Anodic Amalgam Voltammetry with Programmed Current on a Spherical Mercury Electrode
Abstract Generated abstract
This paper develops theoretical relations for anodic amalgam voltammetry with programmed current at a spherical mercury electrode, with comparison to film electrodes. Starting from concentration distributions for reduced and oxidized forms under diffusion-limited conditions, it derives expressions for surface concentrations and transition times when the current follows a power law, with particular attention to constant and linearly varying current. The analysis gives transition-time equations for single and complex amalgams, showing that under linearly varying current the oxidation time of a more electropositive component depends on the concentration of the more electronegative component. It also derives time dependences of electrode potential for reversible and completely irreversible processes at a spherical electrode.
Full Text
ANODIC AMALGAM VOLTAMMETRY WITH PROGRAMMED CURRENT AT A SPHERICAL MERCURY ELECTRODE
V. V. Pnev, M. S. Zakharov
(Presented by the scientific seminar of the Department of Physical and Colloid Chemistry)
In the present work we consider several theoretical questions of the AAV method with programmed current. In [1], under conditions of limited diffusion for the case of a single electrode reaction not complicated by kinetic effects, the following relations were obtained for the concentration distribution of the oxidized \((O)\) and reduced \((R)\) forms of the element at the electrode surface for an arbitrary current form:
\[ C_R(y_0,t)=C_R^0-\frac{1}{y_0}\int_0^t q(\xi)\left\{(2\gamma+2)+2\sum_{n=1}^{\infty}\exp\left[-\mu_n^2\frac{D_R}{y_0^2}\times \right.\right. \tag{1} \]
\[ \left.\left. \times(t-\xi)\right]\right\}\,d\xi=C_R^0-I_1, \]
\[ C_O(y_0,t)=C_O^0+\frac{1}{\pi^{1/2}D_0^{1/2}} \int_0^t q(\xi)\left\{ \frac{1}{\sqrt{t-\xi}} -\frac{a\pi^{1/2}D_0^{1/2}}{y_0} +\frac{2a^2\sqrt{t-\xi}\,D_0}{y_0^2} \right. \]
\[ \left. -\frac{a^3D_0^{3/2}(t-\xi)}{\pi^{1/2}y_0^3} +\ldots\right\}d\xi=C_O^0+I_2, \tag{2} \]
where \(y_0=r_0\) is the radius for a spherical electrode; \(y_0=l\) is the thickness of the mercury film for a film electrode; \(a=\gamma+\frac{1}{2}\), \(\gamma=-1/2\) for a film electrode; \(\gamma=1/2\) for a spherical electrode; \(\xi\) is an auxiliary variable.
Let us consider AAV when the current at the electrode changes according to the law \(i_0(t)=bt^m\), where \(b\) is the proportionality coefficient; \(m\ge 0\). In this case
\[ I_1=\frac{b}{zFy_0}\left[(2\gamma_R+2)\frac{t^{m+1}}{m+1} +\frac{2t^m}{\gamma_R}\sum_{n=1}^{\infty}\mu_n^{-2} +2\Phi(t)\right], \tag{3} \]
where
\[ \gamma_R=\frac{D_R}{y_0^2},\qquad \Phi(t)=2\sum_{n=1}^{\infty}\sum_{k=1}^{m} b(-1)^k \frac{m(m-1)\ldots(m-k+1)} {\left(\gamma_R\mu_n^2\right)^{k+1}} \cdot t^{m-k}. \]
From the general expression (3) one can obtain a number of particular equations for different values of the exponent \(m\). a) \(m=0\). For a film electrode, the expression for the surface concentration of the reduced form of the element will have the form
\[ C_R(l,t)=C_R^0-\frac{bt}{zFl}-\frac{bl}{3zFD_R}. \tag{4} \]
For a spherical electrode, the analogous expression is written as follows:
\[ C_R(r_0,t)=C_R^0-\frac{3bt}{zFr_0}-\frac{br_0}{5zFD_R}. \tag{5} \]
Expressions for the transition time are obtained under the condition \(C_R(r_0,\tau)=0\). b) \(m=1\)—the current changes according to a linear law. In this case, for a film electrode, from equation (3), taking equation (1) into account, one obtains
\[ C_R(l,t)=C_R^0-\frac{b}{zFl} \left[ \frac{t^2}{2}+\frac{2l^2t}{D_R}\frac{\xi(2)}{\pi^2} -\frac{2l^4}{D_R^4}\frac{\xi(4)}{\pi^4} \right], \tag{6} \]
where \(\xi\) is the Riemann zeta function. Since \(\xi(2)=\frac{\pi^2}{6}\), and \(\xi(4)=\frac{\pi^4}{90}\), then
\[ C_R(l,t)=C_R^0-\frac{b}{zFl} \left[ \frac{t^2}{2}+\frac{l^2t}{3D_R}-0.22\frac{l^4}{D_R^4} \right]. \tag{7} \]
The expression for the transition time is obtained under the condition \(C_R(l,\tau)=0\):
\[ C_R^0=\frac{b}{zFl} \left[ \frac{\tau^2}{2}+\frac{l^2\tau}{3D_R}-0.22\frac{l^4}{D_R^4} \right]. \tag{8} \]
For \(\frac{l^2}{D_R}\leqslant 10^{-1}\ \mathrm{sec}^{-1}\) and \(\tau\geqslant 6.7\ \mathrm{sec}\), expression (8), with an error of less than 1%, can be reduced to the form
\[ \tau^2=\frac{2zFlC_R^0}{b}. \tag{9} \]
In an analogous way, for a spherical electrode one can obtain the following relation for the transition time:
\[ C_R^0=\frac{b}{zFr_0} \left( 1.5\tau^2+0.2\frac{r_0^2\tau}{D_R} -0.06\frac{r_0^4}{D_R^2} \right). \tag{10} \]
For a spherical electrode \(\frac{r_0^2}{D_R}\gg 1\), and therefore a simple dependence of the type (9) cannot be obtained here. Let us consider the electrooxidation of a complex amalgam at a linearly varying electrode current*.
Following the method of treating the problem of electrooxidation of complex amalgams proposed in [2], for a film electrode, at
\[ \frac{l^2}{D_R}\leqslant 10^{-1}\ \mathrm{sec}^{-1} \]
and \(\tau\geqslant 6.7\ \mathrm{sec}\), one can write the following relation:
\[ \left(\sum_{m=1}^{m}\tau_m\right)^2= \frac{zFl}{b}\sum_{m=1}^{m} z_m C_{R,m}^0. \tag{11} \]
The expression for the transition time in the electrooxidation from a complex amalgam of the \(m\)-th component will have the form
\[ \left(\sum_{m=1}^{m}\tau_m\right)^2- \left(\sum_{m=1}^{m-1}\tau_m\right)^2 = \frac{zFl}{b}z_m C_{R,m}^0. \tag{12} \]
* The processes of electrooxidation of metals proceed consecutively.
For the element oxidized second, from formula (12) one can obtain the following expression:
\[ (\tau_1+\tau_2)^2-\tau_1^2=\frac{zFl}{b}\,z_2 C_{R,2}^{0}. \tag{13} \]
In the electrooxidation of a complex amalgam from a spherical electrode, the expression for the transition time is written as follows:
\[ 1.5\left(\sum_{m=1}^{m}\tau_m\right)^2 +0.2\,\frac{r_0^2}{D_R}\sum_{m=1}^{m}\tau_m = \frac{Fr_0}{b}\sum_{m=1}^{m}z_m C_{R,m}^{0} -\frac{0.06r_0^4}{D_R^2}. \tag{14} \]
Here it is assumed that \(D_{R,m}=D_R\). From equations (12–14) it is evident that, in AAV with a linearly varying current, the transition time of the oxidation process of the more electropositive metal from a complex amalgam depends on the concentration (through \(\tau\)) of the more electronegative metal.
With a linearly varying electrode current, equation (2) for determining the concentration of the oxidized form of the element at the electrode surface is reduced to the form
\[ I_2=\frac{bt^{3/2}}{zFD_0^{1/2}} \left[ \frac{4}{3\sqrt{\pi}} -\frac{1}{2}\left(\frac{D_0t}{r_0^2}\right)^{1/2} +\frac{8}{15\sqrt{\pi}}\left(\frac{D_0t}{r_0^2}\right) -\frac{1}{24}\left(\frac{D_0t}{r_0^2}\right)^{3/2} +\frac{16}{105\sqrt{\pi}}\left(\frac{D_0t}{r_0^2}\right) -\ldots \right]. \tag{15} \]
Substituting into the Nernst equation, instead of \(C_R(r_0,t)\) and \(C_0(r_0,t)\), their values from equations (1–2), taking into account (10, 15), and expressing \(C_R\) from equation (11), we obtain the following dependence of the electrode potential on time for reversible processes with a linearly varying current at a spherical electrode:
\[ \varphi=\varphi_{1/2} -\frac{RT}{zF}\ln \frac{b\left[1.5(\tau^2-t^2)+0.2\gamma R(\tau-t)\right]} {zFr_0\left(C_0^{0}-I_2^{1}\right)}. \tag{16} \]
For completely irreversible processes, the dependence of the electrode potential on time will have the form
\[ \varphi=\frac{RT}{\beta zF}\ln\frac{r_0}{k_s} +\ln\frac{t}{1.5(\tau^2-t^2)+0.2\gamma R(\tau-t)}. \tag{17} \]
References
- M. S. Zakharov, V. V. Pnev. Theory and Practice of Amalgam Processes. Abstracts of papers at the All-Union Conference, Alma-Ata, October, p. 33, 1966.
- R. W. Murray, C. N. Reilley, J. Electroanal. Chem., 3, 64, 1962.