Study of the Cementation Reaction on a Stationary Spherical Mercury Electrode
Abstract Generated abstract
This paper examines cementation as an interference in amalgam polarography with accumulation and chronopotentiometric determination of trace substances, focusing on a stationary spherical mercury electrode. A kinetic diffusion model is developed for an irreversible cementation reaction under stirred-solution conditions, yielding general differential equations and explicit forms for selected reaction orders, especially the case first order in the oxidized metal ion. The theory is tested for the reaction between zinc amalgam and thallium ions in potassium nitrate solution, with a convection correction introduced for the measured flux. Experimental data for varying thallium concentrations agree within error with the calculated second-order reaction curve, supporting an electrochemical cementation mechanism and giving a reported rate parameter of 3.44 cm6 per sec per gram atom.
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STUDY OF THE CEMENTATION REACTION ON A STATIONARY SPHERICAL MERCURY ELECTRODE
B. F. Nazarov, Yu. N. Zhikharev
(Presented by the scientific seminar of the Department of Chemical Technology)
In the analysis of small amounts \((10^{-9}\%\) and less) of a substance by the method of amalgam polarography with accumulation (APA) or by the chronopotentiometry method [1], one of the substantial interferences is the oxidation of the amalgam by oxygen present in the solution, as well as the cementation reaction between the impurity being concentrated and the cations of the metal being analyzed. The purpose of the present work is to study the cementation reaction on a stationary spherical mercury electrode under the conditions of the method of amalgam polarography with accumulation.
We shall assume that the chemical reaction at the electrode
\[ a\mathrm{Me}_{\mathrm{I}}^{0}+b\mathrm{Me}_{\mathrm{II}}^{+z_2} \xrightarrow{k_1} c\mathrm{Me}_{\mathrm{I}}^{+z_1}+d\mathrm{Me}_{\mathrm{II}}^{0} \tag{1} \]
proceeds irreversibly, i.e., in the general case one may write
\[ v=\frac{dQ_{\mathrm{II}}}{dt}=K_{\mathrm{c}}S\left(C_{\mathrm{R}}^{s}\right)^{a}\left(C_{\mathrm{o}}^{s}\right)^{b}, \tag{2} \]
where \(v\) is the rate of the cementation reaction;
\(Q_{\mathrm{II}}\) is the amount of reacted metal \((\mathrm{Me}_{\mathrm{II}})\);
\(C_{\mathrm{R}}^{s}\) is the concentration of the reduced form of metal \(\mathrm{Me}_{\mathrm{I}}\) on the electrode surface;
\(C_{\mathrm{o}}^{s}\) is the concentration of the oxidized form of metal \(\mathrm{Me}_{\mathrm{II}}\) on the electrode surface;
\(t\) is time;
\(K_{\mathrm{c}}\) is the rate constant of reaction (1);
\((a+b)\) is the order of reaction (1);
\(S\) is the electrode surface.
Let us consider the case when the cementation process takes place in a stirred solution and sufficiently slowly, so that
\[ \frac{dC_{\mathrm{R}}}{dx}=0, \tag{3} \]
i.e., the concentrations of substance \(\mathrm{Me}_{\mathrm{I}}\) at the electrode surface and in the interior of the electrode are identical (this condition is readily justified for \(C_{B}>C_{0}\)). In addition, we shall assume that after electrolysis (electrolysis time greater than 5 min) the concentration distribution of \(\mathrm{Me}_{\mathrm{II}}\) obeys the law of semi-infinite spherical diffusion, and the flux of substance to the electrode surface is expressed by the formula [3]
\[ D\,\frac{dC_o}{dx}\bigg|_{x=r_o}=DS\,\frac{1}{r_o}\left(C_o^o-C_o^s\right). \tag{4} \]
The concentration of substance \(\mathrm{Me}_1\) at the surface will be found from the formula
\[ C_R^s=C_R^o-\frac{Z_2}{Z_1}\frac{Q_0}{\vartheta}, \tag{5} \]
where \(C_R^o\) is the initial concentration of substance \(\mathrm{Me}_1\); \(\vartheta\) is the volume of the electrode. From (2), (4), and (5) we obtain an equation in differential form for the rate of cementation:
\[ \frac{d\alpha_t}{dt}=\frac{Z_1}{Z_2}A\left\{C_o+ \left[ \frac{\vartheta^a\left(Q_R^o\right)^{1-a}}{KS(1-\alpha_t)^a} \frac{d(1-\alpha_t)}{dt} \right]^{1/b} \right\}, \tag{6} \]
where \(\alpha\) is the fraction of metal \(\mathrm{Me}\) that has reacted in time \(t\). \(A=\dfrac{SD}{rQ_R^o}\).
Equation (6) is valid for a cementation reaction of any order. However, in explicit form we were able to obtain solutions only for certain values of \(a\) and \(b\).
The simplest solutions are obtained for the values \(b=1\). For \(b\ne 1\), equation (6) becomes nonlinear and apparently can be solved only by numerical methods.
For the values \(b=1\), after integration, equation (6) for \(a=1\) takes the following form:
\[ \alpha_t=\frac{Z_2}{Z_1}AC_o t+A\frac{\vartheta}{KS}\ln(1-\alpha_t). \tag{7} \]
For any values of \(a\), at \(b=1\) we shall have
\[ \alpha_t=\frac{Z_2}{Z_1}AC_o t+ A\frac{\vartheta^a}{KS}\left(Q_R^o\right)^{1-a} \frac{\left[(1-\alpha)^{1-a}-1\right]}{1-a}. \tag{8} \]
The experiment was carried out in an ordinary electrolyzer [4]. A \(1\,N\,\mathrm{KNO_3}\) solution was chosen as the indifferent electrolyte. At the electrode, with the circuit open, the reaction proceeded
\[ \mathrm{Zn}^o+2\,\mathrm{Tl}^+\longrightarrow 2\,\mathrm{Tl}^o+\mathrm{Zn}^{++}. \tag{9} \]
The concentration of \(\mathrm{Zn}^{++}\) in the cell was \(1\cdot10^{-7}\) g-ion/mm and \(8\cdot10^{-7}\) g-ion/ml. The accumulation time was 5 minutes, after which the contacts were opened and reaction (9) proceeded.
Electrolysis was carried out in a stirred solution. After 5 minutes the concentration distribution of thallium in the solution may be considered as corresponding to semibounded spherical diffusion. Indeed, at \(t=300\) sec in the equation
\[ q=SD^{1/2}C_o\frac{1}{\sqrt{\pi t}}+SDC_o\frac{1}{r_o}, \tag{10} \]
where \(q\) is the flux of substance, the first term may be neglected, i.e., a stationary state will be reached. However, we were unable to eliminate convection. The experimental value of the flux was \(\theta=1.44\) times greater than that calculated from equation (10). Nevertheless, the stationarity of the flux of substance to the electrode was not disturbed; consequently, in order to bring the experimental data and the theoretical equations into agreement, it is necessary to multiply the right-hand side of equation (4) by the quantity \(\theta\); then the final equation for a second-order reaction will have the form
\[ \alpha_t=\theta\,\frac{Z_2}{Z_1}AC_o t+\theta A\frac{\vartheta}{KS}\ln(1-\alpha_t). \tag{11} \]
The concentration of thallium in the solution for different series varied
from \(5\cdot 10^{-3}\) to \(1\cdot 10^{-4}\) g-ion/l, i.e., by a factor of 20. The cementation time for the different series was varied from 30 to 600 sec. The experimental data are presented in Fig. 1.
Fig. 1. Change with time in the fraction of metal remaining in the mercury drop for different concentrations of thallium ions in solution:
1 — \(5\cdot 10^{-3}\), 2 — \(1\cdot 10^{-3}\), 3 — \(5\cdot 10^{-4}\), 4 — \(1\cdot 10^{-4}\) g-ion/l.
The cementation rate constant was found from the experimental data of the second curve according to the formula
\[ KS= -\frac{\theta A V \ln(1-\alpha_t)} {\alpha_t-\theta \dfrac{Z_2}{Z_1}\, A C_0 t}. \tag{12} \]
The value of \(KS\), found from formula (12), is
\[ 3.44\,\frac{\mathrm{cm}^6}{\mathrm{sec}\cdot \mathrm{g\text{-}at}}}. \]
It follows from equation (11) that, for a second-order reaction, the dependence of \(\alpha_t\), or \(1-\alpha_t\), on \(C_0 t\) at a constant value of \(A\) or \(Q_R^0\) (the initial concentration of zinc) must be described by a single curve. Figure 2 presents the theoretical curve (equation 11 was calculated by the method of successive approximations) and the experimental data.
Fig. 2. Theoretical dependence of the change in the fraction of metal remaining in the mercury drop.
As is evident from Fig. 2, the deviation of the experimental points from the theoretical curve lies within the experimental error; i.e., reaction (10) proceeds as an electrochemical reaction according to the scheme [2].
\[ \mathrm{Tl}^{+}+e \rightarrow \mathrm{Tl}^{0}, \]
\[ \mathrm{Zn}^{0}-2e \rightarrow \mathrm{Zn}^{+2}. \]
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