A. G. STROMBERG

ON THE COMPARATIVE SENSITIVITY OF MERCURY FILM AND HANGING MERCURY DROP ELECTRODES IN THE METHOD OF AMALGAM POLAROGRAPHY WITH ACCUMULATION

(Presented by the scientific-methodological seminar of the Faculty of Chemical Technology)

An expression has been derived for the dependence of the minimum impurity content determinable by the method of amalgam polarography with accumulation (APA) in a high-purity material on various factors. It is shown that, under optimal conditions, less than \(10^{-8}\) wt. % impurity can be determined on a hanging drop electrode in a high-purity material, whereas in practice the sensitivity of the APA method usually does not exceed \(10^{-6}\) wt. %. Ways of increasing the sensitivity of the APA method are considered.

An expression has been derived for the comparative sensitivity of mercury film and hanging mercury drop electrodes. It is shown that the reason for the increase in sensitivity of the APA method observed in practice on a film electrode is the narrowing of the peak and the reduction of the influence of background fluctuations and oxygen traces in the solution on the anodic peak on the film electrode as compared with the drop electrode.

The question of increasing the sensitivity of analytical methods for determining \(10^{-5}\)—\(10^{-6}\) percent and less of impurities in high-purity materials is one of the most important questions of analytical chemistry. The question of increasing the sensitivity of the method of amalgam polarography with accumulation (APA) has already been discussed in the literature [1, 2]. Igolinsky proposed [3] a mercury film electrode, consisting of a silver wire coated electrolytically with a thin layer (2—10 microns) of mercury. As experience has shown [4], the mercury film electrode makes it possible to increase the sensitivity of the method of amalgam polarography with accumulation by one to one and a half orders of magnitude and to determine certain impurities (zinc, lead, cadmium, etc.) at concentrations down to \(10^{-10}\) M. Earlier we [5] expressed certain considerations regarding the possible reason for the increase in sensitivity on the film mercury electrode. Namely, it was indicated that the reason for the increase in sensitivity of the APA method when using this electrode is the greater value of the ratio of the electrode surface to the volume of mercury, as compared with a mercury drop.

The purpose of the present article is to derive an expression for the dependence of the minimum impurity content determinable with a given electrode

in a material of high purity and a comparative evaluation of the sensitivity of mercury film and hanging mercury drop electrodes.

The amount of metal deposited on the electrode during preliminary electrolysis can be expressed by the formula

\[ \frac{q}{zF}=\gamma C_1^0 V\cdot 10^{-3}, \tag{1} \]

where \(q\) is the amount of electricity that has passed through the electrode, coulombs;
\(z\) is the number of electrons participating in the discharge process of one ion;
\(F\) is the Faraday constant;
\(C_1^0\) is the initial concentration of ions in the solution, gram-ion per liter;
\(V\) is the volume of the solution, \(\mathrm{cm}^3\);
\(\gamma\) is the fraction of the substance deposited from the solution, which is given by the expression

\[ \gamma=1-e^{-b}; \tag{2} \]

\[ b=\frac{\kappa_1}{zF}\,S\frac{\tau}{V}, \tag{3} \]

where \(b\) is a dimensionless parameter;
\(S\) is the electrode surface area, \(\mathrm{cm}^2\);
\(\tau\) is the duration of the preliminary electrolysis, sec;
\(\kappa_1\) is the proportionality coefficient, A cm gram-ion\(^{-1}\), in the formula \(i_e=\kappa_1 S C_1\cdot 10^{-3}\); \(i_e\) and \(C_1\) are the electrolysis current (ampere) and the concentration of ions in the solution (gram-ion per liter) at the given moment during electrolysis.

The depth of the anodic peak in the AIP method on a stationary mercury electrode (hanging drop or film) with a linearly time-varying potential is given by the expression [1]:

\[ I=\kappa_2 S C_2\cdot 10^{-3}, \tag{4} \]

\[ C_2=\frac{q\cdot 10^3}{zFV}, \tag{5} \]

where \(C_2\) is the concentration of atoms in the amalgam, gram-atom per liter;
\(V\) is the volume of mercury in the electrode, \(\mathrm{cm}^3\);
\(\kappa_2\) is a proportionality constant, A cm gram-atom\(^{-1}\), independent of the concentration of metal atoms and of the size of the mercury electrode, but dependent on the shape of the electrode (drop or film). We shall call the ratio of the area under the peak (in coulombs) to the peak current (ampere) the effective half-peak width \((\delta^*, \text{volt})\). From (4) and (5) we obtain

\[ \frac{\mu q}{I}=\frac{\delta^*}{w}=\frac{zF}{\kappa_2}\frac{V}{S}, \tag{6} \]

where \(\mu\) is the fraction of metal released from the amalgam in the process of anodic dissolution at a linearly varying potential, relative to the amount of metal that entered the electrode during electrolysis; \(w\) is the rate of change of potential, volts per sec. As experience shows, with a slow change of potential on recording polarographs (0.1–1.0 volt/min), the numerical value of \(\mu\) is close to unity. In what follows we shall regard the change of potential as slow and take \(\mu=1\). For the half-peak width \(\delta\) one may write

\[ \delta^*=\nu\delta, \tag{7} \]

where \(\nu > 1\) is an empirical coefficient; for the mercury drop electrode it is about 1.4, and for the mercury film electrode about 1.2. It is easy to see that if the anodic peak had the form of a regular triangle, then we would have \(\nu = 1\) and \(\delta^* = \delta\).

From formulas (1), (6), and (7) we obtain

\[ C_1^0=\frac{10^3\nu\delta}{zF\omega V}\frac{I}{\gamma}, \tag{8} \]

\[ \delta=\frac{zF\omega}{\nu}\frac{V/S}{\kappa_2}. \tag{9} \]

The content of an impurity (\(a\), wt. %) in a weighed sample of material (\(g\), grams) is related to the concentration of impurity ions in the solution (\(C_1^0\), gram-ion/liter) by the relation

\[ a=C_1^0\frac{AV}{10g}, \tag{10} \]

(\(A\) is the atomic weight). From (9) and (10) we obtain

\[ a=\frac{100\nu A}{zF\omega g}\,\delta\,\frac{I}{\gamma}. \tag{11} \]

The minimum height of the anodic peak (\(I_m\), amperes) for measuring it with a specified accuracy (for example, 10 rel. %) must be in a definite ratio to the residual-current density (\(j\), ampere/cm\(^2\)):

\[ I_m=mSj, \tag{12} \]

where \(m\) is a coefficient depending on the nature of the residual current and the method of its compensation, on the shape of the peak, and on the type of instrument.

From formulas (11) and (12) we obtain an expression for the theoretical estimate of the minimum content (\(a_m\), wt. %) of an impurity in the material that can be determined by the APP method with a specified accuracy under the given conditions,

\[ a_m=\frac{100\nu A}{zF\omega g}\frac{S}{\gamma}\,\delta\,mj. \tag{13} \]

Let us examine the influence of the electrode surface \(S\) and of the quantity \(\kappa_1 \frac{\tau}{V}\) [which enters the expression for the coefficient \(\gamma\), see formula (2)] on the minimum determinable impurity content \(a_m\). For this purpose we shall write formula (13) in dimensionless form

\[ \frac{a_m}{PS}=\frac{1}{\gamma}\quad(\text{for } S=\mathrm{const}) \tag{14} \]

and

\[ \frac{a_m}{PQ}=\frac{b}{\gamma}\quad\left(\text{for } \kappa_1\frac{\tau}{V}=\mathrm{const}\right), \tag{15} \]

where

\[ P=\frac{100\nu A}{zF\omega g}\,\delta\,mj. \tag{16} \]

and

\[ Q=\frac{zF}{\kappa_1}\frac{V}{\tau}. \tag{17} \]

From Fig. 1 it is seen that, as the value \(k_1 \dfrac{\tau}{v}\) increases (curve 1, formula (14)), the value \(a_m\) decreases and approaches unity at \(b = 3\) \((1/\gamma = 1/0.95 = 1.05)\); as the surface \(S\) increases (curve 2, formula (15)), the value \(a_m\) increases, reaching the value \(b/\gamma = 3\) at \(b = 3\). Obviously,

Fig. 1. Effect of the value \(k_1 \dfrac{\tau}{v}\) (curve 1) and of the surface area of the mercury electrode \(S\) (curve 2) on the “minimum” impurity content in the material, determined by the PNP method, in dimensionless coordinates. Curve 1—\(y = 1/\gamma\); 2—\(y = b/\gamma\). The meanings of \(\gamma\) and \(b\) are given in formulas (2) and (3). Explanations for the figure are given in the text.

it is seen that the smallest value of \(a_m\) will be found at the intersection of curves 1 and 2 at \(b = 1\) and \(\gamma = 0.63\) (see formula (2)); in this case the surface area \(S\) must be as small as possible, and the value \(k_1 \dfrac{\tau}{V}\) must be as large as possible.

Let us estimate the minimum value of \(a_m\) that can be obtained for the following numerical values of the quantities: \(g = 12\); \(V = 1\) ml; \(A = 100\); \(z = 2\); \(\delta/w = 0.12\) volt; \(\nu = 1.4\); \(m = 1\); \(j = C_\gamma w = 1.4 \cdot 10^{-7}\) A/cm\(^2\) \((C_\gamma = 20 \cdot 10^{-6}\) farad/cm\(^2\)—capacitance of the double layer; \(w = 7 \cdot 10^{-3}\) volt/sec); \(S = \dfrac{2}{3} 4\pi r^2 = 0.01\) cm\(^2\) \((r = 0.04\) cm—the radius of the mercury drop; we take \(1/3\) of the surface of the mercury drop to be screened by the end of the glass tube); \(\gamma = 0.63\). Substituting these numerical values of the quantities into formula (13), we obtain

\[ a_m = 2.7 \cdot 10^{-9}\%; \qquad C_m = 2.7 \cdot 10^{-10}\ \text{g-ion/l}. \]

Using formula (2), let us estimate the time $\tau$ required to obtain $\gamma = 0.63$ ($b=1$) under the indicated conditions and with $k_1 = 10^3$. We obtain $\tau = 2 \cdot 10^4$ sec $= 5.5$ hours.

In practice, the minimum impurity content determined with a dropping electrode usually turns out to be two or three orders of magnitude lower, since a shorter electrolysis time is used, and the residual current exceeds the capacitive current. With the following values of these quantities, $\tau = 10$ min ($\gamma = 0.03$) and $i = 10 C_y$, $w = 1.4 \cdot 10^{-6}$ A/cm$^2$, we obtain (with the same values of the other quantities)

\[ a_m = 5 \cdot 10^{-7}\%; \qquad C_m = 5 \cdot 10^{-8}\ \text{g-ion/l}. \]

If we have $g = 0.2$ g; $V = 5$ ml ($\gamma = 0.006$), then the value of $a_m$ increases by another factor of 125.

It is obvious that, in order to increase the sensitivity of the method (to decrease $a_m$) when working with this electrode, it is necessary first of all to reduce the residual current $i$ to the value of the capacitive current, and also to eliminate the capacitive current (for example, by using a vector polarograph), and to intensify stirring of the solution in order to increase the coefficient $k_1$, so as to achieve substantial depletion of the solution ($\gamma = 0.63$) at practically acceptable values of the electrolysis time $\tau$ and the solution volume $V$, and at a minimum value of the electrode surface $S$ (see formula (2)).

Let us now compare the minimum determinable impurity content $a_m$ at mercury film and hanging mercury drop electrodes under otherwise identical conditions. Then from formula (13) we obtain

\[ \frac{a'_m}{a_m} = \frac{S'}{S} \frac{\gamma}{\gamma'} \frac{\nu'}{\nu} \frac{\delta'}{\delta} \frac{m'}{m} \frac{j'}{j}. \tag{18} \]

(The prime sign ($'$) denotes quantities in the experiment with the film electrode.)

Let us assume that the surfaces of both electrodes are the same; then $S = S'$ and $\gamma = \gamma'$ ($k_1$, $\tau$, $V$ in formula (2) are also regarded as identical). It is easy to see that if the surface of the film electrode is increased to the value at which $\gamma = 0.63$ ($S' = 1.7$ cm$^2$ at $\tau = 10$ min), then the value of $a'_m$ will increase, since

\[ \frac{S'}{S}\frac{\gamma}{\gamma'} = \frac{0.7}{0.01}\frac{0.03}{0.63} \simeq 8. \]

Thus, by increasing the electrode surface it is impossible to achieve an increase in the sensitivity of the method, as is clearly seen from curve 2, Fig. 1.

From formula (9) it follows that

\[ \frac{\delta'}{\delta} = \frac{\nu}{\nu'} \frac{k_2}{k'_2} \frac{3d}{r} \tag{19} \]

\[ (\nu'/S') = d; \qquad (\nu/S) = \frac{r}{3}; \]

$d$ is the thickness of the mercury film on the film electrode.

If for both electrodes the constants were $k_2 = k'_2$, then for the numerical values $d = 4 \cdot 10^{-4}$ cm, $r = 0.04$ cm and $\nu/\nu' = 1.2$ we would obtain from formula (19) $\delta'/\delta = 1/28$. But experiment shows that this ratio is only about $1/3$. This means that the width of the film electrode is only 3 times smaller than that of the drop electrode, and not 28 times, as follows from the theo-

...assuming \(k_2 = k'_2\). It follows from this that the ratio \(k_2/k'_2 \simeq 10\). Consequently, the assumption we made earlier [5] concerning the influence of increasing the ratio \(v/S\) of the electrode in order to increase the sensitivity of the method is justified, but the influence of this factor is weakened because of the decrease in the constant \(k_2\) for the film electrode.

For the value of the ratio \(m'/m\) we currently have no precise experimental data. Semiquantitative experiments carried out by us show that for the film electrode the value \(m' < m\). These experimental data can be supported by certain theoretical considerations taken from the theory of “noise” in measuring instruments. It is known [6] that when the frequency range (pass band) in which the noise is determined is narrowed, its magnitude decreases. In exactly the same way, evidently, one should expect by analogy that, when the anodic peak on the film electrode is narrowed threefold, the interfering influence of the residual current should decrease. We shall take, as an approximation, \(m'/m = 1/3\).

If the residual current at the hanging mercury drop exceeds the capacitive current by 10–15 times because of the presence of traces of oxygen in the solution, then on the film electrode, as experience shows, the influence of oxygen is manifested more weakly and the ratio \(j'/j < 1\). Experience indicates that \(j'/j =\) from \(1/3\) to \(1/6\). The reason for this phenomenon is not entirely clear to us. We shall take \(j'/j = 1/3\).

On the basis of the analysis carried out, we obtain from formula (18)

\[ \frac{a'_m}{a_m} \simeq \frac{1}{30}. \]

Experience shows that the use of a film electrode makes it possible to increase the sensitivity of the AAP method by more than an order of magnitude. In addition, a major advantage of the film electrode is the considerable increase in the resolving power of the AAP method due to narrowing of the anodic peaks by a factor of two to three.

Thus, the use of a film electrode, even under the conditions considered above for the hanging drop electrode, makes it possible to solve the problem of determining \(10^{-7}\)—\(10^{-8}\%\) impurities in high-purity materials.

Increasing the intensity of stirring of the solution, deep purification of the nitrogen and of the solution from oxygen, and selection of optimal values of the other quantities in formula (13) make it possible to increase the sensitivity of the AAP method still further. This question will be considered in more detail elsewhere.

REFERENCES

1. A. G. Stromberg. On the question of the influence of electrolysis time, solution volume, and radius of the mercury drop on the depth of the anodic peak in amalgam polarography. Izv. SO AN SSSR, No. 5, 76, 1962.

2. A. G. Stromberg. On increasing the sensitivity of the method of amalgam polarography with accumulation on a stationary mercury drop. Zavodskaya laboratoriya, XXIX, 387, 1963.

3. V. A. Igolinskii, A. G. Stromberg. A highly sensitive method of polarographic analysis with preliminary accumulation on a stationary mercury film electrode. Collection “Methods of Analysis of Chemical Reagents and Preparations,” IREA Publishing House, issue 5–6, 29, 1963.

4. V. A. Igolinskii. Investigations in the field of polarography and amalgam polarography with accumulation on a stationary mercury film electrode. Dissertation. Tomsk Polytechnic Institute, Tomsk, 1963.

5. V. A. Igolinskii, A. G. Stromberg. On the maximum sensitivity of the method of amalgam polarography with preliminary accumulation on a stationary electrode and ways of increasing it. Zavodskaya laboratoriya, XXX, No. 4, 1964.

6. Yu. I. Gribanov. Measurement of weak currents, charges, and high resistances. Gosenergoizdat, p. 6, 1962.