Theory of Anodic Amalgam Voltammetry with an Arbitrary Form of Potential or Current Variation at a Spherical Electrode. Electrode Process Complicated by a Preceding Chemical Reaction

M. S. Zakharov, V. V. Pnev

(Presented by the scientific seminar of the Faculty of Chemical Technology)

The purpose of the present work is to derive the equation of the \(\varphi - t\) curve (AAV at constant current), the \(i - t\) curve (AAV at constant potential), and the \(i - \varphi\) curve (AAV at a linearly varying electrode potential) for the process of electro-dissolution of an amalgam complicated by dissociation of a soluble intermetallic compound. According to literature data [1], some metals in mercury form intermetallic compounds soluble in mercury. The dissociation reaction of the intermetallic compound is a first-order reaction.

Let the following process take place at the electrode:

\[ \mathrm{Y} \underset{k_2}{\stackrel{k_1}{\rightleftarrows}} R + nR_1 \to O, \tag{1} \]

where \(\mathrm{Y}\) is an intermetallic compound soluble in mercury; \(R\) and \(R_1\) are the metal atoms into which the intermetallic compound dissociates.

The solution of the problem posed in this work is carried out under the following conditions: 1) \(\mathrm{Y}\) and \(R_1\) are not oxidized at the potential at which \(R\) is oxidized; 2) Fick’s law is applicable to each component independently; 3) an excess of indifferent electrolyte is present; 4) convection is absent; 5) chemical equilibrium in the mercury electrode exists before the beginning of the electro-dissolution process; 6) adsorption effects are neglected.

The concentrations \(C_R(X,\Theta)\), \(C_{\mathrm{Y}}(X,\Theta)\), and \(C_0(X,\Theta)\) can be found by solving the system of Fick differential equations, which in dimensionless variables have the form

\[ \frac{\partial C_{\mathrm{Y}}(X,\Theta)}{\partial \Theta} = \frac{\partial^2 C_{\mathrm{Y}}(X,\Theta)}{\partial X^2} + \frac{\Gamma}{X} \frac{\partial C_{\mathrm{Y}}(X,\Theta)}{\partial X} - \lambda_1 C_{\mathrm{Y}}(X,\Theta) + \lambda_2 C_R(X,\Theta) C_{R_1}^{\,n}, \quad 0 \leq X \leq 1; \tag{2} \]

\[ \frac{\partial C_R(X,\Theta)}{\partial \Theta} = \frac{\partial^2 C_R(X,\Theta)}{\partial X^2} + \frac{\Gamma}{X} \frac{\partial C_R(X,\Theta)}{\partial X} + \lambda_1 C_{\mathrm{Y}}(X,\Theta) - \lambda_2 C_R(X,\Theta) C_{R_1}^{\,n}, \quad 0 \leq X \leq 1; \tag{3} \]

\[ \frac{\partial C_0(X,\Theta)}{\partial \Theta} = \frac{\partial^2 C_0(X,\Theta)}{\partial X^2} + \frac{\Gamma}{X}\frac{\partial C_0(X,\Theta)}{\partial X}, \qquad 1 \leq X < \infty . \tag{4} \]

Here

\[ \Gamma \]
is the electrode-shape coefficient, \(\Gamma=0\) for a planar electrode and \(\Gamma=2\) for a spherical electrode;

\[ X=\frac{y}{y_0} \]
is the dimensionless coordinate;

\(y_0\) is the thickness of the film electrode or the radius of the spherical electrodes;

\[ \Theta=\frac{Dt}{y_0^2} \]
is dimensionless time;

\[ D=D_{\mathrm r}=D_R \]
is the diffusion coefficient, \(\mathrm{cm^2/sec}\);

\(t\) is time, sec;

\[ \Theta_0 \equiv \frac{D_0t}{y_0^2} \]
is the dimensionless time for substance \(0\);

\(D_0\) is the diffusion coefficient of ions in solution, \(\mathrm{cm^2/sec}\);

\[ \lambda_1=\frac{\kappa_1 y_0^2}{D};\qquad \lambda_2=\frac{\kappa_2 y_0^2}{D};\qquad K_2=K_1 C_R^n . \]

We solve Fick’s equations under the following initial and boundary conditions:

\[ \Theta=0,\quad C_R(X,0)+C_{\mathrm r}(X,0)=C^0; \tag{5} \]

\[ C_{R_1}^{\,n}\frac{C_{\mathrm r}(X,0)}{C_R(X,0)}=K_p'=\frac{\kappa_2}{\kappa_1}; \tag{6} \]

\[ C_0(X,0)=C_0^0; \tag{7} \]

\[ \Theta>0,\quad \lim_{X\to 0}\frac{\partial C_R(X,\Theta)}{\partial X} = \lim_{X\to 0}\frac{\partial C_{\mathrm r}(X,\Theta)}{\partial X} =0; \tag{8} \]

\[ \lim_{X\to\infty} C_0(X,\Theta)=C_0^0; \tag{9} \]

\[ \frac{\partial C_R(1,\Theta)}{\partial X} = \mu\,\frac{\partial C_0(1,\Theta)}{\partial X} = -\lambda; \tag{10} \]

\[ \frac{\partial C_{\mathrm r}(1,\Theta)}{\partial X}=0; \qquad \lambda=\frac{i_0 y_0}{zFD}; \qquad \mu=\frac{D_0}{D}. \tag{11} \]

The solution of equation (4) under conditions (7), (9), and (10) is known [2]. For a film electrode, at the surface

\[ C_0^{\mathrm{pl}}(1,\Theta)=C_0^0+2\lambda_0\sqrt{\frac{\Theta}{\pi}}, \tag{12} \]

where

\[ \lambda_0=\frac{i_0 y_0}{zFD_0}, \]

and for a spherical electrode:

\[ C_0^{\mathrm{sph}}(1,\Theta_0) = C_0^0+\lambda_0\left[1-\exp \Theta\,\operatorname{erfc}\sqrt{\Theta}\right]. \tag{13} \]

Applying the Laplace transform to the system of equations (2–3) under conditions (5, 6, 8, 10, 11), one can obtain for the transform of the con-

centration \(C_R(1,s)\) at the electrode surface, the following expression:

\[ C_R(1,s)=\frac{1}{1+K_p}\left[ \frac{C_R^0}{s} -\lambda\frac{I_\gamma(\sqrt{s})}{s\sqrt{s}\,I_{\gamma+1}(\sqrt{s})} -\lambda K_p\frac{I_\gamma(\sqrt{s+\kappa})}{s\sqrt{s+\kappa}\,I_{\gamma+1}(\sqrt{s+\kappa})} \right], \tag{14} \]

where

\[ I_\gamma(z) \]

is the modified Bessel function of the second kind of order \(\gamma\) \([3,\ \text{p. }139]\);

\(\gamma=-1/2\) for a film electrode and \(\gamma=1/2\) for spherical electrodes;

\[ \kappa=\lambda_1+\lambda_2;\qquad K_p=K_p'C_R. \]

Passing from the solution in images (17) to the original by the expansion theorem, one can obtain for the concentration at the electrode surface the expression:

\[ C_R(1,\Theta)=\frac{C_R^0-\lambda S_1-\lambda K_p S_2}{1+K_p}, \tag{15} \]

where

\[ S_1=\frac{1}{2}\left[ 2(2\gamma+2)\Theta+1-\frac{2\gamma+2}{2\gamma+4} -4\sum_{n=1}^{\infty}\frac{\exp(-\mu_n^2\Theta)}{\mu_n^2} \right]; \tag{16} \]

\[ S_2= \frac{1}{\sqrt{\kappa}}\cdot \frac{I_\gamma(\sqrt{\kappa})}{I_{\gamma+1}(\sqrt{\kappa})} -\frac{2\gamma+2}{\kappa}e^{-\kappa\Theta} -2\sum_{n=1}^{\infty} \frac{\exp[-(\mu_n^2+\kappa)\Theta]}{\mu_n^2+\kappa}; \tag{17} \]

where \(\mu_n\) are the roots of the transcendental equation

\[ I_{\gamma+1}(\mu_n)=0. \tag{18} \]

The stationary dropping electrode is most convenient for studying intermetallic compounds in mercury. Taking in expressions (16, 17 and 18) \(\gamma=1/2\) and bearing in mind the relation of Bessel functions with an index equal to half an odd integer to trigonometric functions \([3,\ \text{p. }142]\), one can obtain

\[ S_1=3\Theta+0.2-2\sum_{n=1}^{\infty}\frac{\exp(-\mu_n^2\Theta)}{\mu_n^2}; \tag{19} \]

\[ S_2= \frac{1}{\sqrt{\kappa}\,\operatorname{cth}\sqrt{\kappa}-1} -\frac{3}{\kappa}e^{-\kappa\Theta} -2\sum_{n=1}^{\infty} \frac{\exp[-(\mu_n^2+\kappa)\Theta]}{\mu_n^2+\kappa}, \tag{20} \]

where \(\mu_n\) are the roots of the transcendental equation \(\operatorname{tg}\mu=\mu\) (21) \([4,\ \text{p. }47]\). The series in expression (19) converges rapidly, since \(\mu_n\) are positive increasing quantities. As was shown in works [5], if \(\Theta>0.14\), the series may be neglected with an error of no more than 1%; then:

\[ S_1\simeq 3\Theta+0.2. \tag{22} \]

It is easy to note that the series in expression (20) converges still more rapidly,

since the quantity \(\kappa=\lambda_1+\lambda_2>0\), consequently, for \(\Theta>0.14\) this series may be neglected; then:

\[ S_4=\frac{1}{\sqrt{\kappa}\,\operatorname{cth}\sqrt{\kappa}-1}-\frac{3}{\kappa}e^{-\kappa\Theta}. \tag{23} \]

The expression for the dimensionless transition time \(\Theta'_k\) at \(\Theta'_k>0.14\), taking into account the expressions for \(S_3\) and \(S_4\) (22–23), is found from equation (15) under the condition \(C_R(1,\Theta')=0\):

\[ \Theta'_k=\frac{C_R^0}{3\lambda}-\frac{0.2}{3} -\frac{K_p}{3\left(\sqrt{\kappa}\,\operatorname{cth}\sqrt{\kappa}-1\right)} -\frac{K_p}{\kappa}e^{-\kappa\Theta'_k}. \tag{24} \]

Using this equation, by trial one can find the value of \((\kappa_1+\kappa_2)\), if \(K_p\), \(C_R^0\), and \(\Theta'_k\) are known. Equation (24) may be rewritten in the following form:

\[ \Theta'_k=\Theta'- K_p\left[ \frac{1}{3\left(\sqrt{\kappa}\,\operatorname{cth}\sqrt{\kappa}-1\right)} -\frac{1}{\kappa}e^{-\kappa\Theta'_k} \right], \tag{25} \]

where \(\Theta'\) is the value of the dimensionless transition time that would be observed if the bulk concentration of substance \(O\) were equal to \(C_R\) in the absence of the kinetic effect. If it is assumed that a change in the transition time down to \(10\%\) can be detected [6], then it follows from equation (25) that the kinetic effect can be detected if

\[ K_p\left[ \frac{1}{3\left(\sqrt{\kappa}\,\operatorname{cth}\sqrt{\kappa}-1\right)} -\frac{1}{\kappa}e^{-0.15\kappa} \right]\geq 0.014. \tag{26} \]

For this estimate the value of \(\Theta'\) has been taken as the minimum possible value for which equation (24) is still valid, namely \(0.14\). For \(\kappa>100\) and \(\Theta'_k>0.15\), \(\operatorname{cth}\sqrt{\kappa}\simeq 1\), and \(\frac{1}{\kappa}e^{-\kappa\Theta'}\simeq 0\). In this case the condition for detecting the kinetic effect (26) reduces to the form:

\[ K_p(3\sqrt{\kappa})^{-1}\geq 0.014. \tag{27} \]

Allowance for faster preceding reactions than is determined by conditions (26) and (27) will lead to complicated calculations with allowance for the terms of the series in expressions (19, 20).

The equation of the \(\varphi-t\) curve for a reversible electrode process is obtained by substitution into the Nernst equation:

\[ \varphi=\varphi_{1/2}-\frac{RT}{zF}\ln\frac{C_R(1,\Theta)}{C_O(1,\Theta)}, \tag{28} \]

at \(D_0=D_R\) of the expressions for \(C_R(r_0,t)\) and \(C_0(r_0,t)\).

For any \(\Theta_k\),

\[ \varphi=\varphi_{1/2} -\frac{RT}{\beta zF} \ln \frac{ \displaystyle \frac{\lambda}{1+K_p}\left(S''_1+K_pS''_2-S_1-K_pS_2\right) }{ \displaystyle C_0^0+\lambda\left(1-\exp\Theta\,\operatorname{erfc}\sqrt{\Theta}\right) }. \tag{29} \]

\(S''_1\) and \(S''_2\) are expressed by equations (19, 20), except that in the latter \(\Theta\) is replaced by \(\Theta'_k\).

At \(\Theta'_k>0.14\),

\[ \varphi=\varphi_{1/2} -\frac{RT}{zF}\ln \frac{ \displaystyle \frac{\lambda}{1+K_p}\left(S'_3-K_pS'_4-S_3-S_4\right) }{ \displaystyle C_0^0+\lambda\left(1-\exp\Theta\,\operatorname{erfc}\sqrt{\Theta}\right) }. \tag{30} \]

\(S'_3\) and \(S'_4\) are expressed by equations (22, 23), only in the latter \(\Theta_k\) is substituted in place of \(\Theta\).

For irreversible processes, the equations of the \(\varphi - t\) curves for an electrode reaction complicated by a preceding reaction in a spherical mercury electrode have the form:

for any \(\Theta'_k\)

\[ \varphi=\varphi^\circ+\frac{RT}{\beta zF}\ln\frac{q_0}{\kappa_s} -\frac{RT}{\beta zF}\ln\frac{\lambda}{1+K_p} \left(S''_1+K_pS''_2-S_1-K_pS_2\right); \tag{31} \]

for \(\Theta'_k \gg 0.14\)

\[ \varphi=\varphi^\circ+\frac{RT}{\beta zF}\ln\frac{q_0}{\kappa_s} -\frac{RT}{\beta zF}\ln\frac{\lambda}{1+K_p} \left(S'_3-K_pS_4-S_3-S_4\right), \tag{32} \]

where

\[ q_0=\frac{i_0}{zFS}; \]

\(\beta\) is the transfer coefficient;

\(\kappa_s\) is the electrode-process constant at the standard potential \(\varphi^\circ\), cm·s\(^{-1}\).

The expression for \(C_R(r_0,t)\) for an arbitrary form of variation of the electrode current (for an arbitrary form of variation of the electrode potential) is obtained from equation (15) using Duhamel’s theorem:

\[ C_R(r_0,t)=\frac{1}{1+K_p} \left(C_R^0-\frac{1}{r_0}I_1-\frac{K_p}{r_0}I_2\right), \tag{33} \]

where

\[ I_1 \equiv \int_0^t q(\tau) \left\{3+2\sum_{n=1}^{\infty}\exp\left[-\frac{\mu_n^2D}{r_0^2}(t-\tau)\right]\right\}\,d\tau; \tag{34} \]

\[ I_2=\int_0^t q(\tau) \left\{3\exp[-\kappa(t-\tau)] +2\sum_{n=1}^{\infty}\exp\left[-\left(\frac{\mu_n^2D}{r_0^2}+\kappa\right)(t-\tau)\right]\right\}\,d\tau; \tag{35} \]

\[ q(t)=\frac{i(t)}{zFS}; \]

From equation (33) it is evident that the integral \(I_2\) is a correction for the kinetic effect for any form of variation of the electrode current (potential).

The expression for \(C_0(r_0,t)\) for an arbitrary form of variation of the electrode current (potential) under conditions of spherical diffusion has the form:

\[ C_0(r_0,t)=C_0^0+\frac{1}{\sqrt{\pi D_0}}\,I_3-\frac{1}{r_0}I_4, \tag{36} \]

where

\[ I_3\equiv \int_0^t q(\tau)\frac{d\tau}{\sqrt{t-\tau}}, \tag{37} \]

\[ I_4=\int_0^t q(\tau) \left[ \exp\frac{D(t-\tau)}{r_0^2}\, \operatorname{erfc}\frac{D^{1/2}(t-\tau)^{1/2}}{r_0} \right]d\tau; \tag{38} \]

It should be noted that equation (33) is obtained here in a form valid for any \(t\). It can be shown that, expanding the function

expanding the Bessel functions in equation (14) into an asymptotic series, one can obtain equations identical to the equations of semi-infinite diffusion.

The expression for the anodic current for a quasi-reversible electrode reaction of the first order has the form:

\[ i=zFS\kappa_s\left[ e^{-\frac{\alpha zF(\varphi-\varphi^0)}{RT}}C_0(r_0,t) - e^{\frac{\beta zF(\varphi-\varphi^0)}{RT}}C_R(r_0,t) \right]. \tag{39} \]

By combining equations (33, 36, and 39), we obtain the equation of the curve for the quasi-reversible process of electro-dissolution of an amalgam, complicated by a preceding first-order chemical reaction, under conditions of restricted spherical diffusion:

\[ |i|=be^{y_2}\left[ \frac{1}{1+K_p} \left( C_R^0-\frac{1}{r_0}I_1-\frac{K_p}{r_0}I_2 \right) \right] - ae^{y_1}\left( C_0^0+\frac{1}{\sqrt{\pi D}}I_3+\frac{1}{r_0}I_4 \right); \tag{40} \]

where

\[ a=zFS\kappa_s\exp\left[ -\frac{\alpha zF(\varphi_i-\varphi_0)}{RT} \right]; \tag{41} \]

\[ b=zFS\kappa_s\exp\left[ \frac{\beta zF}{RT}(\varphi_i-\varphi_0) \right]; \tag{42} \]

\[ \left. \begin{aligned} y_1&=-\frac{\alpha zF}{RT}\omega t,\\ y_2&=\frac{\beta zF}{RT}\omega t \end{aligned} \right\} \quad \begin{array}{l} \text{for anodic amalgam voltammetry}\\ \text{with a linearly varying potential;} \end{array} \tag{43} \]

\[ \left. \begin{aligned} y_1&=-\frac{\alpha zF}{RT}\,[\omega t-zFS\Omega q(t)]\\ y_2&=\frac{\beta zF}{RT}\,[\omega t-zFS\Omega q(t)] \end{aligned} \right\} \quad \begin{array}{l} \text{with allowance for the ohmic resistance } \Omega \text{ of all elements}\\ \text{of the electrolytic cell [7];} \end{array} \tag{44} \]

\[ y_1=y_2=0 \text{ — with a constant electrode potential.} \tag{45} \]

In the latter case, the equation of the \(i-\varphi\) curve becomes the equation of the \(i-t\) curve.

When there is an overvoltage at the electrode greater than \(\dfrac{0.12}{z}\) V, the reverse process (the oxidation process) may be neglected, and an irreversible process will occur at the electrode. Setting \(a=0\) in equation (40), we obtain the equation of the \(i-\varphi\) curve for the irreversible process of electro-dissolution of an amalgam, complicated by a preceding first-order reaction:

\[ |i|=be^{y_2}\left[ \frac{1}{1+K_p} \left( C_R^0-\frac{1}{r_0}I_1-\frac{K_p}{r_0}I_2 \right) \right]. \tag{46} \]

Using the Nernst equation

\[ \frac{C_0(r_0,t)}{C_R(r_0,t)} = \exp\left[ \frac{zF}{RT}(\varphi_i-\varphi^0)\exp(-y) \right], \tag{47} \]

one can obtain the equation for the current of the reversible process of electro-oxidation of an amalgam, complicated by a preceding reaction:

\[ \frac{ C_O^0+\frac{1}{\sqrt{\pi D}}\,I_3+\frac{1}{r_0}\,I_4 }{ \frac{1}{1+K_p}\left(C_R^0-\frac{1}{r_0}\,I_1-\frac{K_p}{r_0}\,I_2\right) } = \exp\left[\frac{zF}{RT}\left(\varphi_i-\varphi^0\right)\exp(y)\right]; \tag{48} \]

where

\[ y=\frac{zF}{RT}\,\omega t \]

—for voltammetry with a linearly varying electrode potential.

\[ y=0 \]

—for voltammetry with a constant electrode potential.

The integrals \(I_1\), \(I_2\), \(I_3\), \(I_4\), by one of the methods of numerical integration, are reduced to algebraic equations, which can be programmed and then used on electronic computers to calculate current values for constructing \(i-\varphi\) or \(i-t\) curves. These questions will be discussed in other works.

Conclusions

For anodic amalgam voltammetry, equations have been obtained for the process of electro-dissolution of an amalgam, complicated by a preceding first-order reaction, at constant current, linearly varying, and constant electrode potential.

References

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