On Accounting for the Asymmetry of the Stationary Mercury Drop Electrode in the Theory of the APN Method
Abstract Generated abstract
This paper addresses how the nonideal shape of a stationary mercury drop electrode, distorted by its support, can be incorporated into the theory of anodic current calculations in the APN method. Because direct treatment of nonstationary diffusion to a truncated spherical segment is mathematically difficult, the authors propose replacing the real electrode by an equal-volume parallelepiped model and derive criteria for choosing its dimensions so that its surface area matches, or minimally differs from, that of the spherical segment. Analysis of the resulting dimensionless relations identifies ranges of segment geometry for which surface equivalence is possible and shows that the maximum surface discrepancy in the intermediate range is about 20 percent. Comparison with a known hemispherical electrode equation suggests that the model can estimate currents with small error when the Fourier parameter is below approximately 0.1 to 0.2.
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On Accounting for the Asymmetry of a Stationary Mercury Drop Electrode in the Theory of the APN Method
E. E. Katyukhin, A. G. Stromberg, A. A. Kaplin.
The known equations for the anodic current for a stationary mercury drop electrode (S.M.D.E.) have been derived on the basis of representations of an ideal sphere. However, it may be assumed that in real electrodes the character of diffusion differs substantially from the ideal one because of the presence of a support (contact), which distorts the shape of the mercury electrode.
The solution of the problem of nonstationary diffusion in a rigorous formulation directly for electrodes in the form of a truncated sphere encounters great mathematical difficulties. Therefore we propose an indirect approach. The problem is solved for a model electrode in the form of a parallelepiped with dimensions \(2a \times 2b \times h\) (\(h\) is the thickness of the mercury film, \(2a \times 2b\) are the dimensions of the support), after which the conditions for transition from an electrode in the form of a truncated spherical segment (with dimensions \(r\) and \(H\)) to the model electrode are considered.
For the model electrode, an equation of the anodic current at constant potential was derived earlier [1]:
\[ i = \frac{512}{\pi^4} z F D C_R^0 \left( \frac{bh}{r} + \frac{ah}{b} + \frac{ab}{h} \right) \exp \left[ - \frac{\pi^2 D t}{4 r^2} \right] \tag{1} \]
Let us consider the relationship between the model electrode and the electrode in the form of a truncated sphere in order to determine the conditions under which, with minimal error in the theoretical equations, one electrode may be replaced by the other.
We shall estimate the surface of the model electrode \((S_m)\), which differs minimally from the surface of an equal-volume spherical segment \((S_{\text{s.s.}})\). The volumes and surfaces of each of the figures are calculated by the formulas:
\[ V_{\text{s.s.}} = \frac{\pi}{6H\left(H^2 + 3r^2\right)} \tag{2} \]
\[ S_{\text{ш.с.}} = 2\pi RH = \pi(H^2 + r^2) \tag{3} \]
\[ V_n = 4a^2h \qquad \text{base square} \tag{4} \]
\[ S_n = 4a^2 + 8ah \tag{5} \]
By the condition \(V_{\text{ш.с.}} = V_n\). Let us estimate at what ratios of \(a\) and \(h\) the parallelepiped has the minimal surface area.
\[ \left(\frac{dS_n}{da}\right)_{V=\text{const}} = 0; \qquad a_m = \sqrt[3]{V/4} \tag{6} \]
Substituting (4) into (6), we obtain \(a_m = h_m\). Thus, for a given volume the minimal surface area is possessed by a parallelepiped whose height \((h)\) is twice the base \((2a)\).
After substituting (2) into (6) and (6) into (5), under the condition \(a_m = h_m\), the surface is
\[ S_n = 3\pi^2(H^2 + 3r^2) \tag{7} \]
In dimensionless quantities
\[ \rho = r/H;\qquad \alpha = a/r;\qquad \beta = h/H \]
whence
\[ \alpha/\rho = a/h \tag{8} \]
\[ \beta/\rho = h/r \tag{9} \]
\[ \beta/\alpha\rho = h/a \tag{10} \]
Let us determine the relative error \(\delta_m = S_n/S_{\text{ш.с.}}\) as a function of the dimensionless parameters, using relations (3), (7), (10); after simple transformations we obtain
\[ \delta_m = \frac{1}{1+\rho^2} \sqrt{\frac{3}{\pi}(1+3\rho^2)^2} \tag{11} \]
Graphical analysis of the dependence obtained shows that at two ratios of the dimensions of the spherical segment \((r_1 = 2.5H_1\) and \(r_2 = 0.14H_2)\), replacing it by an equal-volume parallelepiped with surface \(S_n\), calculated by equation (7), does not lead to a change in surface area. At all other values of \(\rho\), there is a difference between \(S_{\text{ш.с.}}\) and \(S_n\), which in the interval \(0.14 < \rho < 2.5\) reaches a maximum at \(\rho = 1\) (hemisphere).
Let us clarify the general character of the dependence of the ratio \(\delta = S_n/S_{\text{ш.с.}}\) on \(\rho\) for equal-volume parallelepipeds of arbitrary shape. Substituting (3) into (5), taking into account the equality \(2ah = V/2a\), we obtain
\[ \delta = A\alpha^2 + \frac{AB}{\alpha}, \tag{12} \]
where
\[ A = \frac{4}{\pi}\cdot \frac{\rho}{1+\rho^2}; \qquad B = \frac{\pi}{12}(1+3\rho^2)/\rho^2. \]
Analysis of the dependence of \(\delta\) on \(\alpha\) by equation (12) at various \(\rho\) makes it possible, in addition to the known conclusions, to make the following:
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In the range of values \(0.11 > \rho > 2.5\), two parallelepipeds of equal volume to the spherical segment are possible, for which \(S_{\text{sh.s.}} = S_n\).
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In the range of values \(0.14 < \rho < 2.5\), the surface area of the equal-volume parallelepiped is always greater than the surface area of the spherical segment. The maximum ratio of the surfaces is \(1.2\).
Consequently, for a spherical segment of any shape one can choose such an equal-volume parallelepiped whose surface either does not differ from the surface of the segment, or differs by a definite amount. In the first case \((0.14 > \rho > 0.25)\), parallelepipeds are selected for which the parameter \(\alpha\) corresponds to the value \(\delta = 1\); in the second, \(\alpha\) is chosen equal to \(\alpha_m\). This choice is considerably facilitated with the aid of the graph \(\alpha = f(\rho)\).
The applicability of the equation describing the anodic current at the model electrode to the calculation of the current at an electrode in the form of a spherical segment can be checked, for example, by comparison with the equation proposed in the work of Chovnyuk and Vashchenko / 2 / for the case of a hemisphere; the symmetry of diffusion is not disturbed in this case.
Analysis shows that, for \(F_o = D\tau/r^2 > 0.2\), the equation of Chovnyuk and Vashchenko and equation (1) can, with insignificant error, be simplified. After this the following equations are compared:
\[ i_n = \frac{512}{\pi^4}\, 3a z F D C_R^0 \exp\left[-\frac{\pi^2 D\tau}{a^2}\right] \tag{13} \]
\[ i_{\text{sh.s.}} = 4r C_R^0 z F D \pi \exp\left[-\frac{\pi^2 D\tau}{r^2}\right]. \tag{14} \]
For \(\rho = 1\) we determine \(\alpha = 0.81\) and \(a = 0.81r\).
Dividing (14) by (13), we obtain
\[ \frac{i_{\text{sh.s.}}}{i_n} = 0.8\alpha^{-1} \exp\left[ 7.4\,\frac{D\tau}{r^2}\left(\alpha^{-2}-1\right) \right] \tag{15} \]
From calculations by equation (15) (the “Promin’” computer) it follows that the values of the currents calculated from equations (14) and (13) practically coincide in the region where the inequality \(F_o < (0.1 \div 0.2)\) is satisfied. The difference between the values of the currents at other values of this parameter is due mainly to the difference in the character of diffusion of the metal atoms inside the sphere and the model electrode. The influence of the sphericity of the electrode increases all the more noticeably with increasing parameter \(F_o = D\tau/r^2\). Thus, the comparison carried out shows that, by the proposed method, one can calculate the current at an electrode in the form of a hemisphere if the value \(F_o < (0.1 \div 0.2)\). Probably, this
the conclusion can be extended to spherical segments of any cross section, since a hemisphere is the least favorable variant for comparison.
Consequently, the general scheme for using a parallelepiped as a model for calculating the current at a real mercury spherical electrode can be represented as follows.
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The shape of the spherical segment is evaluated, i.e., the value of the parameter $\rho$ is found.
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Using the value of $\rho$, the parameter $\alpha$ is chosen, i.e., a parallelepiped is selected whose surface is either equal to the surface of the spherical segment under study or differs from it minimally.
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For the selected parallelepiped, by formula (I) the current is calculated under the condition $F_0 < (0.1 \div 0.2)$. The anodic current at the electrode in the form of a truncated sphere is taken to be equal to this value.
References
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V. E. Katyukhin, A. A. Kaplin. In: “Chemistry and Chemical Technology,” Tomsk, TTU Press, 1973, p. 57.
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N. V. Chovnyk, V. V. Vashchenko. ZhFKh, 37, 538, 1963.
EFFECT OF A SUBSEQUENT CHEMICAL COMPLEX-FORMATION REACTION ON THE POTENTIAL OF THE REVERSIBLE ANODIC PEAK IN THE APN METHOD AT A STATIONARY Hg DROP ELECTRODE
S. S. Reznikova, Yu. A. Karbainov
In work /1/ the results were presented of a numerical evaluation of the equation of the reversible anodic peak in the APN method at a stationary mercury dropping electrode for an electrode process complicated by a subsequent chemical complex-formation reaction.
By interpolation of these data, expressions were obtained relating the potential of the reversible anodic peak to the parameters of the subsequent chemical reaction. The results of these investigations are discussed below.
As was to be expected, the subsequent chemical complex-formation reaction occurring in solution causes a shift of the potential of the reversible anodic peak toward more negative—