On the Resolving Power of Polarography with Linearly Varying Potential
Abstract Generated abstract
This paper analyzes the resolving power of polarography with linearly varying potential when neighboring cathodic or anodic peaks overlap and hinder quantitative impurity determination. Using an additive peak-current assumption, the author derives expressions for the relative error in measuring a distorted peak and formulates static resolving power for plane electrodes and spherical stationary mercury-drop electrodes as functions of peak-potential separation, concentration or peak-height ratio, and the parameter governing spherical diffusion. The results indicate that cathodic polarography has low resolving power and generally requires correction for mutual peak influence, while anodic electro-oxidation peaks from a stationary mercury drop can provide substantially higher resolution, especially at larger values of the diffusion parameter achieved with smaller electrode radii and lower potential sweep rates.
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ON THE RESOLVING POWER OF POLAROGRAPHY WITH LINEARLY VARYING POTENTIAL
V. E. Gorodovykh
(Presented by the Scientific-Methodological Seminar of the KhTF)
In the practical application of polarography with linearly varying potential for the quantitative determination of impurities in complex objects, it is most expedient to obtain several peaks simultaneously on a single polarogram. However, in the case when several metals having close peak potentials are present in the solution or amalgam, mutual superposition of peaks, or even their complete merging, is observed on the polarogram, making accurate quantitative determinations impossible.
The previously proposed graphical methods for taking into account the mutual influence of peaks \[1, 2\] are laborious and do not always give an accurate result. In this connection, the search for ways to increase the resolving power of polarography with linearly varying potential is of considerable interest.
A quantitative criterion that makes it possible to evaluate the mutual influence of two neighboring peaks may be the value of the static resolving power. According to \[3\], the greatest ratio of the concentrations of two neighboring metals at which their peaks are still separated with a specified accuracy is taken as the measure of resolving power with respect to peak depth. Analogously, as the measure of resolving power with respect to potentials one may take the smallest difference in potentials between two neighboring peaks at which they are still separated.
From this point of view, let us consider the resolving power of polarography with linearly varying potential at plane and spherical electrodes. For simplicity, we shall restrict ourselves to the case when there are two peaks (cathodic or anodic) on the polarogram, and the depth of only one of them is distorted. In what follows, we shall agree to denote by subscript 1 the characteristics of the peak whose depth remains constant in the presence of the second peak, and by subscript 2 the characteristics of the peak whose depth changes in the presence of the first peak.
If it is assumed that the peak currents are added additively \[1\], then the absolute error in determining the depth of the second peak will be equal to the current of the descending branch of the first peak at the potential of the second peak1 (Fig. 1). Then the relative error in measuring the second peak \((\alpha)\) is equal to:
\[ \alpha=\frac{i_{н,1}(\varphi_{п,2})}{I_2}, \tag{1} \]
where \(i_{\mathrm{n},1}(\varphi_{\mathrm{p},2})\) is the current of the descending branch of the first peak at the potential of the second peak; \(I_2\) is the height of the second peak.
The general equation for the height of a reversible peak at plane and spherical electrodes can be written in the following form [4]:
\[ I=\frac{z^{3/2}F^{3/2}}{R^{1/2}T^{1/2}}\,w^{1/2}D^{1/2}SC^0\psi_{\mathrm{p}}(\varepsilon), \tag{2} \]
where \(I\) is the peak height; \(w\) is the rate of change of potential; \(D\) is the diffusion coefficient; \(S\) is the electrode surface; \(C^0\) is the initial concentration of the depolarizer; \(z\) is the number of electrons; \(\psi_{\mathrm{p}}(\varepsilon)\) is a function depending on the parameter \(\varepsilon\), equal to \(\varepsilon=\dfrac{1}{r}\sqrt{\dfrac{D}{zw}}\) (\(r\) is the electrode radius).
Fig. 1. Mutual influence of neighboring anodic peaks (details in the text).
The form and magnitude of the function \(\psi_{\mathrm{p}}(\varepsilon)\) depend on the type of electrode (plane, spherical), the type of electrode process (cathodic, anodic), and on the parameter \(\varepsilon\).
We shall next write the equations for the descending branch of the peak:
\[ i_{\mathrm{n.pl}}=zFSC^0\sqrt{\frac{Dw}{\pi(\varphi-\varphi^*)}}, \tag{3} \]
\[ i_{\mathrm{n.sph.c}}=zFSDC^0\left(\sqrt{\frac{w}{\pi D(\varphi-\varphi^*)}}+\frac{1}{r}\right), \tag{4} \]
\[ i_{\mathrm{n.sph.an}}=\frac{2zFSDC^0}{r}\sum_{n=1}^{\infty} \exp\left[-(n\pi)^2\frac{D}{r^2w}(\varphi-\varphi^*)\right], \tag{5} \]
where \(\varphi\) is the potential.
The potential from which the descending branch of the peak begins to be counted is very close to the peak potential; for example, for cathodic peaks at a plane
electrode, \(\varphi^* - \varphi = 0.013\) V, and therefore, without making a large error, one may take
\[ \varphi^* \simeq \varphi_{\mathrm{p}}. \tag{6} \]
Substituting equations (2)—(5) into equation (1) and putting \(z_1=z_2\) and \(D_1=D_2\), we obtain relations for calculating the static resolving power at plane and spherical electrodes:
\[ \alpha = \frac{1}{\psi_{\mathrm{p},2}} \sqrt{\frac{RT}{\pi zF \Delta\varphi}}\, \frac{C_1^0}{C_2^0}, \tag{7} \]
\[ \alpha = \frac{1}{\psi_{\mathrm{p},2}} \sqrt{\frac{RT}{F}}\, \frac{C_1^0}{C_2^0} \left( \frac{1}{\sqrt{\pi z \Delta\varphi}}+\varepsilon \right), \tag{8} \]
\[ \alpha = \frac{2\varepsilon}{\psi_{\mathrm{p},2}} \sqrt{\frac{RT}{F}}\, \frac{C_1^0}{C_2^0} \sum_{n=1}^{\infty} \exp\left[-(n\pi)^2 \varepsilon^2 z\Delta\varphi\right], \tag{9} \]
where \(\Delta\varphi = |\varphi_{\mathrm{p},2}-\varphi_{\mathrm{p},1}|\) is the difference in potentials between the potentials of two adjacent peaks.
Equations (7) and (8) refer to cathodic peaks at plane and spherical electrodes, respectively, while equation (9) refers to anodic peaks obtained in the electrooxidation of a metal from a stationary mercury-drop electrode.
Since the ratio of the concentrations of two metals is equal to the ratio of the depths of their corresponding peaks, i.e.,
\[ \frac{C_1^0}{C_2^0}=\frac{I_1}{I_2}, \tag{10} \]
equations (7)—(9) make it possible to calculate directly the ratios of the depths of two adjacent peaks for which the depth of the second peak can be measured with a prescribed accuracy \(\alpha\). The results of calculations of the static resolving power for \(\alpha=0.1\) are given in Fig. 2.
Fig. 2. Dependence of the resolving power for cathodic (curve 1) and anodic (curves 2 and 3) peaks on the value of \(\Delta\varphi\) and the parameter \(\varepsilon\). Curve 4—the resolving power for cathodic peaks at a plane electrode.
Discussion of the results
As is seen from equation (7), the resolving power at a plane electrode depends only on the difference in potentials between the peak potentials (and also on the number of electrons \(z\)), since for the case of linear semi-infinite diffusion the function \(\psi_{\mathrm{p}}\) is constant and does not depend on the experimental conditions1. The resolving power for cathodic and anodic peaks at a stationary mercury-drop electrode, in addition to \(\Delta\varphi\) and \(z\), also depends on the magnitude of the parameter \(\varepsilon\). The value of \(\Delta\varphi\) has the same effect on the resolving power at plane and spherical electrodes: for a prescribed measurement accuracy \(\alpha=\mathrm{const}\) and an unchanged parameter \(\varepsilon\), the static resolving power with respect to peak depth increases as the difference in potentials between the peak potentials increases (Fig. 2). The influence of the parameter \(\varepsilon\) proves to be more complex, since the functions \(\psi_{\mathrm{p}}\) entering the equations depend on its value.
(8) and (9). At a given measurement accuracy $\alpha = \mathrm{const}$ and a constant difference between the peak potentials $\Delta \varphi = \mathrm{const}$, the influence of the parameter $\varepsilon$ on the resolving power for cathodic and anodic peaks on a stationary mercury drop electrode is directly opposite: as $\varepsilon$ increases, the resolving power for cathodic peaks decreases, whereas for anodic peaks it increases (Fig. 2).
It should be noted that the resolving power for cathodic peaks on a plane and spherical electrode is very low. Thus, for a plane electrode, even at $\Delta \varphi = 0.5\ \mathrm{V}$ and $\alpha = 0.1$, the resolving power for divalent metals is only $I_1/I_2 = 0.5$, and for a spherical electrode it is still smaller. Therefore, in cathodic polarography with linearly varying potential, in all cases it is necessary to take into account the mutual influence of the peaks and to introduce the corresponding corrections into the measurement results.
The resolving power of the APR method (anodic peaks of electro-oxidation of a metal from a stationary mercury drop electrode) is higher than the resolving power of cathodic polarography and, as is evident from Fig. 2, depends strongly on the magnitude of the parameter $\varepsilon$. Thus, for example, for $r = 4 \cdot 10^{-2}\ \mathrm{cm}$, $D = 1.6 \cdot 10^{-5}\ \mathrm{cm}^2/\mathrm{s}$, $z = 2$, $\alpha = 0.1$ and $\Delta \varphi = 0.2\ \mathrm{V}$, the resolving power of the APR method at $\omega = 300\ \mathrm{mV}/\mathrm{min}$ ($\varepsilon = 1$) is 34, whereas at $75\ \mathrm{mV}/\mathrm{min}$ ($\varepsilon = 2$) it is already about 2770. Thus, in order to increase the resolving power of APR, it is necessary to work at large $\varepsilon$, i.e., to use small radii and low rates of change of the potential.
References
- Ya. P. Gokhshtein. Doctoral dissertation, GEOKhI, Moscow, 1962.
- V. S. Smorodinov, A. G. Stromberg, Collection of Works of IREA, 5--6, 24, 1963.
- S. B. Tsfasman. Electronic Polarographs, Metallurgizdat, Moscow, 1959.
- V. E. Gorodovykh. Candidate dissertation, Tomsk, 1964.
- R. S. Nicholson, I. Shain, Anal. Chem., 36, 706 (1964).