PROCEEDINGS

OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC

INSTITUTE named after S. M. KIROV

Vol. 128　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1965

V. E. GORODOVYKH

THEORY OF COMPLETELY IRREVERSIBLE ELECTRODE PROCESSES AT A STATIONARY SPHERICAL ELECTRODE UNDER A LINEARLY VARYING POTENTIAL

(Presented by the seminar of the Department of Physical Chemistry, TPI)

The method of amalgam polarography with accumulation [1] includes two stages: a) electrolytic concentration, at a constant potential, of metal atoms in the small volume (of the order of 2.5 mm³) of a stationary mercury drop electrode (stationary m.d.e.), and b) anodic dissolution of the amalgam formed, with subsequent application to the electrode of a linearly (or according to another law) varying potential. The first stage is considered in detail in works [2, 3]. The purpose of the present work is to calculate the current at a stationary m.d.e. under the conditions of amalgam polarography for the case of completely irreversible electrode processes at a linearly varying potential.

The problem of finding the dependence of the current at a stationary electrode whose potential varies linearly with time has been solved by a number of authors. Randles [4], Ševčík [5], Berzins and Delahay [6] considered the case of linear semi-infinite diffusion. In the work of Nicholson [7], the theory of semi-infinite diffusion to a cylindrical electrode was developed, while Frankenthal, Chyn, and Reinmuth [8] obtained equations for a spherical electrode.

The development of the method of amalgam polarography with accumulation created the need to obtain the theoretical current–potential dependence under conditions in which the finite dimensions of the electrode play an essential role and, consequently, the laws of semi-infinite diffusion are not applicable. A problem of this kind was first posed and partially solved by Reinmuth [10]. A complete derivation of the anodic-current equation for the case of reversible processes is contained in [11]. Below, a derivation is given of the equations for the cathodic and anodic peaks for completely irreversible electrode processes at a stationary m.d.e.

Equation of the Cathodic Peak

The solution of this problem will be obtained by us under the following assumptions:

a) the electrode process includes only one slow stage;

b) only simple metal ions take part in the electrode process; chemical reactions in the solution and in the amalgam are absent;

c) the electrode process is sufficiently irreversible and the influence of the reverse reaction may be neglected (this assumption is valid at overvoltages greater than \(0.12/n\) V [12]);

d) there is no mass transfer due to migration or convection;

e) the capacitive current may be neglected (low rate of change of potential).

If the electrode reaction is completely irreversible, then the flux of metal ions to the surface of a stationary spherical electrode of radius \(r_0\) is related to the concentration of ions at the electrode surface by the relation

\[ D_0\left(\frac{\partial C_0(r,t)}{\partial r}\right)_{r=r_0} = \kappa_f' C_0(r_0,t), \tag{1} \]

where \(C_0(r,t)\) is the concentration of metal ions in solution at an arbitrary time \(t\);

\(D_0\) is the diffusion coefficient;

\(\kappa_f'\) is the discharge rate constant.

The discharge constant is related to the electrode potential by the relation [12]

\[ \kappa_f'=\kappa_f \exp\left[-\frac{\alpha nF}{RT}E\right], \tag{2} \]

where \(\kappa_f\) is the discharge rate constant at \(E=0\);

\(\alpha\) is the transfer coefficient;

\(n\) is the number of electrons participating in the electrode process;

\(E\) is the electrode potential.

If the potential varies linearly with time,

\[ E=E_i-vt, \tag{3} \]

where \(E_i\) is the initial potential;

\(v\) is the rate of change of potential;

\(t\) is time,

then the discharge constant may be written as follows:

\[ \kappa_f=\kappa_f' \exp(\beta t), \tag{4} \]

where

\[ \beta=\frac{\alpha nF}{RT}v. \tag{5} \]

In order to find \(C_0(r,t)\) for a linearly varying potential, it is necessary to solve Fick’s equation for spherical diffusion

\[ \frac{\partial C_0(r,t)}{\partial t} = D_0\left[ \frac{\partial^2 C_0(r,t)}{\partial r^2} + \frac{2}{r}\frac{\partial C_0(r,t)}{\partial r} \right], \tag{6} \]

under the following initial and boundary conditions:

\[ t=0,\qquad r_0 \le r<\infty,\qquad C_0(r,0)=C_0^*, \tag{7} \]

where \(C_0^*\) is the initial concentration of metal ions in solution;

\[ t>0,\qquad r=r_0,\qquad C_0(r_0,t)=f(t), \]

\[ \lim_{r\to\infty} C_0(r,t)=C_0^*. \tag{8} \]

The result of solving this problem may be represented in the form\(^1\)

\(^1\) This and the following problems were solved by us by the operational method.

\[ C_0(r,t)=\frac{r_0}{r}\int_0^t f'_\tau(\tau)\,\operatorname{erfc}\frac{r-r_0}{2\sqrt{D_0(t-\tau)}}\,d\tau+C_0^* = \]

\[ =\frac{r_0}{r}\int_0^t f'_\tau(t-\tau)\,\operatorname{erfc}\frac{r-r_0}{2\sqrt{D_0\tau}}\,d\tau+C_0^* . \tag{9} \]

The values of the function \(\operatorname{erfc} z\) have been tabulated and can be easily calculated.

It is easy to see that the solution obtained satisfies the initial and boundary conditions of problem (6)—(8)\(^2\).

From equation (9) we find the flux at the electrode surface:

\[ D_0\left(\frac{\partial C_0(r,t)}{\partial r}\right)_{r=r_0} = D_0\int_0^t f'_\tau(t-\tau) \left(\frac{1}{\sqrt{\pi D_0\tau}}-\frac{1}{r_0}\right)d\tau . \tag{10} \]

Thus, in order to find the flux (and hence also the current) at a stationary spherical electrode when a linearly varying potential is applied to it, it is necessary to determine the form of the function \(f(t)\) and its derivative and substitute it into (10).

From Duhamel’s theorem [14] it follows that the function \(f(t)\) represents the expression for the concentration of metal ions at the electrode surface at constant potential \(C_0(r_0,t)_{E=\mathrm{const}}\), in which the discharge constant \(\kappa_f'\) is replaced by \(\kappa_f\) according to equation (4).

Let us find \(C_0(r_0,t)_{E=\mathrm{const}}\), for which purpose we solve equation (6) under the following conditions:

\[ \left. \begin{aligned} &t=0,\qquad r_0\leq r<\infty,\qquad C_0(r,0)=C_0^*;\\ &t>0,\qquad r=r_0,\\ &D_0\left(\frac{\partial C_0(r,t)}{\partial r}\right)_{r=r_0} =\kappa_f' C_0(r_0,t);\\ &\lim_{r\to\infty} C_0(r,t)=C_0^*, \end{aligned} \right\} \tag{11} \]

where \(\kappa_f'=\mathrm{const}\) is determined from equation (2).

The solution of equation (6) with the initial and boundary conditions (11) gives the following expression for \(C_0(r,t)_{E=\mathrm{const}}\):

\[ C_0(r,t)_{E=\mathrm{const}} = -\frac{C_0^*\,\kappa_f' r_0}{r\left[\kappa_f'+\dfrac{D_0}{r_0}\right]} \left\{ \operatorname{erfc}\frac{r-r_0}{2\sqrt{D_0t}} -\right. \]

\[ \left. -\exp\left[ \left(\frac{\kappa_f' r_0+1}{r_0D_0}\right)(r-r_0) + \left(\frac{\kappa_f' r_0+1}{r_0\sqrt{D_0}}\right)^2 t \right] \operatorname{erfc}\left( \frac{\kappa_f' r_0+1}{r_0\sqrt{D_0}}\sqrt{t} + \frac{r-r_0}{2\sqrt{D_0t}} \right) \right\} +C_0^* . \tag{12} \]

\(^2\) It should also be noted that as \(r_0\to\infty\), equation (9) goes over into the corresponding equation for a plane electrode,

\[ C_0(x,t)=\int_0^t f'_\tau(t-\tau)\operatorname{erfc}\frac{x}{2\sqrt{D_0\tau}}\,d\tau+C_0^* . \]

Thus, Delahay’s remark [13] that a problem of this kind cannot be solved by the Laplace-transform method is incorrect.

It follows from (12) that the concentration at the electrode surface \((r=r_0)\) is

\[ C_0(r_0,t)_{E=\mathrm{const}} = -\frac{C_0^* \kappa'_f}{\kappa'_f+\dfrac{D_0}{r_0}} \left[1-\exp a^2 \operatorname{erfc} a\right]+C_0^*, \tag{13} \]

where

\[ a=\frac{\kappa'_f r_0+1}{r_0}\sqrt{\frac{t}{D_0}}. \]

If one sets \(r_0=4\cdot 10^{-2}\ \mathrm{cm}\), \(D_0=10^{-5}\ \mathrm{cm^2/sec}\), \(t=10\text{--}30\ \mathrm{sec}\), then \(a\gg 1\), and on the basis of the expansion

\[ \exp a^2 \operatorname{erfc} a = \frac{1}{\sqrt{\pi}} \left( \frac{1}{a} - \frac{1}{2a^3} + \frac{1\cdot 3}{4a^5} -\cdots \right) \]

from (13) we have

\[ C_0(r_0,t)_{E=\mathrm{const}} = -\frac{C_0^* \kappa'_f}{\kappa'_f+\dfrac{D_0}{r_0}} \left( 1-\sqrt{\frac{D_0}{\pi t}}\, \frac{1}{\kappa'_f+\dfrac{1}{r_0}} \right) + C_0^*. \tag{14} \]

Under ordinary conditions of amalgam polarography, the term

\[ \sqrt{\frac{D_0}{\pi t}}\, \frac{1}{\kappa'_f+\dfrac{1}{r_0}} \ll 1 \]

and it may be neglected without large error (less than \(10^{-3}\%\)). Then

\[ C_0(r_0,t)_{E=\mathrm{const}} = \frac{C_0^* D_0}{r_0\left(\kappa'_f+\dfrac{D_0}{r_0}\right)}. \tag{15} \]

Replacing \(\kappa'_f\) in (15) by \(\kappa_f\), differentiating with respect to \(t\), and substituting the result into (10), we obtain the final expression for the flux

\[ D_0\left(\frac{\partial C_0(r,t)}{\partial r}\right)_{r=r_0} = - C_0^* \frac{D_0^2 \beta}{\kappa'_f r_0} \int_0^t \frac{e^{\beta(t-\tau)}} {\left[e^{\beta(t-\tau)}+\dfrac{D_0}{\kappa'_f r_0}\right]^2} \times \tag{16} \]

\[ \times \left( \frac{1}{\sqrt{\pi D_0 \tau}} - \frac{1}{r_0} \right)d\tau . \]

The current at a stationary mercury drop electrode under a linearly varying potential is equal to

\[ i=nFSD_0\left(\frac{\partial C_0}{\partial r}\right)_{r=r_0} = \tag{17} \]

\[ = - \frac{n^2F^2D_0^2 C_0^* v}{RT\kappa'_f r_0} \int_0^t \frac{e^{\beta(t-\tau)}} {\left[e^{\beta(t-\tau)}+\dfrac{D_0}{\kappa'_f r_0}\right]^2} \left( \frac{1}{\sqrt{\pi D_0 \tau}} - \frac{1}{r_0} \right)d\tau . \]

It follows from equation (17) that the height of the cathodic peak is directly proportional to the concentration of metal ions in the solution.

Equation of the anodic peak

The derivation of the equation for the anodic peak is analogous to the solution of the problem for the cathodic peak. The concentration of metal atoms \(C_R(r,t)\) inside the stationary mercury drop electrode at an arbitrary time \(t\) is equal to [11]

\[ C_R(r,t)=\int_0^t \left\{ f'_{\tau}(t-\tau)\left[1+\right.\right. \]

\[ \left.\left. +2\sum_{n=1}^{\infty}(-1)^n\frac{r_0}{n\pi r}\sin\frac{n\pi r}{r_0} \exp\left(-\frac{n^2\pi^2D_c}{r_0^2}\right)\right]\right\}d\tau+C_R^*, \tag{18} \]

where \(f(t)=C_R(r_0,t)\) is the concentration of metal near the surface of the stationary mercury drop electrode;
\(C_R^*\) is the initial concentration of metal in the amalgam.

The current at the electrode is

\[ i=\frac{2nFDS}{r_0}\int_0^t f'(t-\tau) \sum_{n=1}^{\infty}\exp\left(-\frac{n^2\pi^2D_c}{r_0^2}\right)d\tau . \tag{19} \]

To find the function \(f(t)\), as in the preceding case, we solve the problem for a constant potential

\[ \frac{\partial C_R(r,t)}{\partial t} = D\left( \frac{\partial^2 C_R(r,t)}{\partial r^2} + \frac{2}{r}\frac{\partial C_R(r,t)}{\partial r} \right), \tag{20} \]

\[ \begin{aligned} &t=0,\qquad 0\leq r\leq r_0,\qquad C_R(r,0)=C_R^*,\\ &t>0,\qquad r=r_0,\qquad C_R(r_0,t)=k'_b C_R(r_0,t),\\ &t>0,\qquad \lim_{r\to 0}\frac{\partial C_R(r,t)}{\partial r}=0 \end{aligned} \left\} \right., \tag{21} \]

where

\[ k'_b=k_b^\circ \exp\left[\frac{(1-\alpha)nFE}{RT}\right] \]

is the ionization constant.

We represent the solution of the problem (20)—(21) in the form

\[ C_R(r,t)_{E-\mathrm{const}} = \frac{C_R^* k'_b r_0^2}{Dr} \sum_{n=1}^{\infty} A_n \frac{\sin\mu_n\frac{r}{r_0}}{\mu_n} e^{-D\frac{\mu_n^2}{r_0^2}t}, \tag{22} \]

where

\[ A_n=\frac{2\sin\mu_n}{\sin\mu_n\cos\mu_n-\mu_n}, \]

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

\[ \tg \mu=\frac{\mu}{A}. \tag{23} \]

The values \(\mu_n\) satisfying equation (23) can be found graphically, by determining the points of intersection of the tangentoid \(y_1=\tg\mu\) with the straight line

\[ y_2=\frac{\mu}{A}. \]

Substituting into the solution (22) the constant

\[ k_b=k'_b\exp(-\beta t) \tag{24} \]

and, differentiating with respect to \(t\), we find the expression for \(f'(t)\)

\[ f'(t)=\frac{C_R^* r_0^3 k_B}{D}\sum_{n=1}^{\infty} A_n \frac{\sin \psi_n}{\psi_n} e^{-\frac{D\psi_n^2}{r_0^2}t}. \tag{25} \]

At large values of \(t\) (a low rate of change of the potential), or more precisely of the dimensionless parameter \(Dt/r_0^2\), the series in (22) converges rapidly and it is possible to restrict oneself to only the first term

\[ f'(t)=\frac{C_R^* r_0^3 k_B}{D} B e^{-\frac{D\gamma^2 t}{r_0^2}}, \tag{26} \]

where

\[ \gamma=\psi_1, \]

\[ B=A_1\frac{\sin \gamma}{\gamma}. \]

The combination of equations (19) and (25), or (26), represents the dependence of the anodic current at a stationary spherical mercury-drop electrode on a linearly varying potential.

A detailed study and experimental verification of the equations obtained will be given elsewhere.

Conclusions

Equations have been derived for the cathodic and anodic currents at a stationary spherical mercury-drop electrode for completely irreversible electrode processes and a linearly varying potential.

LITERATURE

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2. L. N. Vasil’eva and E. N. Vinogradova. On the distribution of metal concentration inside a mercury drop during its electrolytic deposition on a stationary mercury electrode. Zavod. laborator., 27, No. 9, 1079 (1961).

3. I. Shain, J. Lewinson. Analysis by anodic dissolution using mercury electrodes. Anal. chem., 33, No. 2, 187 (1961).

4. J. E. B. Randles. Cathode-ray polarograph. Trans. Faraday Soc., 44, No. 305, 327 (1948).

5. A. Ševčík. Oscillographic polarography with periodic triangular voltage. Coll. czech. chem. comm., 13, 349 (1948).

6. T. Berzins, P. Delahay. Oscillographic polarographic waves of reversible deposition of metals on solid electrodes. J. Am. Chem. Soc., 75, No. 3, 555 (1953).

7. M. M. Nicholson. Diffusion currents at cylindrical electrodes. J. Am. Chem. Soc., 76, No. 92, 2539 (1954).

8. R. P. Frankenthal, I. Shain. Diffusion currents at spherical electrodes. J. Am. Chem. Soc., 78, No. 13, 2969 (1956).

9. W. H. Reinmuth. Currents controlled by the Nernst equation in hanging-drop polarography. J. Am. Chem. Soc., 79, No. 24, 6358 (1957).

10. W. H. Reinmuth. Theory of stripping voltammetry at spherical electrodes. Anal. Chem., 33, No. 2, 185 (1961).

11. V. E. Gorodovykh. Equation of the anodic peak for reversible electrode processes at an amalgam spherical electrode. Izv. TPI (in press).

12. P. Delahay. New Instruments and Methods in Electrochemistry. Moscow, IL, p. 50 (1957).

13. P. Delahay. Theory of irreversible waves in oscillographic polarography. J. Am. Chem. Soc., 75, No. 7, 1190 (1953).

14. H. S. Carslaw, J. C. Jaeger. Conduction of Heat in Solids. Oxford, p. 124 (1947).