EQUATION OF THE DESCENDING BRANCH OF THE ANODIC PEAK AND THE TIME OF METAL EXIT FROM A STATIONARY MERCURY DROP ELECTRODE

V. E. GORODOVYKH

(Presented by the scientific-methodological seminar of the Chemical-Technological Faculty)

In the method of amalgam polarography with accumulation on a stationary mercury electrode, the kinetic (ascending) branch of the anodic peak contains information on the qualitative and quantitative composition of the solution being analyzed. The descending branch of the peak, by its nature, is diffusional; it distorts closely spaced peaks and lowers the resolving power of the method. In the present work a derivation is given of the equation of the descending branch of the peak, and the factors affecting its steepness are considered.

Fig. 1. Anodic surge current and anodic peak. \(t_n\)—time of reaching the peak potential, \(t_0\)—beginning of the surge current.

It follows from the equation of the anodic peak [1] that in the immediate vicinity beyond the peak maximum (at \(t=t'\), Fig. 1), the concentration of metal atoms near the electrode surface becomes close to zero; consequently, in the subsequent process the decay of the current is determined only by the laws of spherical diffusion from a closed volume. If such conditions are created that at the initial instant of time the metal concentration at the electrode surface immediately becomes equal to zero, then the “surge” current arising in this case will coincide with the descending branch of the peak, beginning from the instant \(t'\) (Fig. 1). Let us find the equation of the surge current at a stationary mercury drop electrode. For this it is necessary to solve Fick’s second equation for spherical diffusion:

\[ \frac{\partial C_R(r,t)}{\partial t} = D_R\left( \frac{\partial^2 C_R(r,t)}{\partial r^2} + \frac{2}{r}\frac{\partial C_R(r,t)}{\partial r} \right), \tag{1} \]

under the following initial and boundary conditions:

\[ \begin{aligned} &t=0,\quad 0\le r\le R,\quad C_R(r,0)=C_R,\\ &t>0,\quad r=R,\quad C_R^s(R,t)=0,\\ &t>0,\quad r=0,\quad \lim_{r\to 0}\frac{\partial C_R(r,t)}{\partial r}=0, \end{aligned} \tag{2} \]

where \(R\) is the radius of the electrode,
\(D_R\) is the diffusion coefficient of the metal in the amalgam,
\(C_R^0\) is the initial concentration of the metal in the stationary mercury drop electrode.

The solution of boundary-value problem (1) and (2) gives the concentration distribution of the metal inside the stationary mercury drop electrode (2):

\[ C_R(r,t)=2C_R^0\frac{R}{r}\sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sin n\pi \frac{r}{R}}{n\pi} \exp\left[-(n\pi)^2\frac{D_R t}{R^2}\right]. \tag{3} \]

It follows from equation (3) that the instantaneous current at the stationary mercury drop electrode is equal to [7]

\[ i=\frac{2zFC_R^0SD_R}{R}\sum_{n=1}^{\infty} e^{-(n\pi)^2\frac{D_R t}{R^2}}, \tag{4} \]

where \(S\) is the electrode area.

Taking into account that the descending branch of the anodic peak will obey equation (4) from the time \(t=t'\), and that the origin of the time count for the instantaneous current is the time \(t=t_0\), we obtain the final expression for the diffusion branch of the peak:

\[ i_{\text{desc.}}=-\frac{2zFC_R^0SD_R}{R}\sum_{n=1}^{\infty} e^{-(n\pi)^2\frac{D_R}{R^2}(t-t_0)}, \tag{5} \]

for \(t \geq t'\).

For values of the dimensionless parameter

\[ \lambda=\frac{D_R}{R^2}(t-t_0)\geq 0.11 \]

in equation (5) one may restrict oneself to the first term of the series

\[ i_{\text{desc.}}=-\frac{2zFC_R^0SD_R}{R} e^{-\frac{\pi^2D_R}{R^2}(t-t_0)}. \tag{6} \]

It follows from equations (5) and (6) that the descending branch decays the faster, the larger the diffusion coefficient of the metal in mercury and the smaller the radius of the stationary mercury drop electrode. Therefore, increasing the radius of the stationary mercury drop electrode leads to an increase in the half-peak width and worsens the resolving power of the method of amalgam polarography with accumulation.

In the case of semi-infinite diffusion, the current at a stationary spherical electrode (at \(C^s=0\)) is inversely proportional to the square root of time [3]:

\[ i=zFC_0^0SD_0\left(\frac{1}{\sqrt{D_0\pi t}}+\frac{1}{R}\right), \tag{7} \]

where \(C_0^0\) is the initial concentration of metal ions in the solution;
\(D_0\) is the diffusion coefficient of metal ions in the solution.

Analysis of equations (5) and (7) makes it possible to draw a number of conclusions important in practical respects: the diffusion branch of the anodic peak under the conditions of amalgam polarography with accumulation decays much more rapidly and, consequently, occupies a smaller range of potentials than the analogous branch of the cathodic peak. As a result, anodic peaks close in potential are less distorted by the influence of the preceding peak,

which increases the resolving power of the method. In other words, the “signal-to-noise” ratio in amalgam polarography is higher than in oscillographic polarography*). As the electrode radius increases, the diffusion branch of the anodic peak becomes more gently sloping and, moreover, because of the increase in the electrode volume, the accumulation coefficient drops sharply [4]. From all that has been said above it becomes clear why mercury electrodes with a large surface area (“pool-electrode”) have not found wide application in the field of ultramicroanalysis.

In [5] it was found experimentally that the time for complete oxidation of the metal from a stationary mercury drop electrode does not exceed 10 minutes. This fact allowed the authors to formulate conditions for recording several polarograms on one and the same drop without replacing it. It is of interest to consider this question theoretically, taking into account the diffusional character of the descending branch of the anodic peak.

An exact analytical expression for the time of release of a specified fraction of metal from a stationary mercury drop electrode in the anodic cycle can be obtained only in the case where the equation of the wave is known. Indeed, if the dependence of the anodic current on time is known explicitly, \(i=f(t)\), then the amount of metal that has passed from the stationary mercury drop electrode into solution is

\[ m(t)=\frac{A}{zF}\int_{0}^{t} i(\tau)\,d\tau, \tag{8} \]

where \(A\) is the atomic weight of the metal.

This method, however, is inconvenient because of the cumbersomeness of the wave equations [1]. In practical work it is sufficient to know the upper time limit after which no more than 1–3% of the metal remains in the drop. This makes it possible to find the dependence of the percentage of metal leaving the drop on time in a form convenient for calculations.

It follows from Fig. 1 that the difference between the times of complete release of the metal from the stationary mercury drop electrode when recording the anodic peak and the square-wave current is equal to \(t_{0}\). Since this difference is small (at a rate of change of potential \(w=400\ \text{mV/min}\) it does not exceed 15–20 sec), for approximate calculations we may neglect it and assume that the time of complete release of the metal when recording the wave and the time of complete release when recording the square-wave current are equal to one another. In exact calculations it is necessary to take into account the value \(t_{0}\).

From equation (4), by integration, it is easy to obtain the following expression for the fraction of metal remaining in the drop at time \(t\):

\[ \eta=\frac{6}{\pi^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\exp\left(-n^{2}\pi^{2}\frac{D_{R}t}{R^{2}}\right), \tag{9} \]

with \(0\leq \eta \leq 1\).

For

\[ \frac{D_{R}t}{R^{2}}\geq 0.11 \]

equation (9) is simplified to

\[ \eta=\frac{6}{\pi^{2}}\exp\left(-\pi^{2}\frac{D_{R}t}{R^{2}}\right). \tag{10} \]

Solving (10) with respect to time, we finally obtain

\[ t=\frac{2.3R^{2}}{\pi^{2}D_{R}}\lg\frac{6}{\pi^{2}\eta}. \tag{11} \]

* In oscillographic polarography, high rates of potential change are used, of the order of several volts/sec, and as a consequence the descending branch of the peak is described by the equation for semi-infinite diffusion.

It follows from (11) that the time for a definite fraction of the metal to leave the stationary mercury drop electrode depends on the radius of the drop and on the diffusion coefficient of the metal atoms in the amalgam, and does not depend on the initial concentration of the metal in the drop. If we set \(R = 0.04\ \text{cm}\), \(D_R = 1 \cdot 10^{-5}\ \text{cm}^2/\text{s}\), then 99% of the metal will pass into the solution in 70 s. Experiment confirms this theoretical conclusion [6].

In conclusion, we consider it our duty to express our gratitude to Prof. A. G. Stromberg for a number of valuable comments during preparation of the manuscript for publication.

Conclusions

1. Equations have been derived for the descending branch of the anodic peak and for the time of complete exit of the metal from a stationary mercury drop electrode.

2. The factors affecting the steepness of the descending branch of the peak and the time of metal exit from the stationary mercury drop electrode have been considered.

References

1. V. E. Gorodovykh. Izvestiya TPI (in press).
2. A. V. Lykov. Theory of Heat Conduction. Moscow, 1952.
3. P. Delahay. New Instruments and Methods in Electrochemistry. IL, Moscow, 1957.
4. A. G. Stromberg, V. E. Gorodovykh. Journal of Inorganic Chemistry, 8, 2355, 1963.
5. M. S. Zakharov, A. G. Stromberg. Zavodskaya laboratoriya, 28, 13, 1962.
6. V. E. Gorodovykh, B. F. Nazarov. Izvestiya TPI (in press).
7. N. G. Chovnyk, V. V. Vashchenko. Journal of Physical Chemistry, 37, 538, 1963.