Anodic amalgam voltammetry at constant current on a spherical mercury electrode. Resolving power of the method
Abstract Generated abstract
This paper examines the resolving power of constant current anodic amalgam voltammetry on a spherical mercury electrode, particularly when equilibrium potentials of different amalgam components are close enough for chronopotentiograms to overlap. The authors define resolving power through the potential difference between the initial equilibrium potential and the potential at the transition time, and derive calculation formulas for both irreversible and reversible electrode processes. Numerical calculations at 20 degrees Celsius show that resolving power decreases as the dimensionless time parameter increases, and for reversible processes it also decreases with lower concentration of the oxidized form in solution. The paper further proposes a practical alternative measure based on the potential interval between 0.01 and 0.99 of the transition time, which is smaller and may better represent usable resolution in chronopotentiograms.
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PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
awarded the Order of the October Revolution and the Order of the Red Banner of Labor, named after S. M. Kirov
Vol. 197 1975
ANODIC AMALGAM VOLTAMMETRY
AT CONSTANT CURRENT ON A MERCURY
SPHERICAL ELECTRODE.
RESOLVING POWER OF THE METHOD
M. S. ZAKHAROV, V. I. BAKANOV
(Presented by the scientific seminar of the Department of Physical and Colloid Chemistry)
In previous works it has been shown that anodic amalgam voltammetry (AAV) at constant current can be used for the analysis of complex amalgams if the equilibrium potentials of the amalgams are sufficiently far apart from one another. Naturally, if the equilibrium potentials of the amalgams lie close to one another, superposition of chronopotentiograms will be observed, and separate determination of the elements will be impossible. In this connection, the present work will discuss the question of the resolving power of AAV with constant current.
From Fig. 1 it is seen that a separate chronopotentiogram of the more electropositive element can be obtained if the potential at the beginning of oxidation of the amalgam (the equilibrium potential) of this element is, at least, no less than the potential at which the transition time ($\tau$) of the more electronegative element is reached. We propose that the resolving power of the AAV method with constant current be understood as the difference between the electrode potential ($\Delta \varphi$) upon reaching the transition time and the equilibrium potential of the amalgam of the given element. The smaller $(\Delta \varphi)$ is, the greater will be the resolving power of the method and the more elements can be determined from a single chronopotentiogram (voltammogram).
The expression for calculating $\Delta \varphi$ theoretically can be obtained from the equation of the chronopotentiogram,
Fig. 1. Schematic plot of a chronopotentiogram. $\tau_1$ is the transition time for oxidation of the 1st element from the amalgam.
From equation (12) of [1] one can obtain the following expressions for \(\Delta\varphi\) at \(\vartheta' \ge 0.14\):
\[ \Delta\varphi=\varphi_{(\tau)}-\varphi_{(0)}= \frac{2.3RT}{\beta zF}\lg \frac{3\vartheta'-\displaystyle\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}} {\displaystyle\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}\exp\left(-\mu_n^2\vartheta'\right)} . \tag{1} \]
where \(\varphi_{(0)}\) and \(\varphi_{(\tau)}\) are the electrode potentials at \(t=0\) and upon reaching \(\tau\);
\[ \sum_{n=1}^{\infty}\frac{2}{\mu_n^2}=0.2. \]
From formula (1) it is clear that the resolving power of AAV at constant current for irreversible processes is a function of the following quantities:
\[ \Delta\varphi=f\left(T,\ z,\ \beta,\ \vartheta'\right). \tag{2} \]
By simple transformations of equation (11) of [1], one can obtain the following expressions for \(\Delta\varphi\) for reversible processes at \(\vartheta'\ge 0.14\):
\[ \Delta\varphi=\varphi_{(\tau)}-\varphi_{(0)} = \frac{RT}{zF}\ln \frac{ \left(3\vartheta'+\displaystyle\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}\right) \left[ C_0^0+\lambda\left(+\lambda\left(1-\exp \vartheta' \operatorname{erfc}\lambda\sqrt{\vartheta'}\right)\right) \right] }{ C_0^0\displaystyle\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}\exp\left(-\mu_n^2\vartheta'\right) }. \tag{3} \]
From formula (3) it is clear that the resolving power of AAV at constant current for reversible processes is a function of the following quantities:
\[ \Delta\varphi=f\left(T,\ z,\ \vartheta',\ C_0^0,\ \lambda\right). \tag{4} \]
At \(20^\circ\mathrm{C}\), for various values of \(\vartheta'\), we calculated the quantities \(\Delta\varphi\) for irreversible and reversible processes. In calculations for reversible processes it must be borne in mind that both \(\lambda\) and \(\vartheta'\) are related to the quantity \(i\) by the following relations [1]:
\[ \lambda=\frac{i_0}{4\pi zFr_0D_R}; \qquad i_0\vartheta'+\frac{1}{15}=\mathrm{const}. \tag{5} \]
Therefore, for a given \(\vartheta'\), the value of \(\lambda\) cannot be specified arbitrarily; it is calculated with allowance for expression (5).
The results of the calculations are presented in Fig. 2. From Fig. 2 it is seen that, for both irreversible and reversible processes, with increasing \(\vartheta'\) the resolving power of the AAV method at constant current on a mercury spherical electrode decreases (\(\Delta\varphi\) increases). For reversible processes the resolving power of the method also decreases with a decrease in the concentration of the oxidized form of the element in solution.
It is of interest to estimate in AAV with constant current the quantity \(\Delta\varphi_1=\varphi_{(0.99\tau)}-\varphi_{(0.01\tau)}\). By simple transformations of equation (12) of [1] for irreversible processes we obtain
Fig. 2. Dependence of \(\Delta\varphi\) on \(\vartheta'\):
1) for reversible processes—curve 1 at \(C_0^0=1\cdot10^{-9}\ \mathrm{mol/cm^3}\), curve 2 at \(C_0^0=1\cdot10^{-11}\ \mathrm{mol/cm^3}\);
2) for irreversible processes—curve 3. Dependence of \(\Delta\varphi_1\) on \(\vartheta'\):
1) for reversible processes at \(C_0^0=1\cdot10^{-9}\ \mathrm{mol/cm^3}\)—curve 4;
2) for irreversible processes—curve 5.
\[ \Delta\varphi_1= \frac{2{,}3RT}{\beta zF}\lg \frac{ 2{,}97\vartheta' + \displaystyle\sum_{n=1}^{\infty}\frac{2}{\mu_n^2} \exp\left(-0{,}01\vartheta'\mu_n^2\right) }{ 0{,}03\vartheta' + \displaystyle\sum_{n=1}^{\infty}\frac{2}{\mu_n^2} \exp\left(-0{,}99\vartheta'\mu_n^2\right) }. \tag{6} \]
From equation (11) of work [1], for reversible processes one can obtain the following relation:
\[ \Delta\varphi_1= \frac{2{,}3RT}{zF}\lg \left\{ \frac{ \left[ 2{,}97\vartheta' + \displaystyle\sum_{n=1}^{\infty}\frac{2}{\mu_n^2} \exp\left(-0{,}01\vartheta'\mu_n^2\right) \right] }{ \left[ C_0^0+\lambda\left(1-\exp 0{,}01\vartheta'\operatorname{erfc}\sqrt{0{,}01\vartheta'}\right) \right] } \times \frac{ \left[ C_0^0+\lambda\left(1-\exp 0{,}99\vartheta'\operatorname{erfc}\sqrt{0{,}99\vartheta'}\right) \right] }{ \left[ 0{,}03\vartheta' + \displaystyle\sum_{n=1}^{\infty}\frac{2}{\mu_n^2} \exp\left(-0{,}99\vartheta'\mu_n^2\right) \right] } \right\}. \tag{7} \]
Using equations (6, 7), we calculated the values of \(\Delta\varphi_1\). The results of the calculations are presented in Fig. 2. From Fig. 2 it is seen that \(\Delta\varphi_1\), at the same \(\vartheta'\), is considerably smaller (by a factor of \(1.5\)--\(2\)) than \(\Delta\varphi\) for irreversible processes, and somewhat smaller for reversible processes.
It should be noted that outside the interval \(\Delta\varphi_1\) the chrono-potentiogram runs almost parallel to the electrode-potential axis, and therefore, in practice, not \(\Delta\varphi\) but \(\Delta\varphi_1\) may be taken as the resolving power of AAV with constant current.
Conclusions
Expressions have been obtained for calculating the resolving power (\(\Delta\varphi\) and \(\Delta\varphi_1\)) in the AAV method at constant current.
Literature
- M. S. Zakharov, V. V. Pnev, Izv. TPI (present collection).