V. E. GORODOVYKH

EQUATION OF THE ANODIC PEAK FOR A REVERSIBLE ELECTRODE PROCESS AT AN AMALGAM SPHERICAL ELECTRODE

(Presented by the seminar of the Department of Physical Chemistry, TPI)

Amalgam polarography with accumulation at a stationary mercury drop electrode (stat. m. d. e.) has recently been applied as a highly sensitive method for the quantitative determination of trace amounts of certain elements in various objects [1], and also for solving a number of physicochemical problems. In the literature, however, there are only isolated works devoted to the theoretical development of this new and very promising method [2—4].

The aim of the present work is to derive the equation of the anodic peak, taking into account the curvature and the finite volume of the stationary mercury drop electrode. A similar problem was solved by Reinmuth [2], but his work contains only the solution of the transformed equation and gives no final expression for the anodic current.

Let us write Fick’s second law for spherical regions

\[ \frac{\partial C_R(r,t)}{\partial t} = D_R\left[ \frac{\partial^2 C_R(r,t)}{\partial r^2} + \frac{2}{r}\frac{\partial C_R(r,t)}{\partial r} \right], \tag{1} \]

where \(C_R(r,t)\) is the concentration of metal atoms inside the stationary mercury drop electrode at any time \(t\) and at a distance \(r\) from the center of the drop.

In the general case, the initial and boundary conditions are written as follows:

\[ C_R(r,0)=\xi(r) \qquad \text{for } t=0,\qquad 0\le r\le r_0, \tag{2} \]

\[ C_R(r_0,t)=f(t) \qquad \text{for } t>0,\qquad r=r_0, \tag{3} \]

where \(r_0\) is the radius of the drop.

In addition, owing to the symmetry of the diffusion field inside an ideal spherical electrode (shielding of the electrode is not taken into account in the present solution), we have

\[ \lim_{r\to 0}\frac{\partial C_R}{\partial r}=0, \qquad \text{for } t>0. \tag{4} \]

To solve this problem we shall use the operational method. The transformed equation for (1) has the form

\[ \frac{d^2\overline{C}_R}{dr^2} + \frac{2d\overline{C}_R}{rdr} - \frac{p}{D_R}\,\overline{C}_R = -\frac{\xi(r)}{D_R}, \tag{5} \]

where \(\overline{C}_R=\overline{C}_R(r,p)\) is a function of the parameter \(p\).
Let us find the general solution of (5)

\[ \overline{C}_R=\frac{A}{r}e^{\sqrt{p/D_R}\,r}+\frac{B}{r}e^{-\sqrt{p/D_R}\,r}+\widetilde{C}_R, \tag{6} \]

where

\[ \widetilde{C}_R= \frac{e^{\sqrt{p/D_R}\,r}}{2r\sqrt{pD_R}} \int_0^r \rho\,\xi(\rho)e^{-\sqrt{p/D_R}\,\rho}\,d\rho + \frac{e^{-\sqrt{p/D_R}\,r}}{2r\sqrt{pD_R}} \int_0^r \rho\,\xi(\rho)e^{\sqrt{p/D_R}\,\rho}\,d\rho, \tag{6*} \]

is a particular solution of the nonhomogeneous equation (5).
Using the boundary conditions (3) and (4), we determine the constants \(A\) and \(B\) and find the general solution of the mapping equation (5)

\[ \overline{C}_R(r,p)= -\frac{ r_0\left[\overline{f}(p)-\widetilde{C}_R(r_0)\right] \operatorname{sh}\!\left(\sqrt{\frac{p}{D_R}}\,r\right) }{ r\,\operatorname{sh}\!\left(\sqrt{p/D_R}\,r_0\right) } +\widetilde{C}_R(r,p), \tag{7} \]

where \(\overline{f}(p)\) is the image of the function \(f(t)\), and \(\widetilde{C}_R\) is determined by formula (6*).

The expression obtained is valid, generally speaking, for any processes (reversible and irreversible) and for any initial distribution of the metal inside a stationary mercury dropping electrode. The type of process determines the form of the function \(\overline{f}(p)\), while the initial distribution of concentration is represented by the function \(\widetilde{C}_R(r,p)\).

Let us find the expression for \(\widetilde{C}_R\) in the case of a uniform initial distribution of the metal in the amalgam (3--4). In this case \(C_R(r,0)=C_R^*=\mathrm{const}\) at \(t=0\), or, passing to images,

\[ C_R^* \doteq \frac{1}{p}C_R^* . \]

Since \(\dfrac{C_R^*}{p}\) is a particular solution of (5), we may substitute it for \(\widetilde{C}_R\) in equation (7)

\[ \overline{C}_R= \frac{ r_0\left[\overline{f}(p)-\dfrac{C_R^*}{p}\right] \operatorname{sh}\!\left(\sqrt{p/D_R}\,r\right) }{ r\,\operatorname{sh}\!\left(\sqrt{p/D_R}\,r_0\right) } +\frac{C_R^*}{p}. \]

Reinmuth [2] also arrived at the same conclusion. Thus, his solution is a particular case of the more general expression (7), which is valid for any initial distribution of the metal in the stationary mercury dropping electrode before the start of the anodic process.

Using the expansion theorem and the convolution operation, we find the original for \(C_R\):

\[ C_R(r,t)= \int_0^t \left\{ f'(\tau) \left[ 1+2\sum_{n=1}^{\infty}(-1)^n \frac{r_0}{r\varphi_n} \sin\frac{r}{r_0}\varphi_n \exp\!\left( -\varphi_n^2\frac{D_R(t-\tau)}{r_0^2} \right) \right] \right\}\,d\tau +C_R^*, \tag{9} \]

where \(\varphi_n=n\pi\).

This expression makes it possible to calculate the concentration at any point of a stationary mercury drop electrode at an arbitrary instant of time (for a known \(\tilde f(t)\)).

Substituting (9) into Fick’s first law, we find the flux at the electrode surface

\[ D_R S\left(\frac{\partial C_R(r,t)}{\partial r}\right)_{r\cdot r_0} = \frac{2D_R S}{r_0} \int_0^t f'(\tau) \sum_{n=1}^{\infty} e^{-\mu_n^2 \frac{D_R(t-\tau)}{r_0^2}} \,d\tau . \tag{10} \]

If the rate of change of the potential is equal to \(a\), the process is reversible, and \(D_R=D_0\), then (5)

\[ f(t)=\frac{(C_0^*+C_R^*)}{2} \left[ 1-\operatorname{th}\frac{nFa}{2RT}(t-t_0) \right], \tag{11} \]

where \(C_0^*\) is the concentration of metal ions in the solution,
\(t_0\) is the time at which the half-wave potential is reached.

Substituting (11) into (10), we finally obtain the expression for the anodic current,

\[ i_a=nFSD\left(\frac{\partial C_R}{\partial r}\right)_{r\cdot r_0} = - \tag{12} \]

\[ - \frac{2\pi n^2F^2Da r_0(C_0^*+C_R^*)}{RT} \int_0^t \frac{ \sum_{n=0}^{\infty} e^{-\mu_n^2\frac{D(t-\tau)}{r_0^2}} }{ \operatorname{ch}^2\frac{nFa}{2RT}(\tau-t_0) } \,d\tau . \]

(In accordance with polarographic practice, the anodic current has a negative value.)

For large \(t\), in (12) one may restrict oneself to a single term of the series

\[ i_a \simeq - \frac{2\pi n^2F^2Da r_0(C_0^*+C_R^*)}{RT} \int_0^t \frac{ e^{-\pi^2\frac{D}{r_0^2}(t-\tau)} }{ \operatorname{ch}^2\frac{nFa}{2RT}(\tau-t_0) } \,d\tau . \tag{13} \]

The graph of the function

\[ \Phi(t)= \int_0^t \frac{ e^{-\pi^2\frac{D}{r_0^2}(t-\tau)} }{ \operatorname{ch}^2\frac{nFa}{2RT}(\tau-t_0) } \,d\tau \]

is given in Fig. 1. The function \(\Phi(t)\) reaches its maximum value at \(t=t_{\max}\), which can be found by graphically solving the equation (Fig. 2).

\[ \frac{\partial}{\partial t}[\Phi(t)] = -\pi^2\frac{D}{r_0^2}\Phi(t) + \frac{1}{ \operatorname{ch}^2\frac{nFa}{2RT}(t-t_0) } =0. \]

It follows from Fig. 2 that the potential of the anodic peak is approximately \(30\ \mathrm{mV}\) more positive than the half-wave potential.

Since under the usual conditions of amalgam polarography \(C_R^*\gg C_0^*\), and \(C_R^*=kC_0^*t_{\mathrm{e}}\) (\(K=\mathrm{const}\), \(t_{\mathrm{e}}\) is the accumulation time), then from

(12) directly imply two fundamental facts established by a number of researchers—the directly proportional dependence of the height of the anodic peak on the concentration of metal ions in the solution and on the accumulation time (in the absence of depletion of the solution during electrolysis and of a number of side phenomena associated with the mutual influence of metals in complex amalgams, the formation of supersaturated amalgams, etc.).

Fig. 1. Graph of the function \(\Phi(t)\).

Fig. 2. Graphical solution of the equation
\[ \frac{\partial}{\partial t}\,[\Phi(t)]=0. \]

It is also seen from (12) that, for reversible systems, the potential of the anodic peak does not depend on the metal concentration (if the activity coefficients remain constant).

REFERENCES

1. A. G. Stromberg and E. A. Stromberg. Determination of ultramicroconcentrations of a substance by the method of amalgam polarography at a stationary dropping electrode with continuously varying potential. Zavod. labor., 27, No. 1, 3 (1961).

2. W. H. Reinmuth. Theory of polarography with anodic dissolution at spherical electrodes. Anal. Chem., 33, No. 2, 185 (1961).

3. I. Shain, J. Lewinson. Analysis by anodic dissolution using mercury electrodes. Anal. Chem., 33, No. 2, 187 (1961).

4. L. N. Vaskyleva and E. N. Vinogradova. On the distribution of the concentration of a metal inside a mercury drop during electrolytic deposition of it on a stationary mercury electrode. Zavod. labor., 27, No. 9, 1079 (1961).

5. A. Ševčík. Oscillographic polarography with periodic triangular voltage. Coll. czech. chem. comm., 13, 349 (1948).