ANODIC AMALGAM VOLTAMMETRY WITH PROGRAMMED CURRENT AT A SPHERICAL MERCURY ELECTRODE

M. S. Zakharov, V. V. Pnev

(Presented by the scientific seminar of the Problem Laboratory of Physicochemical Methods of Analysis)

The method of voltammetry with a specified current is finding application in solving a number of physicochemical and analytical problems. Until recently, the literature dealt only with questions of the method under conditions of semi-infinite diffusion [1—12]. Only a few works have been devoted to consideration of voltammetry under conditions of restricted diffusion [13—16].

In the present work we consider certain theoretical questions of the AAV method with programmed current. Here the delivery of the electroactive component to the reaction surface proceeds under conditions of restricted diffusion. In [18], under conditions of restricted diffusion for the case of a single electrode reaction not complicated by kinetic effects, and for an arbitrary form of current, the following relations were obtained for the distribution of the concentrations of the oxidized \((C_0)\) and reduced \((C_R)\) forms of the element at the electrode surface:

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

\[ C_0(y_0,t)-C_0^0 = \frac{1}{\pi^{1/2}D_0} \int_0^t q(\tau_s) \left\{ \frac{1}{\sqrt{t-\tau_s}} - \frac{a\pi^{1/2}D_0^{1/2}}{y_0} + \frac{2a^2\sqrt{t-\tau_s}-D_0}{y_0^2} - \frac{a^3(t-\tau_s)D_0^{3/2}}{\pi^{1/2}y_0^3} +\ldots \right\} d\tau_s \equiv Y_2, \tag{2} \]

where \(C_0^0\) and \(C_R^0\) are the initial concentrations, respectively, of the oxidized and reduced forms of the element; \(D_0\) and \(D_R\) are the diffusion coefficients of the oxidized and reduced forms of the element; \(y_0=r_0\) is the radius for a spherical electrode; \(y_0=l\) is the thickness of the mercury film for a film electrode; \(a=\gamma+\frac{1}{2}\); \(\gamma=-1/2\) for a film electrode;

\(\gamma = 1/2\)—for a spherical electrode; \(q(t)=\dfrac{i(t)}{zF}\)—the flux density at the electrode surface; \(\psi_n=\tg \psi_n\)—for a spherical electrode; \(\psi_n=n\pi\)—for a mercury film electrode; \(\tau_S\)—an auxiliary variable.

Let us consider AAV when the current at the electrode varies according to the law \(i_0(t)=\beta t^m\), where \(\beta\) is the proportionality coefficient; \(t\) is time, sec; \(i_0\) is the current density, A/cm\(^2\); \(m \ge 0\). In this case

\[ I_1=-\frac{\beta}{zF y_0}\left[ 2(\gamma_R+2)\frac{t^{m+1}}{m+1} +\frac{2t^m}{\gamma_R}\sum_{n=1}^{\infty}\psi_n^2 +2\psi(t) \right], \tag{3} \]

where

\[ \gamma_R=\frac{D_R}{y_0^2};\quad \varphi(t)=2\sum_{n=1}^{\infty}\sum_{k=1}^{m} b(-1)^k \frac{m(m-1)\cdots(m-k+1)} {(\gamma_R\psi_n^2)^{k+1}} t^{m-k}. \]

From the general expression (3) one can obtain a number of particular equations for different values of the exponent \(m\).

a) \(m=0\). For a film electrode, the expression for the surface concentration of the reduced form of the element will have the form:

\[ C_R(l,t)=C_R^0-\frac{\beta t}{zFl}-\frac{\beta l}{3zFD_R}. \tag{4} \]

Equation (4) was previously obtained in works [13, 17]. There, an equation for the transition time was also obtained. For a spherical electrode the analogous expression is written as follows:

\[ C_R(r_0,t)=C_R^0-\frac{3\beta t}{zFr_0}-\frac{\beta r_0}{5zFD_R}. \tag{5} \]

Equation (5) was obtained by another method and analyzed in work [15]. An expression for the transition time was also obtained there.

b) \(m=1\)—the current varies according to a linear law. In this case, for a film electrode, from equation (3), taking equation (1) into account, one obtains:

\[ C_R(l,t)=C_R^0-\frac{\beta}{zFl} \left[ \frac{t^2}{2} +2\frac{l^2 t}{D_R}\cdot\frac{\xi(2)}{\pi^2} -\frac{2l^4}{D_R^4}\frac{\xi(4)}{\pi^2} \right], \tag{6} \]

where \(\xi(p)\) is the Riemann zeta function [19, p. 368]. Since \(\xi(2)=\dfrac{\pi^2}{6}\), and \(\xi(4)=\dfrac{\pi^4}{90}\), then

\[ C_R(l,t)=C_R^0-\frac{\beta}{zFl} \left[ \frac{t^2}{2} +\frac{l^2t}{3D_R} -0.22\frac{l^4}{D_R^4} \right]. \tag{7} \]

The expression for the transition time is obtained under the condition \(C_R(l,\tau)=0\)

\[ C_R^0=\frac{\beta}{zFl} \left[ \frac{\tau^2}{2} +\frac{l^2\tau}{3D_R} -0.22\frac{l^4}{D_R^4} \right]. \tag{8} \]

For \(\dfrac{l^2}{D_R}\le 10^{-1}\ \text{sec}^{-1}\) and \(\tau \ge 6.7\ \text{sec}\), expression (8), with an error of less than 1%, can be reduced to the form:

\[ \tau^2=\frac{2zFlC_R^0}{\beta}. \tag{9} \]

By an analogous procedure, for a spherical electrode one can obtain the following relation for the transition time:

\[ C_R^0=\frac{\beta}{zFr_0}\left(1.5\tau^2+0.2\frac{r_0^2\tau}{D_R}-0.06\frac{r_0^4}{D_R^2}\right). \tag{10} \]

For a spherical electrode, \(\frac{r_0^2}{D_R}\gg 1\); therefore a simple dependence of the type (9) cannot be obtained here.

c) \(m=n\), where \(n>2\).

In these cases, expressions of degree \(n+1\) are obtained with respect to \(\tau\), which can be solved only graphically; therefore we do not present them here.

The electrooxidation of complex amalgams upon application of a constant current to the electrode was considered in [14, 16]. In [17] the electrooxidation of complex amalgams from a film electrode upon application to the electrode of a current varying according to the law \(e^{mt}\) was considered. Here we consider the electrooxidation of a complex amalgam at a linearly varying electrode current.

Following the method for treating the problem of the electrooxidation of complex amalgams proposed by Moore and Reilley [6], for a film electrode at \(\frac{l^2}{D_R}\ll 10^{-1}\ \mathrm{sec}^{-1}\) and \(\tau\gg 6.7\ \mathrm{sec}\) one can write the following relation:

\[ \left(\sum_{m=1}^{m}\tau_m\right)^2 = \frac{2Fl}{\beta}\sum_{m=1}^{m} z_m C_{R,m}^0 . \tag{11} \]

The expression for the transition time in the electrooxidation from a complex amalgam of the \(m\)-th component will have the form

\[ \left(\sum_{m=1}^{m}\tau_m\right)^2 - \left(\sum_{m=1}^{m-1}\tau_m\right)^2 = \frac{2Fl}{\beta}z_m C_{R,m}^0 . \tag{12} \]

For the element oxidized second, from formula (12) one can obtain the following relation:

\[ (\tau_1+\tau_2)^2-\tau_1^2 = \frac{2Fl}{\beta}z_2 C_{R,2}^0, \tag{13} \]

where \(\tau_1\) and \(\tau_2\) are the transition times for the oxidation of the first and second metals, sec; \(C_{R,2}^0\) is the concentration of the second metal in the amalgam, g-atom/ml.

In the electrooxidation of a complex amalgam from a spherical electrode, the expression for the transition time is written as follows:

\[ 1.5\left(\sum_{m=1}^{m}\tau_m\right)^2 + 0.2\frac{r_0^2}{D_R}\sum_{m=1}^{m}\tau_m = \frac{Fr_0}{\beta}\sum_{m=1}^{m}z_m C_{R,m}^0 - \frac{0.06r_0^4}{D_R}. \tag{14} \]

Here \(D_{R,m}=D_R\) is assumed. From equations (12–14) it is evident that, in AAV with a linearly varying current, the transition time of the oxidation process of an electropositive metal from a complex amalgam depends on the concentration through \(\tau_1\) of the more electronegative metal.

At a linearly varying electrode current, equation (2) for calculating the concentration of the oxidized form of the element at the electrode surface is reduced to the form:

\[ I'_2= \frac{\beta t^{3/2}}{zFD_0^{1/2}} \left\{ \frac{4}{3\pi^{1/2}} - \frac{1}{2}\left(\frac{D_0t}{r_0^2}\right)^{1/2} + \frac{8}{15\sqrt{\pi}}\left(\frac{D_0t}{r_0^2}\right) - \frac{1}{24}\left(\frac{D_0t}{r_0^2}\right)^{3/2} + \right. \]

\[ +\frac{16}{105\sqrt{\pi}}\left(\frac{D_0t}{r_0^2}\right)^2-\cdots\right\}. \tag{15} \]

Substituting into the Nernst equation, in place of \(C_R(r_0,t)\) and \(C_0(r_0,t)\), their values from equations (1–2), taking (10, 15) into account and expressing \(C_R^0\) from equation (11), we obtain the following dependence of the electrode potential on time for reversible processes at a linearly varying current of a spherical electrode:

\[ \varphi=\varphi_{1/2}-\frac{RT}{zF}\ln\frac{\beta}{zFr_0}\, \frac{1.5(\tau^2-t^2)+0.2\gamma_R(\tau-t)}{(C_0^0+I_2')}, \tag{16} \]

where \(\varphi_{1/2}\) is the half-wave potential of the reversible polarographic wave.

Having found from the chronopotentiogram the time at which the half-wave potential is reached, one can also calculate the concentration of ions in the solution, analogously to how this was done in [14, 16]:

\[ C_0^0= \frac{\beta}{zFr_0}\left[1.5(\tau^2-t_1^2)+0.2\gamma_R(\tau-t_1)\right] -\frac{\beta t_1^{3/2}}{zFD_0^{1/2}} \times \left[ \frac{4}{3\pi^{1/2}} -0.5\left(\frac{D_0t_1}{r_0^2}\right)^{1/2} +\frac{8}{15\sqrt{\pi}}\left(\frac{D_0t_1}{r_0^2}\right) -\frac{1}{24}\left(\frac{D_0t_1}{r_0^2}\right)^{3/2} +\cdots \right], \tag{17} \]

where \(t_1\) is the time at which the half-wave potential is reached.

For completely irreversible processes, the dependence of the electrode potential on time will have the form:

\[ \varphi=\frac{RT}{\beta zF}\ln\frac{r_0}{k_s} +\ln\frac{t}{1.5(\tau^2-t^2)+0.2\gamma_R(\tau-t)}. \tag{18} \]

REFERENCES

1. M. Sende. Rev. Polarog. (Japan), 4, 89 (1956).
2. W. H. Reinmuth. Anal. Chem., 32, 1509 (1960).
3. H. Hurwitz, L. Gierst, J. Electroanal. Chem., 2, 128 (1961).
4. R. C. Bowers, G. Ward, C. M. Wilson, D. D. De-Ford, J. Phys. Chem., 65, 672 (1961).
5. R. W. Murray, S. E.-S. W. Regional Am. Chem. Soc. Meeting, New Orleans, Dec., (1961).
6. R. W. Murray, C. N. Reilley, J. Electroanal. Chem., 3, 64 (1962).
7. T. Kambara, J. Tachi, J. Phys. Chem., 61, 1405 (1957).
8. I. Takemori, T. Kambara, M. Senda, I. Tecchi, J. Phys. Chem., 61, 968 (1957).
9. H. Hurwitz, J. Electroanal. Chem., 2, 142 (1961).
10. H. Hurwitz. Ibid., 2, 328 (1961).
11. R. W. Murray, C. N. Reilley. Ibid., 3, 182 (1962).
12. H. Hurwitz. Ibid., 7, 368 (1964).
13. M. S. Zakharov, V. I. Bakanov. Izv. vysshikh uchebnykh zavedenii, khimiya i khim. tekhnologiya (in press).
14. M. S. Zakharov, V. I. Bakanov. Zavod. labor. (in press).
15. M. S. Zakharov, V. V. Pnev. Izv. vysshikh uchebnykh zavedenii, khimiya i khim. tekhnologiya (in press).
16. M. S. Zakharov, V. V. Pnev. Zh. analit. khimii (in press).
17. M. S. Zakharov, V. I. Bakanov. Zh. obshchei khimii (in press).
18. M. S. Zakharov, V. V. Pnev. Elektrokhimiya (in press).
19. E. Jahnke, F. Emde. Tables of Functions with Formulae and Curves, Ogiz, Gostekhizdat, Moscow–Leningrad, 47, 1948.