REFINED EQUATION OF A COMPLETELY IRREVERSIBLE POLAROGRAPHIC WAVE

A. G. Stromberg, L. N. Popova

(Presented by the Scientific-Methodological Seminar of the KhTF)

One of us has shown [1] that the equation of an irreversible anodic-cathodic wave, derived in [2] using a number of simplifying assumptions, gives results for the calculation of the exchange current and the transfer coefficient that differ somewhat from the exact, but more complicated and therefore inconvenient for physicochemical calculations, equations of Meiman [3] and Koutecký [4]. The aim of the present article is to introduce into the Stromberg equation [2] for an irreversible wave a corrective semiempirical factor for better agreement of the results of calculating the exchange current and the transfer coefficient by this equation with the exact equations of Meiman and Koutecký [3, 4].

In the equation of a completely irreversible Meiman wave [3], the current \((i,\ \mu\text{A})\) is a complex function of the potential \((\varphi,\ \text{volt})\) through the dimensionless parameter \((\mu_m^M)\):

\[ m=\frac{i}{I_k}=f(\mu_m^M), \tag{1} \]

\[ \mu_m^M=\frac{j_0^M}{B}\,e^{U_m}, \tag{2} \]

\[ B=\sqrt{\frac{7}{3}\,zF\cdot 10^{-3}\left(\frac{D_0}{\tau}\right)^{\frac12}} =148\,z\left(\frac{D_0}{\tau}\right)^{\frac12}, \tag{3} \]

\[ U_m=\frac{\alpha}{b^1}\,(\varphi^\circ-\varphi), \tag{4} \]

where \(I_k\) is the average limiting (diffusion) current of the cathodic wave, \(\mu\text{A}\); \(m\) is the ratio of the average current at any potential to the average limiting current during the lifetime of the mercury drop; \(j_0^M\) is the standard exchange-current density, \(a\cdot\text{cm}^3\,\text{mol}^{-1}\cdot\text{l}\); this is the exchange current per \(1\ \text{cm}^2\) of the average surface of the mercury drop at a concentration of the oxidized and reduced forms of the substance equal to unity \((c_0=c_R=1\ \text{mol}/\text{l}=10^{-3}\ \text{mol}/\text{l})\); \(z\) is the number of electrons participating in the electrode process per ion; \(F\) is the Faraday constant; \(D_0\) is the diffusion coefficient of the oxidized form of the substance in solution, \(\text{cm}^2/\text{s}\); \(\tau\) is the mercury drop time, s; \(\alpha\) is the transfer coefficient in the discharge process; \(\varphi^\circ\) is the standard electrode potential, volt; \(\varphi\) is the poten-

tial, volts; \(b' = RT/zF = 0.059/2.3\,z\) (at \(25^\circ\)C); \(B\) and \(U_m\) are abbreviated designations of expressions (3) and (4).

Equations (1)—(4) were derived by Meiman [3] under the assumption that the polarographic wave is completely irreversible, i.e., that the exchange current is sufficiently small and the rate of the reverse process may be neglected; that the electrochemical reaction is a first-order reaction and that there are no other limiting reaction processes; that the delivery of substance to the electrode is limited only by the diffusion process, i.e., that an excess of indifferent electrolyte is present in the solution. For simplicity we shall take the activity coefficients of both forms of the substance participating in the electrode reaction to be approximately equal to one another. By numerical solution of the complex integrals entering into expression (1), Meiman calculated the values of \(m\) for different \(\psi_m^M\) and presented these calculated values in tabular form [3].

The approximate equation of an irreversible anodic-cathodic wave [2]

\[ \frac{i}{i_0} = \frac{I_k-i}{I_k}\, e^{\frac{\alpha}{b'}\eta} - \frac{I_a+i}{I_a}\, e^{-\frac{\beta}{b'}\eta} \tag{5} \]

in the case of a completely irreversible wave is easily reduced to the form

\[ \frac{I_k-i}{i} = \frac{1-m}{m} = \frac{I_k}{i_0}\, e^{-\frac{\alpha}{b'}\eta}, \tag{6} \]

where \(I_a\) is the anodic limiting current, \(\alpha;\ \beta = 1-\alpha\) is the transfer coefficient in the ionization process; \(\eta\) is the overvoltage, volts; \(i_0 = \bar S j_m^c c_0^\beta c_R^\alpha\) is the exchange current, A; \(\bar S\) is the time-averaged surface of the dropping mercury electrode, \(\text{cm}^2\); \(c_0, c_R\) are the concentrations of the oxidized and reduced forms of the substance, mole/l. Taking \(c_0 = c_R = 1\) mole/l, we obtain:

\[ \frac{1-m}{m} = \frac{I_k^*}{\bar S j_m^c}, \tag{7} \]

where \(j_m^c\) is the effective value of the standard exchange-current density, calculated from the approximate formula (7); at different values of the current \(i\) (and of the quantity \(m\)), when calculated by the approximate formula (7), this calculated value comes out somewhat different [1], although in physical meaning the exchange current calculated from experimental data at different potentials (or currents) should, of course, be the same (if the mechanism of the electrode process does not change); \(I_k^*\) is the standard limiting cathodic current at \(c_0 = 1\) mole/l \(= 10^{-3}\) mole/l, A·mole\(^{-1}\)·l.

In deriving equations (5)—(7) the same assumptions were made as in deriving equation (1). In addition, an additional simplifying assumption was made that the current at any potential is related to the concentration by the relation

\[ i = K_0(c_0 - c_0^s), \tag{a} \]

where \(K_0\) is a constant that enters the Ilkovič equation for the average limiting current. Relation (a) is sufficiently rigorous for reversible processes [5, Chapter 3]. However, for irreversible processes it becomes approximate, since, when averaging over the lifetime of the drop, the current \(i\) in expression (a) is assumed—that the current in the process of drop growth increases with time

(proportional to \(t^{1/6}\); \(t\) is time), while the concentration of the substance \(c_0^s\) at the electrode surface remains constant. But this assumption is in contradiction with the basic equation of chemical kinetics, according to which the rate of the process (the current) must be proportional to the concentration of the substance at the electrode surface (to the first power for a first-order reaction): \(i = K_1 c_0^s\).

Thus, when the current changes with time in the process of drop growth (\(i \sim t^{1/6}\)), the concentration of the substance near the surface in the case of an irreversible polarographic wave cannot remain constant. The assumption that the concentration \(c_0^s\) is constant during the growth of the mercury drop makes equation (5) approximate and not very accurate.

It is easy to see\(^{1)}\) that the ratio

\[ \frac{I_k^*}{s} = HB, \tag{8} \]

where

\[ H = \frac{6}{7}\cdot \frac{5}{3}\pi^{-1/2} = 0.806. \tag{9} \]

The expression for the coefficient \(B\) is given by formula (3). Hence we obtain an approximate equation of an irreversible polarographic wave in dimensionless form [1, 6]:

\[ \frac{1-m}{m}=\frac{H}{\mu_m^c}, \tag{10} \]

\[ \mu_m^c=\frac{j_m^c}{B}e^{U_m}. \tag{11} \]

Or, in the usual form [taking into account formulas (3) and (4)]:

\[ \varphi=\varphi_{1/2k}-\frac{b}{a}\lg\frac{m}{1-m}, \tag{12} \]

\[ \varphi_{1/2k}=\varphi^\circ-2.076\,\frac{b}{a}-\frac{b}{a}\lg j_m^c, \tag{13} \]

where \(2.076=\lg(0.806\cdot 148)\); \(b=2.3\,b'=2.3\,RT/zF\); \(\varphi_{1/2k}\) is the half-wave potential, volts.

From comparison of (2) and (11) it is seen that the dimensionless parameter \(\mu_m^c\) in formula (10) has the same expression as in formula (1), but their numerical values will differ because of the approximate nature of formula (10) and the related inaccuracy in calculating \(j_m^c\).

\(^{1)}\) The expressions for the mean limiting current and the mean surface of the dropping electrode can readily be brought to the form:

\[ I_k^*=\frac{6}{7}\pi^{-1/2}\sqrt{\frac{7}{3}}\,zF\cdot 10^{-3}\left(\frac{D}{\tau}\right)^{1/2}S^* \]

and

\[ \bar S=\frac{3}{5}S^*, \]

where \(S^*\) is the maximum surface area of the mercury drop at the moment of detachment, \(\text{cm}^2\).

From formulas (2) and (11) we obtain:

\[ \delta_m=\frac{\mu_m^c}{\mu_m^M}=\frac{j_m^c}{j_0^M} \tag{14} \]

and

\[ j_m^c=\delta_m j_0^M;\qquad \mu_m^c=\delta_m \mu_m^M. \tag{15} \]

Substituting (15) into (10) and taking logarithms, we obtain:

\[ \lg \mu_m^M=\lg H+\lg \frac{m}{1-m}-\lg \delta_m. \tag{16} \]

At the half-wave potential, instead of (16) we have:

\[ \lg \mu_{1/2}^M=\lg H-\lg \delta_{1/2}. \tag{17} \]

From (16) and (17) we obtain:

\[ \lg \frac{\mu_m^M}{\mu_{1/2}^M} = \lg \frac{m}{1-m} - \lg \frac{\delta_m}{\delta_{1/2}} \tag{18} \]

or

\[ \varphi = \varphi_{1/2} - \frac{b}{a} \left[ \lg \frac{m}{1-m} - \lg \frac{\delta_m}{\delta_{1/2}} \right] \tag{19} \]

and

\[ \varphi_{1/2\mathrm{k}} = \varphi^\circ - \frac{b}{a}\lg \frac{HB}{\delta_{1/2}} + \frac{b}{a}\lg j_0^M. \tag{20} \]

Equation (19) is the refined equation of a completely irreversible wave. This refinement is achieved by introducing into the approximate wave equation (12) two corrections: the term \(\lg \delta_m/\delta_{1/2}\) in the slope coefficient of the wave in semilogarithmic coordinates \(\left(\lg \frac{m}{m-1}\ \text{and}\ 1-\varphi\right)\), and the term \(\frac{b}{a}\lg \delta_{1/2}\) in the half-wave potential. These corrections can be evaluated numerically from formula (14), if the dimensionless parameter \(\mu_m^M\) is taken from the table compiled by Meiman [3], and the parameter \(\mu_m^c\) is calculated by formula (10). It is, however, more convenient to express the coefficient \(\delta_m\) as a function of the argument \(m/1-m\). Figure 1 presents a plot in the coordinates

\[ \lg \delta_m \ \text{and}\ \lg \frac{m}{1-m}. \]

In this case the values of \(\lg \delta_m\) were found from the expression [formula (14)]:

\[ \lg \delta_m=\lg \mu_m^c-\lg \mu_m^M. \tag{21} \]

The values of \(\lg \mu_m^M\) at different \(m\) were taken from Meiman’s table [3], and the values of \(\lg \mu_m^c\) were calculated by formula [see formula (10)]:

\[ \lg \mu_m^c=\lg \frac{m}{1-m}-0.0937, \tag{22} \]

\[ (0.0937=-\lg 0.806). \]

From the plot in Fig. 1 it is seen that the dependence of \(\lg \delta_m\) on \(\lg \frac{m}{1-m}\) in the interval of variation of \(\lg \frac{m}{1-m}\) from 1.0 to 2.0 can be satisfactorily

but can be represented as a straight line, whose equation is given by the following empirical expression:

\[ \lg \delta_m = 0.038 + 0.035 \lg \frac{m}{m-1}, \tag{23} \]

whence

\[ \lg \delta_{1/2}=0.038; \qquad \delta_{1/2}=1.093 \tag{24} \]

and

\[ \lg \frac{\delta_m}{\delta_{1/2}} = 0.035 \lg \frac{m}{m-1}. \tag{25} \]

A certain scatter of the points in the graph in Fig. 1 is apparently due to the inaccuracies that occurred in the numerical evaluation of the complicated integrals in formula (1) in [3]. The known arbitrariness of the assumption that there is a linear dependence between

Fig. 1. Dependence of the ratio of dimensionless parameters in the formulas (10) of A. G. Stromberg and (1) of Meiman on the current flowing through the dropping mercury electrode at any potential, in logarithmic coordinates.

\(\lg \delta_m\) and \(\lg \dfrac{m}{m-1}\) cannot substantially affect the accuracy of calculations by formula (17), since the correction \(\lg \delta_m\) amounts to only 3.5% of the value \(\lg \dfrac{m}{1-m}\). Thus, even an error of 10% in the value of \(\lg \delta_m\) from the graph in Fig. 1 will lead to an error of only 0.35% in the estimate of the quantity \((\varphi-\varphi_{1/2})\) in formula (17). Let us also note that, for values of \(\lg \dfrac{m}{1-m}\) less than 1.0, the dependence of \(\lg \delta_m\) on \(\lg \dfrac{m}{1-m}\) should apparently deviate from a straight line and asymptotically approach the zero value \(\lg \delta_m\) (and \(\delta_m=1\)) (this is shown by the dashed line in Fig. 1). But for practical use of equations (19) or (26) this is of no significance, since, for a numerical estimate of the exchange current and the transfer coefficient, one usually uses experimental points located on the wave in the interval of values

\[ \lg \frac{m}{1-m} \]

from \(-1.0\) to \(+1.0\).

Substitution of relations (24) and (25) into formula (19) makes it possible to give the refined formula of the irreversible wave a very simple form, convenient for practical use:

\[ \varphi=\varphi_{1/2\mathrm{k}}-0.965\,\frac{b}{\alpha}\lg\frac{m}{1-m}, \tag{26} \]

\[ \varphi_{1/2\mathrm{k}}=\varphi^\circ-2.039\,\frac{b}{\alpha} -\frac{b}{2\alpha}\lg\left(z^2\frac{D_0}{\tau}\right) +\frac{b}{\alpha}\lg j_0^M \tag{27} \]

\[ (2.039=\lg(0.806\cdot148)\,1.094). \]

In dimensionless coordinates the approximate and refined formulas of the completely irreversible wave can be represented in the form [see formulas (11) and (15)]:

\[ \lg \mu_m^c=\lg \mu_{1/2}^c+\lg\frac{m}{1-m}, \tag{28} \]

\[ \lg \mu_m^M=\lg \mu_{1/2}^M+0.965\lg\frac{m}{1-m}. \tag{29} \]

Figure 2 presents the graph of a completely irreversible polarographic wave in the coordinates \(\lg \frac{m}{1-m}\) and \(\lg \mu_m\), constructed on the basis of theoretical calculations according to formulas (28) and (29). From the figure it is evident that the refined formula (29) is represented on the graph by a straight line (curve 2), shifted toward positive potentials by a certain amount \(\Delta\varphi_m\) and having a gentler slope \((\operatorname{ctg}\vartheta_2)\) in comparison with curve 1 \((\operatorname{ctg}\vartheta_1)\), given by formula (28). These quantities are determined from the relations:

Fig. 2. Semilogarithmic graph of a completely irreversible polarographic wave in dimensionless coordinates. Curve 1—according to the approximate equation (28); curve 2—according to the refined equation (29). The expression for the dimensionless parameter \(\mu_m\) is given by formula (2).

\[ \Delta\varphi_m=\frac{59}{\alpha z}\left(\lg \psi_m^c-\lg \psi_m^M\right) =\frac{59}{\alpha z}\lg \delta_m =\frac{1}{\alpha z}\left(2.25+2.07\lg \frac{m}{1-m}\right), \tag{30} \]

\[ \operatorname{ctg}\vartheta_2-\operatorname{ctg}\vartheta_1=1-0.965=0.035. \tag{31} \]

Thus, calculation by the simplified formula, as is seen from formula (30), gives an error in determining the half-wave potential of \(2.25/\alpha z\) mV. At \(\lg \dfrac{m}{1-m}=1.0\) (at \(i=0.9 I_k\)) this error, in accordance with formula (30), increases to \(4.32/\alpha z\) mV. A numerical estimate of the transfer coefficient \(\alpha\) from the graph in Fig. 2 when using the approximate formula (12) gives an error of 3.5%, as follows from expression (31). For example, if the true value of \(\alpha\) is equal to 0.50, then calculation by the approximate formula (12) gives a value of about 0.52.

From formula (27) we obtain an expression for calculating the standard exchange-current density:

\[ \lg j_0^m=2.039+\frac{1}{2}\lg z^2\frac{D_0}{\tau} -\frac{\alpha}{b}\left(\varphi^\circ-\varphi_{1/2k}\right). \tag{32} \]

A drawback of calculations by formula (32) is the need to know the numerical value of the standard oxidation–reduction potential \(\varphi^\circ\). In strongly irreversible systems (with a small exchange current), exact determination of the quantity \(\varphi^\circ\) is difficult.

If, for the given electrode process

\[ O+ze \rightleftarrows R \]

it is possible to obtain a completely irreversible anodic–cathodic wave, then for the anodic branch, by analogy with (32), one may write [2]:

\[ \lg j_0^m=2.039+\frac{1}{2}\lg z^2\frac{D_R}{\tau} +\frac{\beta}{b}\left(\varphi^\circ-\varphi_{1/2a}\right). \tag{33} \]

Multiplying equation (32) by \(\beta\) and equation (33) by \(\alpha\) and adding them, we obtain (assuming \(\alpha+\beta=1\)):

\[ \lg j_0=2.039+\frac{1}{2}\lg \frac{z^2}{\tau}D_0^\beta D_R^\alpha -\frac{\alpha\beta}{b}\Delta\varphi_{1/2ak}, \tag{34} \]

where \(\Delta\varphi_{1/2ak}=\varphi_{1/2a}-\varphi_{1/2k}\) is the difference between the anodic and cathodic half-wave potentials.

From experimental data for the difference between the anodic and cathodic half-wave potentials, equation (34) can be used to calculate the standard exchange-current density with approximately the same accuracy as by the rigorous Meiman equation [3].

Conclusions

1. A simple universal semiempirical formula has been proposed for expressing the dependence of the ratio of dimensionless parameters (equal to the ratio of the standard exchange-current densities), calculated by the Meiman and Stromberg formulas, on the relative current of a completely irreversible polarographic wave at different potentials.

2. Introduction of this ratio into the approximate equation of an irreversible polarographic wave makes it possible to increase the accuracy of calculations of the standard exchange-current density and the transfer coefficient. The refined equation of an irreversible polarographic wave in accuracy deter-

the exchange current and the transfer coefficient should not be inferior to the Meiman equation, but is considerably simpler and more convenient for practical calculations.

References

1. A. G. Stromberg, V. A. Igolinskii. Zhurn. fiz. khimii, 36, 2714 (1962).
2. A. G. Stromberg. Zhurn. fiz. khimii, 29, 409 (1955).
3. N. Neiman. Zhurn. fiz. khimii, 22, 1454 (1948).
4. I. Koutecky. Coll. Czech. Chem. Comm., 18, 597 (1953).
5. P. Delahay. New Instruments and Methods in Electrochemistry, Foreign Literature Publishing House, (1957).
6. V. S. Bagotskii. Zhurn. fiz. khimii, 22, 1466 (1948).