complex formation \(\left(\lg \frac{K_f^*}{K_p} \le 3\right)\), where \(K_p\) is the stability constant of the complex), the interpolation formula

\[ \lg \Delta \psi = \left(\lg K_f^*\right)^{0.778} + 0.539 \tag{4} \]

is valid.

In logarithmic coordinates this equation is easily represented as a straight line. The tangent of the angle of inclination of this straight line is 0.778, and the intercept cut off on the ordinate axis is 0.445.

For large values of \(K_f^*\left(\lg \frac{K_f^*}{K_p} > 3\right)\) one may use the formula

\[ \lg \Delta \psi = \frac{\lg K_f^*}{0.442 \lg K_f^* + 0.39} \tag{5} \]

Equation (5), in the coordinates \(\lg K_f^*;\ \lg K_f^*/\lg \Delta \psi\), gives a straight line with slope tangent equal to 0.442. The intercept cut off by this straight line on the ordinate axis is 0.38. Equation (5) is valid only for sufficiently strong complexes, \(\lg K_p > 5\).

Expressions (4) and (5) can be used to estimate the parameters of the chemical reaction. The composition of the complex \(P\) can be found from equation (3).

Literature

1. Yu. A. Karbainov, S. S. Reznikova, A. G. Stromberg. Electrochemistry, 9, 1351 (1973).
2. V. E. Gorodovykh. Izvestiya TPI, 164, II (1967),

ON THE APPLICATION OF NUMERICAL METHODS TO SOLVING DIFFUSION EQUATIONS IN STRIPPING POLAROGRAPHY

A. E. Geineman, M. V. Krasheninnikov, A. A. Kaplin

At the present time, in the method of anodic stripping polarography (ASP), two approaches are used rather widely for solving Fick’s second law with various boundary conditions. The first of them consists in obtaining equations for the anodic current as a function of time in explicit form. In this case the solutions are obtained in the form of series or integrals, whose further analysis requires computations on an electronic computer. Sometimes analysis of the general expressions leads to simple approximate analytical expressions, which is the principal advantage of this approach. The second approach is based on deriving integral equations and subsequently solving them on an electronic computer.

In the present work it is proposed for the first time to carry out the solution of Fick’s second law with various boundary conditions in the PSA method directly on a computer. Such an approach makes it possible to eliminate intermediate stages: the derivation of the anodic-current equation in explicit form, based on the use of a very complicated and cumbersome apparatus of mathematical physics (for example, operational calculus), and the derivation of integral equations.

As an example we have chosen a completely irreversible process during deposition on a mercury-film electrode with a linearly varying potential. The choice is due to the fact that, in this case, obtaining a rigorous analytical solution does not appear possible. The problem is posed as follows:

\[ D\frac{\partial^2 C}{\partial x^2}=\frac{\partial C}{\partial t} \]

under the initial and boundary conditions

\[ \left. \begin{aligned} &C(x,0)=C^0,\qquad D\frac{\partial C}{\partial x}\bigg|_{x=0}=0,\\ &D\frac{\partial C}{\partial x}\bigg|_{x=\ell} =-k_s^1 e^{\frac{\beta zF(wt-\varphi^0)}{RT}}\,C(\ell,t) \end{aligned} \right\} \tag{1} \]

where \(C(x,t)\) is the concentration of the metal in the amalgam, mol/l; \(k_s^1\) is the rate constant of the electrochemical reaction, cm/sec; \(z\) is the number of electrons participating in the electrode reaction; \(D\) is the diffusion coefficient of metal atoms in the amalgam, cm\(^2\)/sec; \(\ell\) is the thickness of the mercury film, cm.

For solving the problem posed, the most suitable method apparently is the sweep method. The convergence of this method for problems of the given type has been proved [1].

For comparison we used the approximate equation for the anodic current for a completely irreversible process [2], derived under the assumption that diffusion of metal atoms in the mercury film has no influence on the magnitude of the anodic current:

\[ i=zFSk_s^1 C^0 e^{\frac{\beta zF(wt-\varphi^0)}{RT}} e^{-\frac{RT k_s^0}{\beta zFw\ell} \left(e^{\frac{\beta zF(wt-\varphi^0)}{RT}}-1\right)} \tag{2} \]

In [2] the condition is given under which (2) is valid:

\[ \frac{zFw}{RT} \ll \frac{\pi^2 D}{4\ell^2} \tag{3} \]

If, as in [3], one assumes that during the dissolution stage the concentration \(C(x,t)\) of the metal in the mercury film obeys the following law:

\[ C(x,t)=C^0-\frac{1}{\ell}\int_0^t g(t)\,dt -g(t)\frac{\ell}{6D}\left[3\left(\frac{x}{\ell}\right)^2-1\right]. \tag{4} \]

where \(g(t)\) is the flux of substance onto the surface of the mercury film, then from (4) for the anodic current of a completely irreversible process one can obtain the following expression:

\[ i = \frac{ z F S k_s C^0 e^{\frac{\beta z F(wt-\psi^0)}{RT}} \left(1+\frac{k_s l}{3D}\right)^a }{ \left[ 1+\frac{k_s l}{3D} e^{\frac{\beta z F(wt-\psi^0)}{RT}} \right]^a }, \tag{5} \]

where

\[ a=\left(1+\frac{3DRT}{l^2\beta z F w}\right), \qquad \text{for } \frac{1}{a}<0.01 \tag{6} \]

which coincides with condition (3), and

\[ \frac{k_s l}{3D} e^{\frac{\beta z F(wt-4\psi)}{RT}}<0.01 \tag{7} \]

expression (5) becomes expression (2). Thus an additional condition is imposed under which (2) is valid. It is obvious that condition (3) is not sufficient, since expression (2) cannot be valid for arbitrarily large \(k_s\). We carried out an integral estimate of expression (5), which showed that, for

\[ \frac{k_s l}{3D}<0.01 \tag{8} \]

equation (5) is valid with an accuracy of up to one percent. It should be noted that the expressions for the peak potential \(\psi_n\), obtained from (2) and (5), coincide.

On the BESM-4 electronic computer we carried out comparative calculations of equations (2) and (5) with the rigorous solution obtained on a computer by direct solution of Fick’s second law for a completely irreversible process in stripping polarography with accumulation (1). The calculations fully confirmed conditions (6), (7), under which equation (2) is valid, as well as condition (8), under which equation (5) is valid.

On the basis of the mathematical model of the completely irreversible process considered by us, an attempt was made to apply one of the methods of nonlinear programming to determine physicochemical parameters, which may be of great importance for studying the kinetics and mechanism of electrode reactions. The following was chosen as the criterion for agreement between the experimental and calculated anodic peak:

\[ I=\sum_{j=1}^{n}\left(i_{\mathrm{e}j}-i_{\mathrm{p}j}\right)^2 \]

where \(n\) is the number of points at which agreement between the experimental and calculated anodic peaks is sought; \(i_{\mathrm{e}}\) is the experimental anodic current, \(i_{\mathrm{p}}\) is the calculated anodic current. Minimization of the \(I\)-criterion was carried out by the simplex method with respect to four parameters \(k_s\),

$B', \ell', \varphi^0$. The choice of method is determined by the complexity of the model (the use of gradient methods is difficult, since the computation of derivatives significantly increases the requirement for machine time).

To test the selected algorithm, an anodic peak obtained on the basis of the model under consideration was taken; for it the physicochemical parameters $K_s$, $B$, $\ell$, $\varphi^0$ are known. In the region of variation of the parameters, an initial approximation $K_s'$, $B'$, $\ell'$, $\varphi^{0'}$ was chosen, which serves as the basis for the search. The time for solving the problem, together with translation of the program, was 10 min. The agreement of the peaks is practically complete.

We have obtained preliminary data in processing the anodic peaks of thallium against the following indifferent-electrolyte backgrounds: $0.5\,N\ H_3PO_4$; $0.1\,N\ C_6H_8O_7$; $0.5\,N\ NH_4F$ in 10% alcohol, $0.1\,N\ NaCl$.

Table 1.

$\varphi^0$ is the standard potential relative to the saturated calomel electrode.

Thus, using the example of a completely irreversible process, it has been shown that in the ASP method it is advisable to solve Fick’s second law directly on a computer, and also to apply nonlinear programming methods for determining physicochemical parameters.

Literature

1. A. A. Samarskii. Introduction to the Theory of Difference Schemes. “Nauka,” Moscow, 1972.

2. V. A. Igolinskii, A. G. Stromberg. Izv. TPI 48, 14–17 (1967).

3. W. T. De Vries and E. van Dalen, J. Electroanal. Chem. 8, 366, 1964.