Theory of Completely Irreversible Anodic Peaks at an Amalgam Film Electrode at Low Potential Scan Rates
Abstract Generated abstract
This paper develops a theoretical expression for fully irreversible anodic peaks at a stationary amalgam film electrode under low potential scan rates. Assuming a thin mercury film with uniform metal concentration throughout dissolution and neglecting the reverse electrode process, the authors derive the current as a function of time, the peak time, peak potential, and peak current for a linearly increasing anodic potential. They then relate the peak potential at the mercury film electrode to the half-wave potential of an irreversible anodic wave at an amalgam dropping electrode. The result provides a basis for applying classical polarographic methods to stationary amalgam film electrodes, which are experimentally simpler to prepare and use.
Full Text
THEORY OF FULLY IRREVERSIBLE ANODIC PEAKS AT AN AMALGAM FILM ELECTRODE AT A LOW RATE OF POTENTIAL CHANGE
V. A. Igolinskii, A. G. Stromberg
(Presented by the scientific-methodological seminar of the Faculty of Chemical Technology)
At the present time, the theory of processes at the amalgam dropping electrode has been developed fairly fully, and equations have been derived for anodic-cathodic waves that make it possible to use the data obtained to solve questions concerning the mechanism of electrode reactions [1—3].
Unfortunately, no analogous theory exists for stationary amalgam electrodes, and therefore their application to the solution of theoretical questions is difficult, although experimentally this method is considerably simpler and more elegant than the method using an amalgam dropping electrode.
In the present work an attempt is made, under certain simplifying assumptions, to derive the equation of the anodic peak for an amalgam film electrode, consisting of a thin (2—6 micron) film of mercury deposited on metallic silver, into which the metal under investigation has been introduced by electrolysis.
We shall assume that throughout the entire process of electro-dissolution of the metal from the amalgam, the distribution of the metal in the mercury remains uniform. This condition is fulfilled, as was shown in (4), at sufficiently low rates of potential change, that is, when
\[ \frac{zFw}{RT} \ll \frac{\pi^2 D_R}{4h^2}, \]
where
\(z\) — number of electrons participating in the reaction;
\(F\) — Faraday number;
\(w\) — rate of change of potential;
\(R\) — gas constant;
\(T\) — absolute temperature;
\(D_R\) — diffusion coefficient of the metal atoms in mercury;
\(h\) — thickness of the mercury film.
Let the electrode potential vary linearly with time in the anodic direction, starting from the equilibrium potential,
\[ \varphi=\varphi_p+wt, \tag{1} \]
where \(\varphi\) is the electrode potential at time \(t\) from the beginning of the process;
\(\varphi_p\) is the equilibrium potential.
We shall regard the process as completely irreversible and shall immediately neglect the rate of the reverse process. The equation of the theory of retarded discharge–ionization for this case may be written in the form [2]
\[ \frac{i}{i_0} = \frac{c}{c_0} e^{-\frac{\beta}{b'}(\varphi_p-\varphi)} = \frac{c}{c_0} e^{\frac{\beta}{b'}wt}, \tag{2} \]
where \(i\) is the anodic current passing through the electrode;
\(i_0\) is the exchange current;
\(\beta\) is the ionization coefficient, \(b' = RT/zF = 0.0257/z\) (at \(25^\circ\)C);
\(c_0\) is the initial concentration of the metal in the amalgam;
\(c\) is the concentration of the metal in the amalgam at time \(t\)*).
By time \(t\), a certain amount of metal \(q\) will have dissolved from the amalgam, which, on the basis of Faraday’s law, may be expressed by the equation
\[ q=\frac{1}{zF}\int_0^t i\,dt. \tag{3} \]
Hence, for the concentration of the metal in the amalgam at time \(t\), we obtain the expression
\[ c=\frac{c_0V-q}{V} = c_0-\frac{1}{zFV}\int_0^t i\,dt, \tag{4} \]
where \(V\) is the volume of the amalgam, equal to the product of the electrode area \(S\) and the thickness of the mercury film \(h\).
Substituting \(c\) from equation (4) into equation (2), we obtain
\[ i=i_0\left(1-\frac{1}{zFc_0V}\int_0^t i\,dt\right) e^{\frac{\beta}{b'}wt}. \tag{5} \]
This equation can be solved for \(i\) by separation of variables. To do this, we differentiate equation (5) with respect to \(t\):
\[ \frac{di}{dt} = -\frac{i_0 i}{zFVc_0}e^{\frac{\beta}{b'}wt} + \left[ i_0\left(1-\frac{1}{zFc_0V}\int_0^t i\,dt\right) e^{\frac{\beta}{b'}wt} \right] \frac{\beta}{b'}w. \tag{6} \]
It is easy to see that the factor in square brackets is equal to \(i\), according to equation (5). We replace this factor by \(i\), separate the variables, and integrate the resulting equation (this integration is carried out elementarily). The constant of integration is found from the condition**):
\[ \text{at } t=0 \qquad i=i_0 \]
*) Instead of \(c\) in equation (2), it would have been necessary to write the concentration of the metal in the amalgam near the electrode surface; however, since we regard the metal as uniformly distributed in the film, the concentration near the surface will be equal to the average concentration of the metal in the amalgam at time \(t\).
**) The equality of the external current at the equilibrium potential to the exchange current, and not to zero, is a consequence of neglecting the rate of the reverse process.
After some algebraic transformations we obtain the equation of a fully irreversible anodic peak at a mercury film electrode in the form*)
\[ i=i_0 e^{\frac{i_0 b'}{zFcV\beta w}\left(1-e^{\frac{\beta}{b'}wt}\right)} e^{\frac{\beta}{b'}wt} \tag{7} \]
This equation is represented by a curve with a maximum. Equating the derivative \(\frac{di}{dt}\) to zero and solving the resulting equation, we find the value of \(t\) at which the current maximum is reached. This value is equal to:
\[ t_{\max}=\frac{b'}{\beta w}\ln\frac{zFC_0\beta w}{i_0b'}, \tag{8} \]
\(t_{\max}\) is the time at which the peak potential \(\varphi_{\mathrm{p}}\) is reached. Substituting the value of \(t_{\max}\) from equation (8) into (1), we obtain the potential at which the peak current is reached (the peak potential)
\[ \varphi_{\mathrm{n}}=\varphi_{\mathrm{p}}+\frac{b'}{\beta}\ln\frac{zFC_0V\beta w}{i_0b'}. \tag{9} \]
The peak current can be obtained by substituting the value of \(t_{\max}\) from equation (8) into the peak equation (7). After substitution we have
\[ i_{\mathrm{n}}=\frac{zFC_0V\beta w}{b'}e^{\frac{zFC_0Vw}{b'i_0}-1}. \tag{10} \]
For the half-wave potential of an irreversible wave at an amalgam dropping electrode we have the expression [2]
\[ \varphi_{1/2}=\varphi_{\mathrm{p}}+\frac{b'}{\beta}\ln\frac{I_a}{i_a}, \tag{11} \]
where \(I_a\) is the limiting anodic current.
Comparing (9) and (11), we find:
\[ \varphi_{1,2}=\varphi_{\mathrm{n}}-\frac{b'}{\beta}\ln\frac{zFC_0V\beta w}{b'I_a} \tag{12} \]
or
\[ \varphi_{1/2\,a}=\varphi_{\mathrm{n}}-\frac{2.3RT}{zF\beta}\lg\frac{z^2F^2V\beta w}{RT\kappa_a}, \tag{13} \]
where \(\kappa_a\) is the Ilkovič constant for metal atoms.
Thus, under the appropriate conditions, the half-wave potential of an irreversible anodic wave at an amalgam dropping electrode can be calculated from the anodic peak potential at a mercury film electrode.
Such a recalculation will make it possible to use the existing mathematical apparatus of classical polarography to determine the composition of complexes directly participating in the electrode process, as well as to solve other physicochemical problems for amalgam electrodes. From the methodological standpoint, preparation of the amalgam and recording
*) An analogous equation was obtained by H. Z. Brainina and co-workers under other simplifying assumptions and for the case of electro-dissolution of solid insoluble films (5).
the anodic polarogram at a stationary amalgam film electrode is much simpler than obtaining the anodic polarogram at a dropping amalgam electrode.
REFERENCES
- A. G. Stromberg. ZhFKh, 27, 1287, 1953.
- A. G. Stromberg. ZhFKh, 29, 409, 1955.
- A. G. Stromberg. ZhFKh, 29, 2152, 1955.
- V. A. Igolinskii. Dissertation, Tomsk Polytechnic Institute, Tomsk, 1963.
- H. Z. Brainina, E. M. Roizenblat, V. B. Belyavskaya. Zav. laborator., 28, 1047, 1962.