THEORY OF THE POTENTIOMETRIC CURVE OF AN AMALGAM OF TWO METALS FORMING A SPARINGLY SOLUBLE INTERMETALLIC COMPOUND

A. G. STROMBERG, Yu. P. BELOUSOV

A theoretical equation and graph have been obtained for the potentiometric curve in the presence of a sparingly soluble intermetallic compound (i.m.c.) of two metals in mercury, in the dimensionless coordinates \(\varepsilon, \delta\). On the basis of a mathematical study of this equation, sufficiently reliable and simple graphical methods are proposed for calculating the composition and solubility product of the i.m.c. of two metals in mercury.

In [1] it was noted that from the form of the potentiometric curve one can judge the nature of the i.m.c. of two metals in mercury, and also determine its composition and thermodynamic parameters (the solubility product or dissociation constant). However, a complete quantitative theory of the potentiometric curve for an amalgam of two metals forming an i.m.c. of arbitrary composition has not yet been created.

Theoretical graphs of the potentiometric curve in the coordinates \(E, \lg \overline{C}_{1}/C_{1}\) in [1] were constructed only for the special case \(m = 1\). To determine the parameters \(m\) and \(L\) from an experimental potentiometric curve when \(m\) is not equal to unity, the literature uses a fitting method, since in this case equations of degrees higher than \(m\) are obtained with respect to \(m\).

The aim of the present work is to obtain an equation for the potentiometric curve of an amalgam of two metals forming sparingly soluble i.m.c. of arbitrary composition, to investigate this equation, and also to develop reliable graphical methods for calculating the composition and solubility product of the i.m.c. of two metals in mercury¹.

Let us introduce the dimensionless variables \(\alpha\) and \(\beta\), which are related to the quantities \(E, C_{1}, C_{2}\), directly determined experimentally, by the relations:

\[ \alpha = C'_{1}/C_{1}; \quad -\lg \alpha = \varepsilon = (E - E_{0})zF/2.3RT; \quad \beta = C_{2}/C_{1}; \quad \lg \beta = \delta, \tag{1} \]

where \(E_{0}\) and \(E\) are the e.m.f. of the concentration amalgam cell, respectively, in the absence and in the presence of the second metal in the amalgam; \(C_{1}\) and \(C_{2}\) are the analytical (gross) concentrations of the metals \(M\) and \(N\) in the amalgam; \(z\) is the number of electrons participating in the potential-determining electrode reaction with the more electronegative metal \(M\); \(C'_{1}\), \(C'_{2}\) are the equilibrium concentrations of the metals \(M\) and \(N\) in the amalgam after formation of the sparingly soluble—

of the soluble I.M.C. \(MN_m\), \(m\) is the composition of the I.M.C.\(^2\). \(C'_1, C'_2\) are related by the equation

\[ L=C'_1\cdot (C'_2)^m, \tag{2} \]

where \(L\) is the solubility product.

Let us obtain the equations of the parametric curve for an amalgam of two metals forming a sparingly soluble I.M.C. with one another. To do this, divide both parts of equation (2) by \(C_1^{m+1}\):

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

or

\[ A\beta/\alpha=[1-m(1-\alpha)/\beta]^m, \tag{4} \]

where

\[ A=L/C_2^{m+1};\quad \alpha=10^{-\varepsilon};\quad \beta=10^\delta. \tag{5} \]

And, taking into account the expression

\[ C'_2=C_2-m(C_1-C'_1), \]

as well as relations (1), we obtain the equation of the potentiometric curve in the dimensionless variables \(\alpha\) and \(\beta\)\(^3\). As is evident from formulas (3) and (4), for analytical study the variables \(\alpha\) and \(\beta\) are more convenient, while for graphical interpretation of the results obtained—taking into account that \(\alpha\) and \(\beta\) vary over several orders of magnitude—the variables \(\varepsilon\) and \(\delta\) are more convenient. The form of the graph in the dimensionless coordinates \(\varepsilon,\delta\), in contrast to the ordinary graph of the potentiometric curve in the coordinates \(E,\lg \overline{C}_1/C_1\), does not depend on the charge \(M^{Z+}\) of the ions of metal \(M\), on the concentration \(C_2\) of metal \(N\), or on the concentration \(\overline{C}_1\) of metal \(M\) in the auxiliary electrode.

Fig. 1. Theoretical graph of the potentiometric curve in the coordinates \(\varepsilon,\delta\) upon formation of a sparingly soluble I.M.C. of two metals in mercury, calculated from equation (4) for parameter values: \(m=1,\ A=10^{-2}\)

Let us carry out an investigation of equations (4) and (3). We find the value of the abscissa \(\delta\) at \(\varepsilon=0\), or \(\alpha=1\). Substituting \(\alpha=1\) into formula (3), we obtain \(\beta_1=0\) and \(\beta_2=1/A\). Hence we find \(\delta_1=-\infty\) and \(\delta_2=-\lg A\). Thus, curve 1 in the graph \(\varepsilon,\delta\) (Fig. 1), at large concentrations \(C_1\), asymptotically (\(\delta_1=-\infty\)) approaches the abscissa axis, while in the right-hand part of the graph, at small values of \(C_1\), the curve intersects the axis at point \(a\) for \(\delta_2=-\lg A\) (below the abscissa axis the curve has no physical meaning). Consequently, having determined the parameter \(A\) from the graph and knowing the composition of the I.M.C., one can calculate the solubility product by formula (5). The study of equation (4) and its graphical construction (in the coordinates \(\varepsilon,\delta\)) are simplified if we consider the limiting relations obtained for the two branches of this graph as \(\varepsilon\to 0\) or \(\alpha\to 1\). Denoting by \(\alpha',\beta'\) and \(\alpha'',\beta''\) the values of the quantities for the limiting relations on the left and right branches of the graph (Fig. 1, curves 2, 3), respectively, we find the following limiting relations:

for the left branch

\[ 1-m(1-\alpha')/\beta'=0 \quad \text{or} \quad \lg(1-10^{-\varepsilon})=\delta'-\lg m, \tag{7} \]

\(^2\) \(m\) may be either an integer or a fractional number. For example, I.M.C. of compositions \(M_2N_3, M_2N\) are written as \(MN_{3/2}, MN_{1/2}\).

\(^3\) The concentration \(C_2\) of the more electropositive metal remains constant when recording the potentiometric curve.

for the right branch

\[ A\beta''/\alpha''=1 \quad \text{or} \quad \varepsilon''=-\lg A-\delta'' . \tag{8} \]

As is evident from formula (7), on the plot in the coordinates \(\varepsilon,\delta\) the left limiting branch has an exponential character and, with increasing \(\varepsilon\) \((a\to 0)\), asymptotically approaches the value \(\lim |\delta'|_{\varepsilon\to\infty}=-\lg m\) (Fig. 1, curve 2). The right limiting branch has the form of a straight line, which, at an angle of \(135^\circ\), intersects the \(\delta\) axis (at \(\varepsilon=0\)) at the value \(\delta''=-\lg m\) (Fig. 1, line 3). The two limiting branches intersect at point \(C\) in Fig. 1 with coordinates

\[ \alpha_c=mA/(1+mA),\quad \beta_c=m/(1+mA). \tag{9} \]

Solving the maximum problem using the device of changing variables\(^4\) leads to the following relations between the variables \(\alpha_b\) and \(\beta_b\) (respectively \(\varepsilon_b\) and \(\delta_b\)) at the maximum of the curve on the plot \(\varepsilon,\delta\) (point \(b\)):

\[ \beta_b/(1-\alpha_b)=m(m+1);\quad (1-\alpha_b)/\alpha_b=qA;\quad \alpha_b=pA\beta_b, \tag{10} \]

where

\[ q=m(m+1)p;\quad p=[(m+1)/m]^m . \tag{11} \]

Hence for the coordinates \(\varepsilon_b,\delta_b\) we obtain the expressions

\[ \alpha_b=qA/(qA+1),\quad \beta_b=m(m+1)/(qA+1). \tag{12} \]

Solving the system of equations (12), we find the values of the parameters \(m\) and \(A\). Thus, from the coordinates of the maximum of curve 1 in the plot of Fig. 1, one can calculate the composition \(m\) and the solubility product \(L\) of the sparingly soluble i.m.c. in mercury. The maximum point of the potentiometric curve has an interesting property. Let us draw in Fig. 1 a straight line parallel to the abscissa axis at the value \(\varepsilon_b\), i.e., through the maximum point. Denote by \(g'\) the segment between the left limiting branch and the maximum on curve 1, and by \(g''\) the segment between the right limiting branch and the maximum. Then we have

\[ \delta_b=\delta'+g' \quad \text{and} \quad \delta_b=\delta''-g''. \tag{13} \]

Denote

\[ g'=\lg \alpha',\quad g''=\lg \alpha''. \tag{14} \]

Taking into account relations (1), we find

\[ \alpha'=\beta_b/\beta';\quad \alpha''=\beta''/\beta_b. \tag{15} \]

Substituting the values \(\beta'\) and \(\beta''\) from formulas (7) and (8), we find

\[ \alpha'=\beta_b/m(1-\alpha_b);\quad \alpha''=\alpha_b/A\beta_b. \tag{16} \]

Substituting into equations (16) the values of \(\alpha_b\) and \(\beta_b\) from formulas (12), we obtain

\[ \begin{aligned} \alpha' + \alpha'' &= (m+1)+q/m(m+1),\\ \alpha'/\alpha'' &= m(m+1)^2/q. \end{aligned} \tag{17} \]

Solving this system of equations, we find

\[ \alpha'=m+1 \quad \text{and} \quad \alpha''=[(m+1)/m]^m, \tag{18} \]

or

\[ g'=\lg(m+1);\quad g''=m[\lg(m+1)-\lg m]. \tag{19} \]

As is evident from relation (19) and Fig. 1, the lengths of the segments \(g'\) and \(g''\) do not depend on the parameter \(A\). The values \(g'\) and \(g''\) for any composition of the i.m.c. \((m)\)

\(^4\) In equation (5), the change of variables \((1-\alpha)/\alpha=W,\quad (1-\alpha)/\beta=V\) is carried out, and the derivative \(dW/dV\) is set equal to zero.

can be found by using the graph in Fig. 2, which presents the dependence of \(g'\), \(g''\), and \((g' + g'')\) on \(\lg m\), calculated by formula (19).

Thus, using relations (19) and Fig. 2, two problems can be solved. The first is to determine accurately the point of the maximum when constructing theoretical graphs, and the second is to find an approximate value of \(m\) on an experimental graph.

Fig. 2. Graph of the dependence of \(g'\), \(g''\), and \((g' + g'')\) on \(\lg m\) in accordance with equation (19). Curve 1—\(g'\); 2—\(g''\); 3—\((g' + g'')\).

Figure 3 shows the form of nine graphs of \(\varepsilon,\delta\) for different compositions of the intermetallic compound \(MN_m\) of two metals in mercury (parameter \(m = 1/3;\ 3\)) and for different values of the solubility product of this intermetallic compound (parameter \(A = 0.01;\ 0.1;\ 1.0\)). Along with the theoretical graphs calculated by formula (5), Fig. 3 also shows such limiting branches of the curves, calculated by formulas (7) and (8). From the figure and formulas (7) and (8) it is seen that the left limiting branch depends only on the parameter \(m\); moreover, its form does not depend on \(m\), but at different \(m\) the left limiting branch is displaced along the \(\delta\) axis proportionally to \(-\lg m\). The right limiting branch depends only on the parameter \(A\); at different values of \(A\), the right branch is displaced proportionally to \(-\lg A\).

Fig. 3. Theoretical graph of nine potentiometric curves in coordinates \(\varepsilon,\delta\) and of the limiting branches of the curves in the presence of sparingly soluble intermetallic compounds of two metals in mercury for three values of the parameter \(m\) \((1/3;\ 1;\ 3)\) and three values of the parameter \(A\) \((10^{-2};\ 10^{-1};\ 1)\).

The values of the parameters \(m\) and \(A\) from the experimental values of the coordinates \(\varepsilon_b,\delta_b\) at the maximum on the \(\varepsilon,\delta\) graph are conveniently found from a nomogram in coordinates \(\varepsilon_b,\delta_b\), constructed according to formula (12).

Each point on this nomogram corresponds to a definite pair of values \(A\) and \(m\), which are determined from the grid plotted on the nomogram.

REFERENCES

1. A. I. Zebreva. Collected volume “Electrodes and Electrolytes.” Proceedings of the Institute of Chemical Sciences, Academy of Sciences of the Kazakh SSR, 15, 54, 1967.

2. A. G. Stromberg, N. A. Mesyats, N. P. Mikheeva. Journal of Physical Chemistry, No. 6, (1521, 1971).

1. One of the authors of the article considered a graphical method [2] for determining the thermodynamic parameters of a sparingly soluble compound, obtained on the basis of the present work, the publication of which was delayed. ↩