STUDY OF THE INFLUENCE OF HYDRODYNAMIC FACTORS ON THE PARAMETERS OF THE ANODIC PEAK OF METAL ELECTRODISSOLUTION FROM AN AMALGAM UNDER A LINEARLY VARYING POTENTIAL

B. F. Nazarov, L. D. Vishtakalyuk

(Presented by the scientific-methodological seminar of the Department of Physical Chemistry)

One way to increase the sensitivity and resolving power of the APA method may be to intensify the removal of electrode-reaction products from the electrode–solution interface. In an unstirred solution, under certain conditions it may be assumed that the concentration of the oxidized form remains constant at any distances from the electrode surface. Consequently, in order to find the equation of the voltammetric curves of electrodissolution of a metal from an amalgam under a linearly varying potential, it is necessary to solve the equation

\[ \frac{\partial C_R(x,t)}{\partial t} = D_R \frac{\partial^2 C_R(x,t)}{\partial x^2} \]

under the following initial and boundary conditions:

\[ t = 0;\quad C_R(x,0)=C_R^0;\quad 0 \leq x \leq \varepsilon; \]

\[ t>0;\quad \left.\frac{\partial C_R}{\partial x}\right|_{x=0}=0;\quad \varphi=\varphi_0+\frac{RT}{zF}\ln\frac{C_0^S}{C_R^S}, \quad \text{where } C_0^S=C_0^0=\text{const}. \]

The posed problem is solved by the operational method, using Duhamel’s theorem.

Let us consider the general expression for the current under a linearly varying potential:

\[ i=\int_0^t i_{\mathrm{pr}}(t-\tau)\frac{dC(\tau,t)}{dt}\,d\tau . \]

The limiting current at a film electrode is

\[ i_{\mathrm{pr}}=zFSD\,\frac{2}{l}\sum_{n=1}^{\infty} e^{-\frac{\mu^2Dt}{l^2}}, \quad \text{where } \mu_n=(2n-1)\frac{\pi}{2}, \]

\(D\) is the diffusion coefficient,

\(C(t,o)\) is the concentration at the electrode surface, unknown. We find it from the Nernst equation; therefore,

\[ \frac{dC_0^S}{dt} = C_0^S\frac{d}{dt} e^{-(\varphi_t-\varphi_0)\frac{zF}{RT}}, \]

Taking into account that

\[ -\varphi_t-\varphi_0=Wt, \]

\[ \frac{dC_R^S}{dt} = -\,C_0^S\frac{zFW}{RT} e^{(\varphi_t-\varphi_0)\frac{zF}{RT}} . \]

The final expression for the current has the form

\[ i= \frac{z^2F^2WSD_R\,2C_R^0}{RTl} \int_0^t \frac{\displaystyle \sum_{n=1}^{\infty} e^{-A(t-\tau)}} {e^{(\varphi_t-\varphi_0)}}\,d\tau, \qquad \text{where } A=\frac{\pi^2D}{4l^2}. \]

Solving this equation, we obtain

\[ i= \frac{z^2F^2}{RT}\,WQ\,2 \sum_{n=1}^{\infty} \frac{\gamma}{\mu_n^2\gamma-1} \left[1-e^{-(\mu_n^2\gamma-1)\Phi t}\right]e^{-\Phi t}, \]

where

\[ \gamma=\frac{RT}{F}\cdot\frac{1}{C^2}\cdot\frac{D}{ZW}, \qquad \Phi t=\frac{ZF}{RT}\,Wt. \]

The equation was calculated on a “Promin” computer for various values of \(\gamma\) (Fig. 1). From the data obtained, we calculated that, using the conditions indicated above, it is possible to obtain peaks for which

Fig. 1. Dependence of the peak current on the potential in dimensionless coordinates: curve 1—\(\gamma=100\), curve 2—\(\gamma=1\)

Fig. 2. Dependence of \(\dfrac{\Delta\varphi^b_{n/2}}{\Delta\varphi_{n/2}}\) on \(\varphi_e\) for a film electrode

\[ \Delta\varphi_{n/2}=\frac{20}{Z}\ \text{mV}. \]

Our results were compared with the data obtained by Igolinsky and de Vries [4, 5]. The comparison showed that the values of \(\Delta\varphi_{n/2}\) obtained by us and by Igolinsky for large \(\gamma\) coincide,

\[ \frac{\Delta\varphi'_{n/2}}{\Delta\varphi''_{n/2}}=3, \]

where \(\Delta\varphi''_{n/2}\) is the half-peak width calculated from our data, and \(\Delta\varphi'_{n/2}\) is the half-peak width calculated from the data of de Vries.

To solve our equation, as one of the boundary conditions we took the Nernst equation, in which the unknown quantity is \(C_R^S\). Therefore, in setting up the experiment, we were faced with the task of making the quantity \(C^S\) constant and equal to \(C_0^0\). This can be achieved by removing the products of the electrode reaction from the electrode surface by stirring, or under the condition that \(C_0^0 \gg \Delta C_0^S\), where \(\Delta C_0^S\) is the increase of the oxidized form of the substance during electro-dissolution.

The use of an electromagnetic stirrer did not yield the desired results. The reason is that, in order to eliminate the diffusion layer \((\delta_d)\), its thickness must be approximately 10 times greater than the thickness of the boundary layer \((\delta_{\mathrm{bnd}})\). And, as is known from the laws of hydrodynamics [1–3], the thickness of the boundary layer depends on the flow velocity. It follows from this that the mixing process must be intensified. For this purpose a vibrating electrode was used. Figure 2 presents the dependence

\[ \left( \frac{\Delta \varphi^{b}_{n/2}}{\Delta \varphi_{n/2}} -\varphi_{e} \right), \]

where \(\Delta \varphi^{b}_{n/2}\) is the half-peak width obtained with a vibrating electrode.

It is seen from this that vibration affects the half-peak width. This is especially noticeable at electrolysis potentials close to the equilibrium potential. The maximum flow velocity that we were able to obtain was \(4\ \mathrm{m/sec}\), whereas in order for \(\delta_d \gg \delta_{\mathrm{bnd}}\), the flow velocity must be increased to \(10\ \mathrm{m/sec}\).

Conclusions

1. An equation for the current of electro-dissolution of a metal from the amalgam of a film electrode under a linearly varying potential has been derived and calculated theoretically.

2. The influence of vibration on the half-peak width has been investigated; it is shown that, with increasing mixing intensity, the half-peak width decreases.

References

1. J. Daily, D. Harleman. Fluid Mechanics. Moscow, “Energiya,” 1971.

2. V. G. Levich. Physicochemical Hydrodynamics. Moscow, Fizmatgiz, 1959.

3. J. C. Geyrovsky, J. Kuta. Principles of Polarography. Moscow, Mir Publishing House, 1965.

4. V. A. Itolinsky. Cand. diss., Tomsk, 1963.

5. W. T. de Wries, J. Electroanal. Chem., 9, 5–6, 448, 1965.