COULOMETRIC MEASUREMENTS IN THE APN METHOD

E. A. ZAKHAROVA

(Presented by the scientific-methodological seminar of KhTF)

At the present time, in connection with the development of the theory of the APN method and its various physicochemical applications, the need for coulometric measurements often arises. At the same time, the material relating to this question is scattered among individual articles and has nowhere been generalized sufficiently fully. The present article attempts briefly to set forth the achievements made in the field of coulometric measurements in the APN method.

The principle of coulometry, in particular anodic coulometry, has been set forth in sufficient detail in a number of articles and reviews [1—10]. Coulometric measurements associated with the process of anodic dissolution of amalgams were previously carried out only for analytical purposes [3, 4]. It was even considered [5] that stationary mercury electrodes are unsuitable for anodic coulometry because, allegedly, in the anodic dissolution of a metal from an amalgam drop the current yield never reaches 100%.

In [11], by analogy with cathodic processes [12], a method was proposed for finding the quantity of electricity that had passed in an anodic cycle, using a stationary amalgam drop under continuous change of potential. The method consists in calculating the curve of the descending branch of the anodic peak according to a simplified exponential equation, strictly valid for linear semi-infinite diffusion to a plane electrode. The method is rather simple, but gives only an approximate (underestimated by 10–15%) value of the amount of metal in the drop. In calculations by this method, the amount of metal dissolved from the drop before the peak maximum is reached is not taken into account; moreover, it is assumed approximately that the peak maximum corresponds to the beginning of the time count and to the moment at which the limiting diffusion current is reached. The approximate nature of these assumptions is shown in Gorodovykh’s work [17].

Beginning in 1962, we began systematically to use coulometric measurements in APN in connection with the study of the influence of various factors on the shape of the anodic tooth. A simple procedure, proposed by Igolinskii [22] and developed by us [13—15] and other investigators [16], made it possible to conduct sufficiently accurate measurements of the quantity of electricity that had passed in an anodic cycle, using a stationary mercury drop and rates of potential change customary for recording polarographs.

This procedure is as follows (Fig. 1). After completion of the process of electrolytic deposition of the metal on the mercury drop

(point \(a\)) anodic dissolution of the amalgam begins at a linearly varying potential. After the peak of the wave has passed, approximately after \(0.1\ \mathrm{V}^{1)}\), when the current reaches the limiting-current value, the change of potential is stopped (point \(b\) on curve 2). In this case the paper tape of the polarograph continues to move uniformly and, thus, the change of the limiting current with time at constant potential is recorded further\(^{2)}\). This curve coincides with the descending branch of the anodic wave (after point \(b\)), since the limiting current does not depend on the potential. However, an important advantage of this procedure is the possibility of arbitrarily long anodic dissolution of the metal from the amalgam, until the current falls practically to the value of the residual current. The time for the practically complete release of the metal from the amalgam, as was shown by Gorodovykh [17], depends on the diffusion coefficient and the radius of the stationary drop. Practically all the metal, after the potential is stopped, leaves in 60–90 sec (at \(r=0.04\ \mathrm{cm}\), \(D=1.5\cdot 10^{-5}\ \mathrm{cm}^{2}/\mathrm{sec}\)). If the potential is not stopped after the limiting current has been reached, then, with further linear change of the potential in the positive direction, other electrode processes begin to occur at the electrode (for example, the potential of anodic dissolution of mercury is reached) (Fig. 1, curve 1), which makes it impossible in practice to achieve complete release of the metal from the amalgam.

Fig. 1. Anodic wave of copper, recorded with a linearly varying potential (curve 1). Curve 2 is the same anodic wave, recorded with stopping of the potential change at point \(b\). During electrolysis (points \(0\)—\(a\)), the electrolysis was recorded by the recorder.

With a linearly varying potential, the limiting-current—potential curve may formally be regarded as a current—time curve. Therefore, integration of the anodic-current curve, with stopping of the change of potential after the limiting current has been reached (or without stopping), gives us the amount of electricity passed:

\[ q=\int_{0}^{t} i(t)\,dt . \]

For brevity of exposition we shall denote the area under the anodic wave with stopping of the potential change by the symbol \(q_{\infty}\), bearing in mind that in practice the stopping time is 60–90 sec. The area under the anodic wave at any time \(t\) will be denoted, respectively, by \(q_t\).

Because of the complexity of the equation for the dependence of the anodic current on a sphere on time [17], it is more convenient to determine the amount of electricity by graphical integration of the current—time curve recorded by the recorder. It must, however, be taken into account that from the area obtained one must exclude the amount of electricity going to side electrode processes. Therefore

\(^{1)}\) For reversible anodic peaks, the current reaches the limiting-current value at \(\dfrac{69}{z}\ \mathrm{mV}\) positive of the potential of the anodic half-wave [17] or at \(\dfrac{34}{z}\ \mathrm{mV}\) positive of the peak potential.

\(^{2)}\) A theoretical expression for the dependence of the limiting current on time at constant potential on a spherical electrode is given in works [17, 18].

first the current–time curve for the residual current is recorded, and then the curve of the main electrode process is recorded against it.

When measuring the area under the anodic peak, there was concern that the inertia of the recording instrument (a recorder or mirror galvanometer) might distort the true area. For this purpose, current–time curves were recorded when the galvanometer was shunted by a large resistance, followed by opening the circuit (a rectangular current pulse). The curves obtained show that, owing to the inertia of the system, the real current–time curve differs in shape from an ideal rectangle. However, as our calculations [15] show, the areas under these curves (real and ideal) agree to an accuracy of about 3%.

The area under the anodic peak (expressed in $\mathrm{mm}^2$) can be measured in any way—by the weighing method, with a planimeter, or by direct calculation. Most often we used the weighing method: for this purpose the residual-current curve is transferred onto thick tracing paper and the anodic-dissolution curve is drawn beneath it. The area of the tracing paper between these curves is cut out and weighed. The weight obtained is then compared with the weight of tracing paper of known area. The accuracy of the weighing method (including the accuracy of transferring the curve to the tracing paper, cutting it out, and the magnitude of fluctuations in the density of the tracing paper) averages 2.5% from 6 measurements. The accuracy of measuring the area with a planimeter is 2.2% (average of 5 measurements of an area of $1390\ \mathrm{mm}^2$). In the latter measurement method, the error increases sharply when small areas are measured.

Knowing the area $q\ \mathrm{mm}^2$, it is then not difficult to calculate the area in coulombs; for this we find what number of coulombs corresponds to $1\ \mathrm{mm}^2$ of area. Obviously,

\[ 1\,\frac{\text{coulomb}}{\mathrm{mm}^2} = 1\,\frac{\text{ampere}}{\mathrm{mm}} \times 1\,\frac{\text{sec}}{\mathrm{mm}}, \]

where the first factor represents the sensitivity of the instrument at which the polarogram was recorded, and the second factor depends on the speed of motion of the recorder paper tape and can be readily measured. The accuracy of measuring these quantities is satisfactory (1–1.5%), so that, overall, the value of the area (in coulombs) is determined with an accuracy of 4–5%.

When carrying out coulometric measurements with stopping of the potential, it is necessary to take into account the dependence of $q_{\infty}$ on various factors. In accordance with theoretical considerations, if solution depletion is neglected, this dependence has the form:

\[ q_{\infty}=\frac{F D_{\mathrm{u}}}{\delta} S C_1 \tau z, \tag{1} \]

where $D_{\mathrm{u}}$ is the diffusion coefficient of the ions in solution, $\mathrm{cm}^2/\mathrm{sec}$; $\delta$ is the thickness of the diffusion layer in the stirred solution, $\mathrm{cm}$;
$S$ is the electrode surface, $\mathrm{cm}^2$;
$C_1$ is the ion concentration, $\mathrm{g\text{-}ion}/\mathrm{cm}^3$;
$\tau$ is the electrolysis time, sec;
$z$ is the number of electrons participating in the anodic-dissolution process per 1 atom of metal.

Using lead and copper as examples, we have demonstrated the validity of equation (1) [13, 15]. It has also been established that the area $q_{\infty}$ does not depend on the rate of change of potential (in the range 10–1000 mV/min) or on the passivation of the mercury drop, which complicates the anodic-dissolution process and causes splitting of the anodic peaks of lead and copper [14, 15]. It should be noted that, under conditions of work with recorders, in the presence of a large ohmic voltage drop in the solution, the area under the anodic peak may give overestimated values of the amount

metal present in the drop [15]. Thus, in order to obtain accurate coulometric results in the APN method, it is necessary to maintain constant stirring and to strive to reduce the ohmic resistance of the solution.

Application of coulometric measurements in the APN method

Knowing the area under the wave in coulombs (at complete yield of the metal), it is easy to find the amount of metal deposited, \(\dfrac{q_\infty}{zF}\) (gram-atom), or the concentration of the metal in a mercury drop of volume \(V\):

\[ C=\frac{q_\infty}{zFV}\,\frac{\text{gram-atom}}{\text{cm}^3}. \tag{2} \]

Measurement of the area \(q_\infty\) makes it possible to measure most accurately the mean electrolysis current flowing during reduction of the metal at low concentrations: \(10^{-8}\)—\(10^{-7}\) M and below. Obviously, \(I_\text{э}=q_\infty/\tau\). It is difficult to measure this current by another method at such small concentrations of metal ions in solution because of the large residual current.

Works [16, 13] give examples showing good agreement between the observed electrolysis current and that calculated from the area under the wave under conditions in which, during anodic oxidation and reduction, the same number of electrons participates per atom or ion and the concentration of metal ions is sufficiently large. Measurement of the area under the wave is useful in calculating various constants used in the APN method [16]. For example, calculation of the anodic-current constant \((K_2)\) is carried out by the formula:

\[ K_2=\frac{q_\infty}{I}\cdot\frac{3}{zFr}, \tag{3} \]

where \(I\) is the depth of the anodic current;
\(r\) is the radius of the electrode.

The electrolysis constant (in the absence of depletion) can be found from the expression:

\[ K_1=\frac{q_\infty}{\tau S C_1}, \tag{4} \]

where \(C_1\) is the concentration of ions in solution. Other methods for calculating \(K_1\) are given in [23].

In [13, 15], coulometric measurements were used to determine the dependence of the yield of metal from the mercury drop \((\rho\%)\) on various factors. For this purpose, areas under the anodic wave with a linearly varying potential were calculated, \(q_t=\int_0^t i\,dt\), which were then related to the area under the anodic wave with stopping of the potential change, \(q_\infty\):

\[ \rho=\frac{q_t}{q_\infty}\cdot 100. \tag{5} \]

Valuable conclusions about the character of the anodic or cathodic process can be drawn on the basis of comparison of the amount of electri-

* For an irreversible polarographic cathodic wave, this value is equal to \(77/\alpha z\) mV when the current is 5% less than the limiting value, and equal to \(118/\alpha z\) mV when the current is 1% less (\(\alpha\) is the electron-transfer coefficient).

substance that has passed in the cathodic and anodic cycles. It is often important to carry out the cathodic process with a 100% current efficiency. For this purpose electrolysis is conducted at the potential of the cathodic limiting current (0.1 V more negative than the cathodic half-wave potential) in a solution carefully purified of oxygen and other impurities. In choosing the electrolysis potential it is also important to eliminate the background reduction current; for this purpose the cathodic and anodic background polarograms are recorded. Coulometric measurements during cathodic deposition of a metal at a controlled potential have been studied in detail by Lingane [1] and described by Delahay [5]. In APN practice it is convenient to measure the quantity of electricity that has passed in the cathodic process \((Q)\) as the area under the electrolysis-current–time curve, the electrolysis current being recorded by a self-recording instrument at the slowest paper-tape speed (Fig. 1). If, during electrolysis, depletion of the solution practically does not occur, then the quantity of electricity passed is \(Q = I_\tau\), where \(I\) is the mean recorded electrolysis current (without residual current). For more accurate measurements we recommend carrying out electrolysis under such conditions that the residual current can be neglected; otherwise, when the residual current is subtracted, the error in measuring \(I_\tau\) increases greatly.

From comparison of the quantities \(I_\tau\) (or \(Q\)) and \(q_\infty\), a number of the most important characteristics of the anodic and cathodic processes can be determined.

1. Determination of the number of electrons participating in the process of anodic dissolution of the metal \((z)\). If the valence \((n)\) of the ion being reduced is known, then \(z\) is determined from the relation

\[ z = \frac{n \cdot q_\infty}{Q} \quad \text{or} \quad z = \frac{n \cdot q_\infty}{I_\tau}. \tag{6} \]

1. Determination of the current efficiency of the metal in the cathodic process can be found from the relation:

\[ \eta = \frac{q_\infty}{I_\tau} \cdot 100\%. \tag{7} \]

1. Determination of the current efficiency of the metal in the anodic process \((\rho)\), if \(\eta = 100\%\). In [17], \(\rho\) was found from the relation

\[ \rho = \frac{q_t}{I_\tau} \cdot 100\%, \tag{8} \]

where \(q_t\) is the area under the peak up to a definite instant of time. Relation (8) is analogous to (5).

1. Comparison of \(I_\tau\) and \(q\) makes it possible to detect certain anomalies in the process of anodic dissolution. For example, if the concentration of the metal in the drop becomes greater than the solubility of the metal in mercury, then the area under the anodic peak with potential arrest (for 60–90 sec) becomes smaller than the quantity of electricity participating in the cathodic cycle. This is due to the low rate of dissolution of the solid phase of the metal in the amalgam. In this case the concentration of the metal calculated from the area under the anodic peak corresponds to the solubility of the given metal in mercury (if equilibrium was attained in the heterogeneous amalgam) [19, 20].

Let us consider in more detail the calculation of the number of electrons \(z\) participating in the anodic oxidation of a metal atom from an amalgam, according to formula (6). The accuracy of determining \(z\) depends on the accuracy of measuring the electrolysis current \(I\) and the area under the anodic peak. If the error in measuring the electrolysis current does not exceed 5% (which can be achieved by reducing the residual current to a minimum and increasing the measured current to microamperes), then the error in measuring \(z\) by formula (6) will be about 8–10%, which is quite suf-

sufficient to give an unambiguous answer: whether 1, 2, or 3 electrons take part in the electrode process.

Table 1 gives the calculation of \(z\) by formula (6) for copper. It is seen that the anodic dissolution of copper on the indicated backgrounds proceeds with the participation of two electrons. The backgrounds chosen are of interest because on all the backgrounds, with the exception of the last, under certain conditions copper gives double peaks on the polarogram [14]. Determination of \(z\) made it possible to assert that the double peaks of copper are not connected with stepwise oxidation of copper from the amalgam.

Table 1

Calculation of the number of electrons in the anodic dissolution of copper on several backgrounds
(by formula (6))

Experimental conditions: \(\tau = 3\) min; \(W = 600\) mV/min; \(r = 0.04\) cm;

\(^{1)}\) \(n\) is the valence of copper in solution.

If it is necessary to determine the number of electrons \(z\) in the dissolution of an amalgam of low concentration and the electrolysis current is too small (less than the residual current), one may proceed as follows. As is seen from equality (6), under the condition of 100% oxidation of the metal, the electrolysis current can be determined from the area under the anodic peak in a solution where the degree of oxidation of the metal (\(z\)) is known. Using this principle, we proposed determining the number of electrons in the oxidation of amalgams by the “standard area” method [14]. In this method, calculation of the amount of electricity required to reduce the ions of a given metal is replaced by measurement of the standard area \((q_{\infty}^{*})\) of the anodic dissolution of the amalgam in another solution, where the degree of oxidation \(z^{*}\) of the given metal is known. After this, the amalgam of the same concentration is transferred into the solution under study, and the area \(q_{\infty}\) under the anodic dissolution curve is again measured. The number of electrons is calculated from the equation

\[ z=\frac{z^{*}\cdot q_{\infty}}{q_{\infty}^{*}}, \tag{9} \]

where \(q_{\infty}^{*}\) is the standard area, and \(z^{*}\) is the number of electrons that participates in anodic dissolution in the second, “standard,” solution.

Table 2 gives the values of the number of electrons, calculated by the “standard area” method, participating in the oxidation of copper from amal-

grams on different supporting electrolytes. As can be seen, on supporting electrolytes containing chloride ions, anodic dissolution of copper proceeds with the participation of one electron, which is apparently associated with the formation of a stable Cu(I) chlorocomplex. As a result, the copper peaks on chloride supporting electrolytes are broader and lower.

Spirin [21] proposed another variant of the “standard-area” method. The number \(z\) for the element sought is also determined from (6), but for comparison the area under the peak of another element is taken—\(q_\infty^*\) with a known \(z^*\). In this case both elements must be present in solution at one and the same concentration, have identical diffusion coefficients, and must not form intermetallic compounds with each other in mercury. These requirements are met, for example, by uranium and zinc. From comparison of the areas under these peaks it was found that uranium is oxidized with the participation of 4 electrons [21].

Table 2

Calculation of the number of electrons participating in the anodic oxidation of copper from an amalgam on different supporting electrolytes by the “standard-area” method
\(C_{\mathrm{Cu}^{2+}} = 1 \cdot 10^{-5}\) g-ion/l; \(V = 5\) ml; \(\tau = 3\) min

There are also other applications of coulometric measurements. From the dependence of the fraction of metal remaining in the drop (calculated from the area under the peak with a potential stop) on the rate of change of potential, the diffusion coefficient of the metal in mercury can be calculated [15]. Measurements of the area of individual parts of a complex (double) peak make it possible to calculate the thickness of the passivating film on the surface of the mercury drop [15]. The ratio \(q_\infty/l\), as was shown by Igolinsky [22], is an important characteristic of an anodic peak, characterizing its shape. For studying the kinetics and nature of the electrode process, coulometric measurements of the area under the anodic peak with a linearly varying potential as a function of the rate of potential change are apparently especially promising. For cathodic processes such studies [24, 25] make it possible to determine the phenomena accompanying the main electrode process—adsorption, passivation, etc.

From the material presented it is evident that coulometric measurements are of great importance in the APN method; accordingly, their wider use in various physicochemical and analytical investigations should be recommended.

Conclusions

1. A procedure is described for coulometric measurements in the APN method, based on measuring the area under the anodic peak with stopping of the potential change.

2. Various applications of coulometric measurements in the APN method are considered.

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