POLAROGRAPHIC STUDY OF THE KINETICS OF ELECTRODE PROCESSES IN SOLUTIONS OF COMPLEX IONS AT MERCURY AND AMALGAM DROPPING ELECTRODES

A. G. STROMBERG, L. N. POPOVA

One of the authors of the article has shown theoretically [1, 2] and confirmed experimentally [3–5] that, from the dependence of the potential difference of the irreversible anodic–cathodic half-wave at an amalgam dropping electrode on the ligand concentration, it is possible to determine the average composition of the complexes directly participating in the electrode process and the average exchange current of the electrochemical reaction. It has also been shown [4–9] that, from the dependence on the logarithm of the ligand concentration of the potential of the anodic and cathodic half-waves in the titanium(III)—titanium(IV) oxidation–reduction system at a mercury dropping electrode, one can determine the composition, with respect to halide and hydroxyl, of the titanium complexes directly participating in the electrode process, assuming that the anodic and cathodic processes proceed by the same mechanism. Later it was shown [10] that the composition of the discharging complexes at potentials near the cathodic half-wave potential may differ from the composition of the complexes formed during anodic dissolution of the amalgam at potentials near the anodic half-wave potential. A different mechanism of the process is also indicated by the significant deviation of the sum of the transfer coefficients of the discharge and ionization processes, determined from the slope of the cathodic and anodic waves, respectively [10]. Molodov and Losev [11], in studying the process of anodic dissolution of zinc amalgam in ammonia solutions by radiochemical and electrochemical methods, independently arrived at the conclusion that the electrode process has different mechanisms at different potentials.

Matsuda and Ayabe [12–14] developed a more rigorous polarographic theory of irreversible and quasi-reversible electrode processes at a mercury dropping electrode, which, however, is distinguished by considerable mathematical complexity. It should be emphasized here that, in contrast to our works [1–11], Matsuda considers the theory only of the cathodic irreversible polarographic wave in a solution of complex ions.

The purpose of the present work is to derive mathematical relations for calculating the standard rate constant of an electrochemical process in a solution of complex ions from polarographic experimental data at a mercury or amalgam dropping electrode, under the assumption that, with a change in potential, the composition

of the complex participating in electron transfer may vary. Thus, it is assumed that the kinetic parameters of the electrochemical process at the potential of the anodic half-wave and at the potential of the cathodic half-wave may differ from one another.

The derivation of the theoretical expressions is carried out within the framework of a simplified model of an irreversible electrode process, which has been developed by one of us since 1955 [1, 2]. As shown by one of us [15], the approximate equation gives, for the exchange current, a numerical value differing by less than 8% from the value calculated by the rigorous theory.

We shall make the same assumptions as before (see, for example, [1]), except for the assumption that the mechanism of the electrode process is independent of potential. Namely: let us assume that the solution contains an excess of indifferent electrolyte; the only slow stages are electron transfer and the supply or removal by diffusion of atoms or ions reacting at the electrode; the adsorption or diffusion (psi-prime) potential is practically zero; the process is completely irreversible at the potential, respectively, of the anodic or cathodic half-wave; only one equilibrium complex is present in the solution in predominant amount and, in the electrode process at the potential considered (in the electron-transfer process), only one intermediate complex participates, and in the discharge process at the potential of the cathodic half-wave and in the ionization process at the potential of the anodic half-wave the composition of the intermediate complex directly participating in electron transfer is different.

Calculation of $\alpha_1$, $q_1$, and $k_{s1}$ from the cathodic wave

Let the cathodic process of discharge of a complex ion with formation of an amalgam of the metal under consideration ($M$) proceed through an intermediate complex:

\[ M L_p^{(py-z)-} \rightleftarrows M L_{q_1}^{(q_1 y-z)-} + (p-q_1)L^{y-} \quad \text{equilibrium} \quad (a) \]

\[ M L_{q_1}^{(q_1 y-z)-} + ze \rightleftarrows M^0(Hg) + q_1 L^{y-} \quad \text{(limiting stage)} \quad (b) \]

where $M^{z+}$ is the metal ion; $L^{y-}$ is the ligand; $p$ is the composition of the complex predominant in solution; $q_1$ is the composition of the intermediate complex directly participating in electron transfer in the cathodic process at potentials near the potential of the irreversible cathodic half-wave; $z$ is the number of electrons participating in the electrode process of discharge of ions at the mercury dropping electrode.

If the process of dissociation of the complex ions predominant in solution, with formation of complexes of intermediate composition, in accordance with the simplifying assumptions made, proceeds rapidly, then the equilibrium at the surface of the mercury dropping electrode will not be disturbed, and we may write (assuming the activity coefficients to be constant¹ and incorporating their numerical values into the dissociation constant):

\[ c_4^s = K_g c_1^s c_3^{s\,(p-q_1)}, \tag{1} \]

where $c_1^s$ is the concentration near the electrode surface of the complexes predominant in solution (of composition $p$), g-ion/cm$^3$; $c_4^s$ is the concentration near the electrode surface of those directly participating in the transfer

of the electron in the discharge process of intermediate complexes (of composition \(q_1\)) in \(g\)-ion/cm\(^3\); \(c_3\) is the ligand concentration, which we regard as two or more orders of magnitude higher than the concentration of ions of the complex-forming metal, mole/cm\(^3\); therefore the change in ligand concentration near the electrode surface during the discharge process may practically be neglected; \(K_g\) is the dissociation constant of the complex of composition \(p\) with formation of an intermediate complex of composition \(q_1\), expressed through ion concentrations.

The rate of the discharge process in accordance with the equation of the electrode process (b) is given by the theory of slow ion discharge [16]:

\[ \vec{i}=K_I c_4^s e^{-\frac{\alpha_1}{b'}\varphi}; \qquad b'=\frac{RT}{zF}, \tag{2} \]

where \(\alpha_1\) is the transfer coefficient of the discharge process \((0<\alpha_1<1)\); \(\varphi\) is the electrode potential; \(K_I\) is the rate constant of the discharge process at \(\varphi=0\), A·cm\(^3\)/g-ion. If the process is completely irreversible\(^1\), then the cathodic current is equal to the rate of the discharge process; \(\vec{i}=\vec{i}-\overleftarrow{i}\simeq\vec{i}\).

The rate constant \(K_I\) is related to the standard rate constant of the discharge process \(K_{s1}\) and to the standard exchange-current density \(j_{01}\) by the relation

\[ K_I=\bar{S} j_{01} e^{\frac{\alpha_1}{b'}\varphi^0} =\bar{S} zF K_f =\bar{S} zF K'_{s1} e^{\frac{\alpha_1}{b'}\varphi^0}, \tag{3} \]

where \(K_f\) is the rate constant of the discharge process at \(\varphi=0\), cm/sec; \(K'_{s1}\) is the standard rate constant of the discharge process of the intermediate complex, cm/sec; this is the rate of the discharge process at \(\varphi=\varphi^0\) and at concentrations of metal atoms and of the intermediate complex equal to \(c_4=c_2=1\) mole/cm\(^3\), and ligand concentration equal to \(c_3=1\) mole/liter\(^2\); \(\bar{S}=5.1\cdot10^{-3}(mt)^{2/3}\) is the mean surface area of the mercury dropping electrode, cm\(^2\); \(\varphi^0\) is the formal electrode potential of the electrochemical reaction\(^2\):

\[ M L_p^{(py-z)-}+ze \rightleftarrows M^0(\mathrm{Hg})+pL^{y-}. \tag{b} \]

\(^1\) It is easy to see that the rate of the reverse process may be neglected under the condition:

\[ (\varphi^0-\varphi)_{\min}\ge b\left(\lg\frac{1+m\gamma}{1-m}-\lg\gamma g+p\lg C_3\right), \]

assuming, for simplicity, \(\alpha_1+\beta_2\simeq1\); \(K_{s1}=K_{s2}\); \(q_1=q_2\); \(D_1=D_2\). Here \(m=i/I_k\); \(\gamma=I_k/I_a\); \(100g\) is the error, in percent, due to neglect of the reverse process.

For \(0.01<\gamma<100\); \(0.1<m<0.9\); \(C_3=1\) mole/liter; \(g=0.01\), we obtain: \((\varphi^0-\varphi)_{\min}\) from \(b\) to \(4b\), i.e., from \(0.06/z\) to \(0.24/z\) V. In this case the values of \(K_{s1}\), in accordance with formula (11), range from \(10^{-3}\) to \(10^{-5}\) cm/sec; (for \(m=0.1\); \(\alpha=0.5\); \(c_3=1\) mole/liter). (See p. 6).

The standard value \(c_3=1\) mole/liter (and not \(c_3=1\) mole/cm\(^3\)) is adopted because tabulated data for the standard potential are given for \(c_3=1\) mole/liter. It should be noted that in thermodynamic calculations of the standard or formal potential, the concentrations of complex ions in solution and of metal atoms in the amalgam are also expressed in mole/liter, in contrast to the dimension mole/cm\(^3\) adopted in calculations of \(K_{s1}\) and \(K_{s2}\) in kinetic calculations. However, this difference does not affect the numerical value, since the thermodynamic expression for \(\varphi^0\) contains the ratio of these concentrations \((c_1/c_2)\).

\(^2\) The values of the formal potential \(\varphi^0\) differ from the value of the standard thermodynamic potential \(\varphi^0_{\text{t.d.}}\) by the amount:

\[ \varphi^0=\varphi^0_{\text{t.d.}}-b\lg\frac{f_1}{f_2 f_3^p}, \]

where \(f_1, f_2, f_3\) are the activity coefficients, respectively, of the complex metal ions, metal atoms, and ligand.

Taking into account [1] that \(c_1^s=I_k-\dfrac{i}{K_{u1}}\) (\(I_k\) is the limiting diffusion cathodic current, a; \(K_{u1}=607 zD_1 m^{2/3}\tau^{1/6}\) is the Il’ković constant; \(D_1\) is the diffusion coefficient of the complex ions (of composition \(p\)) in solution, \(\text{cm}^2/\text{sec}\); \(m\) is the rate of outflow of mercury from the capillary, \(\text{mg}/\text{sec}\); \(\tau\) is the drop time, sec), we obtain from (2), taking into account (1) and (3),

\[ \frac{i}{I_k-i}=\frac{zF\bar S}{K_{u1}}(K'_{s1}K_g)c_3^{-(p-q_1)}e^{\frac{\alpha_1}{b'}(\varphi^0-\varphi)}. \tag{4} \]

Taking also into account that

\[ \frac{zF\bar S}{K_u}=0.81\left(\frac{\tau}{D_1}\right)^{1/2};\qquad K'_{s1}K_g=K_{s1}, \tag{5} \]

we obtain, for an irreversible cathodic wave in a solution of complex ions, the expression:

\[ \frac{i}{I_k-i}=0.81\left(\frac{\tau}{D_1}\right)^{1/2}K_{s1}c_3^{-(p-q_1)}e^{\frac{\alpha_1}{b'}(\varphi^0-\varphi)} \tag{6} \]

where \(K_{s1}\) is the standard rate constant of discharge at a concentration of the complex predominating in solution equal to unity \((c_2=c_1=1\ \text{mol}/\text{cm}^3)\), in contrast to the quantity \(K'_{s1}\); \(D_1\) is the diffusion coefficient of the complex ions predominating in solution, \(\text{cm}^2/\text{sec}\).

After taking the logarithm of formula (6), we obtain

\[ \lg\frac{i}{I_k-i}=A_{1k}+B_{1k}(-\varphi), \tag{7} \]

where

\[ B_{1k}=\frac{\alpha_1}{b};\qquad A_{1k}=\lg K_{s1}-\frac{1}{2}\lg\frac{1.51D_1}{\tau}-(p-q_1)\lg c_3+\frac{\alpha_1}{b}\varphi^0 \tag{8} \]

\((b=2.3\,b')\). From the straight line on a plot in the coordinates \(\lg i/(I_k-i)\), \((-\varphi)\), the coefficients \(A_{1k}\) and \(B_{1k}\) can be determined and then, by formulas (8), the kinetic parameters of the electrochemical reaction \(\alpha_1\) and \(K_{s1}\):

\[ \alpha_1=bB_{1k};\qquad \lg K_{s1}=\frac{1}{2}\lg\frac{1.51D_1}{\tau}+(p-q_1)\lg c_3+A_{1k}-\frac{\alpha_1}{b}\varphi^0. \tag{8a} \]

To calculate \(K_{s1}\) by formula (8), it is necessary to know \(\varphi^0\) and \(q_1\). If \(q_1\) is unknown, one should study the dependence of the cathodic half-wave potential on the ligand concentration. At \(i=\dfrac{1}{2}I_k\) and \(\varphi=\varphi_{1/2}\), we obtain from (6):

\[ (-\varphi_{1/2\,k})=A_{2k}+B_{2k}\lg c_3, \tag{9} \]

where

\[ B_{2k}=(p-q_1)\frac{b}{\alpha_1};\qquad A_{2k}=\frac{b}{\alpha_1}\left(\frac{1}{2}\lg\frac{1.51D_1}{\tau}-\lg K_{s1}\right)-\varphi^0. \tag{10} \]

From the straight line on a plot in the coordinates \((-\varphi_{1/2\,k})\), \(\lg c_3\), we find the coefficients \(A_{2k}\) and \(B_{2k}\) and then, by formulas (10), the kinetic parameters \(q_1\) and \(K_{s1}\):

\[ q_1=p-\frac{\alpha_1}{b}B_{2k};\qquad \lg K_{s1}=\frac{1}{2}\lg\frac{1.51D_1}{\tau}-\frac{\alpha_1}{b}A_{2k}-\frac{\alpha_1}{b}\varphi^0. \tag{11} \]

If the theory and calculations are correct, the values of \(K_{s1}\) calculated by formulas (8) and (11) must be identical. The correctness

theory, and the validity of the simplifying assumptions made at the beginning of the article is also indicated by the linearity of the plots in the coordinates \(\lg i/I_k - i\), \((-\varphi)\) and \((\varphi_{1/2\,k})\), \(\lg c_3\), in accordance with formulas (7) and (9). Nonlinearity of the plots in these coordinates indicates either a change in the composition of the predominant and intermediate complex, or the invalidity, for the system studied, of one or several assumptions made in deriving the mathematical relations.

Calculation of \(\beta_2\), \(q_2\), and \(K_{S2}\) from the anodic wave

Let the process of anodic dissolution of the metal from the amalgam upon obtaining an anodic wave at a dropping amalgam electrode proceed through a second intermediate complex (of composition \(q_2\)):

\[ M^0(\mathrm{Hg}) + q_2 L^{y-} \rightleftarrows ML_{q_2}^{(q_2y-z)-} + ze \qquad \text{(limiting stage)} \tag{г} \]

\[ ML_{q_2}^{(q_2y-z)-} + (p-q_2)L^{y-} \rightleftarrows ML_p^{(py-z)-} \qquad \text{(equilibrium)} \tag{д} \]

As was shown theoretically and confirmed experimentally by one of us [10], at different potentials the composition of the complexes discharging and forming during ionization of the metal from the amalgam may be different if several intermediate complexes can participate in the electrode process.

With complete irreversibility of the anodic process, the anodic current is given by the expression:

\[ i=\overleftarrow{i}=K_{\mathrm{II}}\,c_2^S c_3^{q_2} e^{-\frac{\beta_2}{b'}\varphi^0}, \tag{12} \]

where \(\overleftarrow{i}\) is the ionization rate, \(a\); \(K_{\mathrm{II}}\) is the rate constant of the anodic process at \(\varphi=0\), \(a\cdot\mathrm{cm}^3/\mathrm{g\text{-}ion}\); \(\beta_2\) is the transfer coefficient of the ionization process; \(c_2^S\) is the concentration of metal atoms near the surface of the amalgam drop, \(\mathrm{g\text{-}atom}/\mathrm{cm}^3\).

By analogy with formula (3) we have

\[ K_{\mathrm{II}}=\bar S\cdot i_{02} e^{-\frac{\beta_2}{b'}\varphi^0} =\bar S z F K_b =\bar S z F K_{S2} e^{-\frac{\beta_2}{b'}\varphi^0}, \tag{13} \]

where \(K_b\) is the rate constant of the ionization process at \(\varphi=0\), \(\mathrm{cm}/\mathrm{s}\); \(K_{S2}\) is the standard rate constant of ionization during anodic dissolution of the metal from the amalgam at \(\varphi=\varphi^0\) and ligand concentration \(C_3=1\ \mathrm{mol}/\mathrm{l}\).

Using further the relations

\[ c_2^S=I_a-i/K_{u2};\quad K_{u2}=607\,zD_2^{1/2}m^{2/3}\tau^{1/6};\quad \frac{zF\bar S}{K_{u2}}=0.81\left(\frac{\tau}{D_2}\right)^{1/2} \]

(\(I_a\) is the limiting diffusion anodic current, \(a\); \(K_{u2}\) is the Ilkovič constant of the anodic wave; \(D_2\) is the diffusion coefficient of metal atoms in the amalgam, \(\mathrm{cm}^2/\mathrm{s}\)), we obtain, by analogy with formula (6):

\[ \frac{i}{I_a-i} = 0.81\left(\frac{\tau}{D_2}\right)^{1/2} K_{S2}c_3^{q_2}e^{-\frac{\beta_2}{b'}(\varphi^0-\varphi)} \tag{14} \]

or

\[ \lg \frac{I_a-i}{i}=A_{1a}+B_{1a}(-\varphi), \tag{15} \]

where

\[ B_{1a}=\frac{\beta_2}{b};\quad A_{1a}=-\lg K_{S2}+\frac{1}{2}\lg\frac{1.51D_2}{\tau}-q_2\lg c_3+\frac{\beta_2}{b}\varphi^0, \tag{16} \]

From the straight line on the plot in the coordinates \(\lg \dfrac{I_a-i}{i}\), \((-\varphi)\), one can determine the coefficients \(A_{1a}\) and \(B_{1a}\), and then, from formula (16), the kinetic parameters \(\beta_2\) and \(K_{S2}\):

\[ \beta_2=bB_2;\quad \lg K_{S2}=\frac{1}{2}\lg\frac{1.51D_2}{\tau}-q_2\lg c_3-\lg\frac{I_a-i}{i} +\frac{\beta_2}{b}(\varphi^0-\varphi). \tag{17} \]

To determine \(K_{S2}\) from formula (17), it is necessary to know \(q_2\).

Putting \(i=\dfrac{1}{2}I_a\) and \(\varphi=\varphi_{1/2}\), we transform formula (14) into the form

\[ -\varphi_{1/2}=A_{2a}+B_{2a}\lg c_3, \tag{18} \]

where

\[ B_{2a}=q_2\frac{b}{\beta_2};\quad A_{2a}=-\frac{b}{\beta_2}\left(\frac{1}{2}\lg\frac{1.51D_2}{\tau}-\lg K_{S2}\right)-\varphi^0. \tag{19} \]

From the straight line on the plot in the coordinates \((-\varphi_{1/2,a},\ \lg c_3)\) we find the coefficients \(A_{2a}\) and \(B_{2a}\), and then, from formulas (19), the kinetic parameters \(q_2\) and \(K_{S2}\):

\[ q_2=\frac{\beta_2}{b}B_{2a};\quad \lg K_{S2}=\frac{1}{2}\lg\frac{1.51D_2}{\tau} -\frac{\beta_2}{b}A_{2a} +\frac{\beta_2}{b}\varphi^0. \tag{20} \]

If the theory is correct, the values of \(K_{S2}\) calculated from formulas (17) and (20) must be identical.

The linearity of the plots in the coordinates \(\lg(I_a-i)/i,\ (-\varphi)\) and \((-\varphi_{1/2,k}),\ \lg c_3\) indicates the constancy of the composition of the predominant complexes participating in electron transfer over the potential and ligand-concentration interval under consideration, and also confirms the validity of the simplifying assumptions made in deriving the mathematical relations.

Let us also note that the assumption we made earlier [10] about the possibility of participation in the electrode process of two or more intermediate complexes, both in the discharge process and in the ionization process, is indirectly confirmed by the calculations given above.

Calculation of averaged values of \(\overline{q}\) and \(\overline{K}_S\) from the difference of the potentials \(\Delta\varphi_{1/2,ak}\)

In calculating the kinetic parameters \(K_{S1}\) and \(K_{S2}\) from the cathodic and anodic waves of an irreversible electrochemical process, the value of the formal (or standard) potential \(\varphi^0\) often proves to be unknown. In this case it is expedient to calculate the mean values \(\overline{K}_S\) from the difference between the anodic and cathodic half-wave potentials \(\Delta\varphi_{1/2,ak}\). Eliminating \(\varphi^0\) from formulas (9) and (18), taking into account (10) and (19), we obtain

\[ \Delta\varphi_{1/2,ak}=A_{ak}+B_{ak}\lg c_3, \tag{25} \]

where

\[ B_{ak}=(\overline{\beta}p-\overline{q})\frac{b}{\overline{\beta}\alpha_1};\quad A_{ak}=\frac{b}{\overline{\beta}\alpha_1} \left(\frac{1}{2}\lg\frac{1.51\overline{D}}{\tau}-\lg\overline{K}_S\right). \tag{26} \]

The plot in the coordinates \(\Delta\varphi_{1/2,ak},\ \lg c_3\), in accordance with (25), should be represented by a straight line. From the values of the coefficients \(A_{ak}\) and \(B_{ak}\) determined from the plot, we find the values of \(\overline{q}\) and \(\overline{K}_S\):

\[ \overline{q}=\overline{\beta}p-\frac{\overline{\beta}\alpha_1}{b}B_{ak};\quad \lg\overline{K}_S=\frac{1}{2}\lg\frac{1.51\overline{D}}{\tau} -\frac{\overline{\beta}\alpha_1}{b}A_{ak}. \tag{27} \]

The values of the averaged density of the standard exchange current \(j_0\) and of the averaged exchange current \(\bar i_0\) are found from the formulas:

\[ \bar j_0=zF\bar K_S;\quad \bar i_0=\bar j_0 c_3^{\bar q-\bar\beta p}c_1^{\bar\beta}c_2^{\bar\alpha}. \tag{28} \]

The averaged values \(\bar K_S,\ \bar j_0,\ \bar i_0,\) and \(\bar q,\ \bar D,\ \bar\beta,\ \bar\alpha\) are related to the actual values of these quantities at the cathodic and anodic half-wave potentials by the following relations:

\[ \bar K_S=K_{S1}^{\bar\beta}K_{S2}^{\bar\alpha}=K_f^{\bar\beta}K_b^{\bar\alpha};\quad \bar j_0=zF\bar K_S, \tag{29} \]

\[ \bar i_0=\bar i_{01}^{\bar\beta}\bar i_{02}^{\bar\alpha};\quad \bar j_0=\bar j_{01}^{\bar\beta}\bar j_{02}^{\bar\alpha};\quad \bar\alpha=\frac{\alpha_1}{\alpha_1+\beta_2}, \tag{30} \]

\[ \bar q=\bar\beta q_1+\bar\alpha q_2;\quad \bar D=\bar\beta D_1+\bar\alpha D_2;\quad \bar\beta=\frac{\beta_2}{\alpha_1+\beta_2}. \tag{31} \]

It is easy to see that, for calculating the averaged value, as well as all the other kinetic parameters (averaged and non-averaged), in formulas (27–31) it is not necessary to know the numerical value of the formal (or standard) potential \(\varphi^0\).

Calculation of the effective value of the formal potential

From formulas (9) and (18), taking into account (10) and (19), we obtain:

\[ -\varphi_{1/2}=\bar A+\bar B\lg c_3;\quad \bar\varphi_{1/2}=\bar\alpha\varphi_{1/2\,k}+\bar\beta\varphi_{1/2\,a}, \tag{32} \]

where

\[ \bar A=\left(\varphi^0-b\lg\frac{K_{S1}}{K_{S2}}\right)+\frac{b}{2}\lg\frac{D_1}{D_2};\quad \bar B=(p+q_2-q_1)b. \tag{33} \]

From the plot in coordinates \(-\varphi_{1/2},\ \lg c_3\), which, if the initial assumptions are correct, should have the form of a straight line, we find the coefficients \(\bar A\) and \(\bar B\), and then:

\[ q_1-q_2=p-\frac{\bar B}{b};\quad \varphi^0-b\lg\frac{K_{S1}}{K_{S2}}=\bar A-\frac{b}{2}\lg\frac{D_1}{D_2}. \tag{34} \]

Thus, to calculate \(\varphi^0\) by formula (34), it is necessary to know the ratio \(K_{S1}/K_{S2}\). However, to calculate the quantities \(K_{S1}\) and \(K_{S2}\) (and their ratio) from polarographic data, it is necessary to know the numerical value of the formal potential \(\varphi^0\). Therefore it does not appear possible to calculate simultaneously from polarographic data \(\varphi^0\) and the ratio \(K_{S1}/K_{S2}\).

Let us make the simplifying assumption that \(K_{S1}/K_{S2}=1\). We shall call the value \(\varphi_*^0\) at \(K_{S1}/K_{S2}=1\) the effective formal potential. Neglecting the subtrahend \(\frac{b}{2}\lg(D_1/D_2)\) in formula (34), we obtain:

\[ \varphi_*^0=\bar A. \tag{35} \]

Agreement of the numerical value of \(\varphi_*^0\) with the tabulated value of the formal (standard) potential indicates the correctness of the assumption made, i.e., the closeness of \(K_{S1}/K_{S2}\) to unity.

An experimental verification of the theoretical relationships obtained in this article will be published elsewhere.

Conclusions

1. A theory has been developed for irreversible anodic–cathodic waves at an amalgam dropping electrode in a solution of complex ions in the pre-

assumption that the mechanism of the electrode process (the composition of the intermediate complex) at the cathodic and anodic half-wave potentials is different.

1. Mathematical relations have been derived for calculating the standard rate constant of the electrochemical reaction, the composition of the intermediate complex, and the transfer coefficients from the dependence of the cathodic and anodic half-wave potentials on the logarithm of the ligand concentration.

2. Mathematical relations have been derived that relate the true values of the kinetic parameters of the electrochemical process at the cathodic and anodic half-wave potentials to the averaged values of the kinetic parameters determined in the earlier theory (1955).

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1. Under the conditions of our experiments this assumption is justified, since the experiments were carried out at constant ionic strength. ↩