Anodic amalgam voltammetry with programmed current at a mercury film electrode
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Anodic Amalgam Voltammetry with Programmed Current at a Mercury Film Electrode
M. S. Zakharov, V. I. Bakanov
(Presented by the scientific-methodological seminar of the ChTF)
In the present work, expressions will be obtained for the transition time and for the dependence of the electrolysis potential on time in anodic amalgam voltammetry at a mercury film electrode with programmed current. In addition, some applications of the method in physicochemical studies will be considered.
In anodic amalgam voltammetry with a specified current at a mercury film electrode, the expression for the concentration of the reduced form of the element at the electrode surface has the form [1]:
\[ C_R(l,t)=C_R^0-\frac{1}{zFl}(I_1+2I_2), \tag{1} \]
\[ I_1=\int_0^t i(\tau')\,d\tau', \tag{2} \]
\[ I_2=\int_0^t i(\tau')\sum_{j=1}^{\infty} e^{\gamma_j(t-\tau')}\,d\tau', \tag{3} \]
\[ \gamma_j=\mu_n^2\frac{D_R}{l^2}, \tag{4} \]
where \(C_R^0\) is the concentration of metal atoms in the amalgam after accumulation by electrolysis, \(g\)-atom/cm\(^3\); \(l\) is the thickness of the mercury film, cm; \(t\) is time, sec; \(D_R\) is the diffusion coefficient of metal atoms in mercury, cm\(^2\)/sec; \(\mu_n=n\pi\); \(\tau'\) is an auxiliary variable of integration.
Let the current \(i\) be specified in the form of various functions.
- \(i(t)=bt^m\).
In this case we have
\[ I_1=\int_0^t bt^m\,d\tau'=\frac{bt^{m+1}}{m+1}, \tag{5} \]
\[ I_2=\sum_{j=1}^{\infty}\left[\frac{bt^m}{\gamma_j} +b\sum_{k=1}^{m}(-1)^k \frac{m(m-1)\ldots(m-k-1)}{\gamma_j^{(k+1)}}\cdot t^{m-k}\right]. \tag{6} \]
Here \(t\) is time.
\[ C_R(l,t)=C_R^{\mathrm{b}}-\frac{1}{zFl}\left[\frac{bt^{m+1}}{m+1}+\frac{1}{3}\frac{bl^2}{D_R}t^m+2\Phi(t)\right], \tag{7} \]
where
\[ \Phi(t)=\sum_{j=1}^{\infty}\sum_{\kappa=1}^{m} b(-1)^{\kappa}\, \frac{m(m-1)\ldots(m-\kappa+1)}{\gamma_j^{(\kappa+1)}}\, t^{(m-\kappa)} . \tag{8} \]
Consideration of the general case leads to complicated dependences; therefore we shall consider several special cases.
a) \(m=0\)—constant current.
The expression for the concentration of the reduced form of the element at the electrode surface will have the form:
\[ C_R(l,t)=C_R^{\mathrm{b}}-\frac{bt}{zFl}-\frac{1}{3}\frac{bl}{zFD_R}. \tag{9} \]
The expression for the transition time is obtained under the condition \(C_R(l,\tau)=0\):
\[ \tau=\frac{zFlC_R^{\mathrm{b}}}{b}-\frac{1}{3}\frac{l^2}{D_R}. \tag{10} \]
Equation (10) completely coincides with the expression for constant current obtained in [2]. The equations for the dependence of the electrode potential on time for reversible and irreversible processes for this case were obtained in [4].
b) \(m=1\)—the case of a linear change in current.
From equation (7) we obtain
\[ C_R(l,t)=C_R^{\mathrm{b}}-\frac{b}{zFl} \left[\frac{t^2}{2}+\frac{1}{3}\frac{l^2}{D_R}t-0.022\frac{l^4}{D_R^2}\right]. \tag{11} \]
For the transition time we have
\[ \frac{\tau^2}{2}+\frac{l^2}{3D_R}\tau = \frac{zFlC_R^{\mathrm{b}}}{b} +0.022\frac{l^4}{D_R^2}. \tag{12} \]
- The simplest equations for the transition time and for the \(\varphi-t\) dependence are obtained when the current is specified in the form of the function \(i(t)=be^{mt}\). In connection with this, we shall consider this case in more detail.
The expressions for the integrals will have the form:
\[ I_1=b\int_0^t e^{mt'}\,dt' =b\frac{e^{mt}-1}{m}, \tag{16} \]
\[ I_2=b\int_0^t e^{mt'}\sum_{j=1}^{\infty}e^{-\gamma_j(t-\tau)}\,dt'. \tag{17} \]
For \(\Theta\geq 1.1\) (if \(D_R=1\cdot10^{-5}\,\mathrm{cm^2/sec}\) and \(l=1\cdot10^{-3}\,\mathrm{cm}\), then \(t\geq0.11\,\mathrm{sec}\)), expression (14), with an error of less than 1%, is reduced to the form:
\[ I_2=b\sum_{j=1}^{\infty}\frac{e^{mt}}{m+\gamma_j}. \tag{18} \]
Delahay recommends carrying out studies at a transition time on the order of one to two minutes. Therefore, the discarded sec-
the curve \(\varphi - t\) will be insignificant in comparison with the entire magnitude of the transition time.
Substituting (16) and (18) into (1), we obtain an expression for the concentration of the reduced form of the element at the electrode surface:
\[ C_R(l,t)=C_R^0+\frac{b}{mzFl}-\frac{be^{mt}}{mzFl}(1+2\lambda), \tag{19} \]
where
\[ \lambda=\sum_{j=1}^{\infty}\frac{1}{1+\dfrac{\gamma_j}{m}}. \]
The expression for the transition time has the form:
\[ \tau= \frac{ 2.3\,\lg\dfrac{mzFlC_R^0+b}{b(1+2\lambda)} }{m}. \tag{20} \]
If the value of \(m\) is chosen to be less than 0.49, then for \(D_R>1\cdot 10^{-5}\ \mathrm{cm^2/sec}\), \(l\leq 1\cdot 10^{-3}\ \mathrm{cm}\), the term \(2\lambda\) in equation (20) may be neglected with an error of less than 1%, and the expression for the transition time reduces to:
\[ \tau=\frac{2.3}{m}\lg\frac{mzFlC_R^0+b}{b}. \tag{21} \]
It is of interest to compare the magnitude of the transition time at the same amalgam concentration for chronopotentiometry with constant current, \(\tau_2\), and with a current specified as the function \(i(t)=be^{mt}\), \(\tau_1\). From equations (10, 21) we obtain
\[ \frac{\tau_1}{\tau_2}= \frac{ 2.3\,\lg\left(\dfrac{mzFlC_R^0}{b_1}+1\right) }{ m\left(\dfrac{zFlC_R^0}{b_2}-\dfrac{l^2}{3D_R}\right) }. \tag{22} \]
The results of calculations of the ratios \(\dfrac{\tau_1}{\tau_2}\) for various \(m\) are given in the table.
It is evident from the table that, when \(\tau\) is about 1 sec, the transition time in both types of chronopotentiometry is nearly the same for different values of \(m\). For \(\tau>1\) sec, when \(m>0\), chronopotentiometry with constant current should be used for analytical purposes, whereas when \(m<0\) chronopotentiometry with a current specified in the form of the function \(i(t)=be^{mt}\) should be used.
Reversible processes. Substituting into the Nernst equation the expressions for the concentrations of the oxidized*) and reduced forms of the element at the electrode surface, we obtain the equation for the dependence of the electrode potential on time:
\[ \varphi=\varphi_{1/2}-\frac{RT}{zF}\ln \frac{(1+2\lambda)\left(e^{m\tau}-e^{mt}\right)} {\dfrac{mzFlC_R^0}{b}+\dfrac{m^{1/2}l}{D_0^{1/2}}\exp mt\,\operatorname{erf}(mt)^{1/2}}. \tag{23} \]
Irreversible processes. Since, in deriving the equation for \(C_R(l,t)\), no assumptions were made concerning the kinetics of the electrode—
*) The expression for \(C_0(l,t)\) is given in [3].
processes [2], then in the present case one may use equation (19). Substituting the expression for \(C_R(l,t)\) into the current–potential equation [2], we obtain the dependence of the electrode potential on time:
\[ \varphi=\varphi^0+\frac{RT}{\beta zF}\ln\frac{ml}{k_s(1+2\lambda)} -\frac{RT}{\beta zF}\ln\frac{e^{m\tau}-e^{mt}}{e^{mt}}, \tag{24} \]
where \(k_s\) is the rate constant of the electrode process at the standard electrode potential \(\varphi^0\); \(\beta\) is the transfer coefficient.
Table
Values of the ratios \(\tau_1/\tau_2\) for various \(m\), \(b_1=b_2\), \(D_R=1\cdot10^{-5}\ \mathrm{cm^2/sec}\).
\[ l=1\cdot10^{-3}\ \mathrm{cm} \]
| \multicolumn{2}{c}{\(m=0.1\)} | \multicolumn{2}{c}{\(m=-0.1\)} | \multicolumn{2}{c}{\(m=-0.5\)} |
|---:|---:|---:|---:|---:|---:|
| \(\tau_1,\ \mathrm{sec}\) | \(\tau_1/\tau_2\) | \(\tau_1,\ \mathrm{sec}\) | \(\tau_1/\tau_2\) | \(\tau_1,\ \mathrm{sec}\) | \(\tau_1/\tau_2\) |
| 0.5 | 1.07 | 1 | 1.03 | 1 | 1.3 |
| 1 | 1.03 | 10 | 1.57 | 10 | 5.3 |
| 10 | 0.61 | 20 | 2.3 | 20 | 10.1 |
| 20 | 0.27 | 30 | 3.2 | 30 | 15 |
| 30 | 0.17 | 40 | 4.0 | 40 | 20 |
| 40 | 0.074 | 60 | 6.0 | 60 | 30 |
| 60 | 0.015 | 80 | 8.0 | 120 | 60 |
| 80 | 0.0027 | 120 | 12.0 | | |
From equation (21) it is evident that the plot in the coordinates
\[ \varphi-2.3\lg\frac{\exp(m\tau)-\exp(mt)}{\exp(mt)} \]
is a straight line with slope \(\tan\alpha\) equal to
\[ \frac{RT}{\beta zF}. \]
By constructing this plot from experimental data, one can determine the value of \(\beta\).
It is of interest to dwell on possible applications of the method of anodic amalgam voltammetry with a prescribed current. It was indicated above in this article that the method considered can be applied for analytical purposes, for determining the diffusion coefficients of metals in mercury, and for studying the kinetics of electrode processes.
In addition, AAV with a prescribed current can be applied to determine the solubility of sparingly soluble metals in mercury. The essence of the determination is as follows. At different concentrations of the element (whose solubility in mercury is to be determined) in solution, preliminary electrolysis is carried out for one and the same interval of time, and chronopotentiograms are recorded each time. Until the solubility of the metal in mercury \(C_R'\) is reached, with increasing concentration of the element in solution the transition time will increase. Since the magnitude of the transition time is determined only by the dissolved portion of the metal [5], after solubility is reached, with increasing concentration of the element in solution (and consequently also in the amalgam), the increase in transition time will cease. From the limiting value of the transition time, for example by formula [10], knowing all the other quantities, one can calculate the solubility of the metal in mercury.
The method considered may also be applied to the study of intermetallic compounds in mercury. These studies are essentially analogous to similar studies by the methods of classical amalgam polarography [5] and amalgam stripping polarography with accumulation [6, 7].
A detailed consideration of these questions may be the subject of independent papers.
Conclusions
-
Equations have been obtained for the transition time from the dependence of the electrode potential on time in AAV with a current specified in the form of various functions.
-
Possible applications of the AAV method with programmed current have been discussed.
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