THEORY OF REVERSIBLE ANODIC PEAKS AT A STATIONARY MERCURY DROP ELECTRODE

V. E. GORODOVYKH

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

In the literature there are a number of equations for the current at a changing (usually linearly with time) electrode potential—the equations of Shchevchik–Randles, Delahay, Nicholson, and others [1]. We have considered the most general case of the current at an electrode of arbitrary shape with an arbitrary initial distribution of the depolarizer, any type of electrode process, and any form of the applied potential. The general equation of the voltammetric curve was obtained by us in a rigorous way by means of operational calculus and has the following form:

\[ i=\int_0^t f'_\varphi(\tau)\, i_{\mathrm{pr}}(t-\tau)\, d\tau, \tag{1} \]

where \(t\) is time; \(\varphi\) is potential; \(\tau\) is the variable of integration.

The function \(f(t)\) characterizes the change in the concentration field of the depolarizer near the electrode surface during the process. The factor \(i_{\mathrm{pr}}\), entering into the integrand of formula (1), in its physical meaning represents the nonstationary limiting diffusion current at constant electrode potential and zero boundary concentration. Exact expressions for the limiting current at constant potential are usually available in the literature, and thus the main difficulty in using equation (1) consists in determining the form of the function \(f(t)\).

Starting from the general equation of the voltammetric curve (1), we have further obtained, in explicit form, the equation of the anodic peak at a stationary mercury drop electrode for a linearly changing potential and under the assumption of reversibility of the electrode process. In the most general form, the equation of the reversible anodic peak can be written as follows:

\[ i=-\frac{z^{3/2}F^{3/2}}{R^{1/2}T^{1/2}}\, w^{1/2}D^{1/2}SC^0\psi(\varepsilon), \tag{2} \]

where \(w\) is the rate of change of the potential; \(D\) is the diffusion coefficient; \(S\) is the electrode area; \(C^0\) is the initial concentration in the amalgam.

The function \(\psi(\varepsilon)\), depending on the parameter

\[ \varepsilon=\frac{1}{r_0}\sqrt{\frac{D}{zw}} \]

(\(r_0\) is the radius of the electrode), in the general case has a rather cumbersome form, and therefore we give here only its graph (Fig. 1). The combination of this graph with equation (2) makes it possible to calculate directly—

to calculate the height of the anodic peak under specified experimental conditions. At small values of the parameter \(\varepsilon\) (\(\varepsilon < 0.2\)), the function \(\psi(\varepsilon)\) is related to the parameter \(\varepsilon\) by the following relation:

\[ \psi(\varepsilon)=P-\sqrt{\frac{RT}{F}}\,\varepsilon \Phi, \tag{3} \]

where \(P\) is the Ševčík function [1];

\[ \Phi=\frac{1}{2}\left[1+\operatorname{th}\frac{zf}{2RT}\left(\varphi-\varphi_{1/2}\right)\right] \]

is a dimensionless function; \(\varphi_{1/2}\) is the potential of the reversible polarographic half-wave at the stationary mercury drop electrode.

The equations obtained make it possible to draw a number of important conclusions concerning the influence of certain factors on the height and shape of anodic peaks. From (2) it follows, first of all, that the peak height is directly proportional to the concentration of metal ions in solution and to the time of electrolytic concentration (in the absence of depletion of the solution and of a number of other side phenomena).

The height and shape of the peak further depend on the magnitude of the parameter \(\varepsilon\). With a decrease in this parameter, i.e., with an increase in the electrode radius and in the rate of change of the potential, the anodic peak, in its character, approaches more and more the peak at a plane electrode, which is described by the well-known Ševčík equation. In the limit, as \(\varepsilon \to 0\) (\(r_0 \to \infty\) or \(\omega \to \infty\)), equation (2) becomes identical with the Ševčík equation (Fig. 1).

Fig. 1. Dependence of the function \(\psi(\varepsilon)\) and of the potentials of cathodic and anodic peaks at the stationary mercury drop electrode on the parameter \(\varepsilon\):

1, 3 — plots of the functions \(\dfrac{\psi_p}{\psi_{\mathrm{Sh}}}\) for cathodic and anodic peaks, respectively;
2, 4 — dependence of the potentials of the cathodic and anodic peaks on the parameter \(\varepsilon\).

Fig. 2. Dependence of the resolving power for cathodic (curve 1) and anodic (curves 2 and 3) peaks on the value of \(\Delta\varphi\) and the parameter \(\varepsilon\). Curve 4 is the resolving power for cathodic peaks at a plane electrode.

It is also seen from equation (3) and Fig. 1 that the constant of the anodic peak at the stationary mercury drop electrode is always smaller than the constant of the cathodic peak at a plane electrode and approaches the latter as the parameter \(\varepsilon\) decreases. This circumstance accounts for the higher resolving power of the anodic-peak method in comparison with the resolving power of cathodic oscillographic polarography (Fig. 2).

Since the shape of the peak is determined only by the form of the function $\phi(\varepsilon)$ (equation (2)), the potential of a reversible peak does not depend on the initial concentration of the metal in the electrode (provided the activity coefficients and the diffusion coefficients of the metal ions and atoms remain unchanged), and consequently neither on the concentration of metal ions in the solution nor on the accumulation time. It has been found theoretically that the potential of a reversible anodic peak at a stationary mercury drop electrode lies more positive than the potential of the reversible polarographic half-wave of the given element in the given supporting electrolyte, and, as the parameter $\varepsilon$ increases, shifts in the negative direction, approaching the half-wave potential. Conversely, when the parameter $\varepsilon$ decreases, the reversible peak potential shifts in the positive direction, and in the limit, at $\varepsilon = 0$, it coincides with the peak potential at a planar electrode (Fig. 1). Thus, from the foregoing there follows a very important conclusion: the potential of a reversible anodic peak at a stationary mercury drop electrode, unlike the potential of a cathodic peak at a planar electrode and the potential of the polarographic half-wave at a mercury drop electrode, is not a constant quantity, but depends on the parameter $\varepsilon$, i.e., on the experimental conditions.

Taking all the foregoing into account, it should be considered that the parameter $\varepsilon$ is a very important quantity characterizing the height and shape of the peak, and therefore it should necessarily be reported in studies on amalgam polarography. It should be noted that this parameter has general significance in the quantitative description of reversible processes occurring at electrodes of different types. Thus, the height and shape of cathodic peaks at spherical and cylindrical electrodes depend on its magnitude [2–3]. It is interesting to note that the magnitude of $\varepsilon$ has a directly opposite effect on the height and potential of anodic and cathodic peaks at a spherical electrode (Fig. 1).

A large number of different factors influence the height and shape of anodic peaks. A complete quantitative theory of the influence of factors on anodic peaks does not yet exist. At present there is also no classification of these factors. One possible variant of such a classification, in our opinion, could be the grouping of factors in accordance with the two stages of the method of anodic stripping polarography:
a) the stage of electrolytic concentration of the metal from the solution;
b) the stage of electrooxidation of the amalgam upon application to the electrode of a linearly varying potential.

We have considered in greater detail the effect of the rate of change of potential on the height and shape of reversible peaks. On the basis of an analysis of equations (2) and (3), it is shown that the height of the anodic peak at a stationary mercury drop electrode depends on the rate of change of potential in a more complicated manner than does the height of the cathodic peak at a planar electrode. Moreover, the dependence of the peak height on the rate of change of potential must always be stronger than a square-root dependence. As the parameter $\varepsilon$ increases, this dependence becomes stronger and, in the limit as $\varepsilon \to \infty$ ($r_0 \to 0$ or $w \to 0$), it tends to a linear one. Conversely, when the parameter $\varepsilon$ decreases, the dependence $I$--$w$ gradually approaches the square-root dependence (according to Shevchik). Physically, all these conclusions of the theory can be explained, in our opinion, by the effect of compression of the diffusion field inside the stationary mercury drop electrode as the diffusion front advances into the electrode.

When the rate of change of potential is decreased, the spherical character of the stationary mercury drop electrode acquires ever greater importance; the peak potential shifts in the negative direction and, as a consequence, the steepness of the rising (kinetic) branch of the peak increases. At the same time the steepness of the descending branch of the peak also increases. All this should lead to a decrease in the width of the anodic half-peak and to an increase in the resolu-

1. It has been shown that the anodic-current constant for a planar electrode is always higher than that for a spherical electrode at any values of the parameter $\varepsilon$.

References

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5. V. E. Gorodovykh, A. G. Stromberg, B. F. Nazarov. Dependence of the anodic-peak constant in the APN method on the radius of the mercury drop and the rate of change of potential. Present collection.
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