find an approximate solution of equation (5). For this purpose we expand the Bessel function in an asymptotic series:

\[ I_\gamma(z)=\frac{1}{\sqrt{2\pi z}}e^{+z} \left[ 1+\frac{4\gamma^2-1^2}{1!8z} +\frac{(4\gamma^2-1^2)(4\gamma^2-3^2)}{2!(8z)^2} +\ldots \right]. \tag{16} \]

For small values of \(\Theta_R\), the quantity \(\sqrt{s}\) is large, and the quantity \(\sqrt{s}\,X\) for \(X\to0\) (i.e., for the center of the electrode) is small. Therefore, in order to find the concentration distribution in the vicinity of the central point of the electrode, the function \(I_\gamma(\sqrt{s}\,X)\) should be expanded in a power series. We shall be interested in the concentration distribution at the surface of the electrode \((X=1)\). From the solution in transforms (5), taking (16) into account, in which for small \(\Theta_R\) we restrict ourselves to the first two terms*, one can obtain the following equation for the distribution of the element concentration at the electrode surface for small \(\Theta_R\) [11]:

\[ C_R(I_1\Theta_R)=C_R^0-\frac{\lambda_R}{a}\left(1-\exp a^2\Theta_R\,\operatorname{erfc} a\sqrt{\Theta_R}\right), \tag{17} \]

where \(a=\left(\gamma+\frac{1}{2}\right)\), with \(a=0\) for a film and \(a=1\) for a sphere.

From equation (14), resolving the indeterminacy by l’Hospital’s rule as a special case for a film electrode, we obtain:

\[ C_R(I_1\Theta_R)=C_R^0-2\lambda_R\sqrt{\frac{\Theta_R}{\pi}}. \tag{18} \]

For a spherical electrode (at \(a=1\)), as a special case we obtain the equation

\[ C_R(I_1\Theta_R)=C_R^0-\lambda_R\left(1-\exp\Theta_R\,\operatorname{erfc}\sqrt{\Theta_R}\right). \tag{19} \]

Up to now we have considered a general way of solving the equation of Fick’s second law under conditions of bounded diffusion of any type (spherical, linear) for anodic amalgam voltammetry with a prescribed current. The same approach can also be used to solve the equation of Fick’s second law for chronopotentiometry under conditions of semi-infinite diffusion. To find the distribution of the ion concentration at the surface of an electrode of any type, one must solve equation (1) with the initial and boundary conditions (2,4); condition (3) must be replaced by the condition of semi-infinite diffusion:

\[ \lim_{X\to\infty} C_0(X,\Theta_0)=C_0^0, \tag{20} \]

where \(C_0^0\) is the initial concentration of ions in solution;

\[ \Theta_0=\frac{D_0t}{y_0^2}. \]

Solving the stated boundary-value problem by the operational method, we obtain the following expression for the concentration distribution of the oxidized form of the element at the electrode surface, valid for an electrode of any shape:

\[ C_0(I,\Theta_0)=C_0^0+\frac{\lambda_0}{a}\left(1-\exp a^2\Theta_0\,\operatorname{erfc} a\sqrt{\Theta_0}\right), \tag{21} \]

* The values of \(\Theta\) for which the solution will be justified can be estimated experimentally.

where \(a\) for the plane and the sphere has the values indicated above, and for the cylinder \(a=1/2\).

From equation (18), as special cases, one can obtain expressions for the concentration distribution for linear, cylindrical, and spherical diffusion under conditions of semi-infinite diffusion for chronopotentiometry. These equations coincide completely with the equations obtained in [12], in which questions of chronopotentiometry under conditions of semi-infinite diffusion were considered separately for each type of electrode.

Since in deriving the equations for \(C_R(1,\Theta_R)\) and \(C_0(1,\Theta_0)\) no assumptions were made concerning the kinetics of the process, in order to obtain expressions for the dependence of the electrode potential on time these equations may be substituted both into the Nernst equation (reversible processes) and into the kinetic equation (irreversible processes). This question has already been considered in detail in [13], and we shall not dwell on it here.

Applying Duhamel’s theorem, one can obtain a general equation describing the concentration distribution of the reduced form of the element at the surface, for any type of electrode (film,

spherical) and for any form of the current,

\[ q(t)=\frac{i(t)}{nFs}, \]

(where \(q(t)\) is the flux of substance).

Putting \(X=1\), from equation (6) we obtain:

\[ C_R(1,\Theta_R)=C_R^0-\frac{y_0}{D_R}\int_0^{\Theta_R} q(\Theta_R')\left\{(2\gamma+2)+2\sum_{n=1}^{\infty}\exp[-\mu_n^2(\Theta_R-\Theta_R')]\right\}d\Theta_R' \]
\[ = C_R^0-\frac{y_0}{D_R}I_1(1,\Theta_R). \tag{22} \]

As special cases, the following equations can be obtained for the concentration distribution of the reduced form of the element at the surface, respectively for film and spherical electrodes:

\[ C_R^{\text{fl}}=C_R^0-\frac{1}{l}\int_0^t q(\tau)\left\{1+2\sum_{n=1}^{\infty}\exp\left[-\mu_n^2\frac{D_R}{l^2}(t-\tau)\right]\right\}d\tau, \tag{23} \]

\[ C_R^{\text{sph}}=C_R^0-\frac{1}{r_0}\int_0^t q(\tau)\left\{3+2\sum_{n=1}^{\infty}\exp\left[-\mu_n^2\frac{D_R}{r_0^2}(t-\tau)\right]\right\}d\tau. \tag{24} \]

From equations (17–19) one can obtain expressions for the concentration distribution of the reduced form of the element at small \(\Theta_i\), for any form of the current and for any type of electrode. These equations will coincide with the equations of semi-infinite diffusion, i.e., they will also be valid for cathodic processes (for \(C_0(X,\Theta_0)\)) carried out under conditions of semi-infinite diffusion. In this case it is only necessary to bear in mind that, in accordance with the adopted notation, the quantity \(q(t)\) is positive for cathodic and negative for anodic processes.

Applying Duhamel’s theorem to equation (17), one can obtain

\[ C_0(1,\Theta_0)=C_0^0-\frac{y_0}{D_0}I_2(1,\Theta_0), \tag{25} \]

where

\[ I_2^{\mathrm{pl}}\equiv \frac{1}{\sqrt{\pi}}\int_0^{\Theta_0} \frac{q(\Theta_0')\,d\Theta_0'}{\sqrt{v_0-v_0'}}, \tag{26} \]

\[ I_2^{\mathrm{sph}}\equiv I_2^{\mathrm{pl}}-\int_0^{\Theta_0} q(\Theta_0')\exp(\Theta_0-\Theta_0')\operatorname{erfc}\sqrt{\Theta_0-\Theta_0'}\,d\Theta_0'. \tag{27} \]

To obtain the dependences of the electrode potential on time or of the current on time, equations (22, 25) should be substituted either into the Nernst equation (if the electrode process is reversible) or into the kinetic equation (if the electrode process is irreversible). In doing so it must be borne in mind that, for the cathodic process, \(q(\Theta_i)\geq 0\); consequently, in equations (22, 25) the signs before the integrals change to the opposite ones. The expression for the current in the case of electro-dissolution of the amalgam has the form

\[ \pm q(t)=aC_0^0-bC_R^0\mp \frac{a}{\sqrt{\pi D_0}}I_2(y_0,t)\mp \frac{b}{y_0}I_1(y_0,t), \tag{28} \]

where

\[ a=\kappa_s\exp\left\{-\frac{\alpha nF}{RT}\left[E(t)-E^0\pm \Omega\cdot nFsq(t)\right]\right\}; \]

\[ b=\kappa_s\exp\left\{\frac{\beta nF}{RT}\left[E(t)-E^0\pm \Omega nFsq(t)\right]\right\}; \]

\(\kappa_s\) is the rate constant of the electrode process, \(\mathrm{cm/sec}^{-1}\);

\(E(t)\) is the law of variation of the potential; in voltammetry with a prescribed current it is the quantity sought, while in voltammetry with a prescribed law of variation of the potential the quantity sought is \(q(t)\);

\(\Omega\) is the resistance of all elements of the electrolytic cell, ohm;

\(I_1\) and \(I_2\) are the integrals of equations (22–27), in which the dimensionless variables \(X\) and \(\Theta_i\) are replaced by \(y_0\) and \(t\). The upper signs in equation (28) correspond to the cathodic process, and the lower signs to the anodic process. Equation (28), the so-called Volterra-type equation, is in general solved numerically. For a plane electrode this was carried out in [14].

Conclusions

A general method has been developed for solving diffusion problems in voltammetry with a prescribed current or potential for electrodes of any type (spherical, plane). Substituting the equations found for the concentration distributions in the solution and in the amalgam electrode into the Nernst equation or into the kinetic equation, integral equations were obtained.

LITERATURE

1. P. Delahay. New Instruments and Methods in Electrochemistry, IL, Moscow, 1957.
2. A. V. Lykov. Theory of Heat Conduction, State Publishing House of Technical-Theoretical Literature, Moscow, 1952.
3. E. M. Goldfarb. Scientific Reports of Higher School, 129, 1958.
4. N. N. Lebedev. Special Functions and Their Applications, p. 143, Fizmatgiz, Moscow–Leningrad, 1963.

1. E. Jahnke, F. Emde. Tables of Functions with Formulae and Curves. OGIZ, Gostekhizdat, Moscow—Leningrad, 1948.
2. L. N. Vasil’eva, E. N. Vinogradova. Zavod. labor., 27, 1079 (1961).
3. I. Shain, L. Lewinson. Anal. Chem., 33, 187, 1961.
4. V. A. Igolinskii. Dissertation, Tomsk, 1963.
5. W. T. de Vries, E. van Dalen. J. Electroanal. Chem., 8, 366, 1964.
6. R. W. Murray, C. N. Reilley. Ibid., 3, 182, 1962.
7. V. A. Ditkin, L. P. Prudnikov. Handbook of Operational Calculus. “Vysshaya shkola,” Moscow, 1965.
8. H. D. Hurwitz. J. Electroanal. Chem., 7, 368, 1964.
9. M. S. Zakharov, V. V. Pnev, V. I. Bakanov. Elektrokhimiya (in press).
10. W. T. de Vries. J. Electroanal. Chem., 9, 448, 1965.