STUDY OF EQUILIBRIUM COMPLEXES BY AMALGAM CHRONOPOTENTIOMETRY WITH ACCUMULATION

V. I. BAKANOV, M. S. ZAKHAROV

(Presented by the scientific seminar of the MTP laboratory)

In the present work, the question of determining the composition of complexes and stability constants by the method of amalgam chronopotentiometry with accumulation (ACPA) on a mercury film electrode is considered. We shall assume that the overall electrode reaction

\[ \mathrm{Me}(\mathrm{Hg}) + nX^{-m} \rightleftarrows \mathrm{Me}X_n^{(mn-z)-} + ze^{-} \]

is reversible, and that the ligand concentration is in excess, i.e., the ligand concentration in the bulk of the solution and at the electrode surface may be considered identical and constant. In addition, we shall assume that a mobile equilibrium exists between the cation \(\mathrm{Me}^{+z}\), the ligand \(X^{-m}\), and the complex ion \(\mathrm{Me}X_n^{(mn-z)-}\).

Let the formation of the complex proceed according to the scheme

\[ \mathrm{Me}^{+z} + nX^{-m} = \mathrm{Me}X_n^{(mn-z)} . \]

For reversible electrode processes, the electrode potential in the case under consideration is determined by the Nernst equation:

\[ \varphi = \varphi^0 + \frac{RT}{zF}\ln \frac{C_{\mathrm{Me}^{+z}}(l,t)}{C_{\mathrm{Me}(\mathrm{Hg})}(l,t)} . \tag{1} \]

The concentration of metal ions at the electrode surface \(C_{\mathrm{Me}^{+z}}(l,t)\) will be determined by the stability constant, which is equal to:

\[ K_y = \frac{C_{\mathrm{Me}X_n}^{(mn-z)-}}{C_{\mathrm{Me}^{+z}}\cdot (CX^-)^n} . \tag{2} \]

Let the value of \(K_y\) be such that \(C_{\mathrm{Me}X_n}^{(mn-z)-} \gg C_{\mathrm{Me}^{+z}}\), \(C_k^0 = C_0^0\), where \(C_0^0\) is the concentration of introduced metal ions in the solution; \(C_k^0\) is the concentration of the complex in the bulk of the solution.

The concentration of metal atoms at the surface of the amalgam electrode is determined by the following relation [1]:
for \(V_R \ge 0.35\)

\[ C_R(l,V_R) = C_R^0 - \lambda_R \left(\vartheta_R + \frac{1}{3}\right), \tag{3} \]

where

\[ V_R = \frac{D_R t}{l^2}, \qquad \lambda_R = \frac{il}{zFD_R}. \]

The removal of complex ions occurs under conditions of semi-infinite linear diffusion, and the concentration of the complex at the electrode surface will be determined by the following expression:

\[ C_{\mathrm{Me}X_n}^{(mn-z)-} - C_k^0 = \frac{2it^{1/2}}{\sqrt{\pi z F D_k^{1/2}}}. \tag{4} \]

\(D_k\) is the diffusion coefficient of the complex ion.

Substituting (2), (3), and (4) into equation (1), we obtain an expression for the dependence of the electrode potential on time for anodic oxidation of the amalgam with formation of a complex:

\[ \varphi_k=\varphi^0-\frac{RT}{zF}\ln K_y-\frac{RT}{zF}\ln(C_{X^-})^n -\frac{RT}{zF}\ln \frac{V'_R-V_R}{\left(0.886\,\dfrac{C_k^0}{\lambda_k}+V^{1/2}\right)} -\frac{RT}{zF}\ln\frac{\lambda_R}{\lambda_k}, \tag{5} \]

where

\[ V'_R=\frac{D_R\tau}{l^2},\qquad \tau \text{ is the transition time},\qquad \lambda_k=\frac{il}{zFD_k},\qquad V_k=\frac{D_k t}{l^2}. \]

Usually the diffusion coefficient of the complex ion \(D_k\) does not differ very greatly from the diffusion coefficient of atoms \(D_R\) in the amalgam; therefore the magnitude of the last term may be neglected in comparison with the values of the other terms.

Using equation (5), one can determine the composition of the complex. For this purpose it is necessary to obtain two chronopotentiograms with the same transition time at two ligand concentrations \(C_{X_1^-}\) and \(C_{X_2^-}\):

\[ \Delta\varphi_k=\varphi_1-\varphi_2 = \frac{RT}{zF}\ln \left( \frac{C_{X_1^-}}{C_{X_2^-}} \right)^n, \tag{6} \]

where \(\varphi_1\) and \(\varphi_2\) are the electrode potentials at two ligand concentrations and the same \(\tau\). From equation (6) we obtain an equation for determining the composition \(n\) of the complex:

\[ n= \frac{zF\Delta\varphi_k} {2.3RT\lg\dfrac{C_{X_2^-}}{C_{X_1^-}}}. \tag{7} \]

When the amalgam is oxidized with formation of free metal ions, the dependence of the potential on time is given by the following equation [1]:

\[ \varphi=\varphi_0-\frac{RT}{zF}\ln \frac{V'_R-V_R}{0.886\,\dfrac{C_0^0}{\lambda_0}+\vartheta_0^{1/2}} -\frac{RT}{zF}\ln\frac{\lambda_R}{\lambda_0}, \tag{8} \]

where

\[ \lambda_0=\frac{il}{zFD_0}. \]

\(D_0\) is the diffusion coefficient of the free ions. Usually \(D_0=D_k\).

Substituting in equations (5) and (8) the value \(\varphi=\varphi_{V'_{1/2}}\) and subtracting the second from the first, we obtain:

\[ \varphi_{V'_{1/2}}K-\varphi_{V'_{1/2}} = \frac{RT}{zF}\ln\sqrt{\frac{D_0}{D_k}} -\frac{RT}{zF}\ln K_y -\frac{RT}{zF}\ln(C_{X^-})^n, \tag{9} \]

where \(\varphi_{V'_{1/2}}K\) and \(\varphi_{V'_{1/2}}\) are the chronopotentiometric half-wave potentials, respectively, for complex and free ions.

Equation (9) is analogous to the expression for the difference of polarographic half-wave potentials. Consequently, under identical experimental conditions, at constant current density and the same film thickness, the difference $\varphi V'_{1/2}-\varphi_{1/2}$ will not depend on the concentration of the complexing agent. This means that the dependence $\varphi V'_{1/2}, \lg C_{X-}$ is parallel to the dependence $\varphi_{1/2}, \lg C_{X-}$ and can be used to determine the stability constant, as well as the composition of the equilibrium complexes. This was to be expected, since at $\varphi=\varphi_{1/2}$ the time logarithmic term in equations (5) and (8) is equal to zero.

Equation (9) makes it possible to determine $K_y$ when $n$ is known, or, by plotting the graph $\varphi V_{1/2}-\varphi V_{1/2}, \lg C_{X-}$, to find from the slope the number of ligands $n$, and from the intercept on the ordinate axis, $-K_y$. The considerations presented are well confirmed by experimental data, which will be published separately.

LITERATURE

1. M. S. Zakharov, V. I. Bakanov. Izv. TPI, vol. 164, 1967, p. 86.