Anodic amalgam voltammetry at constant electrode potential. Irreversible processes
Abstract Generated abstract
A general theoretical treatment is developed for anodic amalgam voltammetry at constant electrode potential on stationary mercury film and spherical electrodes when the electrodissolution process is irreversible and first order. The authors solve Fick’s second law with uniform initial metal distribution, no convection in the amalgam, and a kinetic boundary condition, using Laplace transforms and Bessel functions to obtain a unified expression for the concentration distribution and the current, time curve. The formulation yields the known separate characteristic equations and coefficients for planar and spherical geometries, shows that the surface concentration tends to zero at long times, and provides both a rapidly convergent long time series and an approximate short time expression involving the complementary error function.
Full Text
ANODIC AMALGAM VOLTAMMETRY AT A CONSTANT ELECTRODE POTENTIAL. IRREVERSIBLE PROCESSES
M. S. ZAKHAROV, V. V. PNEV
(Presented by the scientific seminar of the problem laboratory of physicochemical methods of analysis)
The theoretical questions of anodic amalgam voltammetry (AAV) at a constant potential on stationary mercury electrodes have previously been considered by a number of authors [1—5]. However, in the cited works the derivation of the basic equation (the equation of the \(i—t\) curve) of the method under consideration was carried out separately for spherical and film mercury electrodes.
In the present work a general approach is proposed for solving the basic problem (obtaining the equation of the \(i—t\) curve) of AAV on spherical and film mercury electrodes for irreversible processes. The problem is solved under the following conditions: 1) the metal in the amalgam is distributed uniformly; 2) convection of metal atoms in the amalgam is absent; 3) an irreversible first-order reaction proceeds at the electrode.
To derive the equation of the \(i—t\) curve it is necessary to find the distribution of the metal in the amalgam when a constant potential is applied to the electrode, by solving the equation of Fick’s second law:
\[ \frac{\partial C_R(x,\vartheta_R)}{\partial \vartheta_R} = \frac{\partial^2 C_R(x,\vartheta_R)}{\partial x^2} + \frac{\Gamma}{x}\frac{\partial C_R(x,\vartheta_R)}{\partial x} \tag{1} \]
under the following initial and boundary conditions:
\[ \vartheta_R = 0 \qquad C_R(x,0)=C_R^0, \tag{2} \]
\[ \vartheta_R > 0 \qquad \lim_{x\to 0}\frac{\partial C_R(x,\vartheta_R)}{\partial x}=0, \tag{3} \]
\[ \frac{\partial C_R(1,\vartheta_R)}{\partial x} = -\lambda C_R(1,\vartheta_R), \tag{4} \]
where \(C_R(x,\vartheta_R)\) is the concentration of the metal in the amalgam, g-atom/ml;
\[ x=\frac{y}{y_0} \]
is the dimensionless coordinate; \(y_0\) is the thickness of the mercury film in the case of a film electrode, cm; \(y_0\) is the radius of the mercury spherical electrode, cm; \(\vartheta_R=\dfrac{D_R t}{y_0^2}\) is the dimensionless time; \(t\) is time, sec; \(D_R\) is the diffusion coefficient of metal atoms in mercury, \(\text{cm}^2\text{sec}^{-1}\); \(\Gamma\) is the coef-
the electrode shape coefficient; set \(\Gamma=2\gamma+1\), where \(\gamma=-1/2\) for a film electrode and \(\gamma=1/2\) for a spherical electrode;
\[ \lambda \equiv \frac{\kappa_b y_0}{D_R};\qquad \kappa_b=\kappa_s \exp\left[\frac{\beta zF}{RT}\left(\varphi-\varphi^\circ\right)\right], \tag{5} \]
where \(\kappa_b\) and \(\kappa_s\) are the rate constants of the electrode process at, respectively, any \((\varphi)\) and standard \((\varphi^\circ)\) electrode potentials, cm sec\(^{-1}\); \(\beta\) is the transfer coefficient; \(C_R(1,\vartheta_R)\) is the concentration of the metal at the electrode surface \((x=1)\). Applying the Laplace transform to equation (1), we obtain:
\[ C_R''+\frac{2\gamma+1}{x}C_R'-SC_R=0. \tag{6} \]
The general solution of equation (6) has the form:
\[ \overline{C}_R(x,S)=A_1(\sqrt{S}x)^{-\gamma}I_\gamma(\sqrt{S}x) +A_2(\sqrt{S}x)^{-\gamma}K_\gamma(\sqrt{S}x), \tag{7} \]
where \(A_1\) and \(A_2\) are constants; \(I_\gamma\) and \(K_\gamma\) are Bessel functions of the first and second kind of order \(\gamma\) with a purely imaginary argument.
\[ I_\gamma(u)\equiv \sum_{\kappa=0}^{\infty} \frac{1}{\kappa!\,\Gamma(\gamma+\kappa)} \left(\frac{u}{2}\right)^{\gamma+2\kappa} \tag{8} \]
\[ K_\gamma(u)\equiv \frac{\pi}{2\sin \pi\gamma}\left[I_{-\gamma}(u)-I_\gamma(u)\right], \tag{9} \]
\(\Gamma(u)\) is the gamma function.
Taking (8, 9) into account, expression (7) is reduced to the form:
\[ C_R(x,s)=\frac{C_R^0}{s} -(x)^{-\gamma} \frac{I_\gamma(\sqrt{s}x)} {s\left[I_\gamma(\sqrt{s})+\frac{1}{\lambda}\sqrt{s}\,I_{\gamma+1}(\sqrt{s})\right]}, \tag{10} \]
Passing to the original, we obtain:
\[ C_R(x,\vartheta_R)= C_R^0 \sum_{n=1}^{\infty} A_n\left(\frac{1}{x}\right)^\gamma \frac{I_\gamma(\mu_n x)}{I_\gamma(\mu_n)} \exp(-\mu_n^2\vartheta_R), \tag{11} \]
where
\[ A_n=\frac{2\lambda}{(\lambda-2\gamma)\lambda+\mu_n^2}, \tag{12} \]
\(\mu_n\) are the roots of the characteristic equation:
\[ \frac{I_\gamma(\mu_n)}{I_{\gamma+1}(\mu_n)} =\frac{1}{\lambda}\mu_n. \tag{13} \]
Taking into account the relation of Bessel functions to trigonometric functions [6, p. 143], from equation (11) it is easy to obtain particular solutions for film and spherical electrodes.
For the concentration at the electrode surface \((x=1)\), equation (11) gives:
\[ C_R(1,\vartheta_R)= C_R^0 \sum_{n=1}^{\infty} A_n \exp(-\mu_n^2\vartheta_R). \tag{14} \]
It follows from equation (14) that \(\lim_{\vartheta_R\to\infty} C_R(1,\vartheta_R)=0\) (\(y_0\) and \(D_R\) are finite), i.e., in irreversible electrodissolution the concentration of the metal
in the amalgam, after the lapse of a certain sufficiently long time, falls to zero.
Taking into account the value of \(\gamma\) for planar and spherical electrodes, we obtain:
\[ A_n^{\mathrm{pl}}=\frac{2\lambda}{(\lambda+1)\lambda+\mu_n^2}; \tag{15} \]
\[ A_n^{\mathrm{sph}}=\frac{2\lambda}{(\lambda-1)\lambda+\mu_n^2}; \tag{16} \]
Bearing in mind that
\[ I_{-\frac{1}{2}}(z)=\left(\frac{2}{\pi z}\right)^{1/2}\cos z;\quad I_{1/2}(z)=\left(\frac{2}{\pi z}\right)^{1/2}\sin z; \]
\[ I_{3/2}(z)=-\left(\frac{2}{\pi z}\right)^{1/2}\left(\sin z+\frac{\cos z}{z}\right) \]
the characteristic equations for the planar and spherical electrodes, respectively, can be written as follows:
\[ \text{for the planar electrode}\quad \operatorname{ctg}\mu=\frac{1}{\lambda}\mu, \tag{17} \]
\[ \text{for the spherical electrode}\quad \operatorname{tg}\mu=-\frac{\mu}{\lambda-1}. \tag{18} \]
The roots of these equations are given in [7, pp. 155, 177]. Using the values \(\mu_n\), it is easy to calculate the coefficients \(A_n\) for the corresponding values of the parameter \(\lambda\).
The expression for the current in the process of irreversible electro-dissolution of the amalgam at an electrode of any shape will have the form:
\[ i(t)=zFSK_s\exp\left[\frac{\beta zF}{RT}(\varphi-\varphi^\circ)\right]C_R^0\sum_{n=1}^{\infty}A_n\exp(-\mu_n^2\vartheta_R). \tag{19} \]
The series in expression (19) converges rapidly for large values of \(\vartheta_R\). It can be shown that, with an error of no more than \(a\%\), in expression (19) one may restrict oneself to the first term of the series when
\[ \vartheta_R>\frac{1}{\mu_2^2-\mu_1^2}\ln\frac{100B_2}{aB_1}, \tag{20} \]
where
\[ B_n=\frac{2\lambda}{\lambda^2-\lambda+\mu_n^2}. \tag{20} \]
For small values of \(\vartheta_R\), calculations by formula (19) are cumbersome, since several terms of the series have to be taken into account. Therefore, for small values of \(\vartheta_R\) it is convenient to find the solution of the problem in another form. At the electrode surface, the solution in transforms (10) can be brought to the form:
\[ \bar{C}_R(1,s)=\frac{C_R^0}{s} -\frac{I_\gamma(\sqrt{s})}{s\left[I_\gamma(\sqrt{s})+\frac{1}{\lambda}\sqrt{s}\,I_{\gamma+1}(\sqrt{s})\right]} = \]
\[ =\frac{C_R^0}{s} -\frac{1}{s\left[1+\frac{\sqrt{s}}{\lambda}\cdot\frac{I_{\gamma+1}(\sqrt{s})}{I_\gamma(\sqrt{s})}\right]}. \tag{21} \]
Since
\[ \frac{I_{\gamma+1}(z)}{I_\gamma(z)} = \frac{ 1-\dfrac{(4\gamma+1)^2-1^2}{1!\,8z} }{ 1-\dfrac{4\gamma^2-1^2}{1!\,8z} +\dfrac{(4\gamma^2-1^2)(4\gamma^2-3^2)}{2!\,(8z)^2} -\ldots }, \]
then
\[ \bar C_R(1,s)=\frac{C_R^0}{s} - \frac{1}{s\left[1+\dfrac{\sqrt{s}}{\lambda}-\dfrac{a}{\lambda}+\ldots\right]}. \tag{22} \]
The original of this image has the form:
\[ C_R(1,\nu_R)\simeq C_R^0-\frac{\lambda}{\lambda-a} \left\{1-\exp\left[(\lambda-a)^2\nu_R\right]\operatorname{erfc}\left[(\lambda-a)\sqrt{\nu_R}\right]\right\}. \tag{23} \]
where \(a=\gamma+\dfrac{1}{2}\), i.e., \(a=0\) for a film electrode and \(a=1\) for a spherical electrode. As can be seen from equation (23), at \(\nu_R=0\) \((t=0)\), \(C_R(1,0)=C_R^0\), which corresponds to the initial condition of the problem.
Equation (23) coincides with the analogous equations obtained in [4, 5].
Possible applications of the AAV method at a constant electrode potential have already been discussed in [4, 5].
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