DEPENDENCE OF THE ANODIC PEAK CONSTANT ON THE RADIUS OF THE MERCURY DROP AND THE RATE OF POTENTIAL CHANGE. I. THEORY

V. E. GORODOVYKH, A. G. STROMBERG, B. F. NAZAROV

(Submitted by the scientific-methodological seminar of the Department of Chemical Technology)

In work [1] a phenomenological theory of the A.P.N. method was developed, according to which the expression for the height of the anodic peak may be written as follows:

\[ I = KSC, \tag{1} \]

where \(I\) is the peak height; \(S\) is the electrode surface; \(C\) is the initial concentration of the metal in mercury; \(K\) is the anodic peak constant.

It was assumed that the anodic peak constant is some function of the rate of potential change and does not depend on the radius of the stationary mercury drop electrode. Such an assumption is approximately justified when working with electrodes of large radius and at sufficiently high rates of potential change [2]. Under these conditions, during the recording of the anodic peak the diffusion front moves away from the electrode surface by a distance small in comparison with the radius of the stationary mercury drop electrode, and thus the curvature and boundedness of the electrode volume have practically no effect on the character of the diffusion process. In other words, the use of electrodes of large radius and high rates of potential change leads to the diffusion process inside the stationary mercury drop electrode being, in character, close to linear semi-infinite diffusion, and the height of the anodic peak may be approximately calculated from the Ševčík–Randles equation:

\[ I = K_0SC, \tag{2} \]

where

\[ K_0 = \frac{z^{3/2} F^{3/2}}{R^{1/2} T^{1/2}} \, w^{1/2} D^{1/2} P, \tag{3} \]

\(z\) is the number of electrons participating in the electrode process;
\(P = 0.446\) is the maximum value of the Ševčík function [3, 4].

At \(25^\circ\text{C}\) the Ševčík–Randles equation takes the form

\[ I = 2.63 \cdot 10^{+5} z^{3/2} w^{1/2} D^{1/2} SC. \tag{4} \]

From equations (2)—(4) it follows that under the above-mentioned conditions (large radius, high rate of potential change) the anodic peak constant does not depend on the electrode radius and is directly proportional to the square root of the rate of potential change\(^1\).

Upon transition to small radii and low rates of potential change, the conditions of diffusion inside the stationary mercury drop electrode will differ significantly from the conditions of linear semi-infinite diffusion, which leads

\(^1\) It should be noted that even at large \(r\) and \(w\) the peak height on the stationary mercury drop electrode is described by the Ševčík–Randles equation only approximately. This is discussed in more detail below.

to the need to take into account the curvature and finiteness of the volume of the mercury dropping electrode (5, 6). For these conditions the equation of the reversible anodic peak, obtained in [6], can be written as follows:

\[ I = K_0 S C \psi(\varepsilon) \tag{5} \]

or

\[ I = KSC, \tag{6} \]

where \(K = K_0 \cdot \psi(\varepsilon)\); \(\psi(\varepsilon)\) is a certain function of the parameter

\[ \varepsilon = \frac{1}{r}\sqrt{\frac{D}{z\omega}} \]

(\(r\) is the radius of the stationary mercury dropping electrode).

The function \(\psi(\varepsilon)\) has a rather complicated form and cannot be expressed through elementary functions. The results of a numerical calculation of the function \(\psi(\varepsilon)\) are given in Table 1. As is seen from this table, when the parameter \(\varepsilon\) decreases, which corresponds to an increase in the radius of the stationary mercury dropping electrode and in the rate of potential change, the value of the function \(\psi(\varepsilon)\) approaches the value of the Ševčík–Randles function (0.446). Conversely, when the parameter \(\varepsilon\) increases, i.e., when the radius of the stationary mercury dropping electrode and the rate of potential change decrease, the function \(\psi(\varepsilon)\) tends to zero.

Let us denote

\[ \xi(\varepsilon)=\frac{K_0}{K}=\frac{P}{\psi(\varepsilon)} . \tag{7} \]

It follows from the above that the quantity \(\xi(\varepsilon)\) must satisfy the relations

\[ \lim_{\varepsilon \to 0} \xi \to 1 \quad \text{and} \quad \lim_{\varepsilon \to \infty} \xi \to \infty . \tag{8} \]

From these relations it is seen that the value of the anodic-peak constant at the stationary mercury dropping electrode depends on the relation between the electrode radius and the rate of potential change: at small values of the parameter \(\varepsilon\) the anodic-peak constant approaches in magnitude the constant of the Ševčík–Randles equation, while at large \(\varepsilon\) it tends to zero.

Table 1

Dependence of the functions \(\psi\) and \(\xi\) on the parameter \(\varepsilon\).

The results of the numerical calculation of the functions \(\psi(\varepsilon)\) and \(\xi(\varepsilon)\) (Table 1) are conveniently represented in the form of the following approximate formulas:

\[ \lg \xi = A \varepsilon^m, \tag{9} \]

\[ \xi - 1 = B \varepsilon^n, \tag{10} \]

where \(A, B, m, n\) are constants. It is easy to see that formulas (9) and (10) satisfy the relations (8). It is expedient to determine the constants \(A, B, m, n\) graphically. For this purpose we write relations (9) and (10) as follows:

\[ \lg \frac{1}{\lg \xi} = \lg \frac{1}{A} + m \lg \frac{1}{\varepsilon}, \tag{11} \]

\[ \lg \frac{1}{\xi - 1} = \lg \frac{1}{B} + n \lg \frac{1}{\varepsilon}. \tag{12} \]

If formulas (11) and (12) correctly describe the results of the calculations of the function \(\xi(\varepsilon)\), then the curves in the coordinates \(\lg \dfrac{1}{\lg \xi} — \lg \dfrac{1}{\varepsilon}\) and \(\lg \dfrac{1}{\xi-1} — \lg \dfrac{1}{\varepsilon}\) must be straight lines with slopes equal to \(m\) and \(n\). The results of the graphical solution of equations (11) and (12) are shown in Fig. 1. The scatter of the points in the plots is apparently due to the unequal degree of accuracy of the numerical calculations of the function \(\xi(\varepsilon)\). On the basis of the values of the constants \(A, B, m, n\) found from Fig. 1, equations (9) and (10) take the form

\[ \lg \xi = 0.29 \varepsilon, \tag{13} \]

\[ \xi - 1 = 1.25 \varepsilon^2, \tag{14} \]

Equations (13)—(14) may be used for approximate calculation of the functions \(\xi\) and \(\psi\) for a given value of the parameter.

Let us estimate, using formulas (13) and (14), the quantities \(\xi\) and \(K\) under conditions of practical operation by the APN method with a stationary mercury drop electrode. Taking \(r = 0.04\ \text{cm}\), \(w = 7 \cdot 10^{-3}\ \text{V/sec}\), \(D = 1.5 \cdot 10^{-5}\ \text{cm}^2/\text{sec}\), \(z = 2\), we obtain \(\varepsilon = 0.82\), \(\xi = 1.77\), and \(K = 0.56K_0\).

With a change of the value of \(\varepsilon\) by a factor of 1.41 (which corresponds to a twofold change of \(w\)), we obtain \(\varepsilon_1 = 0.58\) and \(\varepsilon_2 = 1.16\), and, respectively, \(\xi_1 = 1.48\) and \(\xi_2 = 2.17\), or \(K_1 = 0.67K_0\) and \(K_2 = 0.46K_0\).

Fig. 1. Dependence of the function \(\zeta\) on the parameter \(\varepsilon\): \(a\)—curve calculated according to equation (11); \(b\)—curve calculated according to equation (12).

Thus, when the radius of the stationary mercury drop electrode is increased by a factor of 2 (from \(0.0283\) to \(0.0564\ \text{cm}\)) or when the rate of potential change is increased by a factor of two (from \(3.5 \cdot 10^{-3}\) to \(14 \cdot 10^{-3}\ \text{V/sec}\)), the anodic peak constant also increases from \(0.46K_0\) to \(0.67K_0\), i.e., almost one and a half times. Consequently, the assumption that the anodic peak constant under the conditions of the APN method is practically independent of the radius of the mercury drop, which was made in earlier works [1,2], is approximate. It also follows from the numerical estimates given above and from Table 1 that, under the conditions of the APN method, the anodic peak constant may differ substantially from the value of the constant

\[ K = 2.68\ 10^{+5} z^{3/2} w^{1/2} D^{1/2}, \tag{15} \]

which enters the Ševčík–Randles equation for linear semi-infinite diffusion. Therefore, attempts by some authors [7] to use the Ševčík–Randles equation for calculating the diffusion coefficients of metals in mercury from the heights of anodic peaks obtained on a stationary mercury drop electrode under the ordinary conditions of the APN method cannot be regarded as correct.

APPENDIX

Numerical calculation of the function

The equation of the reversible anodic peak at a stationary mercury drop electrode under a linearly changing potential has the following form (6, 8):

\[ i=\frac{z^{2}F^{2}}{2rRT}\,D\omega SC \int_{0}^{t} \frac{ \sum_{n=1}^{\infty} e^{-\frac{(n\pi)^{2}D}{r^{2}}(t-\tau)} }{ \operatorname{ch}^{2}\frac{zF\omega}{2RT}(t_{1/2}-\tau) }\,d\tau, \tag{16} \]

where \(t_{1/2}\) is the time at which the half-wave potential of the reversible polarographic wave at the mercury drop electrode is reached.

Transform equation (16) to the form

\[ i=\frac{z^{3/2}F^{3/2}}{R^{1/2}T^{1/2}}\,D^{1/2}\omega^{1/2}SC\psi, \tag{17} \]

where

\[ \psi=\frac{\gamma^{1/2}}{2\pi} \int_{0}^{\beta t} \frac{ \sum_{n=1}^{\infty} e^{-(n)^{2}\gamma(\beta t-\beta\tau)} }{ \operatorname{ch}^{2}\frac{1}{2}(\beta t_{1/2}-\beta\tau) }\,d(\beta\tau); \]

\[ \gamma=\frac{\pi^{2}RTD}{zFr^{2}\omega}\ \text{— parameter}; \qquad \beta=\frac{zF\omega}{RT}. \]

The parameter \(\gamma\) is related to the parameter \(\varepsilon\) by the relation:

\[ \gamma=\frac{\pi^{2}RT}{F}\,\varepsilon^{2}. \tag{18} \]

The function \(\psi\), entering equation (17), can approximately be represented in the form of a series

\[ \psi=\frac{\gamma^{1/2}}{2\pi}\,\frac{\beta t}{m} \sum_{\kappa=0}^{m-1} \frac{ \sum_{n=1}^{\infty} e^{-(n)^{2}\gamma\left(\beta t-\frac{\kappa}{m}\beta t\right)} }{ \operatorname{ch}^{2}\frac{1}{2} \left(\beta t_{1/2}-\frac{\kappa}{m}\beta t\right) }, \tag{19} \]

where \(0\leq \kappa \leq m-1\) is an integer.

Expression (19) was obtained by replacing the integral by a sum, with the interval of integration divided into \(m\) sections of width \(\frac{\beta t}{m}\).

In calculating the function \(\psi\) by formula (19), \(m=20\), \(\beta t_{1/2}=10\) were adopted. The parameter \(\gamma\) was varied within the range from 0.1 to 5. For a given value of the parameter \(\gamma\), the sum (19) was calculated for different values of \(\beta t\), and thus the maximum value of the function \(\psi\) was found. In the region of the maximum of the function \(\psi\), the values of \(\beta t\) were varied in steps of 0.1, which corresponds to \(\Delta\varphi=2.5\ \mathrm{mV}\). All calculations were carried out to an accuracy of four decimal places.

Conclusions

1. The function \(\psi\), i.e., the quantity proportional to the constant of the reversible anodic peak, has been theoretically calculated for different values of the parameter \(\varepsilon\).

1. It has been shown that the anodic peak constant depends on the radius of the stationary mercury drop electrode and on the rate of potential change. With an increase in the electrode radius and in the rate of potential change, the anodic peak constant approaches in magnitude the Ševčík constant, whereas with a decrease in these parameters it approaches zero.

2. Simple approximate formulas have been proposed that make it possible to calculate the values of the anodic peak constant for any values of the parameter $\varepsilon$.

REFERENCES

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