SOME QUESTIONS IN THE THEORY OF REVERSIBLE ANODIC PEAKS AT A STATIONARY MERCURY FILM ELECTRODE. COMMUNICATION II

B. F. NAZAROV, V. A. NEMOV

(Presented by the scientific-methodological seminar of the Chemical-Technological Faculty)

At present there are no works in the literature in which, in deriving the equations of voltammetric curves for reversible anodic processes, the limited nature of stationary mercury electrodes (s.m.e.) is taken into account, and there is no quantitative estimate of the limits of applicability of the known approximate equations. In most cases, in solving problems of unsymmetrical diffusion, the boundary condition used is

\[ D_0^{1/2}\cdot C_0(l,t)+D_R^{1/2}\cdot C_R(l,t)=D_0^{1/2}\cdot C_0^0+D_R^{1/2}C_R^0, \tag{1} \]

which is strictly applicable only to processes of symmetrical diffusion [1—4]. In work [5] we obtained a boundary condition strictly valid for unsymmetrical diffusion,

\[ D_0^{1/2}\cdot C_0(l,t)=D_R^{1/2}\cdot C_R(l,t)+D_0^{1/2}\cdot C_0^0+D_R^{1/2}C_R^0-\frac{2D_R^{1/2}(\theta C_R^0-C_0^0)}{\theta+\xi}\times \]

\[ \times \sum_{n=1}^{\infty} K^{n-1}\cdot \operatorname{erfc}\frac{nl}{\sqrt{D_R t}} . \tag{2} \]

The purpose of the present work was, using the new boundary condition (2), to obtain equations of voltammetric curves for a mercury film electrode (m.f.e.) at constant and linearly varying potential, if reversible processes occur at the electrode. The expression for the current at constant potential in the case of unsymmetrical diffusion is obtained if, according to [5], the concentrations \(C_0(l,t)\) and \(C_R(l,t)\) are found at the electrode surface, and the flux

\[ q_{0(R)}(l,t)=D_{0(R)}\cdot \left.\frac{\partial C_{0(R)}(x,t)}{\partial x}\right|_{x=l}. \]

Having carried out the indicated transformations, we obtain

\[ i=z\cdot F\cdot S\cdot \frac{(C_0^0-\theta C_R^0)}{\theta+\xi}\cdot \sqrt{\frac{D_R}{\pi t}} \left[ \sum_{n=1}^{\infty}K^{n-1}\cdot e^{-\frac{(n-1)^2l^2}{D_R t}} - \sum_{n=1}^{\infty}K^{n-1}\cdot e^{-\frac{n^2l^2}{D_R t}} \right]. \tag{3} \]

Expression (3) is the general equation for the anodic or cathodic current at any constant potential of the m.f.e. If the limitation is neglected (which is valid for \(l\to\infty\)), then from the equa-

...of (3), the general equation of the anodic–cathodic current at a constant potential is obtained for symmetric semi-infinite diffusion, the particular case of which, for \(C_O^0 \gg C_R^0\), was considered in detail by Delahay [6].

Indeed, for very large values of \(l\), all terms in the second sum will be zero for any \(n\), beginning with \(n=1\). In the first sum, all terms except the first, equal to 1, will be zero.

From equation (3), the expression for the current at any constant potential in the case of an anodic process is obtained for \(C_R^0 \gg C_O^0\), and in the case of a cathodic process for \(C_O^0 \gg C_R^0\). The concentration ratios given correspond to real experimental conditions. It should be noted that the expression obtained, (3), differs from that proposed earlier in [7] in that equation (3) is valid for a constant, but arbitrary, electrode potential, not one corresponding to the limiting current. In addition, [7] does not take account of diffusion asymmetry.

To find the equation of the voltammetric curve for a linear change in the potential of the stationary mercury electrode, the solution of the nonstationary diffusion equation was found

\[ \frac{\partial C_R(x,t)}{\partial t} = D_R \frac{\partial^2 C_R(x,t)}{\partial x^2} \tag{4} \]

for the initial \((t=0)\)

\[ C_R(x,0)=C_R^0 \tag{5} \]

and boundary \((t>0)\)

\[ \text{a) }\; D_R \left.\frac{\partial C_R(x,t)}{\partial x}\right|_{x=0}=0; \qquad \text{b) }\; C_R(x,t)\big|_{x=l}=f(t) \tag{6} \]

conditions. A reversible electrode process was considered: \(O+ze \rightleftarrows R\) (7), without kinetic complications in the solution and in the amalgam.

The initial condition (5) means that before the dissolution process the concentration of the element being determined in the amalgam is constant and equal to \(C_R^0\). Condition (6,a) means impermeability to diffusion of the stationary mercury electrode backing [8], while condition (6,b) shows that the concentration of substance \(R\) at the electrode–solution boundary is a function of time. The solution of this problem was carried out by the operational method [9]. Omitting the intermediate calculations and taking into account that \(i=zF \cdot S \cdot q(l,t)\), the solution of the problem may be written immediately in the form of an expression for the current:

a) in a form more convenient for thin films:

\[ i = 2 \cdot z \cdot F \cdot D_R \cdot \frac{S}{l} \int_0^t f'(\tau)\cdot \sum_{n=1}^{\infty} e^{-\mu_n^2 \frac{D_R}{l^2}(t-\tau)} \cdot d\tau, \quad \text{where }\; \mu_n=(2n-1)\frac{\pi}{2}; \tag{8} \]

b) in a form more convenient for thick films:

\[ i = z \cdot F \cdot S \cdot \sqrt{D_R}\, \frac{1}{\sqrt{\pi}} \int_0^t f'(\tau)\cdot \frac{d\tau}{\sqrt{t-\tau}}. \tag{9} \]

The notation in equations (8) and (9) was given earlier. As for the function \(f'(\tau)\), it is the derivative with respect to time of the concentration of substance \(R\) at the electrode surface. It is found from boundary condition (2), and its form depends on the law of change of the electrode potential. In solving this problem, the Doetsch theorem [10] was used, since boundary condition (2) was derived for an arbitrary but constant electrode potential. The Doetsch theorem makes it possible to generalize the solution of the boundary diffusion problem at a constant po-

potential into the solution for any form of potential change. In our case, the expression \(f'(\tau)\) for the anodic process is written in the form

\[ \begin{aligned} f'(t)={}&-\frac{\sigma}{2}\cdot \frac{C_R^0}{2\operatorname{ch}^2\frac{\sigma}{2}(t-t_{1/2})} -\frac{\sigma}{2}\cdot\frac{4\theta' C_R^0}{(1+\theta')^2} \cdot\sum_{n=1}^{\infty}K_1^{\,n-1}\cdot \operatorname{erfc}\lambda +\\ &+\frac{\sigma}{2\operatorname{ch}^2\frac{\sigma}{2}(t-t_{1/2})} \cdot\frac{2\theta' C_R^0}{1+\theta'} \cdot\sum_{n=1}^{\infty}K_1^{\,n-1}\operatorname{erfc}\lambda +\frac{\sigma}{2\operatorname{ch}^2\frac{\sigma}{2}(t-t_{1/2})} \cdot\frac{2\theta' C_R^0}{(1+\theta')^2}\times\\ &\times\sum_{n=2}^{\infty}K_1^{\,n-2}\cdot\operatorname{erfc}\lambda -\frac{\lambda}{\pi^{1/2} t^{3/2}}\cdot \frac{2\theta' C_R^0}{(1+\theta')^2} \cdot\sum_{n=1}^{\infty}K_1^{\,n-1}\cdot\lambda\cdot e^{-\lambda^2}, \end{aligned} \tag{10} \]

where

\[ \theta'=e^{\frac{zF}{RT}w(t-t_{1/2})} =e^{\sigma(t-t_{1/2})} =e^{(zF/RT)(\varphi-\varphi_0)}\cdot\xi =\theta\cdot\xi; \]

\(w\) is the rate of change of the potential in V/sec;

\(t_{1/2}\) is the time at which the half-wave potential is reached;

\[ K_1=\frac{1-\theta'}{1+\theta'}=\operatorname{th}\frac{\sigma}{2}(t-t_{1/2}); \qquad \lambda=\frac{nl}{\sqrt{D_R t}}; \qquad \xi=\sqrt{\frac{D_0}{D_R}}, \tag{11} \]

the remaining notation is known.

Expressions (8) and (9), together with (10), constitute the equations of the voltammetric curves under a linear change in the electrode potential, when the diffusion process is asymmetric and the electrode process is reversible.

If the finiteness of the mercury film is neglected, i.e., if one assumes \(l\to\infty\), then from equations (9) and (10) we obtain Shevchik’s equation for the anodic process. Since \(C_R^0=K\cdot C_0^0\cdot t_{\mathrm{el}}\), where \(K\) is the electrolysis constant, \(C_0^0\) is the concentration of ions in the solution, and \(t_{\mathrm{el}}\) is the electrolysis time, it may be noted that the height of the anodic peak depends on the concentration of ions in the solution, the time of preliminary electrolysis, the intensity of stirring, etc.

It should also be noted that expressions (8) and (9), with allowance for (10), are valid both for small rates of potential change and for large ones. In addition, if in the course of the process we stop changing the potential—for example, in order to separate the peaks of two elements having close peak-potential values, i.e., set \(w=0\)—then from (8) and (10) we obtain an expression for the descending branch of the anodic peak at constant potential.

An analogous expression is obtained if \(f'(\tau)\) is found from (2), assuming \(\theta=\mathrm{const}\).

Conclusions

1. An equation for the current at constant potential under conditions of asymmetric diffusion has been obtained, valid for anodic and cathodic processes. It is shown that this equation differs from those known in the literature.

2. An equation for the voltammetric curve under a linear change of potential for asymmetric diffusion has been obtained. The validity of this equation at any rate of potential change and for any film thickness has been shown.

References

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