EXPERIMENTAL VERIFICATION OF THEORIES OF THE ANODIC PEAK IN THE METHOD OF AMALGAM POLAROGRAPHY WITH ACCUMULATION ON A FILM ELECTRODE

M. S. ZAKHAROV, L. F. TRUSHINA

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

The general equation of the current-voltage curve for the anodic dissolution of an amalgam for reversible processes on a film electrode at a linearly varying potential was obtained independently by V. A. Igolinsky [1] and by Vries and Dalen [2]. In addition, V. A. Igolinsky obtained particular equations for low and high rates of change of potential [1]. The author of work [1] studied the effect of the rate of change of potential on the depth of the anodic peak and determined the width of the anodic half-peak on a film electrode. The studies confirmed the theoretical conclusion about the rectilinear dependence of the depth of the anodic peak on the rate of change of the electrode potential with a mercury-film thickness from 2 to 12 μ in the range of rates of potential change 100–9600 mV/min. Other studies on the experimental verification of conclusions of the theories of the anodic peak for reversible processes on a film electrode are absent from the literature.

In the present work we verified the accuracy of the equation of the current-voltage curve obtained by Vries and Dalen, and of the approximate equation for low rates of change of potential obtained by Igolinsky.

Verification of the equations was carried out by comparing the width of the anodic half-peak obtained from experimental data with the theoretical value of the half-peak width. According to work [2], the width of the anodic half-peak of cadmium (a two-electron reversible peak) on electrodes with a film thickness of 2–12 μ is equal to 38 mV.

To calculate the theoretical values of the width of the anodic half-peak in verifying Igolinsky’s equation, we, using this equation, obtained theoretical expressions for calculating the indicated quantities.

The approximate equation at low rates of change of potential for reversible processes [1] has the following form:

\[ I_2=\frac{8 C_R^0 l z^2 F^2 s \omega}{\pi^2 R T} \left( e^{-\frac{zF\omega t}{RT}}-e^{-\frac{\pi^2 D_R t}{4l^2}} \right), \tag{1} \]

where \(C_R^0\) is the concentration of metal atoms in the amalgam before stripping the anodic peak; \(z\) is the number of electrons participating in the electrode process per atom of the substance; \(l\) is the thickness of the mercury film on the electrode; \(\omega\) is the rate of change of potential; \(F\) is Faraday’s number; \(s\) is the electrode area; \(R\) is the gas constant; \(T\) is the temperature;

\(t\) is time. This equation, in the author’s opinion, describes the anodic wave quite correctly if the relation [1]

\[ \frac{zF\omega}{RT}\ll \frac{\pi^2D_R}{4l^2} \tag{2} \]

is satisfied.

It is evident from expression (1) that the form of the current–voltage curve is determined by the expression in brackets. The width of the anodic half-wave in mV can be determined from the value \(\Delta t=t_2-t_1\) (see Fig. 1). For the half-peak, when condition (2) is fulfilled, the value \(t_2\) can be found from the following relation:

\[ \left(e^{-\frac{zF\omega t}{RT}}\right)_{\text{at }t=t_2}=0.5. \tag{3} \]

Fig. 1.

From expression (3) we obtain:

\[ t_2=-\frac{\ln 0.5\,RT}{zF\omega}. \tag{4} \]

When relation (2) is fulfilled, one may take \(t_1\simeq t_{\text{p}}\); \(t_{\text{p}}\) is the time at which the peak of the wave is reached (time is counted from the equilibrium potential \(\varphi_{\text{r}}\)). The expression for calculating \(t_{\text{p}}\) can be found by solving for the maximum the expression in brackets in equation (1). Let us take

\[ y=e^{a\omega t_{\text{p}}}-e^{-bt_{\text{p}}}, \tag{5} \]

where the values of \(a\) and \(b\) are clear from comparison of equation (5) with the expression in brackets in equation (1).

\[ \frac{dy}{dt}=-a\omega e^{-a\omega t_{\text{p}}}+be^{-bt}=0, \tag{6} \]

whence

\[ t_{\text{p}}=\frac{\ln \dfrac{b}{a\omega}}{b-a\omega}. \tag{7} \]

Let us estimate the error introduced into the calculations by taking \(t_1=t_{\text{p}}\). To do this, let us compare \(t_{\text{p}}\) and \(t_2\) for one specific case. Let \(T=293^\circ\), \(z=2\), \(\omega=1.7\cdot10^{-3}\ \text{V/sec}\) (\(\sim 100\ \text{mV/min}\)), \(l=5\cdot10^{-4}\ \text{cm}\); \(D_R=1\cdot10^{-5}\ \text{cm}^2\text{ sec}^{-1}\). Under these conditions \(t_2=2.35\ \text{sec}\) and \(t_{\text{p}}=0.004\ \text{sec}\). Thus, the value of \(t_{\text{p}}\) is only \(0.17\%\) of the value of \(t_2\). Consequently, in practice \(t_2-t_1\simeq t_2-t_{\text{p}}\). From expressions (4) and (7) it follows:

\[ \Delta t=t_2-t_1=-\frac{\ln 0.5\,RT}{zF\omega}-\frac{\ln \dfrac{b}{a\omega}}{b-a\omega}. \tag{8} \]

The expression for calculating the width of the anodic half-wave in mV is written as:

\[ \Delta\varphi=\Delta t\omega= \left( \frac{\ln 0.5\,RT}{zF\omega} -\frac{\ln \dfrac{b}{a\omega}}{b-a\omega} \right)\omega. \tag{9} \]

Taking into account that the second term of the difference in equation (9) is much smaller than the first term, for a reversible anodic process at \(20^\circ\text{C}\) one may write

\[ \Delta\varphi=\frac{40}{z}\ \text{mV}. \tag{10} \]

We shall now determine what the static resolving power of the ASV method with a film electrode should be, if equation (1) is valid. According to Shafsmann [3], the measure of resolving power in terms of peak height is taken to be the largest ratio of the concentrations of two metals at which their peaks are still separated. The quantitative measure of resolving power in terms of potentials is taken to be the smallest potential difference between two peaks at which they are still separated.

To determine the resolving power of the ASV method for reversible processes at a film electrode, as will be seen from the subsequent discussion, it is convenient to use the equation for the descending branch of the anodic peak [1]:

\[ I_2=\frac{8C_R^0\cdot l\cdot A\cdot z^2\cdot F^2\cdot \omega}{\pi^2\cdot R\cdot T} e^{\frac{-zF\omega}{RT}(t-t_p)}. \tag{11} \]

Using equation (11) instead of equation (1), we do not introduce a significant error into the calculations, since in practice \(t_p=0\).

Taking into account the remark made, equation (11) can be written in the following form:

\[ I=I_p e^{-a\omega(t-t_p)}. \tag{12} \]

The relative error \(f\) in determining the second peak will be:

\[ f=\frac{I_1}{I_{2,p}}=\frac{I_{1,p}\cdot e^{-a\omega(t_{p,2}-t_{p,1})}}{I_{2,p}}, \tag{13} \]

where the indices 1,2 refer to the characteristics of the electronegative and electropositive elements, respectively. Obviously, equation (13) can be written in the following form:

\[ f=\beta e^{-a\Delta\varphi}. \tag{14} \]

Experimental Part

In this work the width of the half-peak of the anodic oxidation of lead and cadmium amalgams in the presence of \(0.1\,M\) \(K_2SO_4\) was determined, and its dependence for cadmium on the thickness of the mercury film and on the rate of change of the electrode potential was studied. The investigations were carried out on film electrodes with a mercury-film thickness from 4 to \(20\,\mu\). The rate of change of potential was varied from \(0.002\ \text{V/sec}\) to \(0.016\ \text{V/sec}\). An OH-101 polarograph was used in the work. To eliminate the interfering influence of surface-active substances, which can strongly alter the character of the electrode process [5], a number of measures were taken to remove surfactants from the vessels and reagents used. The salts from which the solutions were prepared were recrystallized from tridistillate and ignited. The electrolytic key connecting the electrolyzer with the calomel electrode (reference electrode) was filled with the supporting electrolyte. The electrolyzer and all its elements were treated with a hot solution of \(K_2Cr_2O_7+H_2SO_4\) for 2–3 hours, then washed with concentrated \(H_2SO_4\), and before use washed with tridistillate obtained by distillation with \(KMnO_4\). The results of the studies to determine the width of the anodic half-peak of the electrooxidation of cadmium and lead amalgams on an electrode with a mercury film \(4\,\mu\) thick, at a rate of change of potential of \(0.006\ \text{V/sec}\), are presented in Tables 1 and 2.

It is seen from the tables that, for the electrode processes studied, the width of the anodic half-peak is \(40\ \text{mV}\). This value agrees well with the theoretical conclusion of De Vries and differs by a factor of two from

Table 1

Results of statistical processing of values of the width of the anodic half-peak for the process of electrooxidation of lead amalgam. Experimental conditions: $\tau_{\mathrm{el}} = 2$ min; $V_{\text{soln}} = 5$ ml; $\varphi_{\mathrm{el}} = -1.0$ V (sat. cal. el.)

Table 2

Results of statistical processing of values of the width of the anodic half-peak for the process of electrooxidation of cadmium amalgam. Experimental conditions: background $0.1\ M\ \mathrm{K_2SO_4}$; $\tau_{\mathrm{el}} = 5$ min; $V_{\text{soln}} = 5$ ml; $\varphi_{\mathrm{el}} = -1.2$ V (sat. cal. el.)

where $s_{\bar{\delta}}$ is the root-mean-square error of the arithmetic mean,
$t$ is Student’s coefficient at confidence level $\alpha = 0.95$;
$\varepsilon$ is the absolute error; $\varepsilon$, % is the relative error; $\delta = \Delta \varphi$.

with the theoretical value of the half-peak width obtained from Igolinsky’s theory.

Fig. 2. Dependence of the width of the cadmium anodic half-peak on the rate of change of the electrode potential for mercury films of various thicknesses.
$1$—$l=4\mu$; $2$—$6\mu$; $3$—$8\mu$; $4$—$10\mu$; $5$—$12\mu$; $6$—$14\mu$; $7$—$16\mu$; $8$—$18\mu$; $9$—$20\mu$. Background—$0.1\ M\ \mathrm{K_2SO_4}$.

The results of studies on the influence, on the width of the anodic half-peak, of the rate of change of the electrode potential and of the thickness of the mercury film are presented, respectively, in Figs. 2 and 3. The experimental data

agree well with the theoretical conclusions obtained in [2].

Fig. 3. Dependence of the width of the anodic half-wave of cadmium on the thickness of the mercury film at different rates of change of the electrode potential:
1—$\omega = 0.002$ V/sec; 2—0.004 V/sec; 3—0.006 V/sec;
4—0.009 V/sec; 5—0.012 V/sec; 6—0.016 V/sec.
Background—0.1 M $K_2SO_4$.

Conclusions

The experimental material obtained in this work on determining the width of the anodic half-wave in the ASV method with a mercury film electrode for reversible electrode processes, when the experiment is carried out on a self-recording polarograph, agrees well with the theoretical conclusions of [2] and disagrees with those of [1].

References

1. V. A. Igolinskii. Investigations in the field of polarography and amalgam polarography with accumulation on a film electrode. Candidate’s dissertation, Tomsk, 1964.
2. W. de Vries, E. Van Dalen, J. Electroanal. Chem., 8, 366, 1964.
3. S. B. Tsfasman. Electronic Polarographs. Metallurgizdat, p. 87, 1960.
4. A. G. Stromberg. Theory and practice of polarography and amalgam polarography in particular. Doctoral dissertation. Sverdlovsk, 1951.
5. L. F. Trushina, M. S. Zakharov. Journal of Analytical Chemistry (in press).