Parametric Theory of Amalgam Stripping Polarography

A. G. Stromberg

The development of many branches of new technology (semiconductor technology, the atomic industry, rocket technology, quantum generators, etc.) requires materials of especially high purity, with impurity contents of less than millionths of a percent. In connection with this, analytical chemistry faces the task of developing methods for determining such negligible (nanogram) amounts of a substance. This task is solved in analytical chemistry by means of various physical and physicochemical methods of analysis. Among physicochemical methods of analysis, one of the most promising is the method of amalgam stripping polarography with accumulation (ASPA) [30, 32]. However, the sensitivity of ASPA procedures achieved at the present time still does not correspond to the possibilities of this method.

The problem of increasing the sensitivity of the ASPA method can be solved on the basis of developing the theory of this method, i.e., on the basis of developing the theory of the influence of various factors on the maximum current of the electro-dissolution of a metal from an amalgam (abbreviated: on the anodic peak current). Unfortunately, the rigorous theory of ASPA is rather complicated, which makes it difficult to use rigorous theoretical expressions for solving the practical problem of identifying optimal conditions for carrying out analysis and ways of increasing the sensitivity of the method. Therefore, over a number of years (since 1961) we have been developing an approximate parametric theory of the ASPA method.

The aim of this article is to summarize the results of our work on developing the parametric theory of amalgam stripping polarography with accumulation and its experimental verification.

Previously we [1] considered two relationships containing two coefficients that must be found from experiment. If we initially assume that preliminary electrolysis is carried out at a potential at which the limiting current has already been reached, that the solution is stirred (i.e., the process is stationary), and that there is an excess of indifferent electrolyte (supporting background) in it, then we may write the relationship:

\[ I_e = K_e S C_i, \tag{1} \]

where \(I_e\) and \(C_i\) are the current, amperes, and the concentration of metal ions in the solution, g-ion/cm\(^3\), at the given moment of preliminary electrolysis;

$S$ is the electrode surface, $\mathrm{cm}^{2}$; $K_e$ is the electrolysis coefficient¹. This coefficient does not depend on the electrode surface or on the concentration of ions in solution, but depends on the rate of delivery of metal ions from the solution to the surface of the solution, i.e., on temperature, on the viscosity of the solution, on the intensity of stirring of the solution, on the diffusion coefficient of the metal ions in the solution, and on the charge of the metal ions. The second relation is assumed in the form:

\[ I_a = K_a S C_a, \]

where $I_a$ is the anodic-peak current, amperes; $C_a$ is the average concentration of metal atoms in the amalgam, $\mathrm{g\text{-}atom}/\mathrm{cm}^{3}$, by the end of electrolysis and before anodic dissolution of the metal from the amalgam²; $K_a$ is the anodic-peak coefficient. This coefficient does not depend on the concentration of metal atoms in the amalgam, but depends on the shape of the mercury electrode (drop or film), on the diffusion coefficient of the metal in mercury, on the reversibility of the electrode process, on temperature, and on the intensity of stirring of the amalgam solution. At first we shall assume that $K_a$ is approximately independent of the size of the mercury drop.

The coefficients $K_e$ and $K_a$ must be found directly from the results of experiments in amalgam polarography with accumulation. Knowing the coefficients $K_a$ and $K_e$, one can obtain a number of important relations for the dependence of the anodic-peak current on various factors. In applying the parametric theory chiefly to the mercury-drop and mercury-film electrode. For definiteness, we shall consider the application of the parametric theory mainly to the mercury drop electrode.

Taking into account the obvious relations

\[ C_a v = \gamma C_i V; \qquad S/v = 3/r, \tag{2} \]

we obtain

\[ \tag{3} \]

\[ I_a = a\gamma; \qquad a = 3K_a C_i \frac{V}{r}, \tag{4} \]

\[ \gamma = 1 - C_i'/C_i; \tag{4a} \]

where $V$ and $v$ are, respectively, the volume of the solution and the volume of the amalgam, $\mathrm{cm}^{3}$; $r$ is the radius of the amalgam drop, $\mathrm{cm}$; $C_i$ is the initial concentration of metal ions in the solution, $\mathrm{g\text{-}ion}/\mathrm{cm}^{3}$; $\gamma$ is the degree of depletion of the solution by the end of electrolysis. An expression for the degree of depletion $\gamma$ of the solution can be obtained by substituting into formula (4a) the value of $C_i'/C_i$ found by comparing the right-hand sides of expressions (1) and (5a) for the electrolysis current:

\[ I_e = -ZF\frac{d(C_i'V)}{dt}, \tag{5a} \]

\[ \gamma = 1 - e^{-B}, \qquad B = \frac{4\pi}{F}\,\frac{K_e}{Z}\,\frac{r^{2}t}{V} \tag{5} \]

($t$ is the duration of electrolysis).

It follows from (7) that, with complete depletion of the solution, the anodic-peak current $I_a$:

\[ I_a = 3K_a C_i \frac{V}{r}\left(1-e^{-\frac{4\pi K_e}{F}\frac{z r^2 t}{V}}\right), \tag{6} \]

From expression (6) there immediately follows the proportionality between the current \(I_a\) of the anodic peak and the initial concentration \(C_i\) of the metal ions in solution.

The dependence of \(I_a\) on \(t\), \(V\), and \(r\) is more complex. Let us first consider two limiting cases. With almost complete depletion of the solution, from formula (6) we obtain:

\[ \text{for } B>3;\quad \gamma>0.95;\quad I_a=a=3K_a C_i\frac{V}{r}. \tag{7} \]

It is seen from (7) that, with complete depletion of the solution, the anodic peak current is proportional to the volume of the solution, inversely proportional to the radius of the amalgam drop, and does not depend on the duration of the preliminary electrolysis. If the given amount of metal is present in different volumes of solution, then in equations (6) and (7) the product \(C_iV=\mathrm{const}\), and from formula (7) it follows that \(I_a\) does not depend on the volume of solution in which the given amount of the element being determined is dissolved.

With practically no depletion of the solution, from (6) we have:

\[ \text{for } B<0.05;\quad \gamma \simeq B;\quad I_a=aB=\frac{12\pi}{F}K_aK_eC_irt. \tag{8} \]

It is seen from (8) that, in the absence of depletion of the solution, the anodic peak current is proportional to the radius of the amalgam drop and to the time of preliminary electrolysis and does not depend on the volume of the solution.

It is advisable to obtain plots of the dependence of the anodic peak current \(I_a\) on the factors \(t\), \(v\), and \(r\) in dimensionless form. It is easy to see that if the time \(t\) of electrolysis is considered as the variable factor, then the plot should be represented in dimensionless coordinates

\[ y_t=\frac{I_a}{a},\quad x=B \]

(Fig. 1, curve 1); if the volume \(V\) of the solution is considered as the variable factor, then the plot should be represented in the coordinates

\[ y_V=\frac{I_a}{aB},\quad x=\frac{1}{B} \]

(Fig. 1, curve 2); and if the variable quantity is the radius \(r\) of the mercury drop, then in the coordinates

\[ y_r=\frac{I_a}{aB^{1/2}},\quad x=B^{1/2} \]

(Fig. 1, curve 3). From the plot (Fig. 1) it is seen that the dimensionless anodic peak current increases with increasing duration of the preliminary electrolysis and with the volume of the solution and reaches the limiting value of unity (curves 1 and 2). With increasing radius of the amalgam drop, the curve \(y,x\) (curve 3) passes through a maximum at \(x_m=1.12\); \(B_m=x_m^2=1.26\); \(y_m=0.638\).

Fig. 1. Theoretical plot of the dependence of the anodic peak current on electrolysis time in dimensionless coordinates \(\frac{I_a}{a}\)–\(B\) (curve 1), on the volume of the solution in coordinates \(\frac{I_a}{aB}\), \(\frac{1}{B}\) (curve 2), and on the radius of the mercury drop in coordinates \(\frac{I_a}{aB^{1/2}}\), \(B^{1/2}\), in accordance with formula (7).

The numerical values of the coefficients \(K_e\) and \(K_a\) can easily be found from the curve \(I_a,t\) by means of the following device [4]. They choose

on the abscissa of the plot \(I_a, t\) two points corresponding to electrolysis times equal to \(t_1\) and \(2t_1\). We then obtain:

\[ a=\frac{I_1}{2-I_2/I_1};\quad B_1=-2.3\lg(1-I_1/a)=B_t t_1, \tag{9} \]

where \(I_1\) and \(I_2\) are the heights of the anodic peak at electrolysis times \(t_1\) and \(2t_1\), respectively. From (4) and (5) we find

\[ K_a=\frac{1}{3}\frac{r}{C_iV}\cdot a;\quad \frac{1}{z}K_e=\frac{F}{4\pi}\frac{V}{r^2t_1}\cdot B_1. \tag{10} \]

Formula (6) may be represented in the form:

\[ -\lg(1-I_a/a)=0.434\,B. \tag{11} \]

It follows from formula (11) that the experimental points on a plot in the coordinates \(-\lg(1-I_a/a), B\) for all three dependences \((I_a, t; I_a, V; I_a, r)\) should lie on one and the same straight line passing through the origin and having a slope equal to 0.434.

Fig. 2. Comparison of theory with experiment for the dependence of the anodic peak current on electrolysis time in the coordinates \(\dfrac{I_a}{a}, B\) (curve 1) and on solution volume in the coordinates \(\dfrac{I_a}{aB}, \dfrac{1}{B}\) (curve 2). Curve 3 is the universal straight line for both dependences in the coordinates \(-\lg(1-\dfrac{I_a}{a}), B\), with slope 0.434 (see formulas (7) and (14)). Experimental conditions for the dependence of \(I_a\) on \(t\): \(5\cdot10^{-5}\) g-ion/l \(Pb^{+2}\) + MKop \(V=0.5\) ml; \(r=0.04\) cm; \(K_e=2000\) a·cm/g-eq; \(K_a=90\) a·cm/g-eq; 25°C; \(t\) from 3 to 50 min. Experimental conditions for the dependence of \(I_a\) on \(v\): \(5\cdot10^{-5}\) g-ion/l \(Pb^{2+}\) + 0.1 M KOH; \(t=1800\) sec; \(r=0.04\) cm; \(K_e=800\) a·cm/g-eq; \(K_a=90\) a·cm/g-eq; 25°C; \(V\) from 0.1 to 3.0 ml.

Fig. 3. Dependence of the anodic peak current of cadmium on the radius of a mercury drop. Solution composition: \(5\cdot10^{-5}\) g-ion/l \(Cd^{2+}\) + 0.5 M \(Na_2SO_4\). Curves 1, 2, and 3 are in the coordinates \(j_a B'^{1/2}\) at different values of the rate of potential change: 1 — \(W\to\infty\); 2 — \(W=1\) V/sec; 3 — \(W=0.016\) V/sec = 1000 mV/min. The solid lines for curves 1, 2, and 3 were calculated by formula (18) at \(E=0\). Experimental conditions: \(V=0.1\) ml; \(t=1200\) sec; 25°C; \(r\) from 0.02 to 0.065 cm. Experimental parameter values: \(K_e=550\) a·cm/g-eq; \(K_a=850\,W^{1/2}/(1+1.15\,\varepsilon)\) a×cm/g-atom.

The dependence of the anodic peak current on electrolysis time and on solution volume, as predicted by the parametric theory, is in good agreement with experiment. In Fig. 2 (curves 1 and 3) experimental data are presented for the dependence of the anodic peak current of lead,

as a function of electrolysis time in dimensionless coordinates \(\dfrac{I_a}{a}\), \(B\), and \(-\lg\left(1-\dfrac{I_a}{a}\right)\), \(B\). Curves 2 and 3 present experimental data for the dependence of the current of the anodic lead tooth on the volume of solution in dimensionless coordinates \(\dfrac{I_a}{a}\), \(\dfrac{1}{B}\), and \(-\lg\left(1-\dfrac{I_a}{a}\right)\), \(B\) [4]. The lines in curves 1, 2, and 3 are theoretical curves calculated from formulas (6) and (11). The experimental conditions are given in the caption to Fig. 2.

A more complicated question is the effect of the radius of the amalgam drop on the anodic peak current. It is seen from Fig. 3 that theoretical curve 1 in the plot in coordinates \(\dfrac{I_a}{aB^{1/2}}\), \(B^{1/3}\), calculated from expression (6), agrees poorly with the experimental data on the dependence of the height of the anodic thallium peak on the radius of the amalgam drop.

The reason for this discrepancy between theory and experimental data is that formula (6) did not take into account the dependence of the anodic peak coefficient on the size of the amalgam drop. From theoretical calculations [6–11] for a reversible anodic peak it follows that the anodic peak coefficient \(K_a\) depends on the radius of the amalgam drop through the parameter \(\varepsilon\):

\[ K_a/K_0 = 7.03\,\varepsilon \Phi(\varepsilon);\quad \text{(at }25^\circ\mathrm{C}\text{),} \tag{12} \]

where

\[ \Phi(\varepsilon)= \int_{t_0}^{t_p} \frac{ \sum \exp\left[-(n\pi\varepsilon)^2 ZW(t_p-\tau)\right] }{ c h^2 \dfrac{F}{2RT} ZW(\tau-t_0) } \,ZW\,d\tau; \tag{13} \]

\[ \varepsilon=\frac{1}{r}\left(\frac{D}{ZW}\right)^{1/2}, \tag{14} \]

where \(t_p\) is the time at which the anodic peak is reached; \(t_0\) is the time at which the half-wave potential is reached; \(D\) is the diffusion coefficient of the metal in mercury, \(\mathrm{cm^2/sec}\); \(K_0\) is the limiting value of \(K_a\) as \(\varepsilon \to 0\).

In a plot in coordinates \(K_a/K_0\), \(1/\varepsilon\), the dependence calculated from equation (12) by numerical integration is represented by a curved line passing through the origin and approaching the limiting value of unity as \(1/\varepsilon \to \infty\). In practical work on a recording polarograph, \(\varepsilon = 0.9 \div 1.1\) and

\[ \frac{K_a}{K_0}=0.4 \div 0.6 \]

(for a mercury drop).

The graph of the theoretical dependence of \(K_a/K_0\) on \(1/\varepsilon\) can be satisfactorily represented by the interpolation formula:

\[ \frac{K_a}{K_0} = \frac{1}{1+\sigma\varepsilon} = \frac{1/\varepsilon}{1/\varepsilon+\sigma} \tag{15} \]

(\(\sigma\) is an empirical coefficient, equal to about 1.2 for the metals studied). As \(1/\varepsilon \to 0\), \(K_a/K_0 \to 0\), and as \(\varepsilon \to 0\), \(K_a/K_0 \to 1\), in agreement with the theoretical formula (12). Formula (12) was derived for the case of a reversible electrode process. It may be assumed that relations (12) and (15) will also approximately describe the dependence of \(K_a\) on \(\varepsilon\) for irreversible electrode processes.

The dependence of the anodic-peak coefficient on the radius of the mercury drop is obtained from expression (15), taking into account the expression of \(\varepsilon\) through \(r\):

\[ K_a = K_0 \frac{r}{r+\sigma\varepsilon_*}; \qquad \varepsilon_*=\left(\frac{D}{ZW}\right)^{1/2}. \tag{16} \]

Substituting the expression for \(K_a\) from (15) into (4), we obtain:

\[ a=3K_a C_i \frac{V}{r}=\frac{a_*'}{r+\sigma\varepsilon_*}; \qquad a_*'=3K_0 C_i V. \tag{17} \]

And for the anodic-peak current in dimensionless coordinates we obtain the expression

\[ y_r=\frac{I_a}{aB^{1/2}}=\frac{1-e^{-B}}{B^{1/2}+E}; \qquad E=B^{1/2}\sigma\varepsilon . \tag{18} \]

From the maximum condition for the function \(y_r\) \((dy_r/dB=0)\), we obtain the relation between the parameters \(E\) and \(B\) at the maximum of the curve \(y_r, B\)

\[ 2B_m+2B_m^{1/2}E=e^{B_m}-1. \tag{19} \]

An interpolation formula for this transcendental equation can be represented in the form (for \(E\) from 0 to 2):

\[ B_m=1.26+0.80E^{0.70}. \tag{20} \]

It follows from formula (20) that, as \(E\) increases from 0 to 1.5, the value of \(B_m\) at the maximum of the curve shifts from 1.26 to 2.3; at the same time the height of the anodic peak (relative value) decreases from 0.64 to 0.30. The value of the radius of the amalgam drop at which the maximum on the curve \(I_a, r\) is attained under the given conditions is found from the relation

\[ r_m=\left(\frac{B_m}{B_*}\right)^{1/2}=\frac{\varepsilon_*}{\varepsilon_m}. \tag{21} \]

As is seen from Fig. 3, when the dependence of the anodic-peak coefficient on the size of the amalgam drop is taken into account, agreement is obtained between the parametric theory and the experimental data on the dependence of the anodic-peak height on the radius of the amalgam drop [11].

The theoretical dependence of the anodic-peak coefficient on the parameter \(\varepsilon\) also makes it possible, with the aid of the parametric theory, to explain the effect of the rate of change of the potential on the height of the anodic peak [9, 10, 12]. From formula (15) we obtain:

\[ \frac{K_0}{K_a}-1=\sigma\varepsilon; \qquad K_a=\frac{ZFrI_a}{3q}; \qquad q=ZFVC_a \tag{22} \]

(\(q\) is the area under the anodic peak, coulomb). From equation (15) we further obtain the dependence of the anodic-peak coefficient on the rate of change of the potential:

\[ K_a=\frac{K_*W}{W^{1/2}+H_*}; \qquad K_*=\frac{K_0}{W^{1/2}}; \qquad H_*=\frac{\sigma\varepsilon}{W^{1/2}}=\frac{\sigma}{r}\left(\frac{D}{Z}\right)^{1/2}. \tag{23} \]

Let us note that interpolation formula (23) correctly describes the limiting cases: a linear dependence of \(K_a\) on \(W\) at small \(W\), and a square-root dependence at large \(W\).

Substituting the expression for \(K_a\) from (23), we obtain an expression for the dependence of the anodic-peak current on the rate of change of the electrode potential:

\[ I_a=a_*\frac{W}{W^{1/2}+H_*}\gamma; \qquad a_*=3K_*C_i\frac{V}{r}. \tag{24} \]

And in dimensionless coordinates:

\[ y_{\omega}=\frac{\omega}{1+\omega^{1/2}};\quad \text{or}\quad \frac{\omega}{y_{\omega}}=1+\omega^{1/2}, \tag{25} \]

where

\[ \omega=\frac{W}{H^2}=(\sigma \varepsilon)^{-2};\quad y_{\omega}=\frac{I_a}{a_* H_* \gamma}. \tag{26} \]

In accordance with formula (25), the experimental points for gallium on the plot in the coordinates \(\dfrac{\omega}{y_{\omega}}, \omega^{1/2}\) (Fig. 4) are satisfactorily arranged on a straight line with slope equal to unity and passing through the point with coordinates \((0; 1)\).

Let us next consider some cases of the influence of the electrolysis potential on the anodic peak current. The simplest case is an irreversible electrode process without depletion of the solution [13]. In this case, simple qualitative considerations lead to the conclusion that the electrolysis current and the anodic peak current must depend on the electrolysis potential (at a potential more positive than the potential for attaining the limiting current) in the same way as the current depends on the potential for a polarographic wave. These qualitative considerations can be expressed by means of the following relations (in dimensionless form):

\[ \begin{gathered} j_a=j_e=j_p=\frac{\theta}{\theta+1};\quad j_a=\frac{i_a}{I_a};\quad j_e=\frac{i_e}{I_e};\quad j_p=\frac{i_p}{I_p}; \\ \text{where} \\ \lg \frac{j_a}{1-j_a}=\lg \frac{j_e}{1-j_e}=\lg \frac{j_p}{1-j_p}=\lg \theta, \\ 59\lg \theta=\alpha Z(\varphi_{1/2}-\varphi_e),\ \text{mV (at }25^\circ\mathrm{C}\text{).} \end{gathered} \tag{27} \]

(\(j_a, j_e, j_p\) are, respectively, the anodic peak current, the electrolysis current, and the polarographic current in dimensionless form; \(\varphi_e\) is the electrolysis potential; \(\varphi_{1/2}\) is the half-wave potential). These relations are consistent with experiment.

The dependence of the current of an irreversible anodic peak on the electrolysis potential in the presence of solution depletion during electrolysis has a more complicated form. It can be shown that in this case the electrolysis coefficient and the parameter \(B\) depend on the electrolysis potential as follows [14]:

\[ K_{e\theta}=K_e\frac{\theta}{\theta+1};\quad B_{\theta}=B\frac{\theta}{\theta+1}. \tag{28} \]

Hence, for the dependence of the electrolysis current and the anodic peak current on the electrolysis potential we obtain the expressions:

\[ i_e=\frac{\theta}{\theta+1}K_e S C_i e^{-B\frac{\theta}{\theta+1}};\quad i_a=a\left(1-e^{-B\frac{\theta}{\theta+1}}\right). \tag{29} \]

From formulas (29) and (6) we obtain

\[ j_a=\frac{i_a}{I_a}=\frac{1-e^{-B\frac{\theta}{\theta+1}}}{1-e^{-B}}. \tag{30} \]

For \(B>3\) we may take the denominator in formula (30) to be equal to unity. In this case:

\[ j_a=1-e^{-B\frac{\theta}{\theta+1}}. \tag{31} \]

For \(j_a=0.5\), \(B>3\), we obtain:

\[ \alpha Z(\varphi_{1/2}-\varphi_{e/2})=59\lg\theta_{1/2} =59\lg\frac{0.69}{B_{1/2}-0.69}. \tag{32} \]

(\(\varphi_{e/2}\) is the potential of the midpoint of the curve on a plot in the coordinates \(j_a,\varphi_e\), at \(j_a=0.5\)).

It follows from formula (32) and the graph (Fig. 5) that at large values of \(B\) (\(B=3\div 10\)) the curve \(j_a,\varphi_e\) should shift noticeably toward more positive potential values in comparison with the polarographic wave. This prediction of the theory is confirmed by experimental data [14] on the effect of the electrolysis potential on the current of the anodic peak of zinc at a high degree of depletion of the solution (Fig. 5). It is seen from the figure that, at a high degree of depletion of the solution, the curve \(j_a,\varphi_e\) shifts toward more positive potentials in

Fig. 4. Dependence of the current of the anodic peak of thallium on the rate of change of the electrode potential (formula 26). Composition of the solution: \(10^{-5}\) g-ion/l Tl\(^+\) + \(0.5M\) KNO\(_3\). Experimental conditions: \(t=600\) sec; \(V=5\) ml; \(r=0.04\) cm; \(T=20^\circ\)C; \(W\) from \(4\cdot10^{-4}\) to \(1.5\) V/sec. Experimental values of the coefficients: \(K_e=950\) A·cm/g-ion; \(K_a=850\) A·cm·sec\(^{-1/2}\) g \(B^{1/2}\); \(H_w=0.092\) V\(^{1/2}\) sec\(^{-1/2}\).

Fig. 5. Dependence of the current of the anodic peak on the electrolysis potential for an irreversible electrode process in the presence of depletion of the solution during electrolysis. Solid lines are theoretical curves calculated from formula (30) for different values of the parameter \(B\). Experimental conditions: \(6\cdot10^{-6}\) g-ion/l Zn\(^{2+}\) + \(0.5M\) KOH; \(V=5\) ml for the curve at \(B=0.1\); \(V=0.5\) ml for the curves at \(B=2.15\) and \(4.30\). Electrode—mercury-film: \(S=0.38\) cm\(^2\), \(l=5.3\cdot10^{-4}\) cm; \(S\) is the electrode surface; \(l\) is the thickness of the mercury film; \(W=0.016\) V/sec; \(T=25^\circ\)C.

comparison with the polarographic wave, and quantitative agreement between theory and experiment is observed.

The dependence of the current of the anodic peak on temperature can be obtained if one takes into account the temperature dependence of the coefficients of electrolysis \(K_e\) and of the anodic peak \(K_a\):

\[ K_e=K_e^* e^{-\frac{E_e}{RT}};\qquad K_a=K_a^* e^{-\frac{E_a}{RT}}, \tag{33} \]

where \(E_e\) and \(E_a\) are the effective activation energies of the overall electrolysis process and of the anodic dissolution of the metal from the amalgam; \(K_e^*,K_a^*\) are pre-exponential factors, which may approximately be regarded as independent of temperature.

Substituting the values of \(K_e\) and \(K_a\) from formula (33) into (6), we obtain an expression for the dependence of the anodic peak current on temperature:

\[ I_a=a_* e^{-\frac{E_e}{RT}}\left(1-e^{-B_* e^{-\frac{E_e}{RT}}}\right), \tag{34} \]

where \(a_*\), \(B_*\) do not depend (approximately) on temperature.

In dimensionless form, the dependence of the anodic peak current \(y_T\) on the temperature \(B\) is given by the expression

\[ y_T = B^\alpha \gamma = B^\alpha(1-e^{-B}), \tag{35} \]

where

\[ y_T=\frac{I_a B^\alpha}{a}; \qquad \alpha=\frac{E_a}{E_e}. \]

For small values of the parameter \(B\) (\(B \leqslant 0.5\)) we have \(\lg y_T=(1+\sigma)\lg B\); for large values of \(B\) (\(B \geqslant 3\)), \(\lg y_T=\sigma\lg B\); for intermediate values of \(B\) (\(0.05<B<3\)), \(\lg y_T=\sigma\lg B+\lg(1-e^{-B})\). Figure 6 presents the theoretical dependence of the anodic peak on temperature in the dimensionless coordinates \(\lg y_T\), \(\lg B\). This dependence is confirmed experimentally [33—34] if the electrode process is irreversible.

Fig. 6. Theoretical plot of the dependence of the anodic peak current on temperature in the dimensionless coordinates \(\lg \dfrac{I_a B^\alpha}{a}\), \(\lg B\), in accordance with formula (35)

For reversible electrode processes, the dependence of the anodic peak coefficient on temperature, obtained on the basis of the rigorous theory of reversible anodic peaks, has the form [35]: for a mercury film electrode

\[ K_a=K_{a,T}\frac{T^{-1/2}e^{-\rho \chi}}{d+Te^{-2\rho \chi}}; \tag{36} \]

for a stationary mercury dropping electrode

\[ K_a=m\frac{T^{-1/2}}{n+e^{-\rho \chi}}, \quad \text{where} \tag{37} \]

\[ K_{a,T}=0.248\,\frac{Z^{3/2}F^{3/2}}{R^{3/2}}\cdot \frac{l^{2}w^{1/2}}{D_{R,T}^{1/2}}, \tag{38} \]

\[ d=0.556\,\frac{ZF}{R}\cdot\frac{l^{2}w}{D_{R,T}}, \tag{39} \]

\[ m=0.446\,\frac{z^{3/2}F^{3/2}}{R^{1/2}}\,w^{1/2}D_{R,T}^{-1}, \tag{40} \]

\[ n=66.9\,\frac{R^{1/2}}{z^{1/2}F^{1/2}}\cdot \frac{1}{rw^{1/2}}\cdot D_{R,T}^{-1}, \tag{41} \]

\[ \chi=\frac{E_{D_\mathrm{e}}}{2RT}, \tag{42} \]

\[ \frac{E_{D_R}}{E_{D_\mathrm{e}}}=\rho. \tag{43} \]

In (42) and (43), \(E_{D_\mathrm{e}}\) and \(E_{D_R}\) are the activation energies of the diffusion process of metal ions in solution and metal atoms in the amalgam.

Taking (36) and (37) into account, for the dependence of the height of the reversible anodic peak on temperature over a wide range of rates of change of potential and electrode sizes, one may write: for a mercury film electrode

\[ Y'=\frac{I_a'}{a_T'}= \frac{T^{-1/2}e^{-\rho\chi}\left(1-e^{-B_\chi}\right)} {d+Te^{-2\rho\chi}}; \tag{44} \]

for a stationary mercury dropping electrode

\[ Y=\frac{I_a'}{a_T}= \frac{T^{-1/2}\left(1-e^{-B_\chi}\right)} {n+e^{-\rho\chi}}, \tag{45} \]

where

\[ a_T'=K_{a,T}\frac{s}{v}\,C_iV, \tag{46} \]

\[ a_T=m\frac{3}{r}C_iV, \tag{47} \]

\[ B_\chi=\frac{K_{e,T}}{zF}\,s\,\frac{t}{V}. \tag{48} \]

Expressions (44) and (45) were subjected to experimental verification [35], which showed good agreement with theory.

On the basis of the parametric theory of anodic stripping polarography (ASP), we have developed theoretical considerations [15—21] concerning the limiting sensitivity of ASP and ways of achieving this sensitivity, which made it possible to increase the sensitivity of ASP by several orders of magnitude [22—27]. Theoretical considerations have also been developed concerning the resolving power of the ASP method [28—29].

LITERATURE

1. A. G. Stromberg, Izv. SO AN SSSR, No. 5, 76, 1962.
2. I. Shain, J. Lewinson. Analyt. Chem., 33, 187, 1961.
3. L. N. Vasil’eva, E. N. Vinogradova. “Zavod. labor.”, 27, 1079, 1961.
4. A. G. Stromberg. Izv. TPI, 164, 128, Tomsk, 1967.
5. A. G. Stromberg, A. A. Kaplin. “Zavod. labor.”, 30, 525, 1964.
6. V. E. Gorodovykh. Izv. TPI, 128, 3, Tomsk, 1964.
7. V. E. Gorodovykh, V. F. Nazarov. Ibid., 164, 7, 1967.
8. V. E. Gorodovykh. Ibid., 164, 11, 1967.
9. V. E. Gorodovykh, A. G. Stromberg, B. F. Nazarov. Ibid., 164, p. 15.
10. A. G. Stromberg, B. F. Nazarov, V. E. Gorodovykh. Ibid., 164, p. 20.
11. A. G. Stromberg, B. F. Nazarov. Ibid., 164, p. 25, 1967.
12. V. E. Gorodovykh. Ibid., 164, p. 30.
13. M. S. Zakharov, “Zavod. labor.”, 30, 14, 1964.
14. A. G. Stromberg, L. G. Baletskaya. “Elektrokhimiya”, 5, 20, 1969.
15. A. G. Stromberg. “Zavod. labor.”, 29, 387, 1963.
16. V. A. Igolinskii, A. G. Stromberg. “Zavod. labor.”, 30, 656, 1964.
17. A. G. Stromberg, A. A. Kaplin. Zh. analit. khimii, 20, 1043, 1965.
18. A. G. Stromberg. “Zavod. labor.”, 31, 1175, 1965.
19. A. A. Kaplin, A. G. Stromberg. Izv. TPI, 164, 108, Tomsk, 1967.
20. A. G. Stromberg, M. N. Potapov. Ibid., p. 114.
21. A. A. Kaplin, L. F. Trushina. Ibid., p. 119.
22. A. G. Stromberg, V. I. Kuznetsov, A. V. Kon’kova. Ibid., p. 133.
23. A. A. Kaplin, V. M. Pichugina. Ibid., p. 137.
24. A. V. Kon’kova, A. G. Stromberg. Ibid., p. 147.
25. Yu. A. Karbainov, A. G. Stromberg. Zh. analit. khimii, 20, 769, 1965.
26. A. G. Stromberg, A. A. Kaplin et al. In: “Application of Polarography in the Chemistry and Technology of Polymers,” IREA, issue 5 (11), 109, Moscow, 1966.
27. Yu. A. Karbainov, A. G. Stromberg. Ibid., p. 230.

1. A. G. Stromberg, V. F. Yankauskas. Izv. TPI, 164, 99, Tomsk, 1967.
2. V. E. Gorodovykh. Ibid., p. 104.
3. A. G. Stromberg, E. A. Stromberg. Zavod. laboratoriya, 27, 1, 1961.
4. A. G. Stromberg, E. A. Zakharova. Zavod. laboratoriya, 30, 261, 1964.
5. A. G. Stromberg, E. A. Zakharova. In: Modern Methods for the Analysis of Materials, “Metallurgiya,” 1969.
6. A. G. Stromberg, Yu. A. Karbainov, S. N. Karbainova. Zavod. laboratoriya, 6, 257, 1970.
7. Yu. A. Karbainov, A. G. Stromberg, S. N. Karbainova. Proceedings of Higher Educational Institutions, Chemistry and Chemical Technology, 6, 345, 1970.
8. Yu. A. Karbainov, S. N. Karbainova, A. G. Stromberg. Elektrokhimiya, 7, 1409, 1971.

1. From comparison with Fick’s equation for the stationary process of convective diffusion it follows that $K_e = ZF\dfrac{D}{\delta}$, where $D$ is the diffusion coefficient; $Z$ is the charge of the metal ions; $\delta$ is the effective thickness of the diffusion layer, which depends on the intensity of stirring of the solution, temperature, diffusion coefficient, viscosity of the solution, and other factors. ↩

2. After 5–10 minutes of electrolysis, the distribution of the metal in the mercury drop becomes practically uniform, and therefore $C_a$ is the average concentration of the metal in the amalgam [2, 3]. ↩