Voltammetry with a Constant Electrode Current under Conditions of Semi-Infinite Spherical and Cylindrical Diffusion.

1. Electrode Processes Complicated by a Preceding Chemical Reaction

M. S. Zakharov, V. V. Pishev

(Presented by the scientific seminar of the problem laboratory
of physicochemical methods of analysis)

In the present work, some questions of voltammetry under conditions of semi-infinite spherical and cylindrical diffusion at constant current will be considered for electrode processes complicated by a preceding first-order chemical reaction.

Let us consider the reduction of a substance proceeding according to the scheme:

\[ Y \underset{k_2}{\stackrel{k_1}{\rightleftarrows}} O \xrightarrow{ze} R . \tag{1} \]

The substance \(Y\) is not reduced at the potential at which \(O\) is reduced. To obtain dependences of the electrode potential on time, it is necessary to find expressions for the distribution of the concentrations of substances \(O\) and \(R\) in solution. Expressions for the distribution of the concentration of substance \(R\) in solution under conditions of semi-infinite spherical and cylindrical diffusion are available in the literature. To find an expression for the distribution of the concentration of substance \(O\) in voltammetry with constant current, it is necessary to solve Fick’s second equation with kinetic terms for substances \(O\) and \(Y\):

\[ \frac{\partial C_O(X,\vartheta)}{\partial \vartheta} = \frac{\partial^2 C_O(X,\vartheta)}{\partial X^2} + \frac{\Gamma}{X}\frac{\partial C_O(X,\vartheta)}{\partial X} + \lambda_1\left[C_Y(X,\vartheta)-K_p C_O(X,\vartheta)\right], \tag{2} \]

\[ \frac{\partial C_Y(X,\vartheta)}{\partial \vartheta} = \frac{\partial^2 C_Y(X,\vartheta)}{\partial X^2} + \frac{\Gamma}{X}\frac{\partial C_Y(X,\vartheta)}{\partial X} - \lambda_1\left[C_Y(X,\vartheta)-K_p C_Y(X,\vartheta)\right], \tag{3} \]

with the following initial and boundary conditions:

\[ \vartheta=0 \qquad C_O(X,0)+C_Y(X,0)=C_O^0, \tag{4} \]

\[ \frac{C_Y(X,0)}{C_O(X,0)}=K_p, \tag{5} \]

\[ \vartheta>0 \quad \lim_{X\to\infty}\left[C_O(X,\vartheta)+C_Y(X,\vartheta)\right]=C_O^0, \tag{6} \]

\[ \frac{\partial C_O(1,\vartheta)}{\partial X}=\lambda, \tag{7} \]

\[ \frac{\partial C_Y(1,\vartheta)}{\partial X}=0, \tag{8} \]

where

\[ \lambda=\frac{i_0 r_0}{zFD}; \qquad K_p=\frac{k_2}{k_1}; \qquad X=\frac{r}{r_0}; \qquad \lambda_1=\frac{k_1 r_0^2}{D}. \]

The expression for the concentration at the surface is written in the following form:

\[ C_{0}^{\mathrm{sph}}(1,\vartheta)=\frac{1}{1+K}\left\{C_{0}^{0}-i_{0}\left(1-\exp \vartheta\, \operatorname{erfc}\sqrt{\vartheta}\right)-\right. \]
\[ \left.-K_{p}\frac{\lambda}{\kappa-1}\left[e^{-(\kappa-1)\vartheta}\cdot \operatorname{erfc}\sqrt{\vartheta}+\sqrt{\kappa}\,\operatorname{erf}\sqrt{\kappa\vartheta}-1\right]\right\}, \tag{9} \]

where

\[ \kappa=\lambda_{1}+\lambda_{2},\qquad \lambda_{2}=\frac{\kappa_{2}r_{0}^{2}}{D}. \]

Expanding the function \(\exp \vartheta\, \operatorname{erfc}\sqrt{\vartheta}\) in a Maclaurin series, it can be shown that the correction for sphericity may be neglected with an error up to \(a\%\) under the condition

\[ \vartheta \leq \frac{1.28a^{2}}{10^{4}}. \tag{10} \]

The expression for the concentration at the surface of a cylindrical electrode has the form

\[ C_{0}^{\mathrm{cyl}}(1,\vartheta)=\frac{1}{1+K_{p}}\left(C_{0}^{0}-\lambda S_{1}-\lambda K_{p}S_{2}\right), \tag{11} \]

\[ S_{1}=2\sqrt{\frac{\vartheta}{\pi}}-\frac{\vartheta}{2}+\frac{\vartheta^{3/2}}{2\sqrt{\pi}}-\frac{3\vartheta^{2}}{16}+\cdots . \tag{12} \]

\[ S_{2}=\frac{\operatorname{erf}\sqrt{\kappa\vartheta}}{\sqrt{\kappa}} -\frac{1}{2\kappa}\left[1-\exp(-\kappa\vartheta)\right] +\frac{3}{8}\kappa^{-3/2}\operatorname{erf}\sqrt{\kappa\vartheta}- \]
\[ -\frac{33\sqrt{\vartheta}}{84\pi^{1/2}\kappa}\exp(-\kappa\vartheta) -\frac{3}{8\kappa^{2}}\left[1-(1+k\vartheta)\exp(-k\vartheta)\right]. \tag{13} \]

The correction for cylindrical diffusion in the expression for \(S_{1}\) (equation 12) may be neglected with an error of \(a\%\) when

\[ \vartheta \leq \frac{5.1a^{2}}{10^{4}}. \tag{14} \]

The expression for the transition time in voltammetry with constant current is found from equation (9) under the condition \(C_{0}(1,\vartheta)=0\):

\[ i_{0}\left(1-\exp \vartheta'_{k}\,\operatorname{erfc}\sqrt{\vartheta'_{k}}\right) =\frac{C_{0}^{0}zFD}{r_{0}}-\frac{\kappa_{p}i_{0}}{\kappa-1}\times \]
\[ \times\left[e^{-(\kappa-1)\vartheta'_{k}}\operatorname{erfc}\sqrt{\vartheta'_{k}} +\sqrt{\kappa}\,\operatorname{erf}\sqrt{\kappa\vartheta'_{k}}-1\right], \tag{15} \]

where \(\vartheta'_{k}=D\tau_{k}/r_{0}^{2}\) is the dimensionless transition time for an electrode process complicated by a kinetic effect, \(\tau_{k}\) is the transition time, sec. From equation (15) it is seen that the plot in the coordinates \(i_{0}\left(1-\exp \vartheta'_{k}\operatorname{erfc}\sqrt{\vartheta'_{k}}\right),\, i_{0}\) is a straight line with a slope equal to

\[ -\frac{K_{p}}{\kappa-1}\left\{\exp\left[-(\kappa-1)\vartheta'_{k}\right]\operatorname{erfc}\sqrt{\vartheta'_{k}} +\sqrt{\kappa}\,\operatorname{erf}\sqrt{\kappa\vartheta'_{k}}-1\right\}. \]

Having determined the slope and knowing \(K_{p}\), the values of \(k_{1}\) and \(k_{2}\) can be calculated by the method of successive approximation.

The expression for the transition time under conditions of semi-infinite cylindrical diffusion will be determined by the following relation:

\[ i_{0}S'_{1,k}=\frac{C_{0}^{0}zFD}{r_{0}}-i_{0}K_{p}S'_{2,k}. \tag{16} \]

The values \(S'_{1,k}\) and \(S'_{2,k}\) are determined by equations (12) and (13), except that in the latter \(\vartheta\) must be replaced by \(\vartheta'_k\). From equation (16) it is seen that the plot in the coordinates \(i_0 S'_{1,k}\), \(i_0\) is a straight line with a slope tangent equal to \(-K_p S'_2\). Having determined the slope tangent and knowing \(K_p\), one can calculate \(\kappa_1\) and \(\kappa_2\) by the method of fitting.

The dependences of the electrode potential on time under conditions of semi-infinite spherical and cylindrical diffusion for reversible electrode processes are obtained by substituting into the Nernst equation, in place of the concentrations at the electrode surface, the expressions for \(C_O(1,\vartheta)\) and \(C_B(1,\vartheta)\):

for the sphere,

\[ \varphi=\varphi_{1/2}+\frac{RT}{zF}\ln \frac{\dfrac{1}{1+K_p}\left(P^1-P-Z-Z^1\right)}{P}, \tag{17} \]

where

\[ P=\lambda\left(1-\exp^{\vartheta}\operatorname{erfc}\sqrt{\vartheta}\right), \tag{18} \]

\[ Z=\frac{K_p\lambda}{k-1} \left\{ \exp[-(k-1)\vartheta]\operatorname{erfc}\sqrt{k\vartheta}-1 \right\}, \tag{19} \]

\(P^1\) and \(Z^1\) are described by equations analogous to (18) and (19), except that in the latter \(\vartheta\) is replaced by \(\vartheta'_k\);

for the cylinder,

\[ \varphi=\varphi_{1/2}+\frac{RT}{zF}\ln \frac{ \dfrac{1}{1+K_p} \left(\lambda S'_{1,k}+\lambda K_p S'_{2,k}-\lambda S_{1,k}-\lambda K_p S_{2,k}\right) }{ \lambda\mu S_{1,k} }. \tag{20} \]

For irreversible processes, the equations of the \(\varphi\)–\(t\) curves for spherical and cylindrical electrodes have, respectively, the following form:

\[ \varphi^{\mathrm{sph}}=\varphi^0-\frac{RT}{\alpha zF}\ln\frac{i_0K_s}{zF} +\frac{RT}{\alpha zF}\ln\frac{1}{1+K_p} \left(P-P^1-Z-Z^1\right); \tag{21} \]

\[ \varphi^{\mathrm{cyl}}=\varphi^0-\frac{RT}{\alpha zF}\ln\frac{i_0K_s}{zF} +\frac{RT}{\alpha zF}\ln\frac{1}{1+K_p}\times \]

\[ \times\left(\lambda S'_{1,k}+\lambda K_p S'_{2,k}-\lambda S_{1,k}-\lambda K_p S_{2,k}\right). \tag{22} \]