Constant-current anodic amalgam voltammetry at a film electrode
Abstract Generated abstract
This paper develops a theoretical description of constant-current anodic amalgam voltammetry at a mercury film electrode. By solving Fick diffusion equations with appropriate boundary conditions, the authors derive potential-time relations for reversible, irreversible, and quasi-reversible electrode processes, together with expressions for transition times and for estimating diffusion coefficients and kinetic parameters. The treatment is extended to multicomponent amalgams, showing that under specified dimensionless time conditions the transition time for oxidation of a more electropositive component is independent of the concentration of a more electronegative component. The results indicate analytical uses of the method and its applicability to studies of electrode-process kinetics.
Full Text
Anodic Amalgam Voltammetry at Constant Current on a Film Electrode
M. S. Zakharov, V. I. Bakanov
(Presented by the scientific seminar of the Department of Physical and Colloid Chemistry)
In the present work, equations of $\varphi$-$t$ curves for AAV on a mercury film electrode will be obtained.
Reversible processes. In the case of reversible processes, the electrode potential is determined, as is known, by the Nernst equation:
$$ \varphi=\varphi_{1/2}-\frac{RT}{zF}\ln \frac{C_R(l,t)\cdot D_R^{1/2}}{C_0(l,t)\cdot D_0^{1/2}}, \tag{1} $$
where $\varphi_{1/2}$ is the polarographic half-wave potential, V; $C_R(l,t)$, $C_0(l,t)$ are the concentrations of the reduced and oxidized forms of the element on the electrode surface. From equation (1) it follows that, to determine the value of $\varphi$, it is necessary to find expressions for $C_R(l,t)$ and $C_0(l,t)$.
The latter can be obtained by solving the system of equations of Fick’s second law
$$ \frac{\partial C_R(x,t)}{\partial t} = D_R\frac{\partial^2 C_R(x,t)}{\partial x^2}, \qquad 0\le x\le l; \tag{2} $$
$$ \frac{\partial C_0(x,t)}{\partial t} = D_0\frac{\partial^2 C_0(x,t)}{\partial x^2}, \qquad 0\le x<\infty. \tag{3} $$
under the following initial and boundary conditions:
$$ C_R(x,0)=C_R^0,\qquad 0\le x\le l; \tag{4} $$
$$ C_0(x,0)=C_0^0,\qquad l\le x<\infty; \tag{5} $$
$$ \left. \frac{\partial C_R(x,t)}{\partial x} \right|_{x\to 0} =0; \tag{6} $$
$$ C_0(x,t)_{x\to\infty}=C_0^0; \tag{7} $$
$$ \left. \frac{\partial C_R(x,t)}{\partial x} \right|_{x=l} = -q_R = -\frac{i_0}{zFD_R} = \mathrm{const}; \tag{8} $$
$$ D_0\frac{\partial C_0(x,t)}{\partial x} - D_R\frac{\partial C_R(x,t)}{\partial x} =0; \qquad \text{at } x=l, \tag{9} $$
where $C_R^0$ and $C_0^0$ are the initial concentrations of atoms $R$ in mercury and of ions in solution; $i_0$ is the current density, A/cm$^2$.
The formulated boundary-value problem is solved by the operational method. The equations describing the distribution of concentrations at the electrode surface have the following form:
$$ C_R(l,t) = C_R^0 - q_R l \left[ \vartheta_R+\frac{1}{3}+\lambda(\vartheta_R) \right]; \tag{10} $$
\[ C_0(l,t)=C_0^0+\frac{2l}{\pi^{1/2}}q_0\vartheta_0^{1/2}; \tag{11} \]
\[ \vartheta_R=\frac{D_Rt}{l^2};\qquad \mu_n=n\pi;\qquad q_0=-\frac{i_0}{zFD_0}; \]
\[ \lambda(\vartheta_R)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{\mu_n^2}\exp(-\mu_n^2\vartheta_R). \]
For \(\vartheta_R\geq 0.35\) (at \(D_R=1\cdot10^{-5}\ \mathrm{cm^2\,sec^{-1}}\) and \(l=10^{-3}\ \mathrm{cm}\), \(t\geq 0.035\ \mathrm{sec}\)), with an error of less than 1%, the quantity \(\lambda(\vartheta_R)\) may be neglected in comparison with the other two terms in the brackets. Expression (10) in this case takes the form
\[ C_R(l,t)=C_R^0-q_Rl\left(\vartheta_R+\frac{1}{3}\right). \tag{12} \]
The expression for the electrode potential is obtained by combining equations (1), (10), (11) \((D_R=D_0)\):
\[ \varphi=\varphi_{1/2}-\frac{RT}{zF}\ln \frac{C_R^0-ql\left[\vartheta+\frac{1}{3}+\lambda(\vartheta)\right]} {C_0^0+\frac{2l}{\pi^{1/2}}q\vartheta^{1/2}}. \tag{13} \]
The transition time is found from the condition \(C_R(l,t)=0\). The expression for the transition time in the general case is obtained by solving the equation
\[ C_R^0=q_Rl\left[\vartheta_R' + \frac{1}{3}+\lambda(\vartheta_R')\right], \tag{14} \]
where \(\vartheta_R'=\dfrac{D_R\tau}{l^2}\); \(\vartheta_R'\) is the dimensionless transition time; \(\tau\) is the transition time, sec. Equation (14) is transcendental with respect to \(\tau\). For \(\vartheta_R'\geq 0.035\), with an error of less than 1%, equation (12) may be written as:
\[ C_R^0=q_Rl\left(\vartheta_R'+\frac{1}{3}\right). \tag{15} \]
In this case the expression for \(\tau\) is written as follows:
\[ \tau=\frac{zFC_R^0l}{i_0}-\frac{l^2}{3D_R}. \tag{16} \]
The transition time depends on the concentration of the amalgams; this regularity can be used for analytical purposes.
From experimental data \((C_R^0,\ l,\ i_0,\ \tau)\), the diffusion coefficient of the metal atoms in mercury can be calculated from the equations:
\[ D_R=\frac{i_0l^2}{3zFC_R^0l-3i_0\tau}; \tag{17} \]
\[ D_R=\frac{l^2(i_0''-i_0')}{3(\tau'i_0'-\tau''i_0'')}, \tag{18} \]
where \(\tau'\) and \(\tau''\) are the transition times at current densities \(i_0'\) and \(i_0''\). The equations of the \(\varphi\)—\(t\) curves will have the following form \((D_0=D_R)\):
(for any \(\vartheta'\) and \(\vartheta\))
\[ \varphi(\vartheta)=\varphi_{1/2}-\frac{RT}{zF}\ln \frac{\left[(\vartheta'+\lambda(\vartheta')-\vartheta-\lambda(\vartheta))\right]ql} {C_0^0+\frac{2lq}{\pi^{1/2}}\vartheta^{1/2}}, \tag{19} \]
(for \(\vartheta'\geq 0.035\) and any \(\vartheta\))
\[ \varphi(\vartheta)=\varphi_{1/2}-\frac{RT}{zF}\ln \frac{ql\left[\vartheta'-\vartheta-\lambda(\vartheta)\right]} {C_0^0+\frac{2lq}{\pi^{1/2}}\vartheta^{1/2}}. \tag{20} \]
The plot in the coordinates
\[ \lg \frac{q\,[\vartheta'-\vartheta-\lambda(\vartheta)]} {C_0^0-\dfrac{2lq}{\pi^{1/2}}\vartheta^{1/2}},\ \varphi \]
is a straight line whose reciprocal slope is equal to \(2.3\,RT/zF\).
Irreversible processes. In the case of irreversible processes the electrode potential is determined by the relation:
\[ \varphi_{(\vartheta)}=\varphi^0-\frac{RT}{\beta zF}\ln k_s +\frac{RT}{\beta zF}\ln\frac{i_0}{zF} -\frac{RT}{\beta zF}\ln C_R(l_1t), \tag{21} \]
where \(\beta\) is the transfer coefficient; \(k_s\) is the rate constant of the electrode process at the standard potential \(\varphi^0\), cm·sec\(^{-1}\). Solving equations (10) and (21) simultaneously, we obtain:
\[ \text{(for any } \vartheta'_R \text{ and } \vartheta_R\text{)} \]
\[ \varphi_{(\vartheta)}=\varphi^0-\frac{RT}{\beta zF}\ln\frac{k_sl}{D_R} -\frac{RT}{\beta zF}\ln\left[\vartheta'_R+\lambda(\vartheta'_R)-\vartheta_R-\lambda(\vartheta_R)\right], \tag{22} \]
\[ \text{(for } \vartheta'_R\geq 0.35 \text{ and any } \vartheta_R\text{),} \]
\[ \varphi_{(\vartheta)}=\varphi^0-\frac{RT}{\beta zF}\ln\frac{D_R}{k_sl} -\frac{RT}{\beta zF}\ln\left[\vartheta'_R-\vartheta_R-\lambda(\vartheta_R)\right]. \tag{23} \]
A plot in the coordinates \(\lg[\vartheta'_R-\vartheta_R-\lambda(\vartheta_R)]\), \(\varphi\) is a straight line, the reciprocal slope of which is equal to \(2.3\,RT/\beta zF\). Thus, \(\beta z\) can be found from the plot. Having this plot and knowing \(\varphi^0\) of the electrode, one can calculate the value of the rate constant of the electrode process \(k_s\). At small \(\vartheta_R\), the equation of the \(\varphi-t\) curve (for \(\vartheta'\geq 0.35\)) has the form
\[ \varphi_{(\vartheta)}=\varphi^0-\frac{RT}{\beta zF}\ln\frac{k_sl}{D_R} -\frac{RT}{\beta zF}\ln\left(\vartheta'_R+\frac{1}{3}-\frac{2\vartheta_R^{1/2}}{\pi^{1/2}}\right). \tag{24} \]
If all parameters of equation (30) are known (except \(k_s\)), then at \(\vartheta=0\) this equation can be used to determine the value of \(k_s\).
Quasi-reversible processes. For these processes the electrode potential is related to the current by the following relation:
\[ \frac{i_0}{zF}=-k_s\left\{ \exp\left[\frac{\beta zF}{RT}(\varphi-\varphi^0)\right]C_R(l_1t) -\exp\left[-\frac{\alpha zF}{RT}(\varphi-\varphi^0)\right]C_0(l_1t) \right\}. \tag{25} \]
The dependence of the electrode potential on time (in implicit form) is determined in the same way as described above. In particular, for \(\vartheta'_R\geq 0.35\) and any \(\vartheta_R\), the function \(\varphi-t\) has the following form:
\[ \frac{i_0}{zF}=-k_s\left\{ \exp\left[\frac{\beta zF}{RT}(\varphi-\varphi^0)\right] q_Rl\,[\vartheta'_R+\lambda(\vartheta'_R)-\vartheta_R-\lambda(\vartheta_R)] -\exp\left[-\frac{\alpha zF}{RT}(\varphi-\varphi^0)\right] \left[C_0^0+\frac{2l}{\pi^{1/2}}q_0\vartheta_0^{1/2}\right] \right\}. \tag{26} \]
The determination of the kinetic parameters of a quasi-reversible electrode process is described in [1].
The electrode reactions during the oxidation of a multicomponent amalgam may be written in the following form:
\[ C_{R,i}\to C_{0,i}+z_ie. \]
To obtain an expression for the transition time, it is first necessary to find an expression for the concentration of the reduced form of the elements on the electrode surface, \(C_{R,i}(l,t)\). For this purpose, it is necessary to solve \(i\) equations of Fick’s second law for all elements dissolved in the mercury film electrode [2].
For the concentration of metal atoms on the electrode surface the following expression is obtained:
\[ \sum_{i=1}^{m} z_i C_{R,i}(l,t) = \sum_{i=1}^{m} \left\{ z_i C_{R,i}^{0} - \lambda_i \left[ \vartheta_i+\frac{1}{3}+\Phi(\vartheta_i) \right] \right\}, \tag{27} \]
where
\[ \vartheta_i=\frac{D_{R,i}\cdot t}{l^2}; \qquad \lambda_i=\frac{ql}{D_{R,i}}. \]
The last sum in equation (27) converges rapidly, and at \(\vartheta \ge 0.35^*\), with an error of less than 1%, it may be neglected. In this case equation (27) reduces to the form
\[ \sum_{i=1}^{m} z_i C_{R,i}(l,t) = \sum_{i=1}^{m} \left\{ z_i C_{R,i}^{0} - \lambda_i \left( \vartheta_i+\frac{1}{3} \right) \right\}. \tag{28} \]
From equation (28) we obtain:
\[ \sum_{i=1}^{m} \left( \tau_i+\frac{l^2}{3D_{R,i}} \right) = \frac{lF}{i_0} \sum_{i=1}^{m} z_i C_{R,i}^{0}, \tag{29} \]
\[ \sum_{i=1}^{m}\tau_i - \sum_{i=1}^{m-1}\tau_i = \frac{z_i F l C_{R,i}^{0}}{i_0} - \frac{l^2}{3D_{R,i}}. \tag{30} \]
The expression for the transition time of the most electronegative element dissolved in the amalgam (the first to dissolve) was obtained above. For the element dissolving second, from equation (30) the following expression for the transition time can be obtained:
\[ (\tau_1+\tau_2)-\tau_1 = \frac{z_2 F l C_{R,2}^{0}}{i_0} - \frac{l^2}{3D_{R,2}}. \tag{31} \]
It is evident from equation (31) that, at \(\vartheta' \ge 0.35\) and in the absence of complicating factors, \(\tau_2\) does not depend on the concentration of the element that is oxidized at more negative potentials, whereas under conditions of semi-infinite diffusion \(\tau_2\) depends on the concentration of the substance reduced at less negative potentials. This regularity of the method of anodic amalgam voltammetry at constant current is very convenient for analytical purposes.
Conclusions
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Equations of \(\varphi\)—\(t\) curves in anodic amalgam voltammetry at a prescribed current with a film electrode have been obtained theoretically for reversible, irreversible, and quasi-reversible processes.
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The possibility of applying the method considered for analytical purposes and for studying the kinetics of electrode processes has been shown.
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An equation has been obtained for the transition time in the method considered during oxidation of a complex amalgam.
\[ \text{*If } \vartheta \ge 0.35,\ \text{then at } D_R=1\cdot10^{-5}\ \mathrm{cm^2/sec} \text{ and } l=1\cdot10^{-3}\ \mathrm{cm},\ t\ge 0.035\ \mathrm{sec}. \]
- At \(\vartheta' \ge 0.35\), the transition time for the oxidation process of the electropositive element does not depend on the concentration of the more electronegative element; this regularity is very convenient for analytical application of the method.
References
- M. S. Zakharov, V. V. Pnev. Izv. TPI (the present collection).
- R. W. Murray, C. N. Reilley, I. Electroanal. Chem., 3, 64, 1962.