Methods for Determining the Composition of Complexes Directly Involved in the Electrode Process

L. N. Pogova

(Presented by the scientific-methodological seminar of KhTF)

At present, the theory and experimental methods for studying electrode reactions involving metal complexes have undergone considerable development.

In the present work, some of these methods are discussed, which make it possible to determine the composition of complexes directly involved in an irreversible electrode process.

Gerischer [1–3] was the first to show that the composition of the complexes present in solution differs from the composition of the complexes participating in the electrode process, and that the study of the influence of ligand concentration on irreversible electrode processes makes it possible to determine the composition of the complexes discharged at the electrode, as well as other kinetic parameters.

Let us consider a complex-forming system, assuming that in solution one complex of composition \([MeLp]^{(py-z)-}\) predominates. This complex diffuses to the electrode surface and dissociates to the complex \([MeLq]^{(qy-z)-}\).

\[ [MeLp]^{(py-z)-} \rightleftarrows [MeLq]^{(qy-z)-} + (p-q)L^{y-}, \tag{1} \]

which, in turn, is discharged at the electrode:

\[ [MeLq]^{(qy-z)-} + ze \rightleftarrows Me + qL^{y-}. \tag{2} \]

Here \(p\) is the composition of the complexes predominating in solution, \(q\) is the composition of the complexes participating in the electrode process. The exchange current of this process is determined by the equation [4]:

\[ i_0 = K_1 c_k e^{-\frac{\alpha}{b'}\varphi_p} = K_2 c_M c_L e^{\frac{\beta}{b'}\varphi_p}, \tag{3} \]

where
\(i_0\) — exchange current;
\(K_1, K_2\) — rate constants of the forward and reverse reactions at a potential equal to zero;
\(c_k\) — concentration of complex ions;
\(c_M\) — concentration of the metal;
\(c_L\) — concentration of the ligand;
\(\alpha, \beta\) — transfer coefficients of the theory of slow discharge–ionization, \(b'=\dfrac{RT}{nF}\); \(\varphi_p\) — equilibrium potential.

Under conditions of an excess of ligand and absence of adsorption on the electrode surface:

\[ \frac{\partial \lg i_0}{\partial \lg c_L} = q+\frac{\beta}{b}\frac{\partial \varphi_{\mathrm{p}}}{\partial \lg c_L}. \tag{4} \]

From the thermodynamic expression for the equilibrium potential we obtain

\[ \frac{\partial \varphi_{\mathrm{p}}}{\partial \lg c_L}=-bp. \tag{5} \]

From equations (4) and (5) we obtain:

\[ \frac{\partial \lg i_0}{\partial \lg c_L}=q-bp. \tag{6} \]

Gerischer [1—3], in order to measure the exchange current as a function of the ligand concentration, used the alternating-current method and the potentiostatic method.

The composition of the complexes participating in the electrode process can also be established with the aid of polarization measurements on a dropping amalgam electrode [4—7].

The equation of an irreversible polarographic wave, derived [8, 9] on the basis of the theory of slow discharge–ionization [10], has the form:

\[ \frac{i}{i_0} = \frac{I_{\mathrm{k}}-i}{I_{\mathrm{k}}}e^{\frac{\alpha}{b'}\eta} - \frac{I_{\mathrm{a}}-i}{I_{\mathrm{a}}}e^{-\frac{\beta}{b'}\eta}. \tag{7} \]

If \(i_0\) is small \((i_0 < 0.01 I)\), then

\[ \lg i_0 = \beta \lg I_{\mathrm{k}}+\alpha \lg I_{\mathrm{a}} -\frac{\alpha\beta}{b}\Delta\varphi_{1/2,\mathrm{a},\mathrm{k}} \tag{8} \]

and

\[ \left(\frac{\partial \lg i_0}{\partial \lg c_L}\right)_{I_{\mathrm{k}},I_{\mathrm{a}}} = \frac{\alpha\beta}{b} \left(\frac{\partial \Delta\varphi_{1/2,\mathrm{a},\mathrm{k}}}{\partial \lg c_L}\right)_{I_{\mathrm{k}},I_{\mathrm{a}}}. \tag{9} \]

From equations (6) and (9) we obtain:

\[ \frac{\partial \Delta\varphi_{1/2,\mathrm{a},\mathrm{k}}}{\partial \lg c_L} = \frac{b}{\alpha\beta}(bp-q). \tag{10} \]

Equations (8), (9), and (10) are valid only for irreversible processes.

Equations (9) and (10) can be used in the case where the composition of the complex does not change in the discharge–ionization process.

If the composition of the discharging complex \((q_1)\) and that of the complex formed upon ionization \((q_2)\) are different, then \(q_1\) and \(q_2\) can be determined from the dependence of the half-wave potentials of the anodic and cathodic waves on the ligand concentration [11]. In this case, the angular coefficients of the straight lines in the coordinates \(\varphi_{1/2,\mathrm{k}}-\lg c_L\) and \(\varphi_{1/2,\mathrm{a}}-\lg c_L\), respectively, are equal to:

\[ \gamma_{\mathrm{k}} = \frac{\partial \varphi_{1/2,\mathrm{k}}}{\partial \lg c_L} = (p-q_1)\frac{b}{\alpha}, \tag{11} \]

\[ \gamma_{\mathrm{a}} = \frac{\partial \varphi_{1/2,\mathrm{a}}}{\partial \lg c_L} = -q_2\frac{b}{\beta}. \tag{12} \]

\(\alpha\) and \(\beta\) are determined from the slope of the cathodic and anodic waves in semilogarithmic coordinates.

An expression similar to (11), for determining the composition of complexes undergoing irreversible discharge at a dropping mercury electrode, was...

was obtained by Matsuda [12]. Unfortunately, this method of analysis of irreversible polarographic waves does not make it possible to determine the composition \(p\) of the complexes predominating in solution. Other methods are used to determine \(p\).

In the case of a quasi-reversible process, the composition of the discharging complexes and of the complexes predominating in solution can be determined polarographically [12].

The reversibility of the electrode process is determined from the behavior of the polarographic curves at constant [12] and alternating [19] current. Quasi-reversible processes include processes whose rates lie between the rates of completely reversible and completely irreversible processes \((10^{-5}\ \text{cm/sec} < K_S < 10^{-1}\ \text{cm/sec}\), where \(K_S\) is the rate constant at the normal potential, related to the exchange current \(i_0\) by the relation \(i_0 = 10^{-3} z F K_S c_O^{\beta} c_R^{\alpha})\). In the case of a quasi-reversible process, the rate of the reverse process must be taken into account \((i = \vec{i} - \overleftarrow{i})\).

If the electrode process proceeds reversibly, then only the composition of the equilibrium complexes can be determined polarographically; no information about the mechanism of discharge can be obtained [12]. Other methods are used to study fast reactions, for example, the double-pulse method [3].

If a whole series of complexes exists simultaneously in solution, in equilibrium with one another [13, 14],

\[ [MeL_p]^{(py-z)-} \rightleftarrows [MeL_{p-1}]^{[(p-1)y-z]-} + L^{y-}\quad (p = 1, 2,\ldots, N) \tag{13} \]

and all the complexes can discharge at the electrode

\[ [MeL_p]^{(py-z)} + ze \rightleftarrows Me(\mathrm{Hg}) + pL^{y-}\quad (p = 1, 2,\ldots, N), \tag{14} \]

then kinetic analysis of the polarographic curve in the case of a quasi-reversible process makes it possible to determine the composition of all complexes present in solution and their dissociation constants, and the composition of the discharging complexes with the corresponding transfer coefficients \((\alpha)\) and rate constants of the electrode process [14].

With the aid of this method, zinc acetate complexes have been investigated. It was shown that of the five complexes present in solution, only three discharge (see the table).

Several complex-forming systems were studied by Koryta using the polarographic method [26–28]. In particular, he explained four waves of reduction of cadmium cyanide complexes [26]. It turned out that the first (more positive) wave is due to the reversible reduction of cadmium complexes, the next two are kinetic in character, and the fourth wave, appearing at a fourfold excess of cyanide relative to the concentration of cadmium ions, is due to the delayed discharge of the complex \(\mathrm{Cd(CN)}_3^-\).

In oxidation–reduction systems, the composition of the complexes directly participating in the electrode process and predominating in solution can be determined\(^1\) polarographically, if the electrode process is completely irreversible [15–18, 29, 30]. Thus, for example, it was shown that the complexes of Ti(III) and Ti(IV) predominating in HCl solution have the compositions \(\mathrm{Ti(OH)}_2\mathrm{Cl}_2\) and \(\mathrm{Ti}^{3+}\), whereas the complexes \(\mathrm{TiOHCl}^{2+}\) and \(\mathrm{TiOHCl}^+\) participate directly in the electrode process. The polarographic behavior was studied analogously

\(^1\) More precisely, from polarographic measurements one can determine the difference in the compositions of the oxidized and reduced forms both for the complexes participating in the electrode process and for those predominating in solution.

Table

\(^{a}\) Alternating-current method — ACM.

Continuation of Table

Continuation of table

the behavior of \(Ti(III) \to Ti(IV)\) in \(HBr\) and \(H_2SO_4\) solutions [18, 29]. This method makes it possible to estimate the equilibrium potential and the standard exchange-current density. Slow electrode reactions can also be studied by means of other polarization methods: galvanostatic and potentiostatic [20–21]. These methods make it possible to study the regularities of the discharge–ionization process of metal complexes under conditions where the influence of diffusion is absent. Let us consider the general case in which the complex predominant in solution is partially dissociated into intermediate complexes containing fewer ligands, and these also participate in the electrode process. The cathodic process is:

\[ [MeL_p]^{(py-z)-} \rightleftarrows [MeL_q]^{(qy-z)-} + (p-q)L^{y-}\ (q<p), \]

\[ [MeL_q]^{(qy-z)-} + ze \to Me + qL^{y-}. \]

The cathodic current density is given by the equation:

\[ i_- = zFK_- [MeL_q]^{(qy-z)-} \simeq zFK_- \frac{[Me_{\text{total}}]}{[L]^{(p-q)}} . \]

The anodic process:

\[ Me + qL^{y-} \to [MeL_q]^{(qy-z)-} + ze \]

is the reverse of the cathodic one. Then we have:

\[ i_+ = zFK_+ [Me][L^{y-}]^q, \]

\[ \left(\frac{\Delta \log i_-}{\Delta \log [c_L]}\right)_{u=\mathrm{const}} = q-p, \]

\[ \left(\frac{\Delta \log i_+}{\Delta \log [c_L]}\right)_{u=\mathrm{const}} = q, \]

\[ \left(\frac{\Delta \log i_0}{\Delta \log [c_L]}\right) = q + \alpha \frac{\Delta u_0}{\Delta \log [c_L]} \cdot \frac{zF}{2.3RT} = \]

\[ = (p-q) - \beta \frac{\Delta u_0}{\Delta \log [c_L]} \cdot \frac{zF}{2.3RT}. \]

These equations represent the ideal case of an electrode process, and in practice deviations may be observed; for example, two processes may occur simultaneously on the electrode surface. Another complicating factor encountered in practice is the nonconstancy of the transfer coefficients \(\alpha\) and \(\beta\) when the potential or the concentration of the active complex is changed, as well as the presence of adsorption on the electrode surface, distorting the values of \(\alpha\), \(\beta\) (obviously, all these remarks also apply to the polarographic method). Sources of error may include the ohmic voltage drop in the electrolyte (in the potentiostatic method) and capacitive effects caused by the charge of the electrical double layer (in the galvanostatic method). But often the capacitive effects are insignificant, and galvanostatic measurements can give good results without compensating for these effects. Therefore, in some cases the galvanostatic method has advantages over the potentiometric one (especially when the ohmic voltage drop is large). Another advantage of the galvanostatic method is that it makes it possible to detect slow chemical reactions preceding the discharge stage [22].

To elucidate the kinetic regularities and the mechanism of the processes of anodic dissolution and electrodeposition of metals, a method is used that is based on a combination of radiochemical and electro-

chemical methods [23]. This method was used to study the effect of ammonia on the kinetics of the electrode process on zinc amalgam [24]. The number of ammonia molecules participating in the anodic process is determined from the dependence $\lg i_+ - \lg C_{\mathrm{NH}_3}$.

It is of interest to use, for studying the kinetics of the discharge-ionization of complex metal ions, the method of amalgam polarography with accumulation (APA) [33, 34], since in this case the amalgam is obtained directly in the experiment itself and there is no need for its special preparation, which considerably simplifies the procedure. The composition of the complexes predominating in solution and participating in the electrode process, as determined by the methods listed above, is given in Table 1.

The kinetic parameters (transfer coefficients, the standard plane of the exchange current, and activation energy) of many irreversible electrode processes (involving chiefly simple metal ions) are given in Tanaka’s tables [35]. But these tables do not give the composition of the complexes participating in the electrode process and predominating in solution.

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