Effect of the Rate of Change of Potential on the Depth and Shape of Reversible Anodic Peaks at a Stationary Mercury Drop Electrode

V. E. Gorodovykh

(Presented by the Scientific-Methodological Seminar of the Faculty of Chemical Technology)

The depth and shape of anodic peaks are affected by a large number of different factors. A complete quantitative theory of the influence of factors on anodic peaks does not yet exist. At present there is likewise 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 anodic stripping voltammetry method: a) the stage of electrolytic concentration of the metal from solution and b) the stage of electrooxidation of the concentrated metal when a linearly varying potential is applied to the electrode.

The first group of factors includes the concentration of metal ions in solution, the accumulation potential, the concentration time, the volume of the solution, the dimensions and shape of the stationary electrode, the diffusion coefficient and charge of the metal ions in the given electrolyte, and, finally, the hydrodynamic conditions under which accumulation is carried out. This group of factors affects mainly the peak depth, since the concentration of metal in the amalgam depends on them, and does not affect the peak shape¹.

At the second stage, the following factors affect the anodic peaks: the concentration of metal in the amalgam, the rate of change of potential, the degree of irreversibility of the electrode process, the dimensions and shape of the electrode, the diffusion coefficient of the metal in mercury, the number of electrons participating in the electrochemical reaction, and the hydrodynamic conditions under which electrooxidation is carried out. This group of factors affects both the depth and the shape of the peaks. It should be noted that the dimensions and shape of the electrode enter into both groups of factors and, as a consequence, their influence on the anodic peak is especially complex² [1, 2].

If the influence of such factors as the concentration of metal ions and atoms is fairly obvious, the influence of most other factors is often difficult to predict without carrying out a special investigation. In the present work the influence of the rate of change of potential on the depth and shape of reversible anodic peaks is examined in more detail, since this question has practically not been covered in the literature.

Earlier [3] we showed that the depth of a reversible peak is equal to

\[ I = K w^{1/2}\psi(\varepsilon), \tag{1} \]

where \(I\) is the peak depth; \(K\) is a coefficient; \(w\) is the rate of change of the potential; \(\psi(\varepsilon)\) is a certain function depending on the parameter

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

The presence of a functional dependence of the function \(\psi\) on the parameter \(\varepsilon\) leads to the fact that anodic peaks at a stationary mercury drop electrode do not obey the known regularity given by the Ševčík equation [4]. In the case considered by us, a strictly square-root dependence of the peak depth on the rate of change of the potential should not be observed. Indeed, if \(I_1\) is the peak depth obtained at a rate of change of the potential \(w_1\), and \(I_2\) is the peak depth obtained at another rate \(w_2\), then

\[ \frac{I_2}{I_1} = \frac{w_2^{1/2}}{w_1^{1/2}} \frac{\psi(\varepsilon_2)}{\psi(\varepsilon_1)} = \kappa \frac{w_2^{1/2}}{w_1^{1/2}}, \tag{2} \]

where

\[ \kappa = \psi(\varepsilon_2)/\psi(\varepsilon_1). \]

Taking into account that, when the parameter \(\varepsilon\) decreases, the value of the function \(\psi\) increases [3, 5], we find that \(\kappa > 1\) when \(\varepsilon_2 < \varepsilon_1\), or, in particular, when \(w_2 > w_1\). Figure 1 shows a theoretical curve in the coordinates \(\lg I\)—\(\lg w\), constructed for small values of the parameter \(\varepsilon\) (\(\varepsilon < 0.2\)). From relation (2) and Fig. 1 the following conclusions may be drawn:

1. The depth of the anodic peak at a stationary mercury drop electrode depends on the rate of change of the potential in a more complex way than does the depth of the cathodic peak at a plane electrode. In this case the dependence of the anodic peak depth on the rate of change of the potential must always be stronger than the square-root dependence.

2. The dependence of the anodic peak depth on the rate of change of the potential gradually becomes stronger with increasing parameter \(\varepsilon\), i.e., with decreasing rate of change of the potential and/or radius of the stationary mercury drop electrode. It can be shown [3] that, as \(\varepsilon\) tends to infinity, the dependence of the peak depth on the rate of change of the potential should tend to a linear one. Conversely, as the parameter \(\varepsilon\) decreases, this dependence gradually approaches a square-root dependence, and in the limit at \(\varepsilon = 0\) a purely square-root dependence of the peak depth on the rate of change of the potential should be observed.

Fig. 1. Theoretical dependence of the depth of the anodic peak at a stationary mercury drop electrode on the rate of change of the potential.

Physically, these conclusions of the theory may, in our opinion, be explained as follows. As the rate of change of the potential increases, the concentration gradient near the electrode surface also increases, since, because of the slowness of the diffusion rate, the decrease in the ve-

the substance in the near-electrode space does not have time to be compensated by its supply to the electrode surface. Consequently, the reason for the increase in peak depth with increasing rate of potential change is the finite diffusion rate of the depolarizer.

The specific form of the dependence of peak depth on the rate of potential change is determined by the diffusion parameters of the system (semi-infinite or finite diffusion, the size and shape of the electrode, etc.). Thus, in the case of linear semi-infinite diffusion the following relation holds:

\[ \frac{K'}{w^{1/2}}=\mathrm{const}, \tag{3} \]

where \(K'\) is the constant of the cathodic or anodic peak obtained under conditions of linear semi-infinite diffusion, determined by the equation \(I_i=K_i' C_i\).

In the case of spherical diffusion, relation (3), as was shown above, is not fulfilled, and

\[ \frac{K''}{w^{1/2}}=f(w)\ne \mathrm{const}, \tag{4} \]

where \(K''\) is the constant of the cathodic peak at a stationary dropping mercury electrode, or the constant of the anodic peak obtained upon electro-dissolution of the metal from a stationary dropping mercury electrode.

Physically, relation (4) reflects the fact of expansion (spherical semi-infinite diffusion) or compression (finite spherical diffusion) of the diffusion field as the diffusion front moves into the bulk of the solution or into the stationary drop electrode. In this case, the lower the rate of potential change, the farther the diffusion front has time to move away, and the more strongly the effects of compression or expansion of the diffusion field will be manifested. Under the conditions of the AIP method, the effect of compression of the diffusion field leads to a more rapid equalization of the metal concentration inside the electrode because of depletion of the amalgam and, as a consequence, to a decrease in the concentration gradient and hence also of the current. This circumstance causes a stronger dependence of peak depth on the rate of potential change than in the case of linear semi-infinite diffusion.

Furthermore, since the compression effect depends on the distance of the diffusion front from the electrode surface, which in turn depends on time, there must obviously exist a certain variable dependence of the peak current on the rate of potential change. This dependence must evidently approach the square-root dependence as the rate of potential change increases, owing to the decrease in the compression effect of the diffusion field and to the approximation of the experimental conditions to those of linear semi-infinite diffusion.

Let us now consider qualitatively the influence of the rate of potential change on the shape of the anodic peak at a stationary dropping mercury electrode. We previously showed [3] that, when the rate of potential change is decreased, the potential of the anodic peak at a stationary dropping mercury electrode shifts in the negative direction and, as a consequence, the steepness of the ascending branch of the peak increases. At the same time, from the equation for the descending branch of the peak [3],

\[ i=\frac{2zFSDC}{r}\,e^{-\frac{\pi^2D}{r^2}\frac{(\varphi-\varphi^*)}{w}} \tag{5} \]

it follows that, when the rate of potential change is decreased, the steepness of the descending branch also increases. All this should lead

to a decrease in the half-width of the anodic peak and to an increase in the resolving power of the ASP method when the rate of potential change is decreased.

The combined influence of curvature and of the limited volume of the stationary mercury drop electrode, together with the smallness of the working rates of potential change usually used in ASP practice, leads to a narrowing of the anodic peaks in comparison with cathodic peaks obtained at high rates of potential change (on the order of several V/sec). Thus, oscillographic polarography¹ is inferior to the ASP method both in sensitivity (because of the absence of accumulation) and in resolving power (because of the high rates of potential change).

All these conclusions of the theory have previously been verified by us experimentally using reversible peaks of lead, cadmium, and thallium as examples [3].

Conclusions

1. The influence of the rate of potential change on the depth and shape of reversible anodic peaks has been considered theoretically. It is shown that, under the conditions of amalgam polarography with accumulation at a stationary mercury drop electrode, the dependence of peak depth on the rate of potential change must always be stronger than a square-root dependence. When the rate of potential change is decreased, this dependence must approach a linear one. It has been shown qualitatively that, when the rate of potential change is decreased, the half-width decreases and the resolving power of the ASP method increases.

2. The influence of the effect of compression of the diffusion field on the nature of the dependence of the depth of the anodic peak at a stationary mercury drop electrode on the rate of potential change has been considered.

REFERENCES

1. A. G. Stromberg, Izv. SO AN SSSR, 5, 76, 1962.
2. A. G. Stromberg, Zavod. labor., 29, 387, 1963.
3. V. E. Gorodovykh. Candidate dissertation, Tomsk, 1964.
4. O. Sevčik, Coll. Czech. Chem. Commun., 13, 349, 1948.
5. V. E. Gorodovykh, A. G. Stromberg, B. F. Nazarov, Izv. TPI, (in press).

1. What is meant is the polarographic determination of impurities directly from dilute solutions by their cathodic peaks and using high rates of potential change. ↩↩

2. Here we consider “pure peaks” and do not take into account such complicating phenomena as the mutual influence of metals in complex amalgams, the formation of insoluble films on the electrode surface, the formation of supersaturated amalgams, etc. ↩