DEPENDENCE OF THE ELECTROLYSIS CONSTANT ON THE ION DIFFUSION COEFFICIENT IN THE EAPN METHOD

Yu. A. Karbainov, A. G. Stromberg

In accordance with the theoretical considerations developed by A. G. Stromberg [1, 2, 3], one of the most important ways of increasing the sensitivity of the method of amalgam polarography with accumulation is a sharp increase in the electrolysis constant. In practical work, it is important to know and to evaluate the influence of various factors on the electrolysis constant. Such factors are, first of all: the influence of the intensity of stirring of the solution, the influence of the number of electrons participating in the electrode process, and, finally, the influence of the diffusion coefficient of the metal ions in solution. As for the first two factors, there are already at present a large number of interesting works on this question [4–10]. M. S. Zakharov, using aqueous solutions as an example, studied the dependence of the electrolysis constant on the diffusion coefficient of metal ions in solution. It was shown that, with other conditions constant, the relation

\[ K_{\mathrm{e}} = \mathrm{const}\, D_i^{0.38}. \]

is valid. In a number of cases, the values of the exponent for the diffusion coefficient were obtained as \(n = 0.39 \div 0.42\). Thus, these data are contradictory. In our work we therefore set ourselves the task of clarifying this dependence.

Proceeding from A. G. Stromberg’s parametric theory, the electrolysis current \((i_{\mathrm{e}})\), depending on the concentration of metal ions in solution \((C_1)\) and on the surface area of the stationary mercury drop \((S)\) in a stirred solution, is determined by the relation [11]

\[ i_{\mathrm{e}} = K_{\mathrm{e}} \cdot S \cdot C_1, \tag{1} \]

where \(K_{\mathrm{e}}\) is the rate constant of electrochemical concentration of the metal in the mercury drop, or the electrolysis constant. The electrolysis constant characterizes the intensity of transport of the substance to the electrode and depends mainly on the rate of stirring of the solution during electrolysis. Its physical meaning, on the basis of relation (1), is as follows: the electrolysis constant is the electrolysis current at a mercury-drop surface area equal to \(S = 1\ \mathrm{cm}^2\), and a concentration of metal ions in solution equal to \(C_1 = 1\ \mathrm{g\text{-}ion}/\mathrm{cm}^3\). The numerical value of the electrolysis constant is found from the area under the anodic peak:

\[ K_{\mathrm{e}} = \frac{q}{\tau_{\mathrm{e}} \cdot S \cdot C_1}, \tag{2} \]

where \(q\) is the area under the anodic tooth, in coul.;
\(\tau_{\mathrm{e}}\) is the electrolysis time, sec.

On the other hand, from the Nernst–Levich theory, the expression for the electrolysis current in the presence of stationary diffusion in a stirred solution can be written in the form

\[ i=zF\cdot S\cdot D\cdot \left(\frac{dc}{dx}\right)_{x=0}, \tag{3} \]

where \(D\) is the diffusion coefficient of the metal ions in solution, \(\mathrm{cm}^{2}/\mathrm{sec}\).

Comparing expressions (1) and (3), it is evident that the electrolysis constant \(K_{\mathrm{e}}\) is directly proportional to the number of electrons participating in the electrode process; that is, with all other conditions constant, the relation

\[ K_{\mathrm{e}}=\mathrm{const}\cdot z \tag{5} \]

is valid.

On the other hand, the electrolysis current depends on the diffusion coefficient of the metal ions in solution; consequently, for the electrolysis constant one may write:

\[ K_{\mathrm{e}}=\mathrm{const}\cdot z\cdot D^{n}. \tag{6} \]

Taking the logarithm of expression (6), we obtain

\[ \lg K_{\mathrm{e}}=\lg(\mathrm{const}\, z)+n\lg D. \]

A plot in the coordinates \(\lg K_{\mathrm{e}}=f(\lg D)\) should give a straight line, whose slope tangent is \(\tan\gamma=n\), while the intercept cut off on the ordinate axis is \(\lg(\mathrm{const}\cdot z)\). Thus, the exponent \(n\) for the diffusion coefficient can be determined semiempirically.

How the electrolysis constant is estimated from the area under the anodic tooth was described above.

The values of the ion diffusion coefficients can in principle be calculated by various methods. In the case of a reversible electroreduction process, the ion diffusion coefficient can be estimated from the Ševčík equation for a reversible cathodic tooth on a planar electrode [12]:

\[ I_{\mathrm{p.k.}}^{\mathrm{rev}}=2.72\cdot 10^{5}\cdot z^{3/2}\cdot D^{1/2}\cdot W^{1/2}\cdot S\cdot C_{1}. \]

Knowing the experimental value of the tooth height \((I_{\mathrm{k,p}})\), one can calculate \(D\). One may also use the Cottrell equation [13] for the dependence of the diffusion current on time in the case of linear semi-infinite diffusion:

\[ I=zF\cdot S\cdot C\cdot \sqrt{\frac{D}{\pi}}\cdot \frac{1}{\sqrt{\tau}}; \]

If a plot is constructed in the coordinates \(I=f\left(\dfrac{1}{\sqrt{\tau}}\right)\), then \(D\) can be calculated from the slope, provided the electrode surface \(S\) is specified. However, the use of these equations for calculating \(D\) is limited (reversibility, high rate of potential change, etc.). With a certain accuracy, the diffusion coefficient of metal ions in solution can be calculated by means of the diffraction micromethod [14–16]. In our case, for all the elements studied (copper, lead, cadmium), distinct polarographic waves without maxima were obtained, and for calculating the diffusion coefficients we used the Ilkovič equation [17].

The dependence of the electrolysis constant on the diffusion coefficients of metal ions in solution was studied using the example of three elements of the same—

covalence (copper, lead, and cadmium) on various supporting electrolytes: 1% oxine in CHCl₃ + 0.2 N NH₄NO₃ in CH₃OH; \(5 \cdot 10^{-3}\)% dithizone in CHCl₃ + 0.2 N NH₄NO₃ in CH₃OH; 2% diethyldithiocarbamate in CHCl₃ + 0.2 N NH₄NO₃ in CH₃OH (1:1).

Special attention was paid to ensuring that the experimental conditions (in particular, the stirring regime during electrolysis) remained constant in all experiments. For this purpose, to stir the solution during electrolysis we used a magnetic stirrer, the rotation rate of which in the experiments was kept constant and equal to 420 rpm. The cathode was a mercury drop obtained electrolytically on a platinum contact (diameter 0.15 mm) sealed into a glass tube, at a current of 30 mA for 1 min. The radius of the mercury drop was then \(r_k = 0.044\) cm. Mercury at the bottom of the electrolyzer served as the anode. Measurements were carried out on a Hungarian OH-101 polarograph.

The experimental procedure was as follows. One of the above supporting solutions was prepared (the extractant and electrolyte were prepared separately, after which they were mixed in a 1:1 ratio). The supporting solution, in an amount of 10 ml, was transferred to the polarographic cell; a definite addition of standard solution was introduced into it, and the mixture was purged with nitrogen for 25–30 minutes (to remove oxygen from the solution). After this the nitrogen flow was stopped, the magnetic stirrer was switched on, and electrolysis was carried out. The experiment was repeated 7–8 times and in each case the value of the electrolysis constant was calculated and the mean value was taken. The area under the anodic peak used in calculating \(K_{\mathrm{el}}\) was determined with a planimeter. On the dropping mercury electrode the cathodic wave of the element under study was recorded on the given supporting electrolyte, and the value of the diffusion coefficient was calculated from the Ilkovič equation. In the same way, the values of the electrolysis constants and diffusion coefficients on the given supporting electrolyte were found for the two other elements, after which a plot was constructed in the coordinates \(\lg K_{\mathrm{e}} = f(\lg D)\), and the exponent \(n\) was determined. The results of the investigations are presented in Fig. 1. It is evident from the figure that this dependence is expressed by a straight line, whose slope is equal to 0.46–0.47; thus, to within 4% one may take \(n = 0.5\). Consequently, the electrolysis constant as a function of the diffusion coefficient of the ions in solution is expressed by the relation

\[ K_{\mathrm{e}} = \mathrm{const} \cdot z \cdot D^{1/2}. \]

Fig. 1. Dependence of \(\lg K_1\) on \(\lg D\).
1 — supporting electrolyte: 1% oxine in CHCl₃ + 0.2 N NH₄NO₃ in CH₃OH;
2 — supporting electrolyte: \(5 \cdot 10^{-3}\)% dithizone in CHCl₃ + 0.2 N NH₄NO₃ in CH₃OH;
3 — supporting electrolyte: 2% sodium diethyldithiocarbamate in ethyl acetate + 0.2 N NH₄NO₃ in CH₃OH.

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