TIME OF COMPLETE METAL EMERGENCE FROM A STATIONARY MERCURY DROP ELECTRODE

V. E. Gorodovykh, B. F. Nazarov

(Presented by the Scientific-Methodological Seminar of the KhTF)

In the practice of amalgam polarography with accumulation, it is most convenient to carry out successive determinations on one and the same electrode without replacing it. Such a working procedure increases the reproducibility of the results of analysis, since in this case one source of error is eliminated—the nonconstancy of the electrode area. In addition, frequent replacement of the electrode lengthens the analysis time, which is especially disadvantageous when working by express methods.

In the literature, as far as we know, there is only one paper [1] devoted to the empirical selection of conditions under which it is possible to carry out a series of analyses on one and the same electrode. In the present work an attempt is made to consider this question theoretically, taking into account the diffusion character of the descending branch of the anodic peak [2].

It is physically clear that one of the conditions for repeated use of the electrode is the practically complete emergence of the metal concentrated in it during the preceding anodic cycle. An exact analytical expression for the emergence time of a definite fraction of metal from a stationary mercury drop electrode in an anodic cycle can be obtained only if the peak equation is known. Indeed, if the dependence of current on time is known explicitly, \(i = f(t)\), then the amount of metal that has passed from the electrode into the solution is equal to

\[ m(t)=\frac{A}{zF}\int_0^t i(\tau)\,d\tau, \tag{1} \]

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

This method, however, is inconvenient because of the cumbersomeness of the equations of the anodic peak [3,4]. In practical work it is sufficient to know the upper limit of time after which no more than 1–3% of the metal remains in the mercury drop. This makes it possible to find the dependence of the percentage of metal emergence from a stationary mercury drop electrode in a form convenient for calculations.

From Fig. 1 it is seen that the difference between the time of complete emergence of the metal from a stationary mercury drop electrode when recording the anodic peak and the pulse current is equal to \(\Delta \tau\). Since this difference is small (at a rate of potential change \(w = 400\ \text{mV/min}\) it does not exceed 15–20 sec), for approximate calculations we may neglect it and assume that the time of complete emergence of the metal when recording the peak and the time of complete emergence

when recording the limiting current are equal to one another. In precise calculations it is necessary to take the value \(\Delta \tau\) into account¹.

Let us find the time of emergence of a certain fraction of the metal for the case of the limiting current. For this purpose we form the difference

\[ m(\infty)-m(t)= \]

\[ =\frac{1}{zF}\left(\int_{0}^{\infty} i(\tau)\,d\tau-\int_{0}^{t} i(\tau)\,d\tau\right)= \]

\[ =\frac{1}{zF}\int_{t}^{\infty} i(\tau)\,d\tau, \tag{2} \]

where \(m(t)\) is the amount of metal (in \(g\)-atom) that has emerged from the stationary mercury drop electrode by the time \(t\); \(m(\infty)\) is the amount of metal that has emerged from the stationary mercury drop electrode after an infinitely long interval of time; \(i(t)\) is the equation of the descending branch of the peak.

Substituting into (2) the equation of the descending branch of the peak² [4], after simple transformations we find:

\[ m(\infty)-m(t)=- \]

\[ -\frac{2CSD}{r}\left(\frac{r^{2}}{\pi^{2}D}\sum_{n=1}^{\infty}\frac{1}{n^{2}}e^{-\frac{(n\pi)^{2}Dt}{r^{2}}}\right)_{t}^{\infty}= \]

\[ =\frac{8Cr^{3}}{\pi}\sum_{n=1}^{\infty}\frac{1}{n^{2}}e^{-\frac{(n\pi)^{2}Dt}{r^{2}}}. \tag{3} \]

Fig. 1. Schematic representation of the anodic peak and limiting nonstationary current at a stationary mercury drop electrode: \(t_p\)—time at which the peak potential is reached; \(t^*\)—beginning of the time count for the descending branch of the peak; \(t'\)—time from which the surface concentration of the metal becomes equal to zero.

In equation (3) it is assumed that \(S=4\pi r^{2}\). Since we are interested in sufficiently large times, when no more than \(1\)--\(3\%\) of the metal remains in the stationary mercury drop electrode, the parameter \(\lambda=Dt/r^{2}>0.11\) (for \(D=1\cdot10^{-5}\ \mathrm{cm^{2}/sec}\), \(r=4\cdot10^{-2}\ \mathrm{cm}\), this corresponds to \(t>17\ \mathrm{sec}\), and in equation (3), with an accuracy of up to \(1\%\), one may restrict oneself to the first term of the series):

\[ m(\infty)-m(t)=\frac{8Cr^{3}}{\pi}e^{-\pi^{2}\frac{Dt}{r^{2}}}. \tag{4} \]

Dividing equation (4) further by \(m(\infty)\) and taking into account that \(m(\infty)\) is equal to the amount of metal initially present in the stationary mercury drop electrode,

\[ m(\infty)=Cv=\frac{4}{3}\pi r^{3}C, \tag{5} \]

where \(v\) is the volume of the stationary mercury drop electrode,

we obtain

\[ \eta_1=\frac{6}{\pi^2} e^{-\pi^2 \frac{Dt}{r^2}}, \tag{6} \]

where \(\eta = 1-\dfrac{m(t)}{m(\infty)}\) is the fraction of metal remaining in the stationary mercury drop electrode at time \(t\).

It is interesting to note that equation (6), which we obtained on the basis of the equation for the limiting anodic current, coincides with the expression given in Crank’s monograph [5]; however, the author obtained it by another route—by integration over the volume for the average concentration of the substance inside the sphere.

Solving (6) with respect to time, we finally obtain

\[ t=\frac{2.3r^2}{\pi^2D}\lg\frac{6}{\pi^2\eta} \tag{7} \]

or, in dimensionless form,

\[ \lambda=0.233\lg\frac{0.608}{\eta}. \tag{8} \]

It follows from equation (7) that the time for the emergence of a given fraction of metal from a stationary mercury drop electrode does not depend on the initial concentration of the metal and is greater the larger the electrode radius and the smaller the diffusion coefficient of the metal in mercury.

Experimental Part

The experimental verification of formula (7) was carried out by us using the example of the reversible currents of thallium, lead, and cadmium. The choice of these elements was also due to the fact that, according to literature data, they have appreciably different diffusion coefficients in mercury. The anodic peaks of lead were recorded against a background of \(0.1\) M KOH, and those of thallium and cadmium against a background of \(1\) M KNO\(_3\). All solutions were prepared from reagents tested for purity, using triply distilled water.

The procedure for recording anodic current–time curves was as follows. After electrolytic accumulation had been carried out, the mercury electrode was disconnected from the polarograph, the potential applied to the cell from the setting rheochord was lowered by \(0.2\)—\(0.3\) V in the positive direction relative to the peak potential, and the circuit was closed again. In doing this we did not fear a decrease in the limiting currents, since under our conditions cementation and self-dissolution of the amalgam were absent. On the contrary, breaking the contact created favorable conditions for producing a uniform distribution of metal atoms inside the stationary mercury drop electrode [4].

The electro-dissolution potential was chosen so that, when it was applied to the electrode, the concentration of the metal near the electrode surface immediately fell to zero. We have previously shown that this condition is practically fulfilled when the applied potential is \(0.069/z\) V more positive than the half-wave potential of the reversible polarographic wave of the given element in the given supporting electrolyte [4].

The anodic peaks were recorded in the usual manner. The concentration of metal ions in the solution was \(500\ \mu\text{g}/\text{ml}\). The accumulation time was \(2\) min. The volume of the solution was \(20\) ml. Depletion of the solution during accumulation was absent, since the solution was not stirred during electrolysis.

In all cases, to determine the amount of metal transferred to the electrode during the accumulation time, cathodic \(i—t\) curves were recorded. All experiments were carried out at \(25^\circ\mathrm{C}\) and were repeated from 4 to 10 times.

Before proceeding to a direct verification of formula (7), the completeness of metal emergence from the stationary dropping mercury electrode was checked when recording peaks and anodic \(i—t\) curves. For this purpose, the areas under the cathodic \(i—t\) curve were compared with the area of the peak and with the anodic \(i—t\) curve. The magnitude of the area was determined by weighing.

Table 1

Experimental determination of the completeness of lead emergence from the stationary mercury drop electrode when recording the anodic peak and the anodic limiting current.
Experimental conditions: \(5\cdot10^2\ \mu\mathrm{g}/\mathrm{ml}\) lead in \(0.1\ M\ \mathrm{KOH}\); \(v_p = 20\ \mathrm{ml}\); \(t_e = 1\ \mathrm{min}\); \(r_0 = 0.04\ \mathrm{cm}\); \(t^\circ = 25^\circ\mathrm{C}\)

Designations: \(S_1\)—area under the anodic peak; \(S_2\)—area under the cathodic \(i—t\) curve; \(S_3\)—area under the anodic \(i—t\) curve.

It is seen from Table 1 that, within the accuracy of the measurements, in the anodic cycle the metal emerges from the stationary mercury drop electrode practically completely.

In order to find the amount of metal that had left the electrode by a definite moment of time, the area of the anodic peak was divided into several parts and the area of each part was compared with the total area of the peak. The same procedure was applied to the area under the anodic \(i—t\) curve.

Statistical treatment of the results of these experiments for lead, thallium, and cadmium is given in Table 2. The dependence of the percentage of metal emergence on time for lead is also given in Fig. 2. From the data presented it is seen that all consequences of formula (7) are confirmed experimentally: the greatest emergence time is observed for thallium, which also has the smallest diffusion coefficient in comparison with lead and cadmium. The emergence time of a definite fraction of the initially concentrated metal increases with increasing electrode radius.

It is seen in Fig. 2 that the experimental data agree satisfactorily with the theoretical ones. Deviations upward for the anodic \(i—t\) curves, for which formula (7) is quite exact, are associated with shielding of the electrode and, in connection with this, an increase of the effective radius in comparison with the geometric one [4].

Table 2

Study of the kinetics of electroreduction of lead, cadmium, and thallium from a stationary mercury drop. Conditions of the experiments are given in the text.

The divergence of the theoretical and experimental curves for anodic peaks at short times is explained by the fact that, for anodic peaks, equation (7) is approximate. However, after the concentration of metal at the electrode surface becomes practically equal to zero, the process of metal emergence from the stationary mercury drop electrode begins to obey equation (7), and the error rapidly decreases.

As can be seen from Fig. 2, in agreement with the theory, the time for the emergence of 99% of the metal from a stationary mercury drop electrode of not very large size \((r = 0.04\ \text{cm})\) does not exceed 1.5 min. Consequently, if, after recording the peak, the electrode is held at a small negative potential for 1.5 min, less than one percent of the initial amount of metal will remain in it, and this electrode can be used again with accuracy sufficient for practice. The time during which the same electrode can be used without a noticeable deterioration in reproducibility apparently depends only on the concentration of surface-active substances and oxygen in the solution, which poison its surface. In the case of an electrode suspended on a metallic contact, this time is also limited by the amount of metal that has passed from the surface of the contact into the bulk of the electrode.

Fig. 2. Dependence of the percentage emergence of lead from a stationary mercury drop electrode on time: 1—theoretical curve; 2 and 3—experimental curves for the limiting nonstationary current and the anodic peak, respectively.

Conclusions

1. An approximate formula is proposed that makes it possible to determine the time of complete metal emergence from a stationary mercury drop electrode when recording an anodic peak.

2. Experimental data on determining the completeness of emergence of lead, thallium, and cadmium from a stationary mercury drop electrode agree satisfactorily with the theory. The reasons for the discrepancies between the theoretical and experimental percentage-emergence–time curves are discussed.

3. It is shown that under ordinary conditions \((r = 0.04\ \text{cm})\) the time for emergence of 99% of the metal initially concentrated in a stationary mercury drop electrode does not exceed 1.5 min.

LITERATURE

1. M. S. Zakharov, A. G. Stromberg, Zavodskaya laboratoriya, 28, 13, 1962.
2. V. E. Gorodovykh. Izv. TPI, (in press).
3. V. E. Gorodovykh. Izv. TPI, (in press).
4. V. E. Gorodovykh. Candidate dissertation, Tomsk, 1964.
5. W. Kemula. Advances in Polarography, Vol. I, p. 105, 1960.

1. The time of emergence of a certain fraction of the metal is always smaller when recording the limiting current than when recording the peak, since in the latter case the concentration of metal atoms near the electrode surface becomes equal to zero only at \(t>t'\). ↩

2. Since we neglect the value \(\Delta \tau\) in the equation of the descending branch, the beginning of the time count \(t^*\) may be set equal to zero. ↩