ANODIC AMALGAM VOLTAMMETRY WITH A SET CURRENT AT A MERCURY FILM ELECTRODE

M. S. ZAKHAROV, V. I. BAKANOV

(Presented to the scientific seminar of the Problem Laboratory of Physicochemical Methods of Analysis)

Questions of voltammetry with constant current (chronopotentiometry) under conditions of semi-infinite diffusion have been considered in numerous works. A list of the latter is given in [1, 2]. In works [3–7], questions of chronopotentiometry with a current specified as a function of \(t^{1/2}\) were considered; Kambara and Tachi [8] obtained a general integral equation for an arbitrary form of current variation with time; in works [9, 10], questions of chronopotentiometry with a linear variation of current were considered. Murray and Reilley carried out a theoretical calculation of chronopotentiograms with current pulses varying with time according to the law \(i=\beta t^2\) [11], and in [12] the theory of cyclic chronopotentiometry was developed. Murray [13] obtained equations for the dependence of the electrode potential on time for chronopotentiometry with a current specified as a function \(\alpha e^{\beta t}\) for cases of reduction of one and of an arbitrary number of substances, and carried out an experimental verification of certain theoretical regularities. Zakharov, Bakanov, and Pnev considered a number of questions of anodic amalgam voltammetry (AAV) at mercury film and spherical electrodes [1, 2, 14, 15]. The essence of this method consists in the preliminary accumulation of an element in an amalgam electrode and the subsequent dissolution of the amalgam with a specified form of current at the electrode.

In the present work, expressions will be obtained for the transition time and for the dependence of the electrode potential on time in anodic amalgam voltammetry at a mercury film electrode with programmed current. In addition, application of the method in physicochemical studies is considered. The problems will be considered under the conditions indicated in [1].

In anodic amalgam voltammetry with a set 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 [16]:

\[ C_R(0,t)=C_R^0-\frac{1}{zFl}(I_1+2I_2); \tag{1} \]

\[ I_1=\int_0^t i(\tau)\,d\tau; \tag{2} \]

\[ I_2=\int_0^t i(\tau)\sum_{j=1}^{\infty} e^{\gamma_j(t-\tau)}\,d\tau; \tag{3} \]

\[ \gamma_j=\mu_n^2\frac{D_R}{l^2}, \tag{4} \]

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 the metal atoms in mercury, cm\(^2\)/sec; \(\mu_n=n\pi\); \(\tau\) is an auxiliary integration variable.

Let the current \((i)\) be specified in the form of various functions.

1. \(i(t)=bt^m\).

In this case we have:

\[ I_1=\int_0^t bt^m\,d\tau=\frac{bt^{m+1}}{m+1}; \tag{5} \]

\[ I_2=b\int_0^t \sum_{j=1}^{\infty} t^m e^{-\gamma_j(t-\tau)}\,d\tau \]

\[ =\sum_{j=1}^{\infty}\left[ \frac{bt^m}{\gamma_j} + b\sum_{k=1}^{m}(-1)^k \frac{m(m-1)\cdots(m-k+1)}{\gamma_j^{k+1}}\,t^{m-k} \right]. \tag{6} \]

Here \(t\) is time.

\[ C_{R(l,t)}=C_R^0-\frac{1}{zFl} \left[ \frac{bt^{m+1}}{m+1} + \frac{1}{3}\frac{bl^2}{D_a}t^m + 2\Phi(t) \right], \tag{7} \]

where

\[ \Phi(t)= \sum_{j=1}^{\infty}\sum_{k=1}^{m} b(-1)^k \frac{m(m-1)\cdots(m-k+1)}{\gamma_j^{k+1}}\,t^{m-k}. \tag{8} \]

Consideration of the general case leads to complex 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:

\[ C_{R(l,t)}=C_R^0-\frac{bt}{zFl}-\frac{bl}{3zFD_R}. \tag{9} \]

The expression for the transition time \((\tau)\) is obtained under the condition \(C_{R(l,\tau)}=0\):

\[ \tau=\frac{zFlC_R^0}{b}-\frac{l^2}{3D_R}. \tag{10} \]

Equation (10) coincides completely with the expression for constant current obtained in [1]. The equations for the dependence of the electrode potential on time for reversible and irreversible processes for this case were obtained in [1].

\(m=1\)—the case of a linear change in current. From equation (7) we obtain

\[ C_{R(l,t)}=C_R^0-\frac{b}{zFl} \left( \frac{t^2}{2} + \frac{b^2l}{3D_R} - 0.022\frac{l^4}{D_R^2} \right). \tag{11} \]

For the transition time we have

\[ \frac{\tau^2}{2}+\frac{l^2}{D_R}\tau = \frac{zFlC_R^0}{b} + 0.022\frac{l^4}{D_R^2}. \tag{12} \]

1. \(i(t)=\sin bt\). In this case the expressions for the integrals will have the following form:

\[ I_1=\frac{1}{b}-\frac{1}{b}\cos bt; \tag{13} \]

\[ I_2=\sum_{j=1}^{\infty} \frac{\gamma_j\sin bt-b\cos bt+b}{\gamma_j^2+b^2}. \tag{14} \]

The sum of the second integral is valid for \(\vartheta \geqslant 1.1\),

where

\[ \vartheta = \frac{D_R t}{l^2}. \]

Substituting (13) and (14) into (1), we obtain the following expression for determining the concentration of the reduced form of the element at the electrode surface:

\[ C_{R(l,t)} = C_R^0 - \frac{1}{bzFl} + \frac{1}{zFl}\left(\frac{1}{b}\cos bt - 2I_2\right). \tag{15} \]

The expression for the transition time is found from the condition

\[ C_R(l,\tau)=0. \]

1. The simplest equations for the transition time and the dependence \(\varphi - t\) 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 greater detail. The expressions for the integrals will have the form:

\[ I_1 = b\int_0^t e^{m\tau_s}\,d\tau_s = b\frac{e^{mt}-1}{m}; \tag{13} \]

\[ I_2 = b\int_0^t e^{m\tau_s}\sum_{j=1}^{\infty} e^{-\gamma_j(t-\tau_s)}. \tag{14} \]

For \(\vartheta \geqslant 1.1\) (if \(D_R=1\cdot10^{-5}\ \mathrm{cm^2/sec}\) and \(l=1\cdot10^{-3}\ \mathrm{cm}\), then \(t \geqslant 0.11\ \mathrm{sec}\)), expression (14), with an error of less than 1%, is reduced to the form:

\[ I_2 = b\sum_{j=1}^{\infty}\frac{e^{mt}}{m+\gamma_j}. \tag{15} \]

Delahay recommends carrying out investigations at a transition time on the order of one to two minutes. Therefore, the discarded section of the \(\varphi - t\) curve will be insignificant in comparison with the entire magnitude of the transition time.

Substituting (13) and (15) into (1), we obtain an expression for the concentration of the reduced form of the element at the electrode surface:

\[ C_{R(l,t)} = C_R^0 + \frac{b}{mzFl} - \frac{be^{mt}}{zmFl}(1+2\lambda), \tag{16} \]

where

\[ \lambda = \sum_{j=1}^{\infty}\frac{1}{1+\dfrac{\gamma_j}{m}}. \]

The expression for the transition time \(\tau\) is determined from the condition \(C_R(l,\tau)=0\)

\[ \tau = \frac{ 2.3\lg\dfrac{mzFlC_R^0+b}{b(1+2\lambda)} }{m}, \tag{17} \]

If the value of \(m\) is chosen to be less than 0.49, then for \(D_R=1\cdot10^{-5}\ \mathrm{cm^2/sec}\), \(l \leqslant 1\cdot10^{-3}\ \mathrm{cm}\), the term \(2\lambda\) in equation (17) may be neglected with an error of less than 1%, and the expression for the transition time will have the form:

\[ \tau = \frac{ 2.3\lg\dfrac{mzFlC_R^0+b}{b} }{m}. \tag{18} \]

It is of interest to compare the value of the transition time at the same amalgam concentration for chronopotentiometry with a constant current \((\tau_2)\) and with a current specified in the form of the function \(i(t)=be^{mt}\), \((\tau_1)\). From equations (10, 18) we obtain:

\[ \frac{\tau_1}{\tau_2} = \frac{ 2.3 \lg \dfrac{m z F l C_R^0 + b_1}{b_1} }{ m\left(\dfrac{z F l C_R^0}{b_2}-\dfrac{l^2}{3D_R}\right) }. \tag{19} \]

The results of calculations of the ratios \(\dfrac{\tau_1}{\tau_2}\) for various \(m\) are given in Table 1.

Table 1

Values of the ratios \(\dfrac{\tau_1}{\tau_2}\) for various \(m\) \((b_1=b_2)\), \(l=1\cdot10^{-3}\ \mathrm{cm}\), \(D_R=1\cdot10^{-5}\ \mathrm{cm^2/sec}\)

It is evident from the table that at \(\tau\) of about 1 sec the transition time in both types of chronopotentiometry is almost the same for different values of \(m\). For \(\tau>1\) sec and \(m>0\), for analytical purposes one should use chronopotentiometry with a constant current, whereas for \(m<0\) one should use chronopotentiometry with a current specified in the form of the function \(i(t)=be^{mt}\).

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,

\[ \varphi = \varphi_{1/2} - \frac{RT}{2F} \ln \frac{(1+2\lambda)\left(e^{m\tau}-e^{mt}\right)} {\dfrac{m z F l C_R^0}{b} + \dfrac{m^{1/2} e^{mt}\operatorname{erf}\sqrt{mt}}{D_0^{1/2}}}. \tag{20} \]

Irreversible processes. Since, in deriving the equation for \(C_R(l,t)\), no assumptions were made about the kinetics of the electrode processes [1], in the present case one may use the equa-

*) The expression for \(C_O(l,t)\) is given in [13]:

\[ C_O(l,t) = C_O^0 \frac{b}{m^{1/2} z F D_O^{1/2}} e^{mt}\operatorname{erf}\sqrt{mt}. \]

... equation (1). Substituting the expression for \(C_R(l,t)\) into the current–potential equation [1], we obtain the dependence of the electrode potential on time:

\[ \varphi=\varphi^0+\frac{RT}{\beta zF}\ln\frac{ml}{K_s(1+2\lambda)} -\frac{RT}{\beta zF}\ln\frac{e^{m\tau}-e^{mt}}{e^{mt}}, \tag{21} \]

where \(K_s\) is the rate constant of the electrode process at the standard electrode potential \((\varphi^0)\); \(\beta\) is the transfer coefficient.

It follows from equation (21) that a plot in the coordinates
\(\varphi,\; 2.3\lg \dfrac{e^{m\tau}-e^{mt}}{e^{mt}}\)
is a straight line with slope ( \(\tg\alpha\) ) equal to \(-\dfrac{RT}{\beta zF}\). By constructing this plot from experimental data, one can determine the value of \(\beta\).

To determine \(K_s\), it is convenient to obtain the expression \(\varphi-t\) for \(t\to 0\). At \(t\to 0\), the following equation can be obtained for \(C_R(l,t)\):

\[ C_R(l,t)=C_R^0-\frac{b}{m^{1/2}zFD_R^{1/2}}e^{mt}\operatorname{erf}\sqrt{mt}. \tag{22} \]

In this case the dependence of the electrode potential on time will be described by the expression

\[ \varphi=\varphi^0-\frac{RT}{\beta zF}\ln\frac{K_s}{D_R^{1/2}l} -\frac{RT}{\beta zF}\ln \frac{ D_R^{1/2}\left[(1+2\lambda)e^{m\tau}-1\right]-m^{1/2}be^{mt}\operatorname{erf}\sqrt{mt} }{ me^{mt} } \tag{23} \]

Having determined \(\varphi\) at \(t=0\), from formula (23) at \(t=0\) and with known other quantities, one can calculate \(K_s\).

Oxidation of multicomponent amalgams. The electrochemical reactions may be represented in the following form:

\[ A_1-Z_1e\to B_1, \]

\[ A_2-Z_2e\to B_2, \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

\[ A_k-Z_ke\to B_k. \]

To obtain an expression for the transition time of the oxidation process of component \(A_k\), it is necessary to find the equation for the concentrations of the reduced forms of all elements dissolved in the amalgam at the electrode surface. For this purpose it is necessary to solve the boundary-value problem posed in [15]. The equation for the concentration of the reduced form of any component of a complex amalgam at the electrode surface, when the current is specified in the form of the function \(i(t)=be^{mt}\), has the following form:

\[ \sum_{\kappa=1}^{k} Z_\kappa C_{R_\kappa}(l,t) = \sum_{\kappa=1}^{k} z_\kappa C_{R_\kappa}^{0} +\frac{b}{mFl} -\frac{be^{mt}}{mFl}(1+2\lambda_\kappa), \tag{24} \]

where

\[ \lambda_\kappa=\sum_{j=1}^{\infty} \frac{ml^2}{ml^2+\mu_n^2\sum_{\kappa=1}^{k}D_{R_\kappa}}. \]

The expression for the transition time of any of the oxidized components of amalgams, which is obtained under the condition \(C_{R_k}(l,\tau_k)=0\), will have the form:

\[ \exp\left[m\left(\sum_{\kappa=1}^{k}\tau_\kappa\right)\right] - \exp\left[m\left(\sum_{\kappa=1}^{k-1}\tau_\kappa\right)\right] = \frac{mZ_kFlC_{R,k}^{0}+b}{b(1+2\lambda_k)}, \tag{25} \]

where

\[ \lambda'_k=\sum_{j=1}^{\infty}\frac{ml^2}{ml^2+\mu_n^2D_{R,k}}. \]

If \(m \le 0.49\), \(D_{R,k}=1\cdot 10^{-5}\ \text{cm}^2/\text{s}\), \(l \le 1\cdot 10^{-3}\ \text{cm}\), then, with an error of less than 1%, expression (25) may be written as follows:

\[ \exp\left[m\left(\sum_{k=1}^{\kappa}\tau_k\right)\right] -\exp\left[m\left(\sum_{k=1}^{\kappa-1}\tau_k\right)\right] = \frac{mZ_kFlC^0_{R,k}}{b}+1. \tag{26} \]

For the first element to be oxidized, equation (26) reduces to equation (18). For the second element we obtain

\[ e^{m\tau_1}\left(e^{m\tau_2}-1\right) = \frac{mZ_2FlC^0_{R,2}}{b}+1. \tag{27} \]

It is seen from equation (27) that the transition time of the oxidation process of the second element depends on the concentration of the element oxidized earlier, whereas at constant current \([i(t)=b]\) the transition time of the element oxidized second does not depend on the concentration in the amalgam of the element oxidized first [15].

It is of interest to consider possible applications of the method of anodic amalgam voltammetry with a set current. It has already been pointed out above in this article and in works [14, 15] that the method under consideration 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 set 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 being determined) in solution, preliminary electrolysis is carried out for one and the same period of time, and chronopotentiograms are recorded each time. Until the solubility of the metal in mercury 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 [18], after the solubility is reached, with increasing concentration of the element in solution (and, consequently, 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 \((C_k^0)\) of the metal in mercury.

The method considered can also be applied to the study of intermetallic compounds in mercury. These studies are, in essence, analogous to similar studies by the methods of classical amalgam polarography [18] and amalgam polarography with accumulation [19, 20].

A detailed consideration of these questions may be the subject of independent studies.

Conclusions

Equations have been obtained for the transition time and for the dependence of the electrode potential on time in AAV with a current set in the form of various functions.

*) It was shown in [17] that AAV with a constant current can determine metals in mercury at concentrations down to \(7.5\cdot 10^{-5}\) g-atom/l.

Possible applications of the AAV method with a set current have been discussed.

REFERENCES

1. M. S. Zakharov, V. I. Bakanov. Izvestiya vysshikh uchebnykh zavedenii, khimiya i khim. tekhnologiya (in press).
2. M. S. Zakharov, V. V. Pnev, ibid. (in press).
3. M. Senda. Rew. Polarog. (Japan), 4, 89 (1956).
4. H. Hurwitz, L. Gierst, J. Electroanal. Chem., 2, 128 (1961).
5. H. Hurwitz. ibid., 2, 142 (1961).
6. H. Hurwitz. ibid., 2, 328 (1961).
7. W. H. Reinmuth. Anal. Chem., 32, 1509 (1960).
8. T. Kambara, J. Tachi, J. Phys. Chem., 61, 1405 (1957).
9. R. C. Rowers, G. Ward, C. M. Wilson, D. D. De-Ford, J. Phys. Chem., 65, 672 (1961).
10. H. Hoffman, W. Jaenicke. Z. Anal. Chem., 186, 93 (1962).
11. R. W. Murray, C. N. Reilley, J. Electroanal. Chem., 3, 64 (1962).
12. R. W. Murray, C. N. Reilley. ibid., 3, 182 (1962).
13. R. W. Murray, Anal. Chem., 35, 1784 (1963).
14. M. S. Zakharov, V. V. Pnev. Zh. analit. khimii (in press).
15. M. S. Zakharov, V. I. Bakanov. Zavod. labor. (in press).
16. M. S. Zakharov, V. I. Bakanov. Izvestiya vysshikh uchebnykh zavedenii, khimiya i khim. tekhnologiya (in press).
17. M. S. Zakharov, V. V. Pnev, V. I. Bakanov. Zavod. labor. (in press).
18. M. T. Kozlovsky. Mercury and Amalgams in Electrochemical Methods of Analysis. Publishing House of the Academy of Sciences of the Kazakh SSR, Alma-Ata, 1956.
19. A. G. Stromberg, V. E. Gorodovykh. Zh. neorganicheskoi khimii, 8, 2355 (1963).
20. W. Kemula, Z. Galus, Z. Kublik. Nature, 182, 1228 (1958).