STUDY OF INTERMETALLIC COMPOUNDS IN MERCURY BY THE APN METHOD. 1. THEORY. DEPENDENCE OF THE DEPTH OF ANODIC PEAKS ON THE CONCENTRATION OF METAL IN AN AMALGAM

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

(Presented by the Scientific-Methodological Seminar of the KhTF)

One of us [1] has proposed a new method for determining the composition and solubility product of intermetallic compounds (i.m.c.) between two metals in mercury, using the method of amalgam polarography with accumulation (APN), under the assumption that the i.m.c. is a solid phase in mercury, that is, a sparingly soluble compound. However, the question of the nature of the i.m.c. in liquid amalgam has still not been resolved and remains controversial. Some authors [2] regard i.m.c. as sparingly soluble compounds; others [3] regard them as intermolecular (complex) compounds soluble in mercury between atoms of two metals (or between a metal and mercury).

The purpose of the present work is: 1) a theoretical consideration of certain consequences that follow from the APN theory for the mutual influence of two elements in an amalgam on the depth of anodic peaks, under the two above-mentioned assumptions concerning the nature of i.m.c., and the establishment of criteria that would make it possible to choose between these two assumptions; 2) the derivation of mathematical relations for determining, from experimental data, the composition of an i.m.c. and its solubility product or equilibrium constant; 3) a discussion of complications associated with disturbance of equilibrium in the amalgam during the recording of the anodic polarogram, and of methods for eliminating their influence on the results of calculations.

Let the components (metals) \(A\) and \(B\) in the amalgam form an i.m.c. with each other:

\[ mA + pB = A_mB_p \tag{1} \]

\((p > m)\). For simplicity, let us assume that the i.m.c. contains only one atom of component \(A\). Then

\[ A + nB = AB_n, \tag{2} \]

where

\[ n = \frac{p}{m}. \tag{3} \]

Let the concentrations of components \(A\) and \(B\) in the amalgam in equilibrium with the intermetallic compound be \(c_1\) and \(c_2\). Then

\[ L_p = c_1 \cdot c_2^n, \tag{4} \]

\[ K_g = \frac{c_1 \cdot c_2^n}{c_3}, \tag{5} \]

where \(L_p\) is the solubility product of the intermetallic compound, on the assumption that the intermetallic compound is sparingly soluble; \(K_g\) is the dissociation constant of the intermetallic compound, on the assumption that the intermetallic compound is soluble; \(c_3\) is the concentration of the intermetallic compound \(AB_n\) in equilibrium with \(A\) and \(B\).

If the initial concentrations of \(A\) and \(B\) in the amalgam were respectively \(c_1^0\) and \(c_2^0\), then after formation of the intermetallic compound according to equation (2) (irrespective of its nature) we have the obvious relation between the concentrations:

\[ n(c_1^0 - c_1) = c_2^0 - c_2. \tag{6} \]

Hence

\[ n = \frac{c_2^0 - c_2}{c_1^0 - c_1}. \tag{7} \]

Knowing \(c_1^0\), \(c_2^0\), \(c_1\), and \(c_2\), one can calculate \(n\) from (7), independently of the assumption concerning the nature of the intermetallic compound.

We assume that both metals give anodic waves during anodic dissolution of metals from the amalgam, while the intermetallic compound does not give anodic waves (as occurs, for example, for the complex amalgam Cu—Zn—Hg). Then, from the theory of APN, we obtain the following relations [4, 5]:

\[ c_i = \frac{q_i}{z_i F \cdot V}, \tag{8} \]

\[ q_i = I_i \cdot \frac{\delta_i}{W}, \tag{9} \]

\[ c_i = \frac{I_i}{a_i}, \tag{10} \]

\[ a_i = W F V \cdot \frac{z_i}{\delta_i}, \tag{11} \]

where \(c_i\) is the concentration of the \(i\)-th component (the 1st or 2nd) in the amalgam, g-atom/cm\(^3\); \(z_i\) is the number of electrons participating in the anodic process of dissolution of the metal (the \(i\)-th component) from the amalgam, per atom; \(V\) is the volume of the amalgam, cm\(^3\); \(W\) is the rate of potential change, volt/sec; \(F\) is the Faraday constant; \(q_i\) is the area under the anodic wave of the \(i\)-th component, equal to the amount of electricity (coulomb) expended in the process of anodic dissolution of this component (metal from the amalgam; it is assumed that the yield of the metal from the amalgam in the process of recording the anodic polarogram is 100%); \(I_i\) is the height of the anodic wave of the \(i\)-th component, A; \(\delta_i/W = q_i/I_i\) is the effective width of the anodic half-wave of the \(i\)-th component, volt; it depends on the nature of the metal, but does not depend on the concentration of metal atoms in the amalgam. The coefficient \(a_i\) in formula (11) depends on the nature of the metal, but does not depend on its concentration in the amalgam. The coefficients \(a_1\) and \(a_2\) for both metals are readily determined from experiments with these metals taken separately.

Thus, the depth of the anodic notch \((l_1\) and \(l_2)\), or the area under the anodic notch \((q_1\) and \(q_2)\), as is seen from formulas (8) and (10), is proportional to the equilibrium concentration of free metal atoms in the amalgam containing the i.m.c., provided that, in the process of anodic dissolution of the metal, the equilibrium does not have time to shift appreciably. This makes it possible to calculate \(L_p\) at \(K_d\) directly from formulas (4) and (5), if the composition of the i.m.c. \((n)\) is known. For simplicity of calculation it is more convenient to use the experimental data on the depth of the anodic notch \((l_1, l_2)\) and formula (10).

Substituting the values \(c_i\) from (10) into (7), we obtain the expression

\[ n=\frac{l_2^0-l_2}{l_1^0-l_1}\,\beta, \tag{12} \]

\[ \beta=\frac{a_1}{a_2}, \tag{13} \]

where \(l_1^0\) and \(l_2^0\) are the depths of the anodic notches of the two metals at the same initial concentration \((c_1^0\) and \(c_2^0)\), but when formation of the i.m.c. is absent, i.e., in an experiment with the separate metals.

Expression (12) is very convenient for calculating the composition of the i.m.c., since it is valid for both soluble and insoluble i.m.c.

If \(n\) is obtained as a fraction (a proper fraction), then, expressing it as a ratio of integers, we find from formula (3) the values of \(m\) and \(p\).

More accurate values can be obtained from a graph, if formula (12) is written in the form:

\[ (l_2^0-l_2)=\frac{n}{\beta}(l_1^0-l_1), \tag{14} \]

where \(l_1\) and \(l_2\) are the depths of the anodic notches of components \(A\) and \(B\) when they are present together in the amalgam; \(l_1^0\) and \(l_2^0\) are the same quantities when they are present separately in the amalgam at the same initial concentration.

On a graph in the coordinates \((l_2^0-l_2)\) and \((l_1^0-l_1)\), irrespective of the nature of the i.m.c., a straight line passing through the origin should be obtained, with slope \(n/\beta\). The coefficient \(\beta\) is easily calculated from experimental data using formulas (13) and (11).

In order to make sure that the results of calculating \(n\) have not been appreciably affected by the shift of equilibrium (2) in the process of anodic dissolution of the metal, experiments and calculations according to formula (12), or from the graph according to equation (14), must be carried out at different values of the rate of change of the potential \(W\). If, over a wide interval of variation of \(W\) (by 2–3 orders of magnitude), the value of \(n\) does not change, then the composition of the i.m.c. determined in this way may be considered sufficiently reliable.

Let us now consider a number of mathematical relations that are obtained in recording anodic polarograms of complex amalgams under two different assumptions concerning the nature of the i.m.c. in the amalgam.

Case 1,a. The i.m.c. is in the precipitate, i.e., is a sparingly soluble compound. The initial concentration of component \(A\) in the amalgam remains constant \((c_1^0=\mathrm{const})\), while the concentration of component \(B\) changes \((c_2^0\) is a variable quantity). Then, from formula (4), taking (10) into account, we obtain:

\[ \lg l_1=\lg M-n\lg l_2, \tag{15} \]

where

\[ M=L_p\cdot a_1\cdot a_2^{\,n}. \tag{16} \]

If the first assumption about the nature of the i.m.c. is correct (the i.m.c. is a sparingly soluble compound), then in the coordinates \(\lg I_1\) and \(\lg I_2\) the plot should be a straight line. From the plot we find \(n\) and \(\lg M\), and then from (16):

\[ L_p=\frac{M}{a_1 a_2^n}. \tag{17} \]

Case 1,b. The same, but \(c_1^0\) varies, while \(c_2^0=\text{const}\). Then, instead of (15), we obtain

\[ \lg I_2=\frac{1}{n}\lg M-\frac{1}{n}\lg I_1. \tag{18} \]

From the plot in the coordinates \(\lg I_2\) and \(\lg I_1\) we find \(n\) and \(\left(\frac{1}{n}\lg M\right)\), and from (16) we obtain

\[ \lg L_p=\left(\frac{1}{n}\lg M\right)n-\lg a_1-n\lg a_2. \tag{19} \]

In order to make sure that the shift of equilibrium (2) during the recording of the anodic polarogram of the complex amalgam has not affected the results, the experiments should be repeated, the plots constructed, and the calculations of \(n\) and \(L_p\) made at different rates of change of the potential \(W\). If, over a wide range of variation of \(W\), calculations by formulas (15)—(17) or (18)—(19) give identical values of \(n\) and \(L_p\), then the results of these calculations may be considered sufficiently reliable.

If, for all values of \(W\), curved lines are obtained on plots in the coordinates \(\lg I_1\) and \(\lg I_2\), or \(\lg I_2-\lg I_1\), then this experimental fact does not agree with the theoretical equations (15) and (18) and may be regarded as indicating that the assumption of low solubility of the i.m.c. is incorrect (provided there is confidence that this deviation from a linear dependence has not been caused by any other factors not taken into account in the experiments, for example, by a shift of equilibrium).

It is also necessary to take into account the possibility of a slow establishment of equilibrium in the case of formation of a precipitate of the i.m.c. Therefore, after the end of electrolysis and before recording the anodic polarogram, a sufficiently long holding period should be allowed so that a state close to equilibrium has time to be established between the precipitate of the i.m.c. formed and the liquid amalgam.

In order to select the required holding time, it should be recommended to carry out experiments with different intervals of holding time for the amalgam drop after electrolysis. If the depth of the anodic notches is the same for different holding times, this may be regarded as an indication of the establishment of equilibrium in the system after electrolysis.

Case 2,a. The i.m.c. is present in the solution in the form of molecules (or complexes) \(AB_n\), in equilibrium with the components \(A\) and \(B\) according to the reaction equation (2). We assume that \(c_1^0=\text{const}\) and \(c_2^0\) is variable. Taking into account that

\[ c_3=c_1^0-c_1, \tag{20} \]

we obtain from (5) and (10)

\[ \lg \frac{I_1}{I_1^0-I_1}=\lg N-n\lg I_2, \tag{21} \]

where

\[ N=K_z\cdot a_2^n. \tag{22} \]

It follows from formula (21) that, if the initial assumptions about the nature of the intermetallic compound are correct and there are no systematic errors, a straight line should be obtained on a plot in the coordinates \(\lg I_1/I_1^0 - I_1\) and \(\lg I_2\). From the plot we find the composition of the intermetallic compound and its dissociation constant:

\[ K_{\partial}=\lg N-n\lg a_2. \tag{23} \]

If \(c_2^0=\mathrm{const}\) and \(c_1^0\) is variable, then, taking into account

\[ c_3=\frac{1}{n}(c_2^0-c_2), \tag{24} \]

we obtain from (4) and (10)

\[ \lg\frac{I_1}{I_2^0-I_2}=\lg N' - n\lg I_2, \tag{25} \]

\[ N'=K_{\partial}\cdot a_2^{\,n-1}\cdot a_1. \tag{26} \]

From the straight line on the plot of \(\lg \dfrac{I_1}{I_2^0-I_2}\) and \(\lg I_2\) we find \(n\) and \(K_{\partial}\):

\[ \lg K_{\partial}=\lg N'-(n-1)\lg a_2-\lg a_1+\lg n. \tag{27} \]

In the absence of a noticeable shift of equilibrium (2) during the recording of the anodic polarogram, repetition of the experiments at different \(W\) should give coincident results when calculating \(n\) and \(K_{\partial}\) by formulas (21)—(27).

In addition, coincident results should be obtained when calculating \(n\) by formulas (12)—(14), (21), and (25), and when calculating \(n\) by formulas (23) and (27).

If, however, curved lines are obtained on plots in the coordinates \(\lg I_1/I_1^0-I_1\), \(\lg I_2\), or \(\lg I_1/I_2^0-I_2\), \(\lg I_2\), then (after checking at different \(W\) for the absence of a shift of equilibrium) this may serve as an indication that the assumption that the intermetallic compound is a compound insoluble in mercury is incorrect.

It should be noted that, when soluble intermetallic compounds are present in the solution, one may expect a more noticeable shift of equilibrium during recording of the anodic polarogram than in the case of sparingly soluble intermetallic compounds, since the rate of dissolution of the solid precipitate of the intermetallic compound in the amalgam is in general a slower process than the rate of establishment of homogeneous equilibrium in the case of a soluble intermetallic compound.

It may even be anticipated, as a limiting case, that soluble intermetallic compounds may not be detected at all by the APN method if the equilibrium between the intermetallic compound and the free metals according to the chemical-reaction equation (2) is reestablished very rapidly, especially if the investigation is carried out in the region of low rates of change of the potential. Therefore, if an intermetallic compound is detected by the potentiometric method but is not detected by the APN method, this may serve as a first preliminary indication that such an intermetallic compound is most likely present in the amalgam in solution.

Summarizing the material presented above, one may conclude that, by means of the method considered here (linearity or nonlinearity of the plot in certain coordinates), it is possible, with the aid of the APN method, to obtain indirect indications of the nature of an intermetallic compound between two metals, and also to calculate the equilibrium parameters for this intermetallic compound.

From an experimental point of view, the study of intermetallic compounds by the APN method is considerably simpler than by the potentiometric method and thus is a valuable supplement to that method.

Conclusions

1. A method is proposed for calculating the composition of intermetallic compounds of two metals in mercury from experimental data of the APN method, without requiring assumptions about the nature of the intermetallic compound.

2. Theoretical relationships have been obtained between the depths of the anodic peaks of two metals in mercury during their anodic dissolution from an amalgam in the APN method in the case of formation between them of an intermetallic compound that is insoluble or soluble in mercury; from these relationships one can obtain indirect indications of the nature of the intermetallic compound in mercury, and also calculate the composition of the intermetallic compound and its solubility product or dissociation constant.

3. The possibility is discussed of obtaining erroneous data on the composition and equilibrium constant for an intermetallic compound because of a shift of equilibrium during anodic dissolution of metals from an amalgam, as well as the importance of carrying out experiments at different rates of potential change in order to eliminate these errors.

References

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2. A. I. Zebreva. In: Electrochemistry of Solutions and Metallic Systems. Proceedings of the Institute of Chemical Sciences, Academy of Sciences of the Kazakh SSR, Alma-Ata, 9, 53 (bibliography on the study of intermetallic compounds in amalgams is given), 1962.

3. H. Hartmann, K. Shoizel. Z. Phys. Chem., N.F. 9, 106, 1956.

4. A. G. Stromberg. Izv. TPI, 128, 1965.

5. E. A. Zakharova. The present collection.