DETERMINATION OF THE COMPOSITION OF COMPLEXES DIRECTLY INVOLVED IN THE ELECTRODE PROCESS BY THE PPN METHOD

A. I. KARTUSHINSKAYA, A. G. STROMBERG, N. A. KOLPAKOVA

To clarify the regularities of the discharge–ionization process of complex ions by the polarographic method, only mercury electrodes have so far been used. However, the determination of such elements as gold, silver, and mercury on a mercury electrode is difficult because many of their simple and complex compounds are readily reduced by mercury, or their oxidation potential coincides with the oxidation potential of mercury. Therefore, for determining these elements, electrodes made of a more noble metal or alloy are most often used.

The aim of the present work is to investigate the possibility of using a solid graphite electrode to study the kinetics of slow electrode processes in a solution of complex ions; to further develop the theory of irreversible processes on a solid electrode for the case of discharge–ionization of complex ions; and to experimentally study, by the PPN method, bromide complexes of gold and to determine from experimental data the composition of the complexes directly involved in electrode processes and other kinetic parameters.

The mechanism of the electrode process may be represented as follows, if it is assumed that the metal ion contains one type of addend ion \(A^-\):

\[ \mathrm{Me}^{0}+rA^{-}\rightleftarrows \mathrm{Me}A_{r}^{r-} \tag{1} \]

(equilibrium of metal atoms with the addend on the electrode surface).

\[ \mathrm{Me}^{0}A_{r}^{r-}+(g-r)A^{-}\rightleftarrows \mathrm{Me}A_{g}^{(g-n)-}+ne^{-}. \tag{2} \]

(ionization of metal atoms on the electrode, the rate-determining stage);

\[ \mathrm{Me}A_{g}^{(g-n)}+(p-g)A^{-}\rightleftarrows \mathrm{Me}A_{p}^{(p-n)}, \tag{3} \]

(equilibrium of two complex metal ions in the bulk of the solution) or the overall reaction at the electrode:

\[ \mathrm{Me}^{0}+pA \rightleftarrows \mathrm{Me}A_{p}^{(p-n)}+ne^{-}, \tag{4} \]

The indices \(p\), \(g\), \(r\) indicate the composition of the complex species prevailing in solution and participating in the electrode reaction.

In accordance with the equation of the rate-determining reaction (2) and on the basis of the theory of slow discharge–ionization (1, 2), assuming that the process is completely irreversible, the following expression may be written for the rate of discharge–ionization:

\[ \vec{i}=nFSK_{b}^{0}C_{2}^{S}C_{A}^{(g-r)}l^{\frac{\beta\varphi}{\beta'}}, \tag{5} \]

\[ \vec{i}=nFSK_f^o C_1^s l^{-\alpha\varphi/b^1}, \tag{6} \]

where \(C_1^s\) and \(C_2^s\) are the concentrations of the predominant complex ions and metal atoms near the electrode surface;

\(q_1\) is the composition of the complexes participating in the anodic process¹;

\(K_f^o\) and \(K_b^o\) are the rate constants of the cathodic and anodic processes at \(\varphi=0\);

\(C_A\) is the addend concentration.

Since the addend concentration is many times greater than the concentration of the ion being determined, we take the addend concentration at the electrode surface to be equal to its concentration in the bulk of the solution; \(\alpha\) and \(\beta\) are the transfer coefficients of the anodic and cathodic processes; \(\varphi\) is the electrode potential, \(b^1=\dfrac{RT}{nF}\); \(\vec{i}\) and \(\overleftarrow{i}\) are the cathodic and anodic currents.

Assuming that equilibrium in the solution between the different complex particles near the electrode surface is established rapidly, we express the experimentally unknown concentrations \(C_1^s\) and \(C_2^s\) through the equilibrium constants of reactions (1) and (3):

\[ C_1^s=K_1C_1C_A^{-(p-g_2)} \tag{7} \]

\[ C_2^s=K_2^1 a_\tau C_A^r \tag{8} \]

where \(C_1\) is the concentration of the complex ions predominant in the solution (of composition \(p\));

\(a_\tau\) is the activity of the deposited metal on the electrode;

\(K_1\) and \(K_2\) are the equilibrium constants of reactions (1) and (3); \(K_2^1=\dfrac{1}{K_2}\).

Assuming that the heights of the anodic and cathodic teeth are directly proportional to the concentration of the deposited metal on the electrode and of the ions in solution, respectively, we may write:

\[ a_\tau=\frac{I_{na}}{nFSK_{\mathrm{II}}} \quad\text{and}\quad C_1=\frac{I_{nk}}{nFSK_{\mathrm{I}}} \tag{9} \]

where \(K_{\mathrm{I}}\) and \(K_{\mathrm{II}}\) are the constants of the cathodic and anodic teeth;

\(I_{nk}\) and \(I_{na}\) are their heights.

Substituting expressions (7–9) into (5–6) and taking \(\vec{i}=I_{nk}\), \(\overleftarrow{i}=I_{na}\), \(\varphi=\varphi_{na}\) or \(\varphi_{nk}\), after logarithmization we obtain:

\[ \varphi_{na}=-\frac{b}{\beta}\lg\frac{K_b^oK_2}{K_{\mathrm{II}}}-g_1\frac{b}{\beta}\lg C_A \tag{10} \]

\[ \varphi_{nk}=-\frac{b^1}{\alpha}\ln\frac{K_f^oK_1}{K_{\mathrm{I}}} -(p-g_2)\frac{b^1}{\alpha}\ln C_A, \tag{11} \]

To find the constants \(K_1\) and \(K_2\), \(K_{\mathrm{I}}\) and \(K_{\mathrm{II}}\), we shall use the equations known in the literature for the potentials of the anodic and cathodic peaks of a completely irreversible electrode process. An equation that takes into account the change in the activity of the solid phase during the electrodissolution of a thin metal film from the surface of an indifferent electrode was derived by Kh. Z. Brainina [5] (without allowance for the complex-formation process):

\[ \varphi_{na}=\varphi_p+\frac{RT}{\beta nF}\ln \frac{\beta w}{\gamma S K_{s1}a_\infty RT\exp\left[\frac{\beta nF}{RT}(\varphi_p-\varphi^0)\right]}, \tag{12} \]

where \(\varphi_p\) is the equilibrium potential, \(w\) is the rate of change of the potential, \(\gamma\) is a certain proportionality coefficient, coulomb\(^{-1}\); \(S\) is the electrode area, \(K_s\) is the standard rate constant of discharge–ionization:

\[ K_s=K_f^0\exp\left[\frac{-\alpha nF\varphi^0}{RT}\right] =K_b^0\exp\left[\frac{\beta nF\varphi^0}{RT}\right], \tag{13} \]

\(\varphi^0\) is the standard electrode potential.

In equation (12), instead of a number of terms one may introduce the exchange current:

\[ S K_{s1}a_\infty\gamma\exp\left[\frac{\beta nF}{RT}(\varphi_p-\varphi^0)\right] =\frac{i_0}{nFQ_{\mathrm{н}}}, \tag{14} \]

where \(Q_{\mathrm{н}}\) is the initial amount of metal in the film, coulomb.

The exchange current, taking into account the complex-formation process on the basis of equations (5) and (9), and the activity of the solid phase \(a_{\mathrm{т}}=\gamma\cdot a_\infty Q_{\mathrm{н}}\), can be written as:

\[ i_0=i=nF K_{s1}K_2Sa_\infty\gamma Q_{\mathrm{н}} \exp\left[\frac{\beta nF}{RT}(\varphi_p-\varphi^0)\right], \tag{15} \]

Substituting expressions (14) and (15) into equation (12), we obtain

\[ \varphi_{na}=\varphi^0+\frac{\varepsilon^1}{\beta}\ln \frac{\beta w}{RTS K_{s1}K_2^1a_\infty\gamma} -\frac{\varepsilon^1}{\beta}g\ln C_A . \tag{16} \]

Let us compare the obtained equation (16) with equation (12). From the comparison it follows that the anodic-peak constant is equal to

\[ K_{\mathrm{п}}=\frac{\beta w}{RTSa_\infty\gamma}, \]

and the value of the rate constant of the electrode process is the product of the standard rate constant \(K_s\) and the equilibrium constant of the complex \(Me^0Ar^{r--}\):

\[ K_{s1}=K_2K_b^0\exp\left[\frac{\beta nF\varphi^0}{RT}\right]. \]

On the basis of the Matsuda and Ayabe equation [6], A. G. Stromberg and Ya. N. Popova [7] derived an equation relating the potential of the cathodic peak to the concentration of addend ions in solution:

\[ \varphi_{nk}=\varphi^0+\frac{\varepsilon}{2} \left[ \lg K_{s2}-\frac{1}{2}\lg D\alpha nW-(p-g_2)\lg C_A-1.14 \right]. \tag{17} \]

where \(g_2\) is the composition of the complexes discharged in the cathodic process; \(D\) is the diffusion coefficient of the complexes predominating in the solution.

Let us compare equation (17) with equation (11). From the comparison we may conclude that the cathodic-wave constant is equal to

\[ K_1=D^{1/2}\alpha^{1/2}n^{1/2}W^{1/2}10^{1.14}, \]

and the rate constant of the cathodic process is the product of the standard rate constant and the instability constant of the complexes of composition \(p\), predominating in the solution, with formation of an intermediate complex of composition \(g_2\):

\[ K_{s2}=K_1\cdot K_f^0\exp\left[\frac{-\alpha nF\varphi^0}{RT}\right], \]

It follows from equation (16) that, knowing the dependence of \(\varphi_{na}\) on the logarithm of the addend concentration, one can determine the composition of the complex \(g_1\) formed when a thin metal film dissolves from the surface of a solid electrode. From the dependence of \(\varphi_{na}\) on \(\lg W\), \(\beta\) can be determined. Knowing \(g_1\) and \(\beta\), one can calculate \(K_{s_1}\) (if the value of \(\varphi^0\) is known). The transfer coefficient \(\beta\) can also be determined from relations (5):

\[ \beta=\frac{1}{14.4\,W_n}\cdot\frac{dI_{na}}{dQ_{\mathrm{n}}}, \tag{18} \]

\[ \Delta\varphi_{\mathrm{per}}=\varphi_1-\varphi_2=1.87\,\frac{RT}{\beta nF}, \]

where \(\varphi_1\) and \(\varphi_2\) are the potentials of the inflection points of the polarization curve, corresponding to the following values of the current at these points relative to the maximum current: \(i_1=0.72\,I_{na}\), \(i_2=0.53\,I_{na}\). The composition of the complex discharged at the electrode can be determined from the dependence \(\varphi_{nk}-\lg C_A\). In this case the composition of the complex predominating in solution must be known or determined by other methods (for example, potentiometrically from the dependence \(\varphi_p-\lg C_A\)). The coefficient \(\alpha\) is determined from relations (6):

\[ \alpha=-\frac{0.048}{n(\varphi_{nk'}-\varphi_{nk})}, \]

\[ \alpha=\frac{0.029}{n}\cdot\frac{\partial \lg W}{\partial(-\varphi_{nk})}. \tag{19} \]

Knowing the quantities \(p\), \(g_2\), \(\varphi^0\), and \(\alpha\), one can calculate the value of the rate constant of the cathodic process \(K_{s_2}\).

As is evident from equations (16 and 17), the experimental polarographic data do not permit simultaneous calculation of \(\varphi^0\), \(K_{s_1}\), and \(K_{s_2}\). Therefore, in calculations of \(K_{s_1}\) and \(K_{s_2}\) from the potentials of the anodic and cathodic peaks, the value of the standard potential \(\varphi^0\) often remains unknown.

If the mechanism of the discharge—ionization process is the same, then the rate constant and \(\varphi^0\) in this case can be calculated on the basis of expressions (20, 21), taking \(K_{s_1}=K_{s_2}\); \(g_1=g_2\); \(\alpha+\beta=1\)

\[ \lg K_s=\alpha \lg \frac{\beta W}{S\alpha_{\infty}\gamma RTC_A^g} +\beta\left[\frac{1}{2}\lg D\alpha nW+1.14+(p-g)\lg C_A\right] -\frac{\beta\alpha}{\theta}\Delta\varphi_{nak} \tag{20} \]

where \(\Delta\varphi_{nak}\) is the difference between the anodic and cathodic peaks.

\[ \varphi_x^{0}=\alpha\varphi_{nk}+\beta\varphi_{na} -\theta\lg\frac{\beta W^{1/2}}{S\alpha_{\infty}\gamma(D\alpha n)^{1/2}} +\theta p\lg C_A\,\frac{0.268}{n}. \tag{21} \]

Experimental Part

The relations given above were used to investigate the mechanism of the discharge—ionization of gold bromide complexes. The work was carried out in a cell of conventional design [8] using an OH-101 polarograph. The graphite electrode was prepared according to a procedure by a co-worker of the Donetsk Branch of IREA [9]. The ohmic voltage drop in the cell did not exceed \(0.003\) V and therefore could be neglected. The salts used for preparing the solutions were calcined for several hours at \(350\)--\(500^\circ\mathrm{C}\); the water was distilled three times. To remove oxygen from the solution and to stir the solution during preliminary electrolysis, nitrogen purified of oxygen was used. The concentration of depolarizer in the solution for obtaining anodic peaks was \(2\cdot10^{-6}\) g-ion/l; in the study of cathodic peaks, \(1\cdot10^{-3}\) g-ion/l. The electrolysis potential was chosen in the region of the limiting

of the cathodic current and was equal to \(-0.2\) V vs. S.C.E. In order to maintain a constant ionic strength, the studies were carried out against the background \((xM\ \mathrm{KBr} + (2-x)M\ \mathrm{KNO}_3)\).

Both monovalent and trivalent gold can participate in electrochemical reactions. During cathodic polarography of 1 M potassium bromide solutions containing trivalent gold, two well-defined peaks are observed (Fig. 1). Coulometric analysis of these peaks showed that the more positive wave is due to the discharge of trivalent gold. The second peak is due to the discharge of monovalent gold. When the concentration of bromide ions in the solution is decreased, the waves come closer together, and at a KBr concentration below 0.5 M only one very broad wave is observed, corresponding to the simultaneous discharge of mono- and trivalent gold. During anodic dissolution of gold from the surface of a graphite electrode, only one peak was obtained; its potential shifts into the region of negative values with increasing concentration of bromide ions. Coulometric analysis showed that this peak corresponds to the process of formation of the monovalent gold ion. The formation of trivalent gold apparently occurs at more positive potentials and is masked by the discharge of the supporting electrolyte. The values of the charge-transfer coefficients for tri- and monovalent gold were determined from the tangent of the slope angle of the straight lines in the coordinates: \(\varphi_{pk} — \lg W\) (Fig. 2).

Fig. 1. Polarization curves of gold against the background of 1 M KBr

Fig. 2. Dependence of the potentials of the cathodic and anodic peaks of gold on \(\lg W\) against the background of 2 M KBr

\[ \frac{\partial \varphi_{pk1}}{\partial \lg W}=-0.036 \qquad \frac{\partial \varphi_{pk2}}{\partial \lg W}=-0.07; \]

\[ n=2 \qquad \alpha=0.41 \qquad n=1 \qquad \alpha=0.42. \]

It turned out that the transfer coefficient \(\alpha\) is the same for the discharge of trivalent and monovalent gold. A similar value of \(\alpha\) was obtained from an analysis of the difference between the potentials of the cathodic half-peaks and peaks (Fig. 1). In this case the difference between the potentials \(\varphi_{pk\,?}-\varphi_{pk}\) does not depend on the concentration of bromide ions

\[ \frac{\Delta \varphi_{pk1}}{2}-\varphi_{pk1}=0.05 \qquad n=2 \qquad \alpha=0.48 \]

\[ \Delta \varphi_{pk2\,?}-\varphi_{pk2}=0.1 \qquad n=1 \qquad \alpha=0.48 \]

Figure 2 also gives the dependence of the potential of the anodic gold peak on the logarithm of the rate of change of potential. The tangent of the angle of inclination of this straight line is \(-0.1\). Hence \(\beta\), in accordance with equation (16), is equal to 0.59.

The dependence of the gold peak current on the area under the anodic tooth is presented in Fig. 3.

Fig. 3. Dependence of the maximum current of gold electrodissolution on the area under the anodic tooth against the background of 1 M KBr. Experimental conditions: \(W=0.009\ \mathrm{V/sec}\), \(\tau=60\ \mathrm{sec}\), \(\varphi_{\mathrm{e}}=-0.7\ \mathrm{V}\) relative to the c.e.

\[ \frac{\partial I_{pa}}{\partial Q_a}=0.075 \]

In accordance with equation (18)

\[ \beta=0.58 \]

The difference between the potentials of the inflection points of the anodic gold teeth does not change when the concentration of bromide ions is varied and is equal to \(0.08\ \mathrm{V}\). Hence \(\beta=0.6\), which is close to the values of \(\beta\) obtained from other dependences.

To determine the composition of the complexes participating in the electrode process, the dependence of the potentials of the anodic and cathodic gold peaks on the logarithm of the concentration of bromide ions was studied (Fig. 4). Lines I and IV correspond to the potentials of the anodic (I) and cathodic (IV) peaks of monovalent gold; line (III) corresponds to the potentials of trivalent gold. Line II expresses the dependence of the equilibrium potentials of \(\mathrm{Au}^{+}\) relative to \(\mathrm{Au}^{0}\) on the concentration of bromide ions in the solution. As can be seen from Fig. 4, the dependence of the potentials of the cathodic and anodic peaks on the concentration of bromide ions is linear. This indicates that, with a change in ligand concentration, the composition of the complex does not change. The values of the potentials of the anodic and cathodic peaks, as well as of the equilibrium potential of monovalent gold, are determined by the equations:

\[ \varphi_{pk1}=0.4-0.12\lg C_{\mathrm{Br}^{-}} \]

\[ \varphi_{pk2}=0.28-0.17\lg C_{\mathrm{Br}^{-}} \]

Fig. 4. Dependence of the peak potentials of gold on the logarithm of the bromide-ion concentration. Explanations in the text.

\[ \varphi_{na}=0.54-0.08\lg C_{\mathrm{Br}^-} \]

\[ \varphi_p=0.4-0.13\lg C_{\mathrm{Br}^-} \]

Hence, on the basis of equations (16), (17), we obtain for the discharge and ionization of monovalent gold:

\[ g=0.8;\qquad p=2.2;\qquad p-g=1.24;\qquad g_2=0.96. \]

For the discharge of trivalent gold, \(p-g_2=1.75;\ p=4;\ g_2=2.25\). Consequently, in the electrochemical process of discharge—ionization of monovalent gold on the graphite electrode, a complex of composition \(\mathrm{AuBr}\) takes part, whereas in solution the predominant complex is of composition \([\mathrm{AuBr}_2]^+\). The mechanism of the anodic and cathodic processes is the same, as is also indicated by the fact that the sum of the transfer coefficients is close to unity:

\[ \alpha+\beta=0.59+0.44=1.03. \]

If it is assumed that the mechanism of discharge—ionization of trivalent gold is the same and the composition of the trivalent-gold complexes predominating in solution is taken to be \([\mathrm{AuBr}_4]^{-2}\), the overall mechanism of discharge—

² No other composition of the complex is possible, since the compound \([\mathrm{AuBr}_2]^+\) is very unstable, while formation of the complex \(\mathrm{AuBr}_3\) would be accompanied by precipitation \((\mathrm{SP}=10^{-36})\), which we have never observed experimentally when working with gold concentrations in solution of \(10^{-5}\)—\(10^{-3}\) g-ion/l. The existence of a gold complex of composition \([\mathrm{AuBr}_4]^-\) in solutions of bromide salts is indicated in a number of works [10—12]; gold complexes with a larger number of ligands are not known in the literature.

ionization of gold on a graphite electrode can be represented as follows:

1. Discharge–ionization of trivalent gold:

\[ [\mathrm{AuBr}_4]^- \rightleftarrows [\mathrm{AuBr}_2]^+ + 2\mathrm{Br}^- \quad \text{equilibrium in solution,} \]

\[ [\mathrm{AuBr}_2]^+ + 2e \rightleftarrows [\mathrm{AuBr}_2]^- \quad \text{electrode reaction,} \]

\[ [\mathrm{AuBr}_4]^- + 2e \rightleftarrows [\mathrm{AuBr}_2]^- + 2\mathrm{Br}^- \quad \text{overall electrode reaction.} \]

1. Discharge–ionization of monovalent gold

\[ [\mathrm{AuBr}_2]^- \rightleftarrows [\mathrm{AuBr}]^0 + \mathrm{Br}^- \quad \text{equilibrium in solution,} \]

\[ [\mathrm{AuBr}]^0 + e \rightleftarrows \mathrm{Au}^0 + \mathrm{Br}^- \quad \text{electrode reaction,} \]

\[ [\mathrm{AuBr}_2]^- + e \rightleftarrows \mathrm{Au}^0 + 2\mathrm{Br}^- \quad \text{overall electrode reaction.} \]

To calculate the rate constant of the electrode process, we determine the diffusion coefficient of the discharging complexes as a function of the peak current of the cathodic process (equation 17). In accordance with the experimental results given in Table 1, the value of the diffusion coefficient of \(\mathrm{Au}^+\) in a 1 M KBr solution is \(1.33 \cdot 10^{-5}\ \mathrm{cm}^2/\mathrm{s}\). The magnitude of the standard electrode potential of discharge—ionization of the complex \([\mathrm{AuBr}_2]\) is given in the handbook [13].

Table 1

Data for calculating the diffusion coefficient of gold in a 1 M KBr solution

\[ \varphi = 0.96\ \text{V vs. SHE.} \]

Assuming

\[ \varphi_{\mathrm{cal}}^0 = 0.2438\ \text{V vs. NHE.} \]

\[ \varphi^0 = 0.716\ \text{V vs. NHE.} \]

Let us compare this value with the value of the formal standard potential calculated from the experimental data by formula (17). It is seen from Table 2 that the value of the standard potential, determined by formula (17), differs from the theoretical value by 5%, which lies within the experimental error. The value of the standard potential for trivalent gold in a potassium bromide solution was taken from the handbook [13], \(\varphi^0 = 0.82\ \text{V vs. SHE} = 0.576\ \text{V vs. NHE}\). The values of the rate constants of this electrode process, calculated by formulas (16, 17, 20), are given in Table 2. The standard exchange-current densities for discharge—ionization of \(\mathrm{Au}^+\) and \(\mathrm{Au}^{3+}\), respectively, are:

\[ \gamma_{\mathrm{Au}^+}^0 = (2.3 \pm 0.32)\cdot 10^{-4}\ \mathrm{A/mol/cm}^{-2}, \]

\[ \gamma_{\mathrm{Au}^{3+}}^0 = (1.32 \pm 0.09)\cdot 10^{-3}\ \mathrm{A/mol/cm}^{2}. \]

From the data we have obtained it is evident that the process of discharge—ionization of gold in bromide solutions is completely irreversible. This is indicated by the fact that the difference between the potentials of the anodic and cathodic peaks \(\Delta \varphi_{\mathrm{peak}}\) is greater than \(\frac{0.15}{n}\ \mathrm{V}\), and the standard rate constant of the electrode process is less than \(10^{-3}\ \mathrm{cm/s}\). Unfortunately, in the literature there are neither theoretical nor experimental data on the study of the kinetics

Table 2

Results of determination of the kinetic parameters of the discharge–ionization process of gold on a graphite electrode against the background \([xM\ \mathrm{KBr} + (2-x)M\ \mathrm{KNO_3}]\)

of the discharge–ionization process of complex metal ions on a graphite electrode. This work is the first attempt to apply the polarographic method on stationary solid electrodes to the study of the kinetics of slow electrode processes. Owing to the experimental simplicity, high sensitivity, and accuracy of this method, it may be expected that in the future it will find broader application.

REFERENCES

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8. V. I. Kuleshov, A. G. Stromberg. Collected Works of IREA, issue 5–6, 37, (1963).
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1. The composition of the discharging complexes at potentials near the cathodic peak may differ from the composition of the complexes obtained during anodic dissolution of the metal film near the potential of the anodic peak (3–4). ↩