STUDIES ON THE DEPENDENCE OF THE CATHODIC PEAK CONSTANT ON A HANGING MERCURY DROP ON THE POTENTIAL SCAN RATE

A. G. STROMBERG, V. Z. BASHKATOV

(Presented by the Scientific-Methodological Seminar of the CTF)

The theory of electrolysis at a stationary electrode in a stirred solution with a linearly varying potential, and for a reversible electrode process, has recently undergone substantial development both for the case of semi-infinite linear diffusion [1—5] and, to a lesser degree, for spherical diffusion [6—9].

However, a detailed experimental verification of the theoretical dependences obtained for the cathodic peak constant at a spherical electrode has not yet been carried out.

The aim of the present work is to study the dependence of the cathodic peak constant on the rate of change of potential and to compare the experimental data obtained with theoretically calculated values.

The cathodic peak constant is defined by the expression

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

where \(I\) is the peak current, \(\mu\text{A}\); \(c\) is the concentration of ions in solution, \(\text{g-ion}/\text{cm}^3\); \(s\) is the electrode surface area, \(\text{cm}^2\); \(K\) is the cathodic peak constant, \(\mu\text{A}\cdot\text{cm}/\text{g-ion}\). Hence, from experimental data, the anodic current constant is calculated from the expression

\[ K = \frac{I}{sc}. \tag{2} \]

The theoretical value of the current at any potential for a reversible electrode process and spherical semi-infinite diffusion is given by the expression [9]

\[ i = 6.02 \cdot 10^5 z^{3/2} D^{1/2} w^{1/2} \left[\sqrt{\pi}\chi(at) + 0.160\, a\Phi(at)\right]\cdot s\cdot c, \tag{3} \]

where \(z\) is the number of electrons participating in the electrode process per metal ion; \(D\) is the diffusion coefficient of metal ions in solution, \(\text{cm}^2/\text{sec}\); \(w\) is the rate of change of potential, \(\text{volt}/\text{sec}\); \(\sqrt{\pi}\chi(at)\) and \(\Phi(at)\) are complex integral expressions not expressible in elementary functions, but found by numerical integration.

at different values of the parameter \(at\) and given in the table in [9]\(^{1}\)); \(\varepsilon\) is a parameter defined by the expression

\[ \frac{1}{\varepsilon}=B w^{1/2}, \tag{4} \]

\[ B=\frac{r z^{1/2}}{D^{1/2}}, \tag{5} \]

where \(r\) is the radius of the sphere, cm. The first term in formula (3) characterizes linear semi-infinite diffusion, and the second term is the correction for spherical diffusion.

The maximum values of the functions \(\sqrt{\pi}\chi(at)\) and \(\Phi(at)\) in Table 1 of [9] are, respectively, 0.4463 and 0.7516. Hence, instead of formula (3), taking (2) into account, after simple transformations we obtain for the cathodic-current constant \(K\) the expression

\[ \frac{K}{K_0}=1+0.27\varepsilon, \tag{6} \]

where \(K_0\) is the cathodic-current constant in the case of linear semi-infinite diffusion, defined by the expression [1–5]:

\[ K_0=A w^{1/2}, \tag{7} \]

\[ A=2.68\cdot 10^5 z^{3/2} D^{1/2}. \tag{8} \]

From formula (5), taking (4) into account, it follows that as \(\varepsilon\to 0\) (large \(w\) or \(r\)) the ratio \(K/K_0\) tends to unity; as \(\varepsilon\to\infty\) (small \(w\) or \(r\)) we have \(K/K_0\to\infty\).

To verify the validity of formula (5), we carried out experiments to obtain the cathodic peaks of cadmium at a hanging mercury electrode at different rates of potential change. The composition of the solution and the experimental conditions are indicated in the caption to Fig. 1. The electrode (hanging mercury electrode) has the form of a spherical segment. It was obtained as follows. A drop of mercury, flowing out of a mercury-dropping contact of diameter 1 mm, ground flush with the end of a glass tube. The volume of the hanging mercury drop was determined by weighing a definite number of mercury drops that had flowed out of the polarographic capillary. The surface and radius of the spherical segment were determined from known geometrical formulas\(^{2}\). The design of the electrolyzer is described in [10].

Table 1

\(^{1}\) In Table 1 on p. 703 of [9], the values of \((\varphi-\varphi_{1/2})z\) are given as the argument, where \(\varphi_{1/2}\) is the potential of the kinetically reversible cathodic wave at a dropping mercury electrode; \(\varphi\) is any potential.

\(^{2}\) From the formula for the volume of a spherical segment:

\[ v=\pi h\left(\frac{c^2}{8}+\frac{h^2}{6}\right) \tag{a} \]

(\(c\) is the base of the drop, cm; \(h\) is the height of the drop, cm), with known \(v=2.39\cdot 10^{-4}\ \text{cm}^3\) and \(c=0.1\ \text{cm}\), by the method of successive approximations we find \(h=4.7\cdot 10^{-2}\ \text{cm}\). From the formula for the surface of a spherical segment

For the range of scan rates 2–20 mV/sec, a recording polarograph with an OH-101 recorder (Hungary) was used, and for the range 30–150 mV/sec, a modified oscillographic polarograph OP-2 was used.

The electrolyzer was placed in a thermostat, and the experiments were carried out at a temperature of 25°C. The diffusion coefficient of cadmium ions in the solution under study at 25°, calculated from the limiting diffusion current at a dropping mercury electrode by the Il’kovič formula, proved to be \(7.1 \cdot 10^{-6}\ \mathrm{cm^2/sec}\).

For the values of \(\xi\), \(D\), and \(r\) found, instead of (4) and (8) we obtain:

\[ \frac{1}{\xi} = 26.6 \cdot w^{1/2}, \tag{9} \]

\[ K_0 = 20.2 \cdot 10^2 w^{1/2}, \tag{10} \]

Fig. 1. Dependence of the cadmium cathodic-peak constant on the rate of change of potential in the coordinates \(K/K_0,\ 1/\xi\). Curve 1—linear semi-infinite diffusion (theoretical straight line \(K/K_0 = 1\)); 2—spherical semi-infinite diffusion (theoretical curve in accordance with formula (5)); 3—experimental data for a hanging mercury drop. Composition of the solution: \(6.67 \cdot 10^{-4}\) g-ion/liter \(Cd^{2+} + 0.1\) M KCl \(+ 0.005\%\) gelatin. Experimental conditions: diameter of the base of the drop (contact) 1 mm; the radius and height of the spherical segment (hanging mercury drop) are respectively \(5.01 \cdot 10^{-2}\) and \(4.70 \cdot 10^{-2}\) cm; temperature 25°C; diffusion coefficient of cadmium ions in the solution \(D = 7.1 \cdot 10^{-6}\ \mathrm{cm^2/sec}\).

Fig. 2. Dependence of the cadmium cathodic-peak constant on the rate of change of potential in the coordinates \(K/K_0\) and \(\xi\). The meanings of straight lines 1–3, the solution composition, and the experimental conditions are given in the caption to Fig. 1.

and, for calculating \(K\) from the experimental values of \(I\), instead of (2):

\[ K = 1.01 \cdot 10^9 I. \tag{11} \]

The experimental results are given in Table 1 and in Figs. 1 and 2. From Fig. 1 it is seen that, in the coordinates \(K/K_0,\ 1/\xi\), with an increase in \(1/\xi\) (or \(w^{1/2}\)) the ratio \(K/K_0\) decreases, approaching unity (curve 3). The theoretical curve calculated by formula (6) has an analogous form, but is located lower (curve 2). In the coordinates \(K/K_0,\ \xi\), the theoretical curve in accordance with formula (6) should be represented by a straight line with slope 0.27 (Fig. 2, straight—

\[ s = \frac{\pi}{4}\left(c^2 + 4h^2\right) \tag{6} \]

we find \(s = 1.48 \cdot 10^{-2}\ \mathrm{cm^2}\). And by the formula:

\[ s = 2\pi rh \tag{6} \]

we find the radius of the spherical segment: \(r = 5.01 \cdot 10^{-2}\ \mathrm{cm}\).

(Fig. 2). The experimental points on this plot are located higher and on a curved line having the character of a parabola. In the coordinates \(\lg (K/K_0-1), \lg \varepsilon\), the experimental points lie satisfactorily on a straight line, which in analytical form can be written as the equation:

\[ \lg (K/K_0 - 1) = -0.07 + 0.90 \lg \varepsilon \tag{12} \]

or

\[ \frac{K}{K_0} = 1 + 0.85 \varepsilon^{0.9}. \tag{13} \]

One might suppose that the deviation of experimental curve 3 from theoretical curve 2 (Figs. 1 and 2) is associated with an incorrect estimate of the true (active) surface from geometrical calculations. In this case, instead of formula (6) we would obtain the expression

\[ \frac{K}{K_0} = \frac{s_0}{s} + \frac{s_0}{7} \, 0.27 \varepsilon, \tag{14} \]

where \(s_0\) is the true (active) surface; \(s\) is the surface calculated from geometrical dimensions. Line 2 in Fig. 2 corresponds to equation (14) under the condition \(\frac{s_0}{s} = 1.30\). However, this assumption is contradicted by the three lower points on experimental curve 3, which, as \(\varepsilon \to 0\), approach unity (and not \(\frac{s_0}{s} = 1.30\)).

Another, more probable assumption is that the overestimated (in comparison with the theoretical) experimental data are obtained because, during electrolysis, convection of the solution occurs in the solution near the electrode surface; this is not taken into account in the theoretical calculations. At high rates of change of potential \(w\) (small \(\varepsilon\)), the diffusion layer is thin, and convection has no effect on the cathodic peak current. Therefore, at small \(\varepsilon\) (large \(\frac{1}{\varepsilon}\)), the experimental points are close to the theoretical ones and to the value unity (Figs. 2 and 1). At low rates of change of potential, the diffusion layer in the solution, by the time the cathodic peak is reached, has time to move away from the surface by a more considerable distance. Therefore convection exerts a stronger influence, and the cathodic peak current is overestimated. Other authors have also arrived at an analogous conclusion when checking the theoretical expression for the dependence of the limiting current at a spherical electrode on time at constant potential [11].

Conclusions

1. Experiments have been carried out to study the dependence of the height of the cadmium cathodic peak on a hanging mercury drop on the rate of change of potential in the range \(2\)—\(150\) mV/sec.

2. It has been shown that the experimental values of the cathodic peak constant exceed the theoretical values by a factor of 1.3 for the case of hemispherical spherical diffusion under the condition of reversibility of the electrode process.

3. Possible causes of such a discrepancy between the experimental and theoretical values of the cathodic peak constant have been discussed.

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