ON THE QUESTION OF THE INFLUENCE OF THE MERCURY DROP RADIUS ON THE CURRENT OF THE ANODIC PEAK IN THE APN METHOD

A. G. STROMBERG, B. F. NAZAROV

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

In previous works [1—6], in the theoretical consideration of the influence of various factors on the depth of the anodic peak in the method of amalgam polarography with accumulation (APN), it was assumed that the constant of the anodic peak does not depend on the radius of the hanging mercury drop. Recently, from the rigorous theory of the anodic current during dissolution of a metal from an amalgam in the form of a sphere under a linearly varying potential, it has been shown [7—9] that the constant of the anodic current depends on the radius of the mercury drop, and, thus, the assumption made earlier is approximate and sufficiently valid only in a comparatively narrow interval of variation of the radius of the mercury drop. Therefore, a number of consequences from the former theoretical propositions require clarification.

The aim of the present work is to consider the question of the influence of the radius of the mercury drop on the depth of the anodic peak, taking into account the influence of the radius of the mercury drop on the constant of the anodic current.

In work [1] it was shown that the dependence of the depth of the anodic peak \(I\) on the radius of the mercury drop \(r\) and other factors can be represented by the expression [1]:

\[ I = K_2 3CV \frac{1}{r}\gamma, \tag{1} \]

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

\[ b = Br^2, \tag{3} \]

\[ B = \frac{4\pi K_1}{zF}\frac{t}{V}, \tag{4} \]

where \(K_2\) is the constant of the anodic peak, \(a \cdot cm/g\text{-}atom\), determined by the relation \(I = K_2 SC_2\); \(S\) is the electrode surface, \(cm^2\); \(C_2\) is the initial concentration of metal atoms in the amalgam, \(g\text{-}atom/cm^3\); \(K_1\) is the electrolysis constant, \(a \cdot cm/g\text{-}ion\), determined by the relation \(i = K_1 SC\); \(i\) is the electrolysis current; \(C\) is the initial concentration of ions in the solution (before electrolysis), \(g\text{-}ion/cm^3\); \(V\) is the volume of the solution, \(cm^3\); \(t\) is the duration of electrolysis, sec; \(\gamma\) is the degree of exhaustion; \(b\) is a dimensionless parameter; \(B\) is a coefficient independent of the radius of the mercury drop and determined from formula (4); \(z\) is the number of electrons participating in the electrode process; \(F\) is the Faraday constant.

Formula (1), taking into account (2) and (3), can be represented in dimensionless form by the following equation [1]:

\[ j=\frac{I}{A}=\frac{1-e^{-b}}{b^{1/2}}, \tag{5} \]

where

\[ A=K_2SCVB^{1/2}. \tag{6} \]

The dependence of \(j\) on \(b\) gives, in dimensionless form, the dependence of \(I\) on \(r\), provided the other factors \((t, V, C)\) remain constant.

Let us find the maximum of the function \(j\) as a function of \(b\), assuming \(K_2\) to be independent of the radius of the mercury drop (the parameter \(b\)). From the condition \(dj/db=0\), we obtain the following relation for the value \(b_*\) of the parameter \(b\) at the maximum value of the function \(y\):

\[ \frac{dy}{db}=e^{-b}\left(b^{1/2}+\frac{1}{2}b^{-1/2}\right)-\frac{1}{2}b^{-1/2}=0, \tag{7} \]

\[ e^{b_*}-1=2b_*. \tag{8} \]

By the method of successive approximations we find \(b_* = 1.25\) [1,2]. Hence \(\gamma=0.713\), and

\[ j_*=\frac{1-e^{-b_*}}{b_*^{1/2}}=2b_*^{1/2}e^{-b_*}=0.64, \tag{9} \]

\[ I_*=0.64\,A. \tag{10} \]

Let us make a numerical estimate. Taking \(t=20\) min; \(K_1=5\cdot10^3\) a·cm/g-ion; \(V=1\) ml; \(z=2\), and \(b_*=1.25\), we obtained from (3) and (4) that \(r_*=0.05\) cm. In this case the depth of the anodic peak has the maximum value in accordance with formulas (10) and (6). If we increase the radius \(r\) twofold \((b=2.5;\ \gamma=0.97);\ j=0.61\) (or decrease it twofold); \(b=0.62;\ \gamma=0.46;\ j=0.58\), then the depth of the anodic peak, with the other factors \((t;\ V;\ K_1;\ C)\) constant, will decrease, and from formula (5) we obtain \(j_*:j_{2*}:j_{1/2*}=1:0.95:0.91\).

Let us now see what changes are introduced into the above reasoning by taking into account the dependence of \(K_2\) on the radius of the mercury drop. The theoretical expression obtained in [7] for the dependence of the depth of the reverse anodic peak on various factors contains an integral that cannot be expressed in elementary functions. The numerical solution of this integral [8] makes it possible to obtain a graph in the coordinates \(\frac{K_0}{K_2}, \varepsilon\).

The theoretical curve in this graph can be satisfactorily represented by the following empirical (interpolation) formula [7, 8]:

\[ \frac{K_0}{K_2}-1=1.25\,\varepsilon^2, \tag{11} \]

\[ \varepsilon=\frac{1}{r}\left(\frac{D}{zw}\right)^{1/2}, \tag{12} \]

\[ K_0=2.68\cdot10^5\,z^{3/2}D^{1/2}w^{1/2}, \tag{13} \]

where \(\varepsilon\) is the parameter defined by relation (12) [9]; \(K_0\) is the value of the coefficient \(K_2\) for semi-infinite linear diffusion, defined by formula (13) [11]; \(D\) is the diffusion coefficient of the metal atoms in the amalgam, cm\(^2\)/sec; \(w\) is the rate of change of potential, volt/sec.

From formulas (11) and (3), after simple transformations we obtain an expression for the dependence of \(K_2\) on the parameter \(b\):

\[ K_2 = K_0 \frac{b}{b+a}, \tag{14} \]

\[ a = HB, \tag{15} \]

\[ H = \frac{1.25D}{z w}. \tag{16} \]

Substituting (14) into (5) and (6), we obtain, in dimensionless form, the dependence of the depth of the anodic notch on the radius of the mercury drop, taking into account the nonconstancy of the value \(K_2\):

\[ j = \frac{I}{A_0} = \frac{b^{1/2}}{b+a}(1-e^{-b}), \tag{17} \]

\[ A_0 = K_0 3CVB^{1/2}, \tag{18} \]

where \(K_0\) and \(B\) are determined from formulas (13) and (4). Solving the problem of the maximum for the function \(j\) as a function of \(b\) according to formula (17), we obtain, for the value \(b_*\) at the maximum value \(j=j_*\), the expression

\[ 2b_* \frac{b_*+a_*}{b_*-a_*} = e^{b_*}-1 . \tag{20} \]

The dependence between the parameters \(b_*\) and \(a_*\), found from (20) by the method of successive approximations, is presented in the graph (Fig. 1, curve 1). In analytical form this dependence (for \(b_* > 1.25\)) can be represented by the following empirical (interpolation) formula:

\[ a_* = 0.21\, b_*^2 - 0.33 . \tag{21} \]

From (17) and (20), for the maximum depth of the anodic notch we obtain the expression

\[ y_* = \frac{I_*}{A_0} = \frac{b_*^{1/2}}{b_*+a_*}(1-e^{-b_*}) = \frac{2b_*}{b_*-a_*} e^{-b_*}, \tag{22} \]

\[ b_* = (4.75\,a_* + 1.43)^{1/2}, \tag{23} \]

\[ K_{2*} = K_0 \frac{b_*}{b_*+a_*}. \tag{24} \]

The theoretical dependence of \(j_*\) on \(b_*\), calculated by formula (22) and at the corresponding value of \(a_*\) by formula (23), is presented in Fig. 1, curve 2. From Figs. 1 and 2 it is seen that, with increasing \(a_*\), the depth of the anodic notch decreases, while the maximum value \(j_*\) is reached at large values of the parameter \(b_*\). At \(a_* > 1.6\) the maximum value of the notch depth (with an increase in the drop radius \(r\)) is reached at \(b_* > 3\) and \(\gamma > 0.96\), i.e., under conditions of practically complete depletion of the solution.

Figure 3 presents the dependence of the anodic-current constant on the size of the mercury drop (through the parameter \(b\)) for various specified values of the parameter \(a\) in accordance with formula (14), and under the conditions of the maximum current according to formula (24), for values of \(a_*\) and \(b_*\) related to one another by formula (23), corresponding to the maximum depth of the notch (curve 6). It is seen from the figure that the dependence of \(K_{2*}\) on \(b_*\)

(curve 6) has a different character than \(K_2\) as a function of \(b\) at \(a=\mathrm{const}\). This is due to the fact that (as is seen from the figure), under the conditions of maximal \(y_*\), simultaneously with the growth of \(b_*\), the parameter \(a_*\) also increases in accordance with formula (21) or (23)\(^1\).

Let us make a numerical estimate of the quantities. Take: \(D=1.6\cdot10^{-5}\ \mathrm{cm^2/sec}\); \(z=2\); \(w=10^{-2}\ \mathrm{V/sec}\). Estimation of the quantity \(H\) by formula (16) gives \(H=10^{-3}\). Further take: \(K_1=5\cdot10^3\ \mathrm{A\cdot cm/g\text{-}ion}\); \(z=2\); \(t=120\ \mathrm{sec}\); \(v=1\ \mathrm{ml}\). Estimation of the quantity \(B\) by formula (4) gives \(B=780\). By formula (15) we find \(a_*=0.78\), by formula (23) or by curve 1 of Fig. 1, \(b_*=2.27\), and by formula (3) \(r=5.4\cdot10^{-2}\ \mathrm{cm}\). Further, by formulas (22), (13), and (18) we find \(j_*=0.444\), \(K_0=332\), \(A_0=2.54\cdot10^{-7}\) (at \(C=10^{-11}\ \mathrm{g\text{-}ion/cm^3}\)) and \(i_*=A_0j_*=1.13\cdot10^{-7}\ \mathrm{A}\). By formula (14), \(K_{2*}=0.745\), \(K_0=225\).

Fig. 1. Dependence of the maximum value of the depth of the anodic notch \(y_*\) (curve 1) and of the corresponding values of the parameter \(a_*\) (curve 2) and the degree of depletion \(\gamma_*\) (curve 3) on the dimensionless parameter \(b_*\) (see formulas (22), (21), and (2)).

For \(r=2r_*\) and \(r=\dfrac{1}{2}r_*\), from formulas (3) and (17) we obtain \(b=9.1\) and 0.57 and \(j=0.303\) and 0.314 (at the same \(a\)). Hence

\[ y_*:y_{2*}:y_{1/2*}=1:0.70:0.71. \]

Fig. 2. Dependence of the depth of the anodic notch on the size of the mercury drop (parameter \(b\)) at different values of the parameter \(a\). Curve 1—\(a=0\); 2—0.5; 3—1.0; 4—2.0.

Fig. 3. Dependence of the anodic-current constant on the parameter \(b\) (the size of the mercury drop) at different values of the parameter \(a\). Curve 1—\(a=0\); 2—0.1; 3—0.5; 4—1.0; 5—2.0. Curve 6—for values of \(b_*\) and \(a_*\) corresponding to the maximum values of the notches.

Comparison with the calculation made earlier shows that, under the same conditions (\(K_1\); \(t\); \(V\)), taking account of the dependence of \(K_2\) on \(r\) leads to obtaining \(y_*\) at a larger \(r\) (0.054 instead of 0.050 cm) and at a larger \(b\)

\(^1\)

\[ \lim_{b_*\to\infty}\left|\frac{K_0}{K_{2*}}\right|\to\infty. \]

This follows from formulas (14) and (21).

\[ \lim_{b\to\infty}\left|\frac{a_*}{b_*}\right| = \lim_{b_*\to\infty}\left|0.216_*-\frac{0.30}{b_*}\right| =\infty. \]

(2.27 instead of 1.25), while \(I_*\) is smaller (\(1.13 \cdot 10^{-7} a\) instead of \(1.21 \cdot 10^{-7} a\)). In this case, the deviation of \(r\) from the optimal value \(r_*\) has a stronger effect on the depth of the anodic peak.

Conclusions

1. Using a theoretical interpolation formula for the dependence of the anodic-current constant on the radius of the mercury drop, a refined expression has been derived for the maximum depth of the anodic peak as a function of the radius of the mercury drop, with the other conditions held constant.

2. It has been shown that, with an increase in the parameter \(a_*\), the maximum value of the anodic-peak depth is attained at larger values of the parameter \(b_*\). A graphical and analytical (interpolation) formula relating the parameters \(a_*\) and \(b_*\) has been established.

3. A numerical estimate has been made of the value of the radius of the mercury drop at which the maximum depth of the anodic peak is attained under the conditions of the APN method (as well as of other quantities), both without taking into account and taking into account the dependence of the anodic-current constant on the radius of the mercury drop.

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