Experimental Verification of Calculated Data for Quasi-Reversible Electrode Processes
Abstract Generated abstract
Equations are developed for current, voltage curves of quasi-reversible electrode processes under linearly varying potential at mercury film, planar, and spherical electrodes. Using expressions for surface concentrations, the authors derive integral forms for the current and solve them numerically after reduction to algebraic equations, calculating peak currents, peak potentials, and half-peak widths as functions of potential sweep rate and electrode dimensions. The theoretical predictions are compared with experiments on electrooxidation of copper amalgam in 1 M potassium nitrate, with precautions taken to reduce surfactant effects. The calculated and experimental dependences show good qualitative agreement, including linear relations of half-peak width and peak potential shift with sweep rate and film thickness, and proportionality of peak current to sweep rate for selected mercury film electrodes.
Full Text
Experimental Verification of Calculated Data for Quasi-Reversible Electrode Processes
M. S. Zakharov, L. F. Zaichko
(Presented by the scientific seminar of the MP laboratory)
In the present work, equations are proposed for current–voltage curves in AVN under LMP at mercury, film, and spherical electrodes, and an experimental verification of the theoretical conclusions is given.
The equation for the current has the form
\[ i(t)=zFSK_s\left\{ \exp\left[-\frac{\alpha zF}{RT}(\varphi-\varphi_0)\right]C_0(0,t) -\exp\left[\frac{\beta zF}{RT}(\varphi-\varphi_0)\right]C_R(0,t) \right\}, \]
where \(z\) is the number of electrons participating in the electrode process per atom; \(F\) is Faraday’s number; \(S\) is the electrode surface; \(K_s\) is the rate constant of the electrode process at the standard electrode potential; \(\alpha\) and \(\beta\) are transfer coefficients; \(R\) is the gas constant; \(T\) is temperature, K°; \(C_0(0,t)\) and \(C_R(0,t)\) are the concentrations of the oxidized and reduced forms of the element at the electrode surface.
Substituting the expressions for \(C_0(0,t)\) and \(C_R(0,t)\) for any form of potential variation of planar and spherical electrodes, obtained respectively in works [1, 2], we obtain equations for the current–voltage curves for quasi-reversible electrode processes: for a film (planar) electrode
\[ q(t)=\frac{i(t)}{zFS} =ae^{y_1}\left(C_0^0+\frac{I_3}{\sqrt{\pi D_0}}\right) -be^{y_2}\left(C_R^0-\frac{I_1}{\ell}\right), \]
for a spherical electrode
\[ q(t)=\frac{i(t)}{zFS} =ae^{t_1}\left(C_0^0+\frac{I_3}{\sqrt{\pi D_0}}-\frac{I_4}{r_0}\right) -be^{t_2}\left(C_R^0-\frac{I_2}{r_0}\right), \tag{3} \]
\[ b=K_s\exp\left[\frac{\beta zF}{RT}(\varphi_i-\varphi_0)\right], \qquad a=K_s\exp\left[-\frac{\alpha Fz}{RT}(\varphi_i-\varphi_0^0)\right], \]
\[ I_1=\int_0^t q(\xi)\left\{1+2\sum_{n=1}^{\infty} \exp\left[-\mu_n^2-\frac{D_R}{\ell}(t-\xi)\right]\right\}\,d\xi, \]
\[ I_2=\int_0^t q(\xi)\left\{3+2\sum_{n=1}^{\infty} \exp\left[-\mu_n^2\frac{D_R}{r_0}(t-\xi)\right]\right\}\,d\xi, \]
\[ I_3=\int_0^t \frac{q(\xi)}{\sqrt{t-\xi}}\,d\xi;\qquad I_4=\int_0^t q(\xi)\exp\frac{D}{r_0^2}(t-\xi)\sqrt{\frac{D}{r_0^2}(t-\xi)}\,d\xi . \]
Here \(\xi\) is an auxiliary variable; \(D_R, D_0\) are the diffusion coefficients of the reduced and oxidized forms of the element; \(l, r_0\) are the film thickness and the radius of the sphere; \(t\) is time; \(\mu_n\) are the roots of the equations: for a film electrode \(\mu_n=n\pi\), for a spherical electrode \(\tan\mu=\mu\); \(y_1=\dfrac{\alpha zF}{RT}\omega t\); \(y_2=\dfrac{\beta zF}{RT}\omega t\); \(\omega\) is the rate of change of the electrode potential.
Equations (2) and (3), by numerical integration, were reduced to algebraic equations and calculated on an M-20 electronic computer. The calculations were carried out under the following conditions: \(t=25^\circ\), \(D_0=D_R=0.74\cdot10^{-5}\ \mathrm{cm^2/sec}\), \(\alpha=\beta=0.5\), \(z=2\).
The calculated current–voltage curves have a characteristic peak. From the peaks obtained, the following graphs were constructed: \(q_{\max}, \omega\); \(\Delta\varphi, \omega\); \(\Delta\varphi(e,r_0)\); \((\varphi_p-\varphi^\circ)\omega\); \((\varphi_p-\varphi^\circ), e(r_0)\);
\(\varphi_p\) is the potential of the peak maximum; \(\Delta\varphi\) is the half-peak width. The calculated data were checked experimentally using as an example the electrooxidation process of copper amalgam against a background of 1 M \(KNO_3\). Under the conditions studied, for this process the value \(K_s=2\cdot10^{-3}\ \mathrm{cm/sec}.\)* To eliminate the influence of surfactants on the electrode process, all glassware was washed with concentrated sulfuric acid; the water used in the studies was distilled with \(KMnO_4\); the \(KNO_3\) salt was recrystallized.
The results of some studies are presented in Figs. 1, б—4, б. It is evident from the figures that the theoretical (a) and experimental curves agree well with one another in character.
Let us note some characteristic quantities and relationships: as \(\omega\to0\), on the film electrode for the process under study, in accordance with theory, \(\Delta\varphi\approx40\ \mathrm{mV}\), while \(\Delta\varphi\) and \((\varphi_p-\varphi^\circ)\) vary linearly with changes in \(\omega\) and \(e\). On electrodes with mercury films of thickness 4 and 8 \(\mu\), in accordance with theory, \(q_{\max}=K\omega\) (\(K\) is the proportionality coefficient). The remaining characteristics and relationships can be followed in the figures.
Fig. 1. a—theoretical and б—experimental dependences of peak height on \(\omega\) for a film electrode \((e=4;\ 8\ \mu)\)
* Determined by the constant-current ACV method [3].
Figure 2. a—theoretical and b—experimental dependences of \(\Delta\varphi\) on \(\omega\) for a film electrode \((l=4;\ 8;\ 12;\ 25;\ 50\ \mu)\).
Figure 3. a—theoretical and b—experimental dependences of \(\varphi_n\) on \(\omega\) for a film electrode \((l=4;\ 16;\ 25\ \mu)\).
Figure 4. a—theoretical and b—experimental dependences of \(\Delta\varphi\) on \(\omega\) \((r_k=2\cdot10^{-2}\ \mathrm{cm};\ 4\cdot10^{-2}\ \mathrm{cm};\ 7\cdot10^{-2}\ \mathrm{cm})\).
LITERATURE
- De Vries, J. Electroanal. Chem. 9, 448, 1965.
- M. S. Zakharov, V. V. Pnev. Izv. TPI, 124, 1967.
- M. S. Zakharov, V. I. Bakanov. Izv. TPI (in press).