On the Calculation of the Anodic Peak Constant at Mercury Electrodes of Limited Volume
Abstract Generated abstract
This paper examines the calculation of the anodic peak constant for reversible processes at limited-volume mercury electrodes, focusing on the validity of an approximate boundary condition commonly used for spherical and planar electrodes. The authors solve a diffusion boundary-value problem for a planar electrode, derive an expression for the peak constant as a function of a dimensionless parameter, and compare the resulting numerical dependence with exact solutions and experimental data. The comparison indicates that the approximate condition gives acceptable accuracy only for small parameter values, about 1 or less, while errors increase substantially at larger values. The work also clarifies differences between planar and spherical electrodes and argues that reported discrepancies arise from comparisons made at different parameter values.
Full Text
ON THE CALCULATION OF THE ANODIC PEAK CONSTANT AT LIMITED-VOLUME MERCURY ELECTRODES
V. E. GORODOVYKH, B. F. NAZAROV
(Presented by the scientific-methodological seminar of KhTF)
There is already a fairly large number of works in the literature in which equations of voltammetric curves are derived for a linearly varying potential for reversible anodic processes and for various diffusion conditions [1, 2, 4, 7]. Starting from the equations obtained, the authors calculate the values of the peak potentials, the half-peak width, and the value of the quantity \(K_a\), which is determined from the following relation:
\[ I_a = K_a S C_R, \tag{1} \]
where \(K_a\) is the anodic peak constant,
\(I_a\) is the maximum current value,
\(S\) is the electrode surface,
\(C_R\) is the concentration of metal atoms in the amalgam.
However, to date there are no data in the literature comparing the results obtained for different types of electrodes (spherical, planar), and there is no quantitative estimate of the limits of applicability of the approximate equations. For example, in works [2, 3, 4] equations of the anodic wave for a spherical electrode are derived under the following boundary condition:
\[ C_0^0 D_0^{1/2} + C_R^0 D_R^{1/2} = C_0^S D_0^{1/2} + C_R^S D_R^{1/2}, \tag{2} \]
where \(C_0^0\) and \(C_R^0\) are the initial concentrations of the oxidized and reduced forms of the metal,
\(C_0^S\) and \(C_R^S\) are the concentrations of the oxidized and reduced forms of the metal at the electrode surface,
\(D_0\) and \(D_R\) are the corresponding diffusion coefficients.
Boundary condition (2), valid for semi-infinite diffusion, is approximate for limited diffusion.
The purpose of the present work is to estimate the limits of applicability of condition (2) for calculating \(K\) for limited-volume electrodes.
Let us solve the boundary-value problem for a planar electrode under the following initial and boundary conditions:
\[ \frac{\partial C_R}{\partial t} = D_R \frac{\partial^2 C_R}{\partial x^2}, \tag{3} \]
\[ t = 0 \qquad C_R(x,0)=C_R^0 \]
\[ t>0 \qquad C_R(x,t)=f(t), \qquad D_R \left.\frac{\partial C_R}{\partial x}\right|_{x=0}=0. \tag{3a} \]
The value of the function \(f(t)\) is found from the boundary condition (2) and the Nernst equation. The solution of this problem, obtained by the operational method, has the following form:
\[ i=\frac{z^{2}F^{2}\omega D_R S\left(C_0^0+C_R^0\right)}{2RTl} \int_0^t \frac{\displaystyle \sum_{n=1}^{\infty}\exp\left[-\left(\mu_n^2D_R\frac{t-\tau}{l^2}\right)\right]} {\displaystyle \operatorname{ch}^2\frac{zF\omega}{2RT}\left(t_{1/2}-\tau\right)} \,d\tau , \tag{4} \]
where
\(z\) is the number of electrons in the electrochemical reaction,
\(F\) is the Faraday constant,
\(\omega\) is the rate of change of the potential,
\(S\) is the electrode surface,
\(l\) is the film thickness,
\(t_{1/2}\) is the time at which the half-wave potential is reached,
\[ \mu_n=(2n+1)\frac{\pi}{2}. \]
Under the conditions of the amalgam polarography method \(C_0^0 \ll C_R^0\); therefore the quantity \(C_0^0\) may be neglected. Taking this remark into account and introducing the following notation:
\[ \gamma=\frac{RT\pi^2}{zF4}\,\varepsilon^2, \tag{5a} \]
\[ \varepsilon=\frac{1}{l}\sqrt{\frac{D_R}{z\omega}}, \tag{5б} \]
\[ \beta=\frac{zF\omega}{RT}, \tag{5в} \]
we obtain
\[ i=\frac{z^{3/2}F^{3/2}\omega^{1/2}D_R^{1/2}}{R^{1/2}T^{1/2}}\,SC_R^0\psi(\varepsilon), \tag{6} \]
where
\[ \psi(\varepsilon)=\frac{\gamma^{1/2}}{\pi} \int_0^t \frac{\displaystyle \sum_{n=1}^{\infty}\exp-\left(\mu_n^2\gamma(\beta t-\beta\tau)\right)} {\displaystyle \operatorname{ch}^2\frac{1}{2}\left(\beta t_{1/2}-\beta\tau\right)} \,d\tau . \tag{7} \]
The value of the quantity \(\psi(\varepsilon)\) at any time depends on the parameter \(\gamma\), or on \(\varepsilon\), which are related to each other by relation (5a). Knowing the quantity \(\psi_n\) at the maximum point, from equations (6) and (1) it is easy to determine the anodic-current constant \(K_a\) for a known value of the parameter \(\varepsilon\):
\[ K_a=\frac{z^{3/2}F^{3/2}}{R^{1/2}T^{1/2}}\,\omega^{1/2}D_R^{1/2}\psi_n . \tag{8} \]
The numerical solution of expression (7) for various values of the parameter \(\varepsilon\) is presented in Fig. 1 in the coordinates \(\dfrac{K_a}{K_0}\)—\(\varepsilon\).
\[ \left(K_0=2.69\cdot10^5\,z^{3/2}\omega^{1/2}D_R^{1/2}\right) \]
is the value of the anodic-current constant under the condition of semi-infinite linear diffusion.
Since the literature contains an exact equation of the reversible voltammetric curve on a mercury-film electrode [1], it is possible
estimate the range of values of the parameter $\varepsilon$ where the solutions of the exact and approximate problems will coincide with the required accuracy. In Fig. 1 (curve 1) the data of [1] are presented. From a comparison of curves 1 and 2 it is seen that the approximate equation (6) can be used only for values of the parameter $\varepsilon \leqslant 1$, where the error does not exceed 10%. At larger values of the parameter $\varepsilon$ the error increases with increasing $\varepsilon$ and already at $\varepsilon = 6$ reaches 200%.
In the same figure (curve 4) the data of [5], obtained for a spherical electrode with boundary condition (2), are presented. As is evident from the above considerations, curve 4 is approximate, but it is well justified for values $\varepsilon \leqslant 1$. In [3] an experimental verification was carried out of the voltammetric curve obtained with boundary condition (2) at a spherical electrode, and the authors of that work showed that the theoretical and experimental curves coincide. This agreement between the theoretical and experimental curves can be explained by the fact that the error under their working conditions did not exceed 5% ($\varepsilon = 0.4$). Experimental verification of curve 4, which we have performed [6], shows (curve 3) that as the parameter $\varepsilon$ is increased the discrepancy between the theoretical and experimental data increases. This can be explained by the fact that boundary condition (2) does not take into account the limited nature of diffusion in the mercury spherical electrode.
Fig. 1. Dependence of the anodic-current constant on the parameter $\varepsilon$. 1—exact dependence for a plane electrode; 2—approximate dependence for a plane electrode; 3—experimental data for a spherical electrode; 4—approximate dependence for a spherical electrode.
From a comparison of curves 1, 2 and 3, 4 it is seen that the value of the anodic-current constant $K_a$ for a plane electrode is always higher than $K_a$ at a spherical electrode. The information available in the literature that the value of $K_a$ for a film electrode is smaller than for a sphere [7] can be explained by the fact that the values of the anodic-current constants were compared at different values of the parameter $\varepsilon$. Under the usual conditions of amalgam polarography with accumulation, for a spherical electrode $\varepsilon \simeq 1$, while for a film electrode $\varepsilon \simeq 30$.
In conclusion, one may note the universality of the chosen system of coordinates, $\dfrac{K_a}{K_0} - \varepsilon$. For example, in [1] the data from which curve 1 (Fig. 1) was constructed are presented in the form of two graphs with eleven curves.
Conclusions
- An evaluation has been made of the limits of applicability of boundary condition (2); it has been shown that, with sufficient accuracy, it can be applied to a spherical electrode under the conditions of amalgam polarography with accumulation.
... of the resolving power of the APN method when the rate of potential variation is decreased.
The experimental data confirm the theory we have developed for reversible anodic peaks at stationary mercury electrodes [4].
REFERENCES
- P. Delahay. New Instruments and Methods in Electrochemistry. IL, Moscow, 1957.
- R. S. Nicholson, I. Shain, Anal. Chem., 36, 706, 1964.
- M. M. Nicholson, J. Am. Chem. Soc., 76, 2539, 1954.
- V. E. Gorodovykh. Candidate’s dissertation. Tomsk, 1964.