STUDY OF INTERMETALLIC COMPOUNDS IN MERCURY BY THE APN METHOD. II. THEORY. DEPENDENCE OF THE DEPTH OF ANODIC TEETH ON ELECTROLYSIS TIME

A. G. STROMBERG, M. S. ZAKHAROV, N. A. MESYATS

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

In the preceding article [1], methods were considered for calculating the composition and the solubility product or equilibrium constant of intermetallic compounds (i.m.c.) of two metals in mercury by the APN method (in the case when both metals give anodic teeth), and criteria were also proposed for elucidating the nature of the i.m.c. (sparingly soluble or soluble compound), based on the straight-line or curved nature of the graphs in the corresponding coordinates.

In the present article, one more method is proposed for establishing the nature of i.m.c. and calculating their equilibrium parameters \((n, L_p\) or \(K_\partial)\), based on consideration of the dependence of the depth of anodic teeth on the time of preliminary electrolysis in the APN method.

For ease of comparing the material in both articles, the numbering of formulas and figures in this article is continued, and the designations of letters indicated in the preceding article are not repeated.

Let us show that, when a sparingly soluble i.m.c. is formed in mercury, the dependence of the depth of anodic teeth on electrolysis time at a stationary mercury electrode (a hanging mercury drop) in a solution of ions of both metals under study will have the form shown schematically in Fig. 1.

As long as the electrolysis time \(\tau < \tau_*\), the concentration of metal atoms in mercury will be less than is required to attain the value of the solubility product according to formula (4) (see the preceding article), and the depths of the anodic teeth of both metals will increase in proportion to their concentration in the solution.

\[ l_1^0 = K_1 \tau;\quad l_2^0 = K_2 \tau. \tag{28} \]

At time \(\tau > \tau_*\), formation of a precipitate will begin. Let component \(A\) be present in excess in comparison with the equivalent concentration \(\left(c_1^0 > \frac{1}{n} c_2^0\right)\). Then, evidently, the concentration of component \(A\) in the solution (in mercury) will increase (along curve 1a), while the concentration of component \(B\) will decrease (along curve 2a). Let us consider some quantitative relations. Let us note that, from experimental data on curves 1a and 2a (Fig. 1), one can calculate \(n\) and \(L_p\) by formulas (12)—(19), considered in the preceding chapter.

But there are also some special features. Let, after a time \(\tau_0\), the depth \(I_2\) have become less than the corresponding tooth depth \(l_2^0 = K_2 \tau_0\) na—

so much (for example, by a factor of 20) that it may be neglected as being subtracted in the difference \((I_2^0 - I_2)\).

Then from formula (12) we obtain:

\[ I_1 = \kappa_3 \tau, \tag{29} \]

\[ \kappa_3 = \kappa_1 - \frac{\beta}{n}\cdot \kappa_2. \tag{30} \]

This means that for \(\tau > \tau_0\) the peak \(I_1\) will increase proportionally to the time \(\tau\), with coefficient \(\kappa_3\), determined by expression (30).

Fig. 1. Schematic form of the dependence of deep anodic peaks of two metals in an amalgam on the electrolysis time during the formation, between the metals, of a sparingly soluble intermetallic compound in mercury. Curve 1—for metal \(A\) in the absence of \(B\); 2—for metal \(B\) in the absence of \(A\). Curves \(1a\) and \(2a\)—the same in the joint presence of both metals at the same initial concentrations and other experimental conditions. Curves \(1b\) and \(2b\)—the same as \(1a\) and \(2a\), when both metals are present in equivalent amounts

\[ \left(c_1^0 = \frac{1}{n}c_2^0\right). \]

From (30) we obtain a formula for calculating \(n\):

\[ n = \frac{\kappa_2}{\kappa_1 - \kappa_3}\cdot \beta. \tag{31} \]

From the time \(\tau_*\) in Fig. 1 one can calculate the solubility product \(L_p\) by the formula:

\[ L_p = \frac{\kappa_1}{a_1}\left(\frac{\kappa_2}{a_2}\right)^n \tau_*^{\,n+1}. \tag{32} \]

In the case of an insoluble intermetallic compound, the theoretical form of the curves \(I_1,\tau\) and \(I_2,\tau\), with an excess of component \(A\)

\[ \left(c_1^0 > \frac{1}{n}c_2^0\right) \]

after simple transformations is conveniently represented in the following form:

\[ I_1 = \kappa_1(1 - p\alpha)\tau, \tag{33} \]

\[ I_2 = \kappa_2(1-\alpha)\tau = \kappa_2 \frac{P^{n+1/n}}{\left(\dfrac{1}{p}-\alpha\right)^{1/n}}\cdot \frac{1}{\tau^{1/n}}, \tag{34} \]

where \(\alpha = nx/c_2^0\) is the fraction of the reacted component; \(P\) is determined from the relation:

\[ (1-p\alpha)(1-\alpha)^n = \left(\frac{P}{\tau}\right)^{n+1}, \tag{35} \]

\[ P = \left(\frac{a_1}{\kappa_1}\right)^{1/n+1}\cdot \left(\frac{a_2}{\kappa_2}\right)^{n/n+1}\cdot L_p^{1/n+1}, \tag{36} \]

\[ \frac{1}{p} = \frac{nc_1^0}{c_2^0} = \frac{n}{\beta}\cdot \frac{\kappa_1}{\kappa_2}, \tag{37} \]

\[ x = c_1^0 - c_1 = \frac{1}{n}(c_2^0 - c_2), \tag{38} \]

\(1/p\) is the excess of component \(A\) over \(B\) in the amalgam beyond the equivalent ratio

\[ \left(c_1^0 = \frac{1}{n}c_2^0\right). \]

At \(\tau_*\) we have \(\alpha=0\), and from (33) and (34) we obtain:

\[ \tau_*=\frac{l_{1*}^{0}}{\kappa_1}=\frac{l_{2*}^{0}}{\kappa_2}. \]

For \(\tau \gg \tau_*\) we have \(\alpha \to 1,\ I_2 \to 0\) and

\[ I_1 \to \kappa_1(1-p)\tau, \tag{39} \]

which coincides with formulas (29) and (30), taking (37) into account.

In formulas (33), (34), and (35) the unknown variables are \(I_1, I_2\), and \(\alpha\). They are uniquely determined from these three equations if the numerical values \(\kappa_1, \kappa_2, n, p, P\) are known from experimental data.

The coefficients \(\kappa_1\) and \(\kappa_2\) are determined from the graphs \((I_1^0,\tau)\) and \((I_2^0,\tau)\); \(n\) is calculated by formulas (12), (31), or (42). The coefficients \(p\) and \(P\) are calculated by formulas (36) and (37) from experimental data for \(a_1, a_2\), and \(L_p\). \(a_1\) and \(a_2\) are calculated from formula (11), and \(L_p\) from formulas (17), (19), (32), or (43). The form of the graph of formula (35) in the dimensionless coordinates \(\alpha\) and \(\tau/P\) (or \(P/\tau\)) also depends on two parameters, \(p\) and \(n\), which may differ for different systems. It is convenient, for each value of \(n\) (\(n=1,2,3\), etc.), to construct graphs of \(\alpha,\tau/P\) (or \(\alpha,P/\tau\)) with a series of curves at different \(p\) (for example, \(p=1.0;\ 0.5;\ 0.2;\ 0.1;\ 0.05\), etc.). Having determined \(\alpha\) for given \(n,p\), and \(\tau/P\), it is easy from formulas (33) and (34) to calculate the theoretical values \(I_1\) and \(I_{2*}\) at any \(\tau\) for the given specific system.

If the components \(A\) and \(B\) enter the amalgam during electrolysis in an equivalent amount \(\left(c_1^0=\frac{1}{n}c_2^0\right)\), then after the time \(\tau_*\) the depth of the notches of both elements must remain constant (straight lines \(1b\) and \(2b\) in Fig. 1), as follows from formula (4):

\[ c_2=(nL_p)^{1/n+1}=\mathrm{const}\quad \text{and}\quad c_1=\left(\frac{L_p}{n^n}\right)^{1/n+1}=\mathrm{const}. \tag{40} \]

Knowing the constant value of the depth of these notches (\(I_{1*}\) and \(I_{2*}\)), one can calculate \(n\) and \(L_p\) from the formulas:

\[ n=\beta\frac{I_{2*}}{I_{1*}}=\beta\cdot\frac{\kappa_2}{\kappa_1}, \tag{41} \]

\[ L_p=\frac{I_{1*}}{a_1}\cdot\left(\frac{I_{2*}}{a_2}\right)^n. \tag{42} \]

From formulas (35), (33), and (34), for \(p=1\) we obtain:

\[ \left. \begin{aligned} 1-\alpha &= \frac{P}{\tau};\\ I_{1*} &= \kappa_1P;\quad I_{2*}=\kappa_2P \end{aligned} \right\}. \tag{43} \]

Let us now suppose that the intermetallic compound is a compound soluble in mercury. Then a schematic representation of the dependence of the depth of the anodic notches of two metals (separately and together) on the time of preliminary electrolysis can be represented by the curves in Fig. 2.

As can be seen from the figure, in this case the form of the dependence \(I_1\tau\) is entirely different from that in Fig. 1. Thus, from the external appearance of the \(I_1\tau\) curves in the joint presence of two metals in mercury, it is possible, even without quantitative calculations, to draw a conclusion about the nature of the intermetallic compound. In the formation of a sparingly soluble intermetallic compound, one of the \(I_1\tau\) curves (for the metal that is deficient in comparison with the equivalent amount) must exhibit a maximum and then decr—

decrease in the depth of the anodic notch to very small values, whereas for the metal taken in excess the curve will increase monotonically, first along a curved line and then proportionally to time, but with a slope coefficient smaller than in the case of a single metal in mercury. When a soluble intermetallic compound is formed, the depth of both notches always increases with electrolysis time. If the metals are taken in equivalent amounts, then, in the case of formation of a sparingly soluble intermetallic compound, two horizontal straight lines are obtained on the plot of \(I_1\tau\), whereas in the formation of a soluble intermetallic compound (as will be shown below), with an equivalent ratio of the concentrations of both metals, a monotonic increase of both curves will be observed.

Fig. 2. Schematic form of the dependence of the depth of the anodic notches of two metals in an amalgam on the time of preliminary electrolysis when a mercury-soluble intermetallic compound is formed between the metals. Curves 1, 2 and 1a, 2a are the same as in Fig. 1.

This sharp difference in the form of the \(I_1\tau\) plots may serve as a sufficiently convincing argument in favor of one of the two assumptions under consideration, provided there is confidence that, under the conditions of the experiments, the equilibrium ratios are practically not disturbed during recording of the anodic polarogram.

Let us now consider some quantitative relationships and calculations that can be made from the experimental \(I_1\tau\) curves in Fig. 2 for a soluble intermetallic compound. Using the experimental data for \(I_1^0\), \(I_2^0\) and \(I_1\) and \(I_2\) (and other data) from the curves in Fig. 2, one can calculate the composition of the intermetallic compound \(n\) and the dissociation constant \(K_{\mathrm{d}}\) from formulas (12)—(14) and (21)—(27) of the preceding article, and verify the linearity of the plots in the corresponding coordinates, confirming the theoretical equations derived for soluble intermetallic compounds.

In addition, some additional relationships can be obtained. For sufficiently large \(\tau\) \((c_1^0\) and \(c_2^0)\), the change in the depth of the anodic notch \(I_2\) of the metal present in deficit relative to the equivalent concentration ratio becomes weak, and this notch may be regarded as approximately constant, \(I_{2*}=\mathrm{const}\), over a certain interval of electrolysis time. In this case, from the formula

\[ K_{\mathrm{d}}=\left(\frac{I_{2*}}{a_2}\right)^n \cdot \frac{I_1}{I_1^0-I} \tag{44} \]

it follows that, in order for \(K_{\mathrm{d}}\) to remain constant, the depth of the anodic notch of the second metal, taken in excess, must be proportional to the electrolysis time \((I_1=\kappa_3\tau)\). Hence we obtain the expression for calculating \(K_{\mathrm{d}}\):

\[ K_{\mathrm{d}}=\left(\frac{I_{2*}}{a_2}\right)^n \cdot \frac{\kappa_1}{\kappa_2-\kappa_3}. \tag{45} \]

If the metals are taken in equivalent amounts \(\left(c_1^0=\frac{1}{n}c_2^0\right)\) and, moreover, \(I_1 \ll I_1^0\) and \(I_1^0-I_1 \simeq I_1^0\), then it is easy to obtain the relation

\[ \lg I_1=\lg H+\frac{1}{n+1}\lg \tau . \tag{46} \]

In this case, in the coordinates \(\lg I_1,\ \lg \tau\), a straight line should be obtained with slope

\[ \frac{1}{n+1}. \]

Having determined the intercept \(H\) from the graph, \(K_{\mathrm{д}}\) can be calculated from the formula:

\[ K_{\mathrm{д}}=(n+1)\lg H-\lg \kappa_1-n\lg \frac{a_1}{n}. \tag{47} \]

In the case of a soluble intermetallic compound, the theoretical form of the curves \(I_1,\tau\) and \(I_2,\tau\) with an excess of component \(A\)

\[ \left(c_1^0 \gg \frac{1}{n}c_2^0\right) \]

can be represented as:

\[ I_1=\kappa_1(1-p\alpha)\tau, \tag{48} \]

\[ I_2=\kappa_2(1-\alpha)\tau. \tag{49} \]

Thus, in their external form, the expressions for \(I_1\) and \(I_2\) do not differ from the case considered earlier for an insoluble intermetallic compound (compare with (33) and (31)).

However, the expression for \(\alpha\) will have a different form:

\[ \frac{1}{\alpha}(1-p\alpha)(1-\alpha)^n=\left(\frac{Q}{\tau}\right)^n, \tag{50} \]

\[ Q=K_{\mathrm{д}}^{1/n}\cdot \frac{1}{n}\cdot \left(\frac{a_1}{\kappa_1}\right)^{1/n}\cdot \left(\frac{\kappa_2}{a_2}\right)^{\frac{1-n}{n}}. \tag{51} \]

Here, as \(\tau\to 0\), \(I_2\to 0\cdot\infty\). Resolving this indeterminacy shows that, in the case under consideration of a soluble intermetallic compound, \(I_2\) will not tend to zero.

Let us show this for a special case:

\[ n=1,\quad 1/p\gg 1 \quad \text{and} \quad 1/p-\alpha \simeq 1/p. \]

Then

\[ \alpha=\tau/pQ+\tau \]

and

\[ I_2=\kappa_2 pQ\cdot \frac{\tau}{pQ+\tau}. \tag{52} \]

As \(\tau\to\infty\),

\[ pQ+\tau\simeq \tau \]

and

\[ I_{2\infty}=\kappa_2pQ=pa_2\cdot K_{\mathrm{д}}. \tag{53} \]

From formula (53) one can calculate the dissociation constant \(K_{\mathrm{д}}\):

\[ K_{\mathrm{д}}=\frac{I_{2\infty}}{pa_2}, \tag{54} \]

where \(I_{2\infty}\) is the limiting value of the depth of the anodic notch on the \(I_2,\tau\) graph, measured in an experiment with a large ratio \(1/p\) of the concentrations of components \(A\) and \(B\) (for example, \(1/p>10\)).

Thus, even with a large excess of component \(A\), the concentration of component \(B\) \((c_2=I_2)\) tends to the constant value \(pK_{\mathrm{д}}\), and not to zero. If the excess of \(A\) is not so large, then the concentration of \(B\) also will not tend to zero.

For theoretical calculations of the quantities \(I_1\) and \(I_2\), formula (53), as well as (35), is conveniently presented in the form of several graphs for different \(n\), and on each graph as a series of curves in dimensionless-

in coordinates \(\alpha\) and \(Q/\tau\) (or \(\tau/Q\)) for various specified values of the parameter \(p\). Using the values of \(\alpha\) found from the graphs, it is easy to calculate the theoretical form of the graphs \(I_{\tau_1,\alpha}\) or \(I_{\tau_2,\alpha}\) for any particular case.

With an equivalent ratio of the concentrations of components \(A\) and \(B\) in the amalgam \(\left(c_1^0=\dfrac{1}{n}c_2^0,\ p=1;\right)\), from formulas (50), (48), and (49) we obtain:

\[ 1-\alpha=\left(\frac{\alpha}{1-\alpha}\right)^{1/n}\frac{Q}{\tau}, \tag{55} \]

\[ I_1=\kappa_1 Q\left(\frac{\alpha}{1-\alpha}\right)^{1/n}, \tag{56} \]

\[ I_2=\kappa_2 Q\left(\frac{\alpha}{1-\alpha}\right)^{1/n}, \tag{57} \]

\[ I_1/I_2=\kappa_1/\kappa_2. \tag{58} \]

As \(\tau\to\infty\), \(\alpha\to 1\), and \(I_1\) or \(I_2\) will increase. Thus, in the case of a soluble intermetallic compound, the condition of constancy of \(I_1\) and \(I_2\) at \(p=1\) will not be satisfied. In particular, for \(n=1\) we obtain

\[ I_1=\kappa_1(\alpha Q\tau)^{1/2};\quad I_2=\kappa_2(\alpha Q\tau)^{1/2}. \tag{59} \]

As \(\tau\to\infty\) and \(\alpha\to 1\), we obtain:

\[ I_{1\infty}=\kappa_1(Q\tau)^{1/2};\quad I_{2\infty}=\kappa_2(Q\tau)^{1/2}, \tag{60} \]

i.e., at \(p=1\), \(n=1\), and large \(\tau\), \(I_1\) and \(I_2\) will increase in proportion to the square root of the electrolysis time.

In conclusion, it should be noted that one must not overlook the possibility of obtaining inaccurate results in all the calculations indicated above from experimental data obtained by the APN method, owing to a possible shift of equilibrium during the anodic dissolution of metals when recording the anodic polarogram. This remark is of particular importance when soluble intermetallic compounds are present in the amalgam, since in this case the equilibrium between the intermetallic compound and the free metal atoms in mercury must be considerably more labile than when the intermetallic compound is present as a solid phase.

Therefore, an obligatory condition in carrying out studies of intermetallic compounds by the APN method is the repeated performance of experiments at different rates of change of the potential \((W)\) over a sufficiently wide range of variation of \(W\). Only in the case when the numerical values of \(n\) and \(K_d\) (or \(L_p\)), as well as the form of the graphs at different \(W\), remain the same, or at any rate their numerical values approach a constant limit as \(W\) increases, can the results obtained be considered sufficiently reliable and can definite conclusions be drawn on their basis concerning the nature of the intermetallic compound and the numerical values of the parameters \(n\), \(K_d\), or \(L_p\).

When deviations from equilibrium of the reaction of formation of the intermetallic compound are detected in experiments with different \(W\), it is evidently possible, from the magnitude and character of these deviations and their dependence on \(W\), to judge the rate of the processes of formation and dissolution of precipitates of the intermetallic compound, or the rate of formation and dissociation of soluble intermetallic compounds.

Conclusions

1. A new criterion is proposed for solving, by means of the APN method, the question of the nature of intermetallic compounds in an amalgam. In addition to the regularities established earlier, it is shown that the form of the curves in the graph of the dependence of the depth of the anodic teeth of two metals in an amalgam on the time of preliminary electrolysis, obtained by the APN method, can serve as a criterion for judging the nature of intermetallic compounds in the amalgam.

2. Certain new quantitative relations have been established that make it possible, from the curves \(I,\tau\), to calculate the parameters \(n, L_p\), or \(K_d\).

3. Theoretical expressions have been obtained for the dependence of the depth of the anodic teeth on the electrolysis time in the APN method when insoluble or soluble intermetallic compounds are formed between two metals.

4. The significance of conducting studies at different rates of potential change has been discussed for detecting and allowing for deviations from the state of equilibrium in the amalgam in the case of the formation of insoluble or soluble intermetallic compounds when using the APN method.

References

1. M. S. Zakharov, A. G. Stromberg, N. A. Mesyats. This collection, p. 167.