THEORY OF THE AMMONIUM AMALGAM EFFECT IN THE APN METHOD. PART I

A. G. STROMBERG, Yu. A. KARBAINOV

(Presented by the scientific-methodological seminar of the CTF)

As was shown by one of us [1, 2, 3], an important way of increasing the sensitivity of the method of amalgam polarography with accumulation (APN) is to increase the constant of preliminary electrolysis. By using various techniques to accelerate the delivery of the substance to the electrode, it is possible to increase the electrolysis constant by more than an order of magnitude [4, 5, 6].

B. F. Nazarov [7] proposed a new technique for increasing the effective electrolysis constant through the use of the so-called ammonium amalgam effect (AAE). This effect consists in the fact that, when electrolysis of a solution of an ammonium salt is carried out on a stationary mercury electrode at a sufficiently negative potential, ammonium amalgam is formed, which gradually decomposes. At the same time, a foam is formed from the decomposition products of ammonium (ammonia + hydrogen), and the surface of the electrode increases. Because of this, the rate of accumulation in the amalgam of the metal being determined increases; the ions of this metal are discharged from the solution simultaneously with ammonium ions in the course of preliminary electrolysis. When the anodic polarogram is recorded, the volume of the mercury drop rapidly decreases to its initial size as soon as the potential becomes more positive than the discharge potential of ammonium ions from the solution. In this case, the concentration of the metal being determined in the mercury drop is greater than in an experiment without AAE, and the depth of the anodic notch of the element being determined increases by the same factor. In the work mentioned [7], due to AAE the sensitivity was increased by a factor of 10–15.

In one of our preceding works [8], the ammonium amalgam effect was used by us in combination with the effect of increasing the temperature (EIT) to a temperature close to the boiling point of the solution, and, using a nonaqueous solution as an example, the possibility was shown of increasing the effective electrolysis constant and increasing the sensitivity by combining both effects by a factor of 50–70.

The theoretical foundations of AAE and of AAE in combination with EIT have so far been poorly developed. In Nazarov’s work an expression was obtained for the dependence of the radius of an ammonium amalgam drop on time without taking into account the removal of the decomposition products of ammonium amalgam from the dropping electrode, i.e., for the initial intervals of time.

The aim of the present work is to obtain expressions for the dependence of the radius of a mercury drop on time for any intervals of electrolysis time on the basis of a definite model of this amalgam and to op-

certain simplifying assumptions about the regularities of the forward and reverse processes. The work also obtains expressions for the effective electrolysis constant and the depth of the anodic notch of the element being determined when the ammonium amalgam effect is used in the APN method.

We assume that practically all of the ammonium liberated into the mercury drop decomposes according to the equation:

\[ \mathrm{NH}_4 \rightleftarrows \mathrm{NH}_3 + \tfrac{1}{2}\mathrm{H}_2 \]

and is present inside the mercury drop in the form of small gas bubbles (foam).

The process of accumulation of ammonium in the mercury drop consists of two opposite processes: the process of discharge of the ammonium ion from the solution at the electrode (rate \(v_1\)) and the process by which the gaseous products of ammonium decomposition leave through the surface of the mercury drop into the solution (rate \(v_2\)). The rate of accumulation of ammonium in the mercury drop is equal to the difference of these rates:

\[ v=\frac{dm}{d\tau}=v_1-v_2 . \tag{1} \]

Let us make two assumptions. 1. Suppose that a fraction of the surface of the mercury drop is covered with gas bubbles. In this case, obviously, it may be assumed that the rate of discharge of the ammonium ion from the solution at the electrode will be proportional to the fraction of the free electrode surface (\(v_1 \sim 1-\Theta\)), while the rate of removal of the gaseous products of decomposition of the amalgam from the mercury drop is proportional to the fraction of the surface occupied by gas bubbles (\(v_2 \sim \Theta\)). 2. Suppose that the fraction of the occupied surface is proportional to the amount of ammonium decomposition products in the mercury drop (\(\Theta \sim m\)). Then, instead of (1), one may write:

\[ \frac{d\Theta}{d\tau}=\kappa_a(1-\Theta)-\kappa_b\Theta, \tag{2} \]

where \(\kappa_a, \kappa_b\) are coefficients independent of \(\Theta\) and time \(\tau\).

As the amount of ammonium decomposition products inside the mercury drop increases, the fraction \(\Theta\) of the electrode surface covered with gas bubbles will increase, the rate of the forward process will decrease, and that of the reverse process will increase, until, finally, the rates of the forward and reverse processes become equal \((d\Theta/d\tau=0)\). In this case, evidently, the degree of filling \((\Theta_\infty)\) and the radius of the mercury drop \((r_\infty)\) will reach a maximum value, and a stationary state will be established. Then from (2) we obtain:

\[ \frac{\kappa_b}{\kappa_a}=\frac{1-\Theta_\infty}{\Theta_\infty}. \tag{3} \]

Integrating equation (2) within the limits from 0 to \(\tau\), and taking (3) into account, we obtain:

\[ \ln \frac{\Theta_\infty}{\Theta_\infty-\Theta}=(\kappa_a+\kappa_b)\tau . \tag{4} \]

In order to relate the degree of filling \(\Theta_\infty\) of the electrode surface to the radius of the mercury drop \(r\) at time \(\tau\), let us suppose that this degree of filling is proportional to the increase in the surface area of the mercury drop over time \(\tau\):

\[ \Theta=\frac{\kappa_c}{4\pi}(S-S_0)=\kappa_c(r^2-r_0^2). \tag{5} \]

\(r, r_0\) are the radii of the mercury drop at time \(\tau\) and \(\tau=0\); \(S\) and \(S_0\) are the surfaces of the drops in the same time intervals; \(\kappa_c\) is a proportionality coefficient independent of the radius of the mercury drop and of time. After a sufficiently long time \((\tau \to \infty)\), when a stationary state is established, we obtain:

\[ \Theta_\infty=\kappa_c\left(r_\infty^2-r_0^2\right), \tag{6} \]

where \(\Theta_\infty, r_\infty\) are the maximum values of the degree of filling and the radius of the mercury drop after establishment of the stationary state.

Substituting (5) and (6) into (4), we obtain:

\[ \ln \frac{r_\infty^2-r_0^2}{r_\infty^2-r^2} =(\kappa_a+\kappa_b)\cdot \tau . \tag{7} \]

Hence, for the dependence of the radius \(r\) of the mercury drop on time we obtain the expression:

\[ r^2=r_\infty^2\left\{1-\left[1-\left(\frac{r_0}{r_\infty}\right)^2\right]e^{-B\cdot \tau}\right\}, \tag{8} \]

where

\[ B=\kappa_a+\kappa_b . \tag{9} \]

Eliminating the quantity \(\Theta_\infty\) from formulas (3) and (6), we obtain:

\[ r_\infty^2=r_0^2+\frac{\kappa_a}{\kappa_c(\kappa_a+\kappa_b)} =r_0^2+A . \tag{10} \]

It is easy to see that formula (8) satisfies the limiting cases. For \(\tau=0\) we obtain \(r=r_0\), and for \(\tau=\infty\) we have \(r=r_\infty\).

For comparison with experiment it is convenient to represent formula (7) in the following form, taking into account relations (9) and (10):

\[ I=\lg(r_\infty^2-r^2)=\lg A-\frac{B}{2.3}\tau . \tag{11} \]

It follows from formula (11): if, from experimental data for different time intervals, one constructs a graph in the coordinates \(\lg(r_\infty^2-r^2)\) and \(\tau\), then, if the initial assumptions and the developed theory of the ammonium amalgam effect are correct, the experimental points should lie on a straight line in this graph. In this case the slope of this straight line is equal to \(-\dfrac{B}{2.3}\), and the intercept cut off on the ordinate axis at \(\tau=0\) is equal to \(\lg A\). From the found values of \(A\) and \(B\), by formulas (9) and (10), one can calculate two coefficients out of three \((\kappa_a,\kappa_b,\kappa_c)\), if one of them is specified. Taking \(\kappa_c\) as known, we obtain:

\[ \kappa_a=B'\cdot A\cdot \kappa_c;\quad \kappa_b=B'(1-A\cdot \kappa_c),\quad \text{where } B'=\frac{B}{2.3}. \tag{12} \]

The constant \(\kappa_c\) can in principle be calculated from formula (6), if by some independent method (for example, from a photograph of the drop) the degree of coverage \(\Theta_\infty\) of the mercury drop by gas bubbles upon attainment of the equilibrium state is determined.

The experiments were carried out in an electrolyzer of the design described earlier [9]. A mercury drop with radius \(0.05\ \mathrm{cm}\) was prepared electrolytically on a platinum contact (diameter \(0.015\ \mathrm{cm}\)) from a saturated solution of mercurous nitrate (univalent). Electrolysis with production of ammonium amalgam was carried out in a \(0.2\,N\) solution of \(\mathrm{NH_4NO_3}\) in \(\mathrm{CH_3OH}\) at a temperature of \(35^\circ\mathrm{C}\); the volume of the solution was \(5\ \mathrm{ml}\). For

to remove dissolved oxygen from the air, nitrogen purified of oxygen was passed through the electrolyzer. The radius of the growing drop of ammonium amalgam was measured with a horizontal microscope with a scale in the eyepiece (1 scale division equals \(5.9\cdot 10^{-2}\) mm). The change in the radius of the mercury drop with time during electrolysis with formation of ammonium amalgam is shown in Fig. 1. As is seen from the graph, under the conditions of our experiments the radius of the mercury drop in the process of formation of ammonium amalgam (more precisely, of the foam of the gas mixture \(NH_3 + H_2\) in mercury) increased gradually from \(0.05\) to \(0.22\) cm and after 50–60 sec reached an approximately constant value. From Fig. 2 it is seen that, in accordance with the theory, in the coordinates \(\lg(r_\infty^2-r^2)=y\) and \(\tau\) a straight line is obtained. From the graph in Fig. 1, in accordance with formula (11), we find

Fig. 1. Dependence of the radius of a drop of ammonium amalgam on electrolysis time. 0.2 N solution of \(NH_4NO_3\) in \(CH_3OH\). \(\varphi_{\mathrm e}=-2.8\) V, sensitivity \(3\cdot10^{-8}\) A/mm, \(T=35^\circ C\).

Fig. 2. Dependence of \(\lg(r_\infty^2-r^2)=y\) on electrolysis time \(\tau\). 0.2 N solution of \(NH_4NO_3\) in \(CH_3OH\). \(\varphi_{\mathrm e}=-2.8\) V, sensitivity \(3\cdot10^{-8}\) A/mm, \(T=35^\circ C\).

\[ A=17.8;\quad B=0.057. \]

In conclusion, let us make some numerical estimates. The effective electrolysis constant \((\kappa_1^*)\) will increase, owing to the increase in the surface of the mercury drop, by

\[ \rho=\frac{\kappa_1^*}{\kappa_1}=\frac{S_1^*}{S_1}=\left(\frac{r_\infty}{r_0}\right)^2 \tag{13} \]

times, where \(S_1^*\) and \(S_1\) are the surface of the mercury drop, respectively, with the ammonium-amalgam effect (AAE) and without it. Under the conditions of our experiments\(^1\)

\[ \rho=\left(\frac{2.2}{0.5}\right)^2\simeq 20. \]

With practically no depletion of the solution, the depth of the anodic notch \((I_a^*)\) of the element being determined under the conditions of the AAE will also increase by \(\rho\) times, that is:

\[ \frac{I_a^*}{I_a}=\frac{\kappa_1^*}{\kappa_1}=\rho. \tag{14} \]

\(^1\) It is assumed here that the duration of electrolysis is considerably greater than the time required to reach the maximum radius of the mercury drop under AAE conditions, i.e., the surface of the drop under AAE conditions is constant. In our calculations the duration of electrolysis is taken as \(\tau=10\) min, while the maximum size of the drop is reached after \(\tau_\infty=1\) min, i.e., \(\tau\) is approximately 10 times greater than \(\tau_\infty\).

However, it is more advantageous to choose the volume of the solution in such a way that, under AAE conditions, sufficiently strong depletion of the solution occurs. Let us take¹ the degree of depletion of the solution to be

\[ \gamma^* = 1 - e^{-b} = 0.63 . \tag{15} \]

In this case the dimensionless parameter \(b^* = 1\) [10], and the ratio of the anodic teeth is

\[ \frac{I_a^*}{I_a} = \gamma^* \cdot \rho . \tag{16} \]

At \(\rho = 20\) we obtain \(\frac{I_a^*}{I_a} = 12.6\). Although the increase of the anodic peak in this case is smaller than according to formula (14), it is nevertheless more advantageous to work under conditions of considerable depletion of the solution, since in this case a large part of the substance passes into the electrode, and the absolute depth of the anodic teeth is greater, and consequently the sensitivity is higher.

Let us make a numerical estimate of the volume of solution \((V)\) that must be available in order, for given values of \(\kappa_1^*, \tau, r\), to obtain the required degree of depletion \(\gamma^* = 0.63\) (and \(b^* = 1\)).

From the expression for the dimensionless parameter [10]

\[ b = \frac{\kappa_1}{zF} \cdot S \cdot \frac{\tau}{V} \tag{17} \]

we obtain

\[ \frac{V}{v} = \frac{\kappa_1^*}{zF} \cdot \tau \cdot \frac{3}{r}. \tag{18} \]

Taking \(\kappa_1^* = 10^4\), \(\tau = 600\ \text{sec}\), \(r = 0.05\ \text{cm}\), we obtain \(V/v = 1.86 \cdot 10^3\). With \(v = 4/3\pi r^3 = 0.5 \cdot 10^{-3}\ \text{cm}^3\), we have \(V = 0.93\ \text{cm}^3\).

Thus, if under AAE conditions the radius of the mercury drop increases by \(\rho^{1/2} = 4.4\) times, then under practically acceptable conditions (\(V = 1\ \text{cm}^3\), \(\tau = 10\ \text{min}\), etc.) it is possible to achieve practically complete depletion of the solution and to increase the sensitivity of determination by one order of magnitude.

Conclusions

1. On the basis of certain ideas concerning the mechanism of the ammonium amalgam effect (AAE), a theoretical expression has been derived for the dependence of the radius of a mercury drop on time during the electrolysis of an ammonium solution.

2. The experimental data presented by us confirmed this theoretical dependence.

References

1. A. G. Stromberg. Zavodskaya laboratoriya (in press).
2. A. G. Stromberg. Izvestiya TPI, 128, 13, 1965.
3. A. G. Stromberg, A. A. Kaplin. Zhurnal analiticheskoi khimii (in press).
4. E. K. Spirin, V. I. Kuleshov. Zavodskaya laboratoriya (in press).
5. V. I. Kuleshov, A. V. Konkova. Izv. TPI (in press).
6. A. A. Kaplin, V. I. Kuleshov. Izvestiya TPI (in press).
7. B. F. Nazarov, V. I. Slonov. Zhurnal analiticheskoi khimii (in press).
8. Yu. A. Karbainov, A. G. Stromberg. ZhAKh (in press).
9. A. G. Stromberg, E. A. Stromberg. Zavodskaya laboratoriya 27, 3, 1961.
10. A. G. Stromberg. Izvestiya SO AN SSSR, 5, 76, 1962.

1. For practical reasons it is more advantageous to take \(\gamma^* = 0.63\), rather than \(\gamma^* = 0.95\), since then \(b^* = 1\) instead of \(b^* = 3\), and, consequently, the electrolysis time is reduced by a factor of three, while the sensitivity (the depth of the anodic tooth) decreases by \(0.95/0.63 = 1.5\) times. ↩