Proceedings
of the Tomsk Polytechnic Institute named after S. M. Kirov,
Order of the October Revolution and Order of the Red Banner of Labor

Volume 250 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1975

THEORY OF AMALGAM STRIPPING POLAROGRAPHY WITH ACCUMULATION (ASP) AT STATIONARY MERCURY ELECTRODES. REVERSIBLE PROCESSES. COMMUNICATION I. DERIVATION OF A BOUNDARY CONDITION FOR THE CASE OF ASYMMETRIC DIFFUSION AT A MERCURY FILM ELECTRODE

B. F. Nazarov, V. A. Nemov

(Presented by the scientific-methodological seminar of the Chemical-Technological Faculty)

The theory of reversible anodic peaks at stationary mercury electrodes has been developed in a number of papers, of which the works [1–8] are of greatest interest for the ASP method. Thus, Reinmuth showed [1] that the form of the equation relating current and potential in the ASP method depends on the choice of the boundary condition at the electrode–solution interface. The author proposed, as such a boundary condition in describing diffusion processes at film and dropping limited electrodes, the use of the condition of equality of the sums of the concentrations of the oxidized and reduced forms of the substance at the electrode surface, on the one hand, and in the bulk of the solution and amalgam, on the other. This condition is written as:

\[ D_R^{1/2} C_R(l,t) + D_0^{1/2} C_0(l,t) = D_R^{1/2} C_R^0 + D_0^{1/2} C_0^0, \tag{1} \]

where \(D_0\) and \(D_R\) are the diffusion coefficients of the oxidized and reduced forms of the electrochemically active substance, respectively;
\(C(l,t)\) is the concentration at the electrode surface;
\(C_0\) is the concentration in the bulk of the solution or amalgam. As shown by Shevchik [9], condition (1) is strictly fulfilled only for symmetric semi-infinite diffusion, i.e., for the case when both the thickness of the mercury and the thickness of the solution are sufficiently large. Reinmuth noted that condition (1) is approximate in the case of asymmetric (bounded–semi-infinite or semi-infinite–bounded) diffusion, “but this approximation is quite good in some cases” [1].

Reinmuth, and then Igolinsky and Gorodovykh [5, 6], applied condition (1) to solve the boundary-value problem of asymmetric diffusion at film and dropping mercury electrodes. Later, one of the authors of the present work showed that boundary condition (1) is also strictly valid in the case of symmetric bounded diffusion [7]. Thus, up to now no boundary condition has been found that strictly describes asymmetric diffusion; therefore, the purpose of the present work was to determine the possibility of obtaining a boundary condition strictly valid for asymmetric diffusion.

The boundary-value problem of asymmetric diffusion was posed as follows. A reversible electrode process was considered:

\[ O + ze \rightleftarrows R \tag{2} \]

without taking into account kinetic complications in the solution and the amalgam. The system of equations solved was

\[ \frac{\partial C_0(y,t)}{\partial t} = D_0 \cdot \frac{\partial^2 C_0(y,t)}{\partial y^2}; \qquad \frac{\partial C_R(x,t)}{\partial t} = D_R \cdot \frac{\partial^2 C_R(x,t)}{\partial x^2} \tag{3} \]

with the initial \((t=0)\)

\[ C_R(x,0)=C_R^0; \qquad C_0(y,0)=C_0^0 \tag{4} \]

and boundary \((t>0)\) \(0 \le x \le l;\; l \le y < \infty\)

\[ \begin{aligned} \text{a) }\quad & D_R \cdot \left.\frac{\partial C_R(x,t)}{\partial x}\right|_{x=0}=0;\\[4pt] \text{b) }\quad & \lim_{y\to\infty} C_0(y,t)=C_0^0;\\[4pt] \text{c) }\quad & D_0 \cdot \left.\frac{\partial C_0(y,t)}{\partial y}\right|_{y=l} = D_R \cdot \left.\frac{\partial C_R(x,t)}{\partial x}\right|_{x=l};\\[4pt] \text{d) }\quad & C_0(l,t)=\theta \cdot C_R(l,t) \end{aligned} \]

conditions. Here \(C_R(x,t)\) and \(C_0(y,t)\) denote the concentration of substance \(R\) or \(O\) at different times and at different distances from the electrode—solution interface. Condition (4) characterizes the distribution of substances \(R\) and \(O\) before the beginning of the process. Boundary condition (5a) indicates impermeability for the flux of the substrate of the film electrode. Condition (5b) shows that the concentration of the substance is constant at an infinite distance from the electrode surface. Condition (5c) describes equality of the fluxes of substances \(O\) and \(R\) at the electrode surface. Condition (5d) is valid only for reversible processes and is a modified form of the Nernst equation. In this condition \(\theta\) is the dimensionless potential, taking values from zero to infinity. The formulated problem was solved by the operational method [10]. Omitting intermediate calculations that are of no interest, the solutions of equations (3) can be written as expressions for the concentrations of substances \(R\) and \(O\) at the electrode surface at any time \(t\):

\[ C_R(l,t) = C_R^0 - \frac{(\theta C_R^0-C_0^0)}{\theta+\xi} \cdot \left[ \sum_{n=1}^{\infty} K^{n-1}\cdot \operatorname{erfc}\frac{(n-1)l}{\sqrt{D_R t}} + \sum_{n=1}^{\infty} K^{n-1}\cdot \operatorname{erfc}\frac{nl}{\sqrt{D_R t}} \right]; \tag{6} \]

\[ C_0(l,t) = C_0^0 + \frac{(\theta C_R^0-C_0^0)\cdot \xi}{\theta+\xi} \left[ \sum_{n=1}^{\infty} K^{n-1}\cdot \operatorname{erfc}\frac{(n-1)l}{\sqrt{D_R t}} - \sum_{n=1}^{\infty} K^{n-1}\cdot \operatorname{erfc}\frac{nl}{\sqrt{D_R t}} \right]. \tag{7} \]

Here

\[ \xi=\sqrt{\frac{D_R}{D_0}}, \qquad K=\frac{\xi-\theta}{\xi+\theta}, \qquad |K|<1; \]

\(\operatorname{erfc}(\ldots)\) is the function complementary to the error function. (The behavior of this function is considered in detail by Delahay [11]); \(n=1,2,3\ldots\). Multiplying further the right-hand side and the left-hand side of (6) by \(D_R^{1/2}\), and the right-hand side and the left-hand side of (7) by \(D_0^{1/2}\), and adding, respectively, the left- and right-hand sides of equations (6) and (7), we obtain the boundary condition that is strictly fulfilled in the case of asymmetric diffusion:

\[ D_R^{1/2}\cdot C_R(l,t)+D_0^{1/2}\cdot C_0(l,t) = D_R^{1/2}\cdot C_R^0+D_0^{1/2}\cdot C_0^0 - \frac{2D_R^{1/2}\left(\theta C_R^0-C_0^0\right)}{\theta+\xi} \times \]

\[ \times \sum_{n=1}^{\infty} K^{n-1}\cdot \operatorname{erfc}\frac{nl}{\sqrt{D_Rt}} . \tag{8} \]

Comparing equations (8) and (1), one can see that the former differs from the latter by a single term containing a sum. From the formulation of the problem and analysis of the additional term, it may be concluded that the term containing the sum

\[ \sum_{n=1}^{\infty} K^{n-1}\cdot \operatorname{erfc}\frac{nl}{\sqrt{D_Rt}}, \]

characterizes the finite nature of the mercury film electrode. Indeed, if finiteness is not taken into account, i.e., if \(l\to\infty\), then condition (1) can be obtained from (8). Particular attention should be paid to the fact that boundary condition (8) is valid for mercury film electrodes with any film thickness. This is ensured by the rapid convergence of the series to zero in the case of thick films \((l\to\infty)\), owing to the rapid convergence of

\[ \operatorname{erfc}\frac{nl}{\sqrt{D_Rt}} . \]

In addition, the magnitude \(K\) has a significant influence on the convergence of the series. To obtain analogous convergence of the series in the case of thin films, a larger number of terms of the series must be taken into account.

An important feature of the boundary condition (8) presented here is its universality. This condition is valid both for anodic processes, for example, in the ASP method \(\left(\text{then } C_R^0\gg C_0^0 \text{ and } C_0^0 \text{ may be neglected in comparison with } C_R^0\right)\), and for cathodic processes \(\left(\text{then } C_R^0 \text{ is neglected in comparison with } C_0^0\right)\).

In conclusion, it may be noted that at very large values of the process time the boundary condition (8) is transformed into the form

\[ D_R^{1/2}C_R(l,t)+D_0^{1/2}C_0(l,t) = C_0^0D_0^{1/2} \left( \frac{1+\xi\theta}{\xi\theta} \right), \tag{9} \]

where \(\xi=\sqrt{D_0/D_R}\).

Conclusions

1. A boundary condition has been obtained that is rigorously applicable to asymmetric diffusion. This condition differs from those known previously in that it contains an additional term accounting for the finite nature of the mercury film electrode.

2. The validity of the boundary condition obtained has been shown for stationary electrodes with any thickness of the mercury film.

3. It has been shown that for very thick films, i.e., when \(l\to\infty\), condition (8) becomes the boundary condition of symmetric semi-infinite diffusion.

4. It has been shown that the boundary condition obtained in the present work is rigorously satisfied for both anodic and cathodic reversible electrode processes.

5. A condition has been obtained for the case of asymmetric diffusion that describes the state of the system after the process has occurred at the electrode.

REFERENCES

1. W. H. Reinmuth, Anal. chem., 33, 185 (1961).
2. P. S. Nicholson and G. Shain, Anal. chem., 36, 706 (1964).
3. W. T. de Vries and E. van Dallen, G. Electroan chem., 8, 366 (1964).

1. W. T. de Vries, G. Electroanalyt. chem., 9, 448 (1965).
2. V. A. Ilyinsky. Abstract of Candidate’s Dissertation. Tomsk, 1963.
3. V. E. Gorodovykh. Abstract of Candidate’s Dissertation. Tomsk, 1964.
4. B. F. Nazarov. Abstract of Candidate’s Dissertation. Tomsk, 1967.
5. M. S. Zakharov. Doctoral Dissertation. Tomsk, 1967.
6. A. Sevcik, Coll. czech. chem. comm., 13, 349 (1948).
7. G. Doetsch. Guide to the Practical Application of the Laplace Transform. Fizmatgiz, Moscow, 1960.
8. P. Delahay. New Instruments and Methods in Electrochemistry. IL, Moscow, 1957, p. 67.