Anodic Amalgam Voltammetry with a Linearly Varying Potential. Equation of the Descending Branch of the Anodic Peak
Abstract Generated abstract
This paper derives equations for the descending branch of the anodic peak in anodic amalgam voltammetry on a film mercury electrode under a linearly varying potential, extending earlier treatment beyond low potential sweep rates. The authors formulate the problem as nonstationary linear diffusion in a mercury film with depletion at the electrode surface, solve the resulting boundary value problem, and obtain equivalent current expressions for different values of the diffusion time parameter, including limiting forms as functions of time and potential. The analysis predicts different dependencies at low and high sweep rates, including a high rate regime in which the current becomes independent of film thickness. Experiments with cadmium in potassium nitrate on a film electrode give linear logarithmic current versus time plots and are reported to support the derived low rate expression within the studied range.
Full Text
ANODIC AMALGAM VOLTAMMETRY WITH A LINEARLY VARYING POTENTIAL. EQUATION OF THE DESCENDING BRANCH OF THE ANODIC PEAK
M. S. Zakharov, L. F. Trushina
(Presented by the scientific-methodological seminar of KhTF)
Earlier, Igolinskii [1] obtained a simplified equation for the descending branch of the anodic peak in the method of “amalgam polarography with accumulation” (the APA method) on a film mercury electrode at low rates of change of the electrode potential, determined by the inequality
\[ \frac{zF\omega}{RT} \ll \frac{\pi^2 D_R}{l^2}, \]
for reversible processes.
Under these conditions the anodic current of the descending branch is determined by the rate of change of the potential and does not depend on the diffusion coefficient of the metal atoms in the amalgam.
It is of interest to obtain an equation for the descending branch of the anodic peak at any rate of change of the potential. The present work is devoted to this question.
Gorodovykh [2] has shown that, beginning with the potential
\[ \varphi = \varphi_{1/2} + \frac{0.069}{z} b, \]
the current of the anodic peak does not depend on the electrode potential and is determined only by the laws of diffusion, i.e., by the dimensions and shape of the electrode, etc. Beginning with the potential \(\varphi'(t')\), the concentration of metal atoms at the electrode surface becomes equal to zero, and a limiting nonstationary current flows through the electrode; consequently, beginning with \(\varphi'(t')\), the limiting nonstationary current coincides with the descending branch of the anodic peak.
Derivation of the equation for the current of the descending branch of the anodic peak
The expression for the limiting nonstationary current is found from Fick’s first equation:
\[ i(t) = zFSD_R \left( \frac{\partial C_R}{\partial x} \right)_{x=l}, \tag{1} \]
where \(z\) is the number of electrons participating in the electrode process per atom of the element; \(F\) is the Faraday constant; \(S\) is the electrode surface; \(D_R\) is the diffusion coefficient of the metal atoms in mercury; \(C_R\) is the concentration of the dissolved metal in the amalgam; \(l\) is the thickness of the mercury film; \(x\) is the coordinate measured from the mercury substrate. The gradient
concentration \(\left(\dfrac{\partial C_R}{\partial x}\right)\) is found as the solution of Fick’s second equation for linear diffusion:
\[ \frac{\partial C_R(x,t)}{\partial t} = D_R\frac{\partial^2 C_R(x,t)}{\partial x^2} \tag{2} \]
under the following initial and boundary conditions:
\[ t=0 \qquad C_R(x,0)=C_R^0 \qquad 0\le x\le l, \tag{3} \]
\[ t>0 \qquad C_R(l,t)=0 \qquad x=l, \tag{4} \]
\[ \lim_{x\to 0}\left(\frac{\partial C_R}{\partial x}\right)=0. \tag{5} \]
The stated boundary-value problem is solved by the operational method\(^*\). The distribution of metal atoms in the mercury film at any time will be described by the following expression:
\[ C_R(x,t)=C_R^0\frac{4}{\pi} \sum_{m=0}^{\infty} \frac{(-1)^m}{(2m+1)} \exp\left[-\frac{(2m+1)^2\pi^2D_Rt}{4l^2}\right] \cos\frac{(2m+1)\pi x}{2l}. \tag{6} \]
The series of the equation converges rapidly for large values of \(\dfrac{D_Rt}{l^2}\). For small values of \(\dfrac{D_Rt}{l^2}\), the following expression can be obtained for the distribution of metal atoms in the mercury film at any time:
\[ \frac{C_R^0-C_R(x,t)}{C_R^0} = \sum_{m=0}^{\infty} (-1)^m \operatorname{erfc} \frac{(2m+1)l-x}{2\sqrt{D_Rt}} + \sum_{m=0}^{\infty} (-1)^m \operatorname{erfc} \frac{(2m+1)l+x}{2\sqrt{D_Rt}}. \tag{7} \]
For the limiting current at the electrode, we obtain two equations that are different in mathematical form but completely identical, (8, 9), valid respectively for large and small \(\dfrac{D_Rt}{l^2}\). To do this, we differentiate expressions (6) and (7) with respect to \(x\), then set \(x=l\) and substitute the obtained values into Fick’s first equation (1).
\[ i(t)=\frac{2zFSD_RC_R^0}{l} \sum_{m=0}^{\infty} \exp\left[-\frac{(2m+1)^2\pi^2D_R\cdot t}{4l^2}\right], \tag{8} \]
\[ i(t)= \frac{zFSD_R^{1/2}C_R^0}{\pi^{1/2}t^{1/2}} \left[ 1+2\sum_{m=1}^{\infty} (-1)^m \exp\left(-\frac{m^2l^2}{D_Rt}\right) \right]. \tag{9} \]
Taking into account that the descending branch of the anodic peak will obey equations (8, 9) from the time \(t'\) (or \(\varphi'\)) and that the beginning of the re—
\(^*\) An analogous boundary-value problem for heat-conduction conditions was solved in Lykov’s monograph [3], and therefore we shall not give the detailed solution here.
taking as the time origin for the impulse current the instant of time \(t=t^*\) [3], we obtain expressions for the current of the diffusion branch of the anodic wave on a film electrode:
\[ i(t)=2zFSD_R C_R^0 \sum_{m=0}^{\infty} \exp\left[-\frac{(2m+1)^2\pi^2D_R(t-t^*)}{4l^2}\right], \tag{10} \]
\[ i(t)=\frac{zFSD_R^{1/2}C_R^0}{\pi^{1/2}t^{1/2}} \left(1+2\sum_{m=1}^{\infty}(-1)^m \exp\left[-\frac{m^2l^2}{D_R(t-t^*)}\right]\right), \tag{11} \]
where \(t^*\gg t^1\); \(t^1\) is the time after which \(\dfrac{C_R(l,t)}{C_R^0}\ll 0.01\). It is easy to show that, to within 1%, equation (11) for \(\lambda=\dfrac{D_R(t-t^*)}{l^2}\ll 0.20\) and equation (10) for \(\lambda=0.235\) can be written in the following form:
\[ i(t)=\frac{2zFSD_RC_R^0}{l} \exp\left[-\frac{\pi^2D_R(t-t^*)}{4l^2}\right], \tag{12} \]
\[ i(t)=\frac{zFSD_R^{1/2}C_R^0}{\pi^{1/2}(t-t^*)^{1/2}}. \tag{13} \]
Equation (13) (valid for small \(\dfrac{D_Rt}{l^2}\)) coincides with the well-known Gokhshtein equation [5] for linear semi-infinite diffusion for reversible processes.
Taking as the initial potential the potential corresponding to the beginning of the time count, and bearing in mind that \(t=\dfrac{\Delta\varphi}{\omega}=\dfrac{\varphi-\varphi^*}{\omega}\), equations (12) and (13) can be rewritten in the following form:
\[ i(t)=\frac{2zFSD_RC_R^0}{l} \exp\left[-\frac{\pi^2D_R(\varphi-\varphi^*)}{4l^2\omega}\right], \tag{14} \]
\[ i(t)=\frac{zFSD_R^{1/2}C_R^0\omega^{1/2}} {\pi^{1/2}(\varphi-\varphi^*)^{1/2}}. \tag{15} \]
Let us estimate the rates of change of the potential \(\omega\) at which equations (14) and (15) may be used. For the estimate, recall that
\[ \frac{D_Rt}{l^2}=\frac{D_R(\varphi-\varphi^*)}{l^2\omega}, \]
and take
\[ D_R=1.5\cdot10^{-5}\ \text{cm}^2/\text{s},\quad l=10\cdot10^{-4}\ \text{cm}\quad \text{and}\quad \Delta\varphi=0.1\ \text{V}. \]
Calculation shows that, under the selected conditions, equations (14) and (15) may be applied, respectively, at the following rates of change of the potential:
\[ \omega_1\leq 5.431\ \text{V/s}. \]
\[ \omega_2\geq 6.38\ \text{V/s}, \tag{16} \]
Thus, equation (15) can be applied only in studies on oscillographic polarographs.
It follows from equation (15) that, at high rates of change of the potential \((\omega\geq 6.38\ \text{V/s})\), the current of the descending branch of the anodic wave on a film mercury electrode does not depend on the thickness of the mercury film and depends on the diffusion coefficient. At low rates of change of the po-
the current of the descending branch of the potential depends on the diffusion coefficient of the metal atoms in mercury (equation 14).
Experimental part
It is seen from equation (14) that the plot in the coordinates $\lg i, t$ is a straight line with a slope equal to $-\dfrac{\pi^2 D_R}{4l^2}$.
To confirm the validity of this equation, we obtained experimental data, which were interpreted in the indicated coordinates. The studies were carried out using Cd in the background of $0.1\,N\,\mathrm{KNO_3}$ at various rates of potential change under the following conditions:
$l = 10^{-3}\ \mathrm{cm};\quad c = 10^{-10}\ \mathrm{g\text{-}atom/l}.$
The results of the studies are presented in Fig. 1. It is seen from the figure that, at all the rates of potential change studied, the plots $\lg i, t$ have the form of straight lines.
Moreover, it follows from Fig. 1 that the tangent of the angle of inclination of the straight lines in the indicated coordinates increases with increasing rate of change of the electrode potential, approaching the theoretical value equal to $-\dfrac{\pi^2 D_R}{4l^2}$. This fact may be explained as follows. If the anodic current is limited only by diffusion, then the slope of the straight lines does not depend on the rate of potential change and in this case is equal to the theoretical value. This is achieved
Fig. 1. Curve 1—$\omega=0.125\ \mathrm{V/sec}$;
2—$\omega=0.2\ \mathrm{V/sec}$;
3—$\omega=0.5\ \mathrm{V/sec}$;
4—$\omega=1\ \mathrm{V/sec}$;
5—$\omega=2\ \mathrm{V/sec}$;
6—$\omega=4\ \mathrm{V/sec}$.
Curve 6 is shifted along the time axis by $0.45\ \mathrm{sec}$.
at very high rates of potential variation. In the range of potential-variation rates investigated by us (0.125–4 V/sec)* the anodic current of the descending branch is determined both by the rate of potential variation and by concentration polarization; moreover, as the rate of potential variation increases, the share of the influence of concentration polarization increases. This is evidenced by the increase in the slope coefficient with increasing rate of potential variation.
Conclusions
- A theoretical derivation has yielded the equation of the descending branch of the anodic peak in the AAV method on a film electrode with a linear potential sweep for reversible processes. The equation of the descending branch of the anodic peak has been confirmed experimentally.
References
- V. A. Igolinskii. Dissertation. Tomsk, 1964.
- V. E. Gorodovykh. Izv. TPI (in press); V. E. Gorodovykh. Dissertation, 1965.
- A. V. Lykov. Theory of Thermal Conductivity, State Publishing House of Technical-Theoretical Literature, Moscow, 1952.
- Ya. P. Gokhshtein, A. Ya. Gokhshtein. DAN SSSR, 128, 5, 1959.
* Studies at potential-variation rates greater than 4 V/sec were not carried out, since the anodic peaks were strongly distorted by capacitive current.