Stationary Potential and Mixed Current in Complex Oxidation-Reduction Systems
Abstract Generated abstract
This paper develops a polarographic treatment of the stationary potential and mixed current in a nonequilibrium oxidation-reduction system containing an amalgam of a more electronegative metal and ions of a more electropositive metal in solution. Using the theory of irreversible polarographic waves, it analyzes how the overall anodic-cathodic polarogram depends on the ratio of cathodic and anodic limiting diffusion currents and derives quantitative relations linking the stationary potential, mixed current, half-wave potentials, transfer coefficients, concentrations, and diffusion-current constants. The authors show that, depending on this current ratio, the mixed current is limited either by diffusion of the solution ion to the amalgam electrode or by diffusion of metal atoms in mercury, and they give expressions applicable to estimating cementation or amalgam corrosion rates under stationary diffusion-limited conditions.
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STATIONARY POTENTIAL AND MIXED CURRENT IN COMPLEX OXIDATION–REDUCTION SYSTEMS
A. G. STROMBERG, A. I. KARTUSHINSKAYA, Yu. N. ZHIKHAREV
(Presented by the scientific-methodological seminar of the Department of Physical Chemistry)
There are a number of works in the literature on the polarographic study of the stationary potential and the mixed current at the stationary potential [1–5]. However, the quantitative relationship between these quantities in polarography has not been discussed [1–2].
The purpose of the present work is the polarographic interpretation of the stationary potential and of the mixed current at this potential (hereinafter we shall simply call it the mixed current) in a nonequilibrium system: an amalgam of a more electronegative metal \((M_2)\), an ion of a more electropositive metal in solution \((M_1^{z_1+})\), as well as the establishment of quantitative dependences between the stationary potential and the mixed current.
Let us consider the theoretical form of a polarogram in a mixture of irreversible electrochemical systems, when the amalgam dropping electrode contains the more electronegative metal \(M_2\), and the solution contains ions of the more electropositive metal \(M_1^{z_1+}\). The stationary potential and the mixed current arise as a result of the simultaneous occurrence on the electrode of anodic and cathodic processes:
\[ M_2(\mathrm{Hg}) \to M_2^{z_2+} + z_2 e + (\mathrm{Hg}); \tag{1} \]
\[ M_1^{z_1+} + z_1 e + (\mathrm{Hg}) \to M_1(\mathrm{Hg}); \tag{2} \]
the overall process
\[ z_1 M_2(\mathrm{Hg}) + z_2 M_1^{z_1+} \to z_1 M_2^{z_2+} + z_2 M_1(\mathrm{Hg}). \tag{3} \]
Figure 1 presents the theoretical form of the overall polarogram (curve 3) in the nonequilibrium system \(M_2(\mathrm{Hg}), M_1^{z_1+}\) for two different ratios of limiting currents \(I_1/I_2=\gamma\) (\(I_1\) is the cathodic limiting diffusion current during discharge of the ion of the more electropositive metal; \(I_2\) is the anodic limiting diffusion current during electro-dissolution of the more electronegative metal from the amalgam dropping electrode).
If \(I_1<I_2\) and \(\gamma<1\) (Fig. 1,a), then the stationary potential \(\varphi_c\) changes in the potential interval from \(\varphi^*\) to \(\varphi_2^0\) (region II) and shifts along the anodic wave of the amalgam \(M_2(\mathrm{Hg})\) (\(\varphi_2^0\) is the standard potential of the more electronegative metal \(M_2^{z_2+}, M_2\); \(\varphi^*\) is the electrode potential at which the limiting diffusion stage changes for the mixed current at the stationary potential). In this case
the mixed current \(i_c\) is equal to the limiting current \(I_1\) of discharge of the ions \(M_1^{z_1+}\). Thus, the rate of discharge of the ions \(M_1^{z_1+}\) on the amalgam \(M_2\) (Hg) in region II is limited by the rate of diffusion of the ions \(M_1^{z_1+}\) to the surface of the amalgam dropping electrode.
Fig. 1. Schematic representation of the theoretical form of the anodic–cathodic polarogram (curve 3) in a nonequilibrium mixture of ions \(M_1^{z_1+}\) of the more electronegative metal in solution and atoms \(M_2\) of the more electropositive metal in an amalgam dropping electrode. Curve 1 is the anodic wave of electrodissolution of atoms \(M_2\) from the amalgam in the absence of ions \(M_1^{z_1+}\) in solution; curve 2 is the cathodic wave of discharge of ions \(M_1^{z_1+}\) on the mercury dropping electrode, i.e., in the absence of atoms \(M_2\) in the amalgam.
If \(\gamma > 1\) (Fig. 1,b), then \(\varphi_c\) varies in the potential interval from \(\varphi^*\) to \(\varphi_1^0\) (region I) and moves along the cathodic wave of the ions \(M_1^{z_1+}\) (\(\varphi_1^0\) is the standard potential of the more electropositive metal \(M_1^{z_1+} - M_1\)). In this case the mixed current is equal to the limiting anodic current \(I_2\) of the metal \(M_2\), that is, the rate of ionization of atoms of the metal \(M_2\) in reaction (3) is limited by the rate of diffusion of atoms of the metal \(M_2\) in mercury to the surface of the amalgam dropping electrode.
Quantitative relations between the stationary potential \(\varphi_c\) and the mixed current \(i_c\) are obtained from the theory of irreversible polarographic waves [6], as was shown earlier by us for particular examples [5].
In region I \((\varphi_1^0 > \varphi_c > \varphi^*)\), where \(\gamma > 1\), we have the expression for the stationary potential \(\varphi_c\) in terms of the mixed current \(i_c\) at the stationary potential:
\[ \varphi_c = \varphi_{1/2,1} - \frac{b}{\alpha_1 z_1}\lg \frac{i_c}{I_1 - i_c} \tag{4} \]
or
\[ i_c = I_2 = K_2 C_2 = I_1 \frac{\Theta_1}{\Theta_1 + 1}; \qquad \lg \Theta_1 = \frac{\alpha_1 z_1}{b}\left(\varphi_{1/2,1} - \varphi_c\right). \tag{5} \]
In region II \((\varphi^* > \varphi_c > \varphi_2^0)\) at \(\gamma < 1\):
\[ \varphi_c = \varphi_{1/2,2} + \frac{b}{z_2 \beta_2}\lg \frac{i_c}{I_2 - i_c} \tag{6} \]
or
\[ i_c=I_1=K_1 C_1=I_2\,\frac{\Theta_2}{\Theta_2+1}; \qquad \lg \Theta_2=-\frac{\beta_2 z_2}{b}\left(\varphi_{1/2,2}-\varphi_c\right), \tag{7} \]
where \(\varphi_{1/2,1}\) is the cathodic half-wave potential of discharge of ions \(M_1^{z_1+}\); \(\varphi_{1/2,2}\) is the anodic half-wave potential of electro-dissolution of metal atoms from the amalgam; \(K_1\) is the constant of the cathodic diffusion current of ions \(M_1^{z_1+}\); \(K_2\) is the constant of the anodic diffusion current of atoms \(M_2\); \(C_1\) is the concentration of ions \(M_1^{z_1+}\) in the solution; \(C_2\) is the concentration of atoms \(M_2\) in the amalgam; \(\alpha_1\) is the electron-transfer coefficient in the discharge process of the ion \(M_1^{z_1+}\); \(\beta_2\) is the electron-transfer coefficient in the ionization process of the atom \(M_2\) from the amalgam; \(b=2.3RT/F\).
If the electrode process for the more electronegative or more electropositive metal is reversible, then formally one must take in formulas (4) and (5) \(\alpha_1=1\), or in formulas (6) and (7) \(\beta_2=1\).^1
The theoretical dependence of the mixed current at the standard potential on the value of the standard potential in coordinates \(\dfrac{i_c}{I_1}\), \((-\varphi_c)\) (curve 1) and \(\dfrac{i_c}{I_2}\), \((-\varphi_c)\) (curve 2), in accordance with formulas (5) and (7), is schematically represented in Fig. 2.
Fig. 2. Schematic representation of the theoretical dependence of the relative value of the mixed current at the stationary potential on the value of the stationary potential. Curve 1—the dependence of \(\dfrac{i_c}{I_1}\) on \((-\varphi_c)\); curve 2—the dependence of \(\dfrac{i_c}{I_2}\) on \((-\varphi_c)\).
An expression for the density of the mixed current can be obtained from formulas (5) and (7), if one takes into account the known expressions for the diffusion-current constant at a dropping mercury electrode, and also for a dropping electrode [2]. In regions I and II the values of the standard densities of the mixed current will be, respectively,
\[ j_{c,1}^{0}=1.19\cdot 10^{5}\cdot z_2\cdot D_2^{1/2}\tau^{-1/2}; \qquad j_{c,2}^{0}=1.19\cdot 10^{5}\cdot z_1\cdot D_1^{1/2}\tau^{-1/2}, \tag{8} \]
\[ \left(j_{c,1}^{0}\ \text{and}\ j_{c,2}^{0},\ \text{in } \mathrm{cm/mol};\quad D,\ \mathrm{cm^2/sec};\quad \tau,\ \mathrm{sec}\right). \]
The expression for the potential \(\varphi^*\), at which a change of the limiting diffusion stage occurs, has the form
\[ \varphi^*= \frac{Z_1\alpha}{z_1\alpha+z_2\beta}\,\varphi_{1/2,1} + \frac{Z_2\beta}{z_1\alpha+z_2\beta}\,\varphi_{1/2,2}. \tag{9} \]
The mixed current at the standard potential is equal to the rate of the electrochemical reaction in a nonequilibrium mixture of two electrochemical systems in the absence of an external current. Therefore, from formulas (5) and (7) one can calculate the rate of cementation on an amalgam or the rate of cor-
^1 The rate constant of the electrochemical reaction is taken into account by means of the half-wave potential, which enters formulas (4)—(7) as an empirical constant.
rosion of the amalgam, provided that the process is diffusion-limited and stationary.
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