ANODIC AMALGAM VOLTAMMETRY AT CONSTANT CURRENT ON A SPHERICAL ELECTRODE

M. S. ZAKHAROV, V. V. PNEV

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

In the present work, equations will be obtained for the transition time and the $\varphi$—$t$ curves for AAV on a spherical electrode, and a number of applications of the method will be proposed.

Boundary-value problem.

\[ C_B(r,0)=C_B^0 \qquad \text{for } 0\le r\le r_0, \tag{1} \]

\[ C_0(r,0)=C_0^0 \qquad \text{for } r_0\le r<\infty, \tag{2} \]

\[ \frac{\partial C_B(r,t)}{\partial r}=-\frac{q_0}{D_B} \qquad \text{for } r=r_0, \tag{3} \]

\[ D_0\left[\frac{\partial C_0(r,t)}{\partial r}\right]_{r=r_0} = D_B\left[\frac{\partial C_B(r,t)}{\partial r}\right]_{r=r_0}; \tag{4} \]

\[ \left.\frac{\partial C_B(r,t)}{\partial r}\right|_{r\to 0}=0, \tag{5} \]

\[ \left.C_0(r,t)\right|_{r\to\infty}=C_0^0, \tag{6} \]

where $q_0=\dfrac{i_0}{zFS}$; $S$ is the electrode surface, cm$^2$. The posed boundary-value problem is solved by the operational method. For the concentrations on the surface of the spherical electrode, the following relations are obtained:

\[ C_0(r_0,t)=C_0^0+\lambda_0\left(1-\exp\vartheta_0\,\operatorname{erfc}\vartheta_0^{1/2}\right)^*, \tag{7} \]

\[ C_B(r_0,t)=C_B^0-\lambda_B \left[ 3\vartheta_B+0.2-\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}\exp\left(-\mu_n^2\vartheta_B\right) \right], \tag{8} \]

where

\[ \lambda_0=\frac{q_0r_0}{D_0},\qquad \lambda_B=\frac{q_Br_0}{D_B};\qquad \vartheta_0=\frac{D_0t}{r_0^2},\qquad \vartheta_B=\frac{D_Bt}{r_0^2}, \]

$\mu_n$ are the roots of the characteristic equation:

\[ \tan\mu=\mu. \]

The expression for the transition time in the general case must be found from the equation:

\[ C_B^0=\lambda_B \left[ 3\vartheta_B'+0.2-\sum_{n=1}^{\infty}\frac{2}{\mu_n^2} \exp\left(-\mu_n^2\vartheta_B'\right) \right]. \tag{9'} \]

\[ \text{* This equation was also obtained earlier in [5].} \]

Equation (9′) is transcendental with respect to \(\vartheta_R'\). For \(\vartheta_R'\ge 0.14\) (for \(r_0=5\cdot10^{-2}\ \mathrm{cm}\) and \(D_R=1\cdot10^{-5}\ \mathrm{cm^2/sec}\), \(\tau\ge35\ \mathrm{sec}\)), the sum in expression (9′) may be neglected with an error of less than 1%; hence we obtain

\[ \tau=\frac{zFC_R^0 r_0}{3i_0}-\frac{r_0^2}{15D_R}. \tag{9} \]

From equation (9) there follows the relation

\[ i_0\left(\vartheta_R'+\frac{1}{15}\right)\frac{zFD_R C_R^0}{3r_0}=\mathrm{const}. \]

From experimental data \((i_0, C_R^0, r_0)\) one can calculate the value of the diffusion coefficient of the metal atom in mercury from the following equations:

\[ D_R=\frac{i_0 r_0^2}{5zFr_0C_R^0-15i_0\tau}, \qquad D_R=\frac{r_0^2\left(i_0'-i_0''\right)}{15\left(\tau''i_0''-\tau'i_0'\right)}. \tag{10} \]

For \(\vartheta_R'\ge0.14\) (\(\tau\ge35\ \mathrm{sec}\)), the dependence of the electrode potential on time for reversible electrode processes is expressed by the equation (for \(D_0=D_R=D,\ \vartheta_0=\vartheta_R=\vartheta,\ \lambda_0=\lambda_R=\lambda\)):

\[ \varphi=\varphi_1-\frac{RT}{zF}\ln \frac{ 3\lambda(\vartheta'-\vartheta)+ \lambda\displaystyle\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}\exp(-\mu_n^2\vartheta') }{ C_0^0-\lambda\left(1-\exp\vartheta\,\mathrm{erfc}\sqrt{\vartheta}\right) }. \tag{11} \]

It should be noted that the plot in the coordinates \(\varphi,\ \lg \dfrac{C_R(r_0,t)}{C_0(r_0,t)}\) is a straight line with the reciprocal value of the slope coefficient \(2.3\,\dfrac{RT}{zF}\).

The dependence of the electrode potential on time in the case of irreversible processes, for \(\vartheta_R'\ge0.14\) and arbitrary \(\vartheta\), will have the form

\[ \varphi=\varphi^0+\frac{RT}{\beta zF}\ln\frac{q_0}{k_s} -\frac{RT}{\beta zF}\ln\left[ 3\lambda_R(\vartheta_R-\vartheta_R') -\lambda_R\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}\exp(-\mu_n^2\vartheta_R) \right]. \tag{12} \]

The plot in the coordinates \(\varphi,\ \lg C_R(r_0,t)\) is a straight line with the reciprocal value of the slope coefficient equal to \(2.3\,\dfrac{RT}{\beta zF}\), which makes it possible to determine the value \(\beta z\).

The value of the rate constant of the electrode process is found from the value of the electrode potential at \(t=0\) from the equation

\[ \varphi=\varphi^0+\frac{RT}{\beta zF}\ln\frac{q_0}{k_s} -\frac{RT}{\beta zF}\ln\left[\Psi'(\vartheta_R,\vartheta_R')\right], \tag{13} \]

where

\[ \Psi'(\vartheta_R,\vartheta_R')= 3\vartheta_R'-0.8+\exp\vartheta_R\,\mathrm{erfc}\sqrt{\vartheta_R} \tag{14} \]

(for \(\vartheta_R'\ge0.14\) and small \(\vartheta_R\));

\[ \Psi'(\vartheta_R,\vartheta_R')= 3\vartheta_R'-0.8+\exp\vartheta_R\,\mathrm{erfc}_R\sqrt{\vartheta_R} -\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}\exp(-\mu_n^2\vartheta_R') \tag{15} \]

(for arbitrary \(\vartheta_R'\) and small \(\vartheta_R\)).

The dependence of the potential on time (in implicit form) for a quasi-reversible electrode process, for \(\vartheta_R'\ge0.14\) and arbitrary \(\vartheta\), has the form

\[ \frac{i_0}{zFS}=-\kappa_s\left\{\exp\left[\frac{\beta zF}{RT}(\varphi-\varphi_0)\right]\left|3\lambda_R(\vartheta_R-\vartheta_R)+\lambda_R\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}\times\right.\right. \]

\[ \left.\left. \times \exp\left(-\mu_n^2\vartheta_R\right)\right|-\exp\left[-\frac{\alpha zF}{RT}(\varphi-\varphi^0)\right]\left[C_0^0+\lambda_0\left(1-\exp\vartheta_0\,\operatorname{erfc}\sqrt{\vartheta_0}\right)\right]\right\}. \tag{16} \]

Using equation (16) and experimental data, one can determine the kinetic parameters of a quasireversible electrode process. For this purpose, for the system under consideration, a \(\varphi\)—\(t\) plot is constructed from the experimental data. From it, \(\tau\) and two values of the electrode potential \(\varphi_1\) and \(\varphi_2\) at \(t_1\) and \(t_2\) are determined. These data are substituted into expression (16), yielding two equations with two unknowns (\(\kappa_s\) and \(\beta\)). The values of the diffusion coefficients of atoms in mercury and ions in solution must be known. The resulting system of two equations is solved and \(\kappa_s\) and \(\beta\) are calculated.

In the general case, if the amalgam contains several electroactive components, the following expression is obtained for the concentrations of metal atoms at the electrode surface:

\[ \sum_{i=1}^{m} z_i C_{R,i}(r_0,t)=\sum_{i\sim 1}^{m}\left\{z_i C_{R,i}^{0}-\lambda_i\left[3\vartheta_i+0.2-\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}\exp\left(-\mu_n^2\vartheta_i\right)\right]\right\}. \tag{17} \]

For \(\vartheta_i\ge 0.14\), in equation (17) the last sum may be neglected with an error of less than 1%:

\[ \sum_{i=1}^{m} z_i C_{R,i}(r_0,t)=\sum_{i=1}^{m} z_i C_{R,i}^{0}-\lambda_i(3\vartheta_i+0.2). \]

For \(\vartheta_i\ge 0.14\), the expression for \(\tau\) has the form

\[ \sum_{i=1}^{m}\left(\tau_i+\frac{r_0^2}{15D_{R,i}}\right)=\frac{r_0F}{3i_0}\sum_{i=1}^{m} z_i C_{R,i}^{0}. \tag{19} \]

Since

\[ \sum_{i=1}^{m}\frac{r_0^2}{15D_{R,i}}-\sum_{i=1}^{m-1}\frac{r_0}{15D_{R,i}}=\frac{r_0^2}{15D_{R,i}}, \]

the transition time for the \(i\)-th component is expressed by the equation

\[ \sum_{i=1}^{m}\tau_i-\sum_{i=1}^{m-1}\tau_i=\frac{z_i r_0F C_{R,i}}{3i_0}-\frac{r_0^2}{15D_{R,i}}. \tag{20} \]

For the element that dissolves second, from equation (20) one can obtain the following expression for the transition time:

\[ (\tau_1-\tau_2)-\tau_1=\frac{z_2 r_0F C_{R,2}^{0}}{3i_0}-\frac{r_0^2}{15D_{R,2}}. \tag{21} \]

Expressions for the transition time for subsequent elements are obtained in an analogous manner.

It follows from equation (21) that, for \(\vartheta'\ge 0.14\) and in the absence of complicating factors, \(\tau_2\) does not depend on the concentration of the element that is oxidized at more negative potentials.

Experimental Part

We carried out studies to assess the reproducibility of determining the transition time and to verify the dependence of $\tau$ on $\dfrac{1}{i_0}$.

The setup for recording curves of anodic electro-dissolution of amalgams is, in principle, no different from the setups used for recording cathodic $\varphi$—$t$ curves [1, p. 435]. As a source of constant voltage, a “Norma” stabilizer (Austria) was used. The voltage taken from the stabilizer exceeded the voltage at the electrodes by two orders of magnitude or more. At the same time, the resistance of the circuit was approximately two orders of magnitude greater than the resistance of the cell. Under these conditions the current may be regarded as constant with an error of 1%.

As the recording instrument, an OH-101 polarograph (Hungary) was used, with an input resistance increased to $1.2 \cdot 10^7$ ohm. In the work, the electrolyzer described in [2] was used. Saturated calomel electrodes were used as the auxiliary electrode and the reference electrode. Chemically pure reagents were used in the work. During electroaccumulation the solution was stirred with gaseous nitrogen for 15 min; the solution rest time was 30 sec. The accumulation potential was $-1.0$ V; the concentration in solution was $C_{\mathrm{Cd}^{2+}} = 2.76 \cdot 10^{-3}$ mol/l; the solution volume was 5.5 ml. A 2 M solution of ammonium sulfate served as the supporting solution. As a result of the high concentration of the amalgam, the errors introduced by oxidation of the amalgam by oxygen and by the cementation reaction [3] were small. Table 1 gives the results of studying reproducibility at an electrooxidation current equal to $1.07 \cdot 10^{-5}$ A.

Table 1

Reproducibility of the results of measuring the transition time

Here $S_x$ is the rms error, $t_{0.95}$ is Student’s coefficient [4]. Verification of the reproducibility of the results of measuring the transition time was carried out on a single mercury drop.

As follows from equation (9), the plot in these coordinates should be a straight line with a slope numerically equal to $\dfrac{z F C_R^0 r_0}{3}$ C. To verify this conclusion, 4 series of experiments were carried out under the conditions described above. At each current value, no fewer than three experiments were performed. The averaged data are given in Fig. 1. The parameters of the straight lines in Fig. 1 were found by the least-squares method [4, p. 285]. The parameters of the straight lines are given in Table 2, and the errors of determining the quantities are also given there.

It is seen from Fig. 1 that the experimental data are described by a straight line with satisfactory accuracy. More precisely, the hypothesis of linearity was checked by the value of the $F$-criterion [4, p. 260], using the variance $S_0^2$, which characterizes the scatter of the values of $\tau$ relative to the straight line (Table 2), with the values $S^2$ (reproduced in Table 2). Comparing the values of $F$ (Table 2) with the values of $F$ for the selected significance level 0.05 and taking into account the number of degrees of freedom [4, p. 374],

Fig. 1. Dependence of the transition time \(\tau\), sec, on the reciprocal current value \(\dfrac{10^5}{i}\,a^{-1}\).

1 — \(q=1.8\cdot 10^{-3}\);
2 — \(q=1.7\cdot 10^{-3}\);
3 — \(q=1\cdot 10^{-3}\);
4 — \(q=0.3\cdot 10^{-3}\).

we are convinced that they differ insignificantly. Thus, the theoretical conclusion on the linearity of the plot of \(\tau,\ \dfrac{1}{i_0}\) may be accepted.

Table 2 gives the values

\[ \frac{zFSC_R^0 r_0}{3}=zFV_k C_R^0 . \]

From Table 2 it is seen that the error in determining this quantity does not exceed \(\pm 7\%\).

Table 2

Values of the parameters of the straight lines in Fig. 1

Conclusions

1. The equations of the \(\varphi\)—\(t\) curves in anodic amalgam voltammetry at a prescribed current with a spherical electrode have been obtained theoretically for reversible, irreversible, and quasireversible processes.

2. The possibility of applying the method under consideration for analytical purposes and for studying the kinetics of electrode processes has been shown.

3. An equation has been obtained for the transition time in the method under consideration during oxidation of a complex amalgam.

4. At \(\vartheta' \ge 0.14\), the transition time for oxidation processes of the more electropositive element does not depend on the concentration of the more electronegative element; this regularity is very convenient for the analytical application of the method.

1. The reproducibility of measuring the transition-time value in the AAV with CT method was studied using cadmium as an example; its concentration in solution was \(2.73 \cdot 10^{-3}\) mol/l. The reproducibility of the results was \(\pm 3.5\%\).

REFERENCES

1. P. Delahay. New Instruments and Methods in Electrochemistry. Inostrannaya Literatura, Moscow, 1957.

2. V. I. Kuleshov, A. G. Stromberg. In: Chemical Methods of Analysis, issues 5–6. Izd. IREA, Moscow, 1963.

3. H. A. Wincent, E. N. Wise, J. Electroanal. Chem., 11, 54 (1966).

4. V. V. Nalimov. The Application of Mathematical Statistics in the Analysis of Substances. Fizmatgiz, Moscow, 1960.

5. H. Hurwitz, J. Electroanal. Chem., 7, 368, 1964.