On the Problem of Estimating the Bounds of the Numerical Value of the Solubility Product or Instability Constant of Intermetallic Compounds of Two Metals in Mercury, Determined with Sufficient Accuracy by the Potentiometric Method

A. G. Stromberg, Yu. P. Belousov

In previously published works on the study of intermetallic compounds in mercury by the potentiometric method, the question of the influence of the concentration of the electropositive metal on the accuracy of the results has not yet been considered [2]. In the present work, we discuss the conditions necessary for detecting, with sufficient accuracy on the potentiometric curve, a sparingly soluble or soluble intermetallic compound with a given composition \(m\) and solubility product \(L\) or instability constant \(K_{\mathrm{n}}\), as well as the bounds of the numerical values of \(L\) and \(K_{\mathrm{n}}\) that can be detected under various experimental conditions.

On the basis of the dimensionless variables \(\alpha\) and \(\beta\), we previously obtained equations for a sparingly soluble compound [1]:

\[ A\beta/\alpha = [1 - m(1-\alpha)/\beta]^m, \tag{1} \]

where

\[ A = L/C_2^{m+1}, \tag{2} \]

and for a soluble compound [1]

\[ B(1-\alpha)/\alpha = [1 - m(1-\alpha)/\beta]^m, \tag{3} \]

where

\[ B = K_{\mathrm{n}}/C_2^m — \tag{4} \]

an intermetallic compound of two metals in mercury, analysis of which makes it possible to draw conclusions important for practice.

To determine the bounds of the numerical value of \(L\) or \(K_{\mathrm{n}}\), let us find the maximum and minimum values of \(A\) and \(B\). The maximum values of the parameters \(A\) and \(B\) are determined from the specified minimum deviation \(\Delta E_{\min}\) of the potentiometric curve in the presence of an intermetallic compound, which must be no less than some value in order that it can be reliably established and measured with an accuracy acceptable for practice. Let us take this value \(\Delta E_{\min}\) equal to 5 mV and find the quantity \(\varepsilon_{\min}\), which at \(20^\circ\mathrm{C}\) is equal to

\[ \varepsilon_{\min} = \Delta E_{\min} zF/2.3RT = 0.085z. \tag{5} \]

The maximum value of the parameter \(A\) is determined from the specified minimum value \(\Delta E_{\min}\) at the maximum of the potentiometric curve, which, according to [1], is equal to

\[ \varepsilon_{\min} = \lg(1 + 1/qA_{\max}). \tag{6} \]

Taking \(z=2\), from formulas (5) and (6) we find

\[ A_{\max}=2.12/q \quad \text{or} \quad -\lg A_{\max}=\lg q-0.33, \tag{7} \]

and, in accordance with equation (2), we have

\[ -\lg L_{\max}=-\lg A_{\max}-(m+1)\lg C_2. \tag{8} \]

The maximum value of the parameter \(B\), on the basis of [1], is equal to

\[ (\varepsilon_\infty)_{\min}=\lg(1+1/B_{\max}). \tag{9} \]

For \(z=1, 2\), and 3 the quantity \(\lg B_{\max}\) is, respectively, \(+0.66;\ +0.33,\ +0.1\).

And from formula (4) we have

\[ -\lg K_{\mathrm{n},\max}=-\lg B_{\max}-m\lg C_2. \tag{10} \]

The minimum values of the parameters \(A\) and \(B\) can be estimated by a method based on the indications available in the literature [3, 4] that, at concentrations of free metal atoms of \(10^{-5}\)—\(10^{-6}\) g-atoms/l, potentiometric measurements become unreliable¹. If it is assumed that the concentration \(C_1\) of metal \(M\) is not less than a specified value \((C_1=10^{-6}\ \text{g-atoms/l})\), then, according to [1], for the point of intersection of the potentiometric curve with the abscissa axis in the \(\varepsilon,\delta\) plot we have²

\[ A_{\min}=1/\beta-C_1/C_2 \quad \text{or} \quad -\lg A_{\min}=\lg C_2+6 \tag{11} \]

and

\[ -\lg L_{\min}=-m\lg C_2+6. \tag{12} \]

To determine the minimum value of the parameter \(B\), we take the value \(C'_1=10^{-6}\ \text{g-atoms/l}\) for the right-hand edge of the potentiometric curve in the \(\varepsilon,\delta\) plot [2], where the deviation \(\varepsilon_\infty-\varepsilon=h=0.05\). Denote \(h=\lg b\); then \(\alpha_\infty/\alpha_1=b\), or

\[ \alpha_1=b\alpha_\infty=bB/(B+1). \tag{13} \]

Substituting expression (13) into equation (3), we obtain

\[ \beta_1=d/(B+1), \quad \text{where} \quad d=\frac{mB(1-b)+1}{1-\left[\dfrac{B(b-1)+1}{b}\right]^{1/m}}. \tag{14} \]

In addition, \(\alpha_1\) can be represented in the form

\[ \alpha_1=C'_1/C_1=\beta_1\cdot C'_1/C_2. \tag{15} \]

Substituting into expression (15) the value of \(\beta_1\) from formula (14) and taking the logarithm of the resulting equation, we find

\[ -\lg B_{\min}=6.05-\lg d+\lg C_2 \tag{16} \]

and

\[ -\lg K_{\mathrm{n},\min}=6.05-\lg d-(m-1)\lg C_2. \tag{17} \]

The limits of the determined value of the parameter \(A\) are presented in Fig. 1, and of the parameter \(B\) in Fig. 2. The upper inclined line gives the values of \(-\lg A_{\min}\) or \(-\lg B_{\min}\) at different values of \(-\lg C_2\) and three values of \(m\). The horizontal straight lines give the values of \(-\lg A_{\max}\) for three values of \(m\), or \(-\lg B_{\max}\) for three values of \(z\).

Figure 3 presents a plot in the coordinates \(-\lg L;\ -\lg C_2\) for five different values of the composition \(m\), constructed from formulas (8) and (12).

Figure 4 presents a plot in the coordinates \(-\lg K_{\mathrm{n}},\ -\lg C_2\) for different values of \(m\), constructed from formulas (10) and (17).

Fig. 1. Plot of the dependence of \(-\lg A\) on \(-\lg C_2\), constructed from formulas (7) and (11).

Fig. 2. Plot of the dependence of \(-\lg B\) on \(-\lg C_2\), constructed from formulas (9) and (16).

As can be seen from the plots in Figs. 3 and 4, the potentiometric method can be used to determine \(L\) and \(K_{\mathrm{n}}\) within a relatively narrow range of values, when the representative point of the true value of \(L\) or \(K_{\mathrm{n}}\) lies inside

Fig. 3. Plot of the dependence of \(-\lg L\) on \(-\lg C_2\)

Fig. 4. Plot of the dependence of \(-\lg K_{\mathrm{n}}\) on \(-\lg C_2\)

the quadrilateral for the composition \(m\) of the intermetallic compound being studied. If the representative point of the values of \(L\) or \(K_{\mathrm{n}}\) lies on the plot below the quadrilateral, then no deviation will be detected on the experimental potentiometric curve, and an erroneous conclusion may be drawn that there is no interaction between the metals. Conversely, if the representative point in the plot of Fig. 3 or Fig. 4 is located above the quadrilateral for the given \(m\), then a significant deviation of the experimental points will be detected in carrying out the experiment, but the reproducibility of the experiments will be very poor. Thus, depending on the concentration \(C_2\), the representative point may lie inside or outside the quadrilateral.

It follows from this that there are a number of important conclusions for the practice of potentiometric investigations. In order to detect the formation of an intermetallic compound between metals in an amalgam, or to prove its absence, it is necessary to record several potentiometric curves at different values of the concentration $C_2$ of the more electropositive metal.

REFERENCES

1. A. G. Stromberg, Yu. P. Belousov. Elektrokhimiya, 1973, No. 3.
2. M. T. Kozlovskii, A. I. Zebreva, V. P. Gladyshev. Amalgams and Their Applications. Alma-Ata, “Nauka,” 1971.
3. E. V. Leontovich, V. V. Losev, M. A. Dembrovskii. Elektrokhimiya, 5, 32, 1969.
4. R. N. Matakova, A. I. Zebreva. Elektrokhimiya, 6, 732, 1970.

1. By applying prolonged purification of solutions with hydrogen, in work [4] for lead and cadmium amalgams the limit of applicability of the Nernst equation was lowered to \(2\cdot10^{-7}\) g-atoms/l. ↩

2. An analogous value of \(A_{\min}\) can also be obtained under the condition that the concentration \(C'_1\) at the maximum point is not less than the same value \((C'_1=10^{-6}\ \text{g-atoms/l})\). ↩