Voltammetry under conditions of semi-infinite spherical and cylindrical diffusion. The electrode process complicated by a subsequent first-order chemical reaction
Abstract Generated abstract
This paper derives voltammetric relations for electrode processes followed by first-order chemical reactions under semi-infinite spherical and cylindrical diffusion. It formulates surface concentration expressions for oxidized and reduced species, then substitutes them into the Nernst equation to obtain potential, time and current, potential curve equations for reversible, quasireversible, and selected irreversible chemical follow-up cases. The treatment includes constant-current conditions and more general current or potential programs through convolution integrals, with limiting series for cylindrical diffusion and corresponding spherical expressions. The resulting integral equations are proposed for reduction by approximate numerical integration and computer evaluation to generate voltammetric curves and estimate kinetic parameters.
Full Text
Voltammetry under Conditions of Semi-Infinite Spherical and Cylindrical Diffusion. Electrode Process Complicated by a Subsequent First-Order Chemical Reaction
M. S. Zakharov, V. I. Bakanov
(Presented by the scientific seminar of the Department of Physical and Colloid Chemistry)
In the present work, equations of the \(\varphi—t\) and \(i—\varphi\) curves will be obtained for electrode processes complicated by a subsequent first-order chemical reaction under conditions of semi-infinite spherical and cylindrical diffusion.
Let the reaction occurring at the electrode be
\[ O \xrightarrow{ze} R \underset{k_2}{\stackrel{k_1}{\rightleftarrows}} Y, \tag{1} \]
where \(K_1\) and \(K_2\) are the rate constants of the chemical reaction. The substance \(Y\) is neither reduced nor oxidized at the potential at which the substance \(O\) is reduced.
To find the equation of the \(\varphi—t\) curve, expressions are needed for the concentrations at the electrode surface. The chemical reaction proceeding after the electrode process has no effect on the concentration distribution of the substance \(O\). The expressions for \(C_0(1,\vartheta)\) at constant current are as follows:
for the sphere [1]
\[ C_0(1,\vartheta)=C_0^0-\lambda\left[1-\exp \vartheta \operatorname{erfc}\sqrt{\vartheta}\right], \tag{2} \]
for the cylinder at small \(\vartheta\) [2]
\[ C_0(1,\vartheta)=C_0^0-\lambda\left[2\sqrt{\frac{\vartheta}{\pi}}-\frac{\vartheta}{2}+\frac{\vartheta^{3/2}}{2\sqrt{\pi}}-\frac{3\vartheta^2}{16}+\ldots\right], \tag{3} \]
where
\[ \lambda=\frac{i_0 r_0}{zFD}, \]
At \(\vartheta \geq 9\cdot 10^{-4}\), equation (3) describes the concentration distribution for a plane electrode. The expressions for \(C_R(1,\vartheta)\) under conditions of spherical and cylindrical diffusion have, respectively, the following form:
\[ C_R^{\mathrm{sph}}(1,\vartheta)= \frac{\lambda}{1-K} \left( 1-\exp \vartheta \operatorname{erfc} c_1\sqrt{\vartheta} - \frac{K\lambda}{(1+K)(1-z_1)} \right) \times \]
\[ \times \left[ 1-\sqrt{z}\operatorname{erf}\sqrt{z\vartheta} -\exp \vartheta\, \operatorname{erfc} c_1\sqrt{\vartheta}\exp(-z\vartheta) \right]; \tag{4} \]
\[ C_R^{\mathrm{cyl}}(1,\vartheta)= \frac{\lambda}{1+K} \left[ 2\sqrt{\frac{\vartheta}{\pi}}-\frac{\vartheta}{2}+\frac{\vartheta^{3/2}}{2\sqrt{\pi}}+\ldots \right] + \frac{K\lambda}{1+K} \left[ \frac{1}{z}\times \right. \]
\[ \times \operatorname{erf}\sqrt{\chi\vartheta} +\frac{\exp(-\chi\vartheta)}{2\chi} +\frac{3\operatorname{erf}\sqrt{\chi\vartheta}}{8\chi^{3/2}} -\frac{3}{4}\sqrt{\frac{\vartheta}{\pi}}\exp(-\chi\vartheta)\chi^{-1}+\ldots \right]. \tag{5} \]
For \(\vartheta \leqslant 9\cdot 10^{-4}\) and \(\chi\vartheta \geqslant 0.9\), equation (5) is valid for a plane electrode.
Substituting the equations for \(C_R(1,\vartheta)\) and \(C_O(1,\vartheta)\) into the Nernst equation for a reversible electrode process complicated by a subsequent reversible first-order chemical reaction, under conditions of semi-infinite spherical and cylindrical diffusion, we obtain
\[ \varphi^{\mathrm{sph}}_{(\vartheta)} = \varphi_{1/2} +\frac{RT}{zF} \ln\left( \exp\vartheta\,\operatorname{erfc}\sqrt{\vartheta} -\exp\vartheta'\operatorname{erfc}\sqrt{\vartheta'} \right) - \frac{RT}{zF}\ln\frac{1}{1+K} - \frac{RT}{zF}\ln\left\{ \left( (1-\exp\vartheta\,\operatorname{erfc}\sqrt{\vartheta} +\frac{K}{1-\chi}) \right)\right. \]
\[ \left. \times \left[ 1-\sqrt{\chi}\operatorname{erf}\sqrt{\chi\vartheta} -\exp\vartheta\,\operatorname{erfc}\sqrt{\vartheta}\exp(-\chi\vartheta) \right] \right\}, \tag{6} \]
\[ \varphi^{\mathrm{cyl}}_{(\vartheta)} = \varphi_{1/2} +\frac{RT}{zF} \ln \frac{ \left( 2\sqrt{\dfrac{\vartheta'}{\pi}} -\dfrac{\vartheta'}{2} +\dfrac{\vartheta'^{3/2}}{2\sqrt{\pi}} -\dfrac{3\vartheta'^2}{16} +\ldots \right) }{ \dfrac{1}{1+K} \left( 2\sqrt{\dfrac{\vartheta}{\pi}} -\dfrac{\vartheta}{2} +\dfrac{\vartheta^{3/2}}{2\sqrt{\pi}} +\ldots \right) } - \]
\[ \frac{ -2\left( \sqrt{\dfrac{\vartheta}{\pi}} -\dfrac{\vartheta}{2} +\dfrac{\vartheta^{3/2}}{2\sqrt{\pi}} -\dfrac{3\vartheta^2}{16} -\ldots \right) }{ +\dfrac{K}{1+K} \left[ \dfrac{1}{\chi}\operatorname{erf}\sqrt{\chi\vartheta} +\dfrac{\exp(-\chi\vartheta)}{2\chi} +\dfrac{3\operatorname{erf}\sqrt{\chi\vartheta}}{8\chi^{3/2}} -\frac{3}{4}\sqrt{\frac{\vartheta}{\pi}} \frac{\exp(-\chi\vartheta)}{\chi} +\ldots \right] }. \tag{7} \]
Using equations (6) and (7) at \(\vartheta=\dfrac{\vartheta'}{2}\), knowing \(K\), \(\varphi_{\vartheta_{1/2}}\), \(\varphi_{1/2}\), the quantities \(\kappa_1\) and \(\kappa_2\) are determined by the method of fitting. If the electrode process is complicated by a subsequent irreversible first-order chemical reaction \(R \xrightarrow{ze} O \xrightarrow{\kappa_1} Y\), then the expression for \(C_R(1,\vartheta)\) has the form:
for a spherical electrode
\[ C_R(1,\vartheta) = \frac{\lambda}{1-\chi_1} \left[ 1-\sqrt{\chi_1}\operatorname{erf}(\chi_1\vartheta)^{1/2} -\exp(\vartheta-\chi_1\vartheta)\operatorname{erfc}\vartheta^{1/2} \right], \tag{8} \]
where
\[ \chi_1=\frac{\kappa_1 r_0^2}{D}; \]
for a cylindrical electrode
\[ C_R(1,\vartheta) = \lambda\left[ \frac{\operatorname{erf}\sqrt{\chi_1\vartheta}}{\sqrt{\chi_1}} +\frac{\exp(-\chi_1\vartheta)}{2\chi_1} +\frac{3\operatorname{erf}\sqrt{\chi_1\vartheta}}{8\chi_1^{3/2}} -\frac{3}{4}\sqrt{\frac{\vartheta}{\pi}} \frac{\exp(-\chi_1\vartheta)}{\chi_1} +\ldots \right]. \tag{9} \]
Substituting equations (2) and (8) into the Nernst equation, we obtain the equations of the \(\varphi\)—\(t\) curves for an electrode process complicated by a subsequent irreversible first-order chemical reaction:
\[ \varphi^{\mathrm{sph}}_{(\vartheta)} = \varphi_{1/2} +\frac{RT}{zF} \ln\left( \exp\vartheta\,\operatorname{erfc}\sqrt{\vartheta} -\exp\vartheta'\operatorname{erfc}\sqrt{\vartheta'} -\frac{RT}{zF} \right) \times \]
\[ \times \ln \frac{1}{1-\chi_1} \left[ 1-\sqrt{\chi_1}\operatorname{erf}\sqrt{\chi_1\vartheta} -\exp(\vartheta-\chi_1\vartheta)\operatorname{erfc}\vartheta^{1/2} \right], \tag{10} \]
\[ \varphi^{\mathrm{cyl}}_{(\vartheta)} = \varphi_{1/2} +\frac{RT}{zF} \left[ \ln\left( 2\sqrt{\frac{\vartheta'}{\pi}} -\frac{\vartheta'}{2} +\frac{\vartheta'^{3/2}}{2\sqrt{\pi}} -\frac{3\vartheta'^2}{16} +\ldots \right) - \right. \]
\[ -\left(2\sqrt{\frac{\vartheta}{\pi}}-\frac{\vartheta}{2} +\frac{\vartheta^{3/2}}{2\sqrt{\pi}}-\frac{3\vartheta^{2}}{16}+\ldots\right) \left\{-\frac{RT}{zF}\ln\left[\frac{1}{\sqrt{\varkappa_1}}\operatorname{erf}\sqrt{\varkappa_1\vartheta} +\frac{\exp(-\varkappa_1\vartheta)}{2\varkappa_1} -\frac{3\operatorname{erf}\sqrt{\varkappa_1\vartheta}}{8\varkappa_1^{3/2}} -\frac{3}{4}\sqrt{\frac{\vartheta}{\pi}}\, \frac{\exp(-\varkappa_1\vartheta)}{\varkappa_1} +\ldots\right]\right\}. \tag{11} \]
For any form of variation of the electrode current (potential), the expressions for the concentrations in the cases under consideration are as follows: for semi-infinite spherical diffusion:
\[ C_0(1,\vartheta)=C_0^0-\frac{r_0}{\sqrt{\pi D}}\,I_1+\frac{r_0}{D}\,I_2, \tag{12} \]
where
\[ I_1=\int_0^\vartheta \frac{q(\xi)}{\sqrt{\vartheta-\xi}}\,d\xi, \qquad I_2=\int_0^\vartheta q(\xi)\exp(\vartheta-\xi)\operatorname{erfc}\sqrt{\vartheta-\xi}\,d\xi, \]
\[ C_R(1,\vartheta)=-\frac{r_0}{D(1+K)}\,I_3, \tag{13} \]
for an electrode process complicated by a subsequent reversible chemical reaction,
\[ I_3=\int_0^\vartheta q(\xi)\left[\frac{1}{\sqrt{\pi(\vartheta-\xi)}}-\exp(\vartheta-\xi)\operatorname{erfc}\sqrt{\vartheta-\xi}\right]\{1+K\exp[-\varkappa(\vartheta-\xi)]\}\,d\xi; \tag{14} \]
for semi-infinite cylindrical diffusion
\[ C_0(1,\vartheta)=C_0^0-\frac{r_0}{D\sqrt{\pi}}\,I_1+\frac{r_0}{2D}\,I_5, \tag{15} \]
where
\[ I_5=\int_0^\vartheta q(\xi)\left[\sqrt{\pi}-\frac{3}{2}(\vartheta-\xi)^{1/2} +\frac{3}{4}\sqrt{\pi(\vartheta-\xi)}-\ldots\right]d\xi; \tag{16} \]
\[ C_R(1,\vartheta)=-\frac{r_0}{D(1+K)}\,I_6, \tag{17} \]
for an electrode process complicated by a subsequent reversible chemical reaction,
where
\[ I_6=\int_0^\vartheta q(\xi)\left\{ \left[\frac{1}{\sqrt{\pi(\vartheta-\xi)}}-\frac{1}{2} +\frac{3}{4}\sqrt{\frac{\vartheta-\xi}{\pi}}+\ldots\right] +K\exp[-\varkappa(\vartheta-\xi)] \left[\frac{1}{\sqrt{\pi(\vartheta-\xi)}}-\frac{1}{2}+\ldots\right] \right\}d\xi. \tag{18} \]
The equations of the \(i\)–\(\varphi\) curve under conditions of semi-infinite spherical diffusion for a quasireversible electrode process complicated by a subsequent reversible first-order chemical reaction have the form
\[ I_{\mathrm{sph}}= ae^{y_1}\left(C_0^0-\frac{r_0 I_1}{\sqrt{\pi}D}+\frac{r_0 I_2}{D}\right) -be^{y_2}\frac{r_0}{D(1+K)}\,I_3, \]
where
\[ a=zFk_s e^{-\frac{\alpha zF}{RT}(\varphi_i-\varphi^0)}; \qquad b=zFk_s e^{\frac{\beta zF}{RT}(\varphi_i-\varphi^0)} \tag{19} \]
\[ \left. \begin{aligned} y_1&=-\frac{\alpha zF}{RT}\,\omega t,\\ y_2&=-\frac{\beta zF}{RT}\,\omega t, \end{aligned} \right\} \qquad \text{for voltammetry with a linearly varying electrode potential;} \]
\[ y_1=y_2=0 \]
for voltammetry with a constant electrode potential.
The consideration of irreversible electrode processes is of no interest, since in this case the subsequent chemical reaction does not affect the \(i\)—\(\varphi\) curve.
The equations of the \(i\)—\(\varphi\) curves for reversible electrode processes are obtained by substituting the expressions for \(C_R(1,\vartheta)\) and \(C_0(1,\vartheta)\) into the Nernst equation:
for spherical semi-infinite diffusion,
\[ \frac{(1+K)\left(C_0^0\pi^{1/2}D-r_0 I_1+\pi^{1/2}r_0 I_4\right)} {\pi^{1/2}r_0 I_3} =\Theta e^y, \tag{20} \]
where
\[ \Theta=\exp\left[\frac{zF}{RT}\left(\varphi_1-\varphi^0\right)\right], \]
\[ y=-\frac{zF}{RT}\omega t \]
for voltammetry with a linearly varying electrode potential,
\[ y=0 \]
for voltammetry with a constant electrode potential;
for cylindrical semi-infinite diffusion,
\[ \frac{\left(2\pi^{1/2}DC_0^0-2r_0 I_1+\pi^{1/2}r_0 I_5\right)(1+K)} {2\pi^{1/2}r_0 I_6} =\Theta e^y. \tag{21} \]
The integrals \(I_1\)—\(I_6\), by one of the methods of approximate integration, are reduced to algebraic equations, which are programmed and then evaluated on electronic computers to obtain the values \(i(t)\) for plotting the corresponding curves.
REFERENCES
- P. Delahay, C. C. Mattax, T. I. Berzins, J. Am. Chem. Soc., 76, 5319 (1954).
- M. S. Zakharov, V. V. Pnev. Izv. TPI (present collection).