On the Calculation of a Differential Electrochemical Cell with Tungsten Electrodes
Abstract Generated abstract
This paper examines the calculation of the electromotive force of a differential electrochemical cell using tungsten electrodes for acidity measurement. It contrasts the ideal equilibrium interpretation of the tungsten electrode potential with experimental deviations, then treats the electrode as stationary, where hydrogen discharge and tungsten ionization processes occur simultaneously. By simplifying the kinetic balance of partial electrode reactions, the author derives an expression relating cell emf to the pH difference and temperature through the sum of kinetic coefficients. Comparison with experimental temperature dependences is presented as support for using the formula to estimate the emf of such cells when the room-temperature value is known.
Full Text
On the Calculation of a Differential Electrochemical Cell with Tungsten Electrodes
N. V. Podbornov
(Presented by the scientific-methodological seminar of the Faculty of Chemical Technology)
In automation systems for technological processes in chemical production, various electrochemical transducers are widely used. Most of them are constructed according to the differential principle and consist of two half-cells with electrodes, between which there is an electrolytic connection in the form of a porous partition or a micro-orifice. In acidity control, metal-oxide electrodes, including tungsten electrodes, are used as electrodes.
The occurrence of a potential on a tungsten electrode and its dependence on the active concentration of hydrogen ions are explained by taking into account the existence of tungsten hydroxide on the electrode surface. The valence of the hydroxide formed does not affect the form of the dependence. For the reaction of interaction of tungsten with hydroxyl ions according to the equation
\[ \mathrm{W} + 4\mathrm{OH}^{-} \rightleftarrows \mathrm{W}(\mathrm{OH})_{4} + 4\bar{e} \]
the magnitude of the electrode potential will be expressed as follows:
\[ \varphi = \varphi_{0} - \frac{2.3RT}{4F}\lg a_{\mathrm{OH}^{-}}^{4}\cdot \frac{a_{W}}{a_{W(\mathrm{OH})_{4}}}. \]
In view of the low solubility of tungsten and its hydroxide, the values of their activities \(a_{W}\) and \(a_{W(\mathrm{OH})_{4}}\) are constant, and \(a_{\mathrm{OH}^{-}}=\dfrac{K_{W}}{a_{\mathrm{H}^{+}}}\) (\(K_{W}\) is the ionic product of water). We may write:
\[ \varphi = \varphi_{0}' + \frac{2.3RT}{F}\lg a_{\mathrm{H}^{+}} \]
or
\[ \varphi = \varphi_{0}' - \frac{2.3RT}{F}\,\mathrm{pH}. \tag{1} \]
If the magnitude of the potential were determined only by the process described by this equation, then the factor \(\dfrac{\Delta \varphi}{\Delta \mathrm{pH}}\) at \(25^{\circ}\mathrm{C}\) would be \(0.0591\) V/pH unit. In reality, however, the value \(\dfrac{\Delta \varphi}{\Delta \mathrm{pH}}\) obtained ex—
experimentally, does not coincide with the theoretically calculated value. In turn, the experimental value \(\varphi_0\) does not correspond to the values of the standard electrode potentials of tungsten oxides.
Both \(\dfrac{\Delta\varphi}{\Delta \mathrm{pH}}\) and \(\varphi_0\) have different values not only for different electrodes, but also vary somewhat with changes in the composition of the solution.
A. N. Frumkin noted [1] that often at electrodes which we regard as equilibrium electrodes, in reality many electrochemical reactions proceed simultaneously, and such electrodes should be considered as stationary. Thus, in particular, in the dissolution of tungsten the scheme of the process may be written in the form:
\[ \mathrm{H}^{+}+\bar e \to \frac{1}{2}\mathrm{H}_2,\quad \text{rate } \vec i_1 \]
\[ \frac{1}{2}\mathrm{H}_2 \to \mathrm{H}^{+}+\bar e \gg \overleftarrow i_1 \quad \mathrm{W}^{4+}+4\bar e \to \mathrm{W} \gg \vec i_2,\quad \mathrm{W}\to \mathrm{W}^{4+}+4\bar e \gg \overleftarrow i_2, \]
In the stationary state
\[ \vec i_1-\overleftarrow i_1=\vec i_2-\overleftarrow i_2. \tag{2} \]
The rate of self-dissolution of tungsten can be calculated if the kinetic regularities of the individual electrochemical reactions occurring on the electrode surface, the concentrations of the components, and the magnitude of the electrode potential are known.
If it is assumed that the kinetic regularities of the discharge reactions and of metal ionization have the same form as for reactions involving hydrogen, then equation (2) takes the form:
\[ \kappa_1[\mathrm{H}^{+}]\exp\left(-\frac{\alpha_1F\varphi_c}{RT}\right) +\kappa_2[\mathrm{W}^{4+}]\exp\left(-\frac{\alpha_2F\varphi_c}{RT}\right) = \kappa'_1 p_{\mathrm{H}_2}^{1/2}\exp\left(\frac{\beta_1F\varphi_c}{RT}\right) + \]
\[ +\kappa'_2 \exp\left(\frac{\beta_2F\varphi_c}{RT}\right). \tag{3} \]
The exchange current at the tungsten electrode at its equilibrium potential is small, and at potentials more positive than the equilibrium potential the rate of the cathodic process of discharge of tungsten ions \(\vec i_2\) will become still smaller, and to a first approximation it may be neglected. The magnitude of the hydrogen overvoltage on tungsten is small, and its equilibrium potential is shifted in the negative direction; therefore it may be assumed that the rate of ionization is small. Thus, one may write:
\[ \kappa_1[\mathrm{H}^{+}]\exp\left(-\frac{\alpha_1\varphi_cF}{RT}\right) = \kappa'_2 \exp\left(\frac{\beta_2\varphi_cF}{RT}\right). \tag{4} \]
Solving with respect to \(\varphi_c\), we obtain an expression for the value of the stationary potential:
\[ \varphi_c= \frac{RT}{(\alpha_1+\beta_2)F}\ln\frac{\kappa_1}{\kappa'_2} + \frac{RT}{(\alpha_1+\beta_2)F}\ln[\mathrm{H}^{+}]. \tag{5} \]
Applying equation (5) to each half-cell of the differential cell and replacing the concentrations by the corresponding activities, we obtain an expression for the e.m.f. of the cell:
\[ E=\frac{2.3RT}{(\alpha_1+\beta_2)F}\lg\frac{a_{\mathrm{H}:1}}{a_{\mathrm{H}:2}} \]
or
\[ E = \frac{2.3RT}{(\alpha_1+\beta_2)F}\cdot \Delta \mathrm{pH}. \tag{6} \]
The sum of the coefficients \(\alpha_1+\beta_2\) may differ from unity. Here there are two different processes, and there is no connection between \(\alpha_1\) and \(\beta_2\). By analyzing a large quantity of experimental data it was shown that, for the given electrode–solution system, the sum of these coefficients does not depend on temperature. Thus, using equation (6), one can determine the emf of a differential cell at any temperature if the value of the emf at room temperature is known, the knowledge of which is necessary for determining the sum
\[ \alpha_1+\beta_2. \]
How well this approach corresponds to reality is seen from the figure. The points show the experimental values of the emf of the differential cell with tungsten electrodes. The solid lines are the values predicted by equation (6).
Fig. 1. Dependence of the emf of a differential electrochemical cell with tungsten electrodes on temperature (comparison of calculation with experiment).
References
- A. N. Frumkin, V. S. Bagotskii, Z. A. Iofa, B. N. Kabanov. Kinetics of Electrode Processes. Moscow State University Press, 1952.