Physical Chemistry

B. B. Damaskin, G. A. Tedoradze

SURFACE COVERAGE BY AN ORGANIC SUBSTANCE AT THE POTENTIALS OF MAXIMA ON DIFFERENTIAL-CAPACITANCE CURVES

(Presented by Academician A. N. Frumkin, May 21, 1963)

It was shown in (¹) that, when the dependence of the attraction constant \(a\) on the electrode potential is taken into account, the differential-capacitance curve in the presence of tert.-\(\mathrm{C_5H_{11}OH}\) can be interpreted quantitatively by using A. N. Frumkin’s adsorption-isotherm equation (²)

\[ Bc=\frac{\theta}{1-\theta}\exp(-2a\theta), \tag{1} \]

where \(c\) is the concentration of the organic substance; \(\theta=\Gamma/\Gamma_m\), i.e., the ratio of the adsorption \(\Gamma\) to its limiting value \(\Gamma_m\), and

\[ B= \]

\[ =B_0\exp\left[ \frac{ \displaystyle \int_{0}^{\varphi}\!\!\int_{0}^{\varphi} C_0\,d\varphi^2 + C'\varphi\left(\varphi_N-\frac{\varphi}{2}\right) }{A} \right] \tag{2} \]

Fig. 1. Dependence of the coverage at the maximum on the peak potential for different \(a\):
\(1\)—\(a=0\); \(2\)—0.5; \(3\)—1.0; \(4\)—1.5; \(5\)—1.9; dashed line—limiting value of \(\varphi^{\max}\) as \(c\to 0\).

\(B_0\) and \(A\) are constants, with \(A=RT\Gamma_m\); \(C_0\) and \(C'\) are the double-layer capacitances at \(\theta=0\) and \(\theta=1\), respectively; \(\varphi\) is the potential measured from the point of zero charge (p.z.c.) at \(\theta=0\); \(\varphi_N\) is the shift of the p.z.c. on going from \(\theta=0\) to \(\theta=1\). In this case, as follows from (³),

\[ C=C_0(1-\theta)+C'\theta+\left(\frac{d\ln B}{d\varphi}\right) \left(\frac{\partial \varepsilon}{\partial \varphi}\right)_{\varphi} h, \tag{3} \]

where

\[ h=\frac{\theta(1-\theta)}{1-2a\theta(1-\theta)}. \tag{4} \]

To find the dependence of \(a\) on \(\varphi\) from experimental data, it was assumed that the position of the peaks on the \(C,\varphi\)-curves is determined mainly by the magnitude \(h\) (³). In this case the coverage at the maximum is \(\theta^{\max}=0.5\), and the value of \(a\) at the peak potentials \(\varphi^{\max}\) is determined by the methods described in (¹).*

Obviously, the position of the peaks on the \(C,\varphi\)-curves corresponds to the maximum of \(h\) only for not too small values of \(a\), when a sharp dependence of \(h\) on \(\varphi\) is observed (³). In the general case the position of the extrema on the \(C,\varphi\)-curves is determined by the condition \(dC/d\varphi=0\), for which the calculation is possible under the assumption \(C_0=\mathrm{const}\) and \(a=\mathrm{const}\), when

\[ B=B_m\exp[-a(\varphi-\varphi_m)^2], \tag{2a} \]

where \(a=(C_0-C')/2A\), \(\varphi_m=-\varphi_N C'/(C_0-C')\); \(B_m=B_0\exp[(\varphi_N C')^2/2A(C_0-C')]\).

* A detailed theory is set forth in an article to be published in the Journal of Physical Chemistry.

In this case

\[ C=C_0(1-\theta)+C'\theta+4Aa^2(\varphi-\varphi_m)^2h, \tag{5} \]

\[ \frac{dC}{d\varphi}=12Aa^2(\varphi-\varphi_m)h \left\{1-\frac{2a}{3}(\varphi-\varphi_m)^2 \frac{1-2\theta}{[1-2a\theta(1-\theta)]^2}\right\}. \tag{6} \]

The condition \(dC/d\varphi=0\) is satisfied in the following cases:

1. \(h=0\), i.e., according to (4), \(\theta=0\) or \(\theta=1\). This condition is practically not realized, since it follows from equations (1) and (2a) that \(\theta=0\) at \(c\ne0\) corresponds to \((\varphi-\varphi_m)\to\infty\), and \(\theta=1\) corresponds to \(c\to\infty\).

2. \(\varphi=\varphi_m\)—this is the condition for a minimum on the \(C,\varphi\)-curves at the potential of maximum adsorption.

3. The expression in braces is equal to zero, which gives the conditions for maxima on the \(C,\varphi\)-curve:

\[ \varphi^{\max}= \varphi_m\pm \sqrt{\frac{3}{2a}\, \frac{1-2a\theta^{\max}(1-\theta^{\max})} {\sqrt{1-2\theta^{\max}}}} . \tag{7} \]

Fig. 2. Dependence of peak potentials on \(\lg c\) for different \(a\):
\(1\)—\(a=0\); \(2\)—1.0; \(3\)—1.9; \(4\)—the same under the condition \(\theta^{\max}=0.5\); \(5\)—limiting values of \(\varphi^{\max}\) as \(c\to0\).

Calculation by equation (7) of the dependence of \(\theta^{\max}\) on \((\varphi^{\max}-\varphi_m)\) for different \(a\) at \(C_0=20\), \(C'=5\), and \(A=1\) is shown in Fig. 1. It is evident from the figure that at \(a<1\) there is a considerable deviation of \(\theta^{\max}\) from 0.5. With decreasing concentration of the organic substance, when \(\theta^{\max}\to0\), the position of the maxima for any \(a\) tends to the value \(\varphi_m\pm\sqrt{3/2a}\), which is obtained under the condition that adsorption of the organic substance obeys the Henry equation (4). Finally, at \(a>0.5\), one and the same position of the maximum corresponds to two values of \(\theta^{\max}\). This means that, as the concentration of the organic substance decreases in this case, the peak potentials first approach one another and then diverge again, asymptotically approaching the value \(\varphi_m\pm\sqrt{3/2a}\).

Indeed, Fig. 2 gives the dependence of the quantity

\[ Q=\lg(B_mc)+\frac{a}{2.3} = \frac{a}{2.3}(\varphi^{\max}-\varphi_m)^2 + \lg\frac{\theta^{\max}}{1-\theta^{\max}} + \frac{a}{2.3}(1-2\theta^{\max}) \tag{8} \]

on \(\varphi^{\max}-\varphi_m\), calculated from equations (7) and (8) for \(a=0\), \(a=1\), and \(a=1.9\). The parabola, with which the curve for \(a=1.9\) coincides over most of its length, corresponds to the condition \(\theta^{\max}=0.5\) \((^3)\), when

\[ Q=\frac{a}{2.3}(\varphi^{\max}-\varphi_m)^2 . \tag{9} \]

The calculation shows that at \(a<1\), even at high concentrations of the organic substance, a noticeable deviation is observed of the \(Q\)—\((\varphi^{\max}-\varphi_m)\) curves from the parabolic dependence corresponding to the condition \(\theta^{\max}=0.5\). Thus, the previously obtained conclusion about the quadratic depend-

dependence of \(\varphi^{\max}\) on \(\lg c\) \((^3)\) retains its validity only under the condition \(a \gg 1\). At \(a=0\), corresponding to the Langmuir isotherm, as is seen from Fig. 2, over a wide interval of \(Q\) (a change of \(c\) by 2 orders of magnitude) an approximately linear dependence of \(\varphi^{\max}\) on \(\lg c\) is obtained, as was pointed out in \((^{5,6})\). Consequently, the conclusion of a quadratic dependence of \(\varphi^{\max}\) on \(\lg c\), obtained in \((^{7,8})\) from an analysis of the Langmuir equation, cannot be regarded as correct*, and the experimentally found linear relation of \(\lg c\) to \((\varphi^{\max}-\varphi_m)^2\) is apparently due to the fact that, for the organic substances studied, \(a \gg 1\) and their adsorption on mercury does not obey the Langmuir equation.

From Fig. 2 it is seen that at \(a=1.9\), in a certain rather narrow concentration region, three extrema (two maxima and one minimum) correspond to a single value of \(Q\) on each of the branches of the \(C,\varphi\)-curve. The cathodic branch of the \(C,\varphi\)-curve, calculated at \(a=1.9\) and \(Q=0.05\), illustrating this regularity, is shown in Fig. 3. This rather rare effect was observed on the cathodic branch of the \(C,\varphi\)-curve in a solution of \(0.5\,N\) CsCl \(+10^{-3}M\) \(\mathrm{C}_{12}\mathrm{H}_{25}\mathrm{SO}_3\mathrm{Na}\) (see Fig. 5 in \((^9)\)).

Fig. 3. Cathodic branch of the \(C,\varphi\)-curve, calculated under the condition \(a=1.9\); \(Q=0.05\); \(C_0=20\); \(C'=5\); \(A=1\)

Using equations (5), (7), and (8), one can calculate the dependence of the peak height \(C^{\max}\) on \(Q\) and compare it with the linear dependence

\[ C^{\max}=\frac{C_0+C'}{2}+2.3\,\frac{C_0-C'}{2-a}\,Q, \tag{10} \]

Fig. 4. Dependence of the peak height on \(\lg c\) for different \(a\):
\(1\)—\(a=0\); \(2\)—\(1.0\); \(3\)—\(1.5\). Straight lines—calculation under the condition \(\theta^{\max}=0.5\) for the corresponding values of \(a\)

* In principle, a quadratic dependence of \(\varphi^{\max}\) on \(\lg c\) under the condition \(a=0\) is possible, but only at such high concentrations of the organic substance that \(|\varphi^{\max}-\varphi_m|\gg 0.8\ \mathrm{V}\). In practice this case is not realized.

following from the condition \(\theta^{\max}=0.5\) (3). Such a comparison is made in Fig. 4 for \(a=0\), \(a=1\), and \(a=1.5\). It is seen from the figure that the resulting dependence of \(C^{\max}\) on \(Q\), over a considerable range of \(Q\), is well approximated by straight lines, although the slope of these lines at \(a<1\) differs appreciably from the value \(2.3\dfrac{C_0-C'}{2-a}\), predicted by equation (10).

Table 1

The relations obtained here cannot be compared quantitatively with experimental data, since a necessary condition for the calculation was the assumption \(C_0=\mathrm{const}\) and \(a=\mathrm{const}\), which does not correspond to experiment (1). Nevertheless, such a calculation gives the limits of applicability of the methods previously described by one of us (1) for determining the magnitude of \(a\). The values of \(a\) found by these methods from theoretically calculated \(C,\varphi\)-curves under the condition \(C_0=20\), \(C'=5\), \(A=1\), and \(\varphi^{\max}-\varphi_m=0.7\), for \(a=1\) and \(a=0.5\), are given in Table 1. As is seen from the table, the average error in determining \(a\) at \(a=1\) is 5%, whereas at \(a=0.5\) it is already 42%. Thus, the noted methods for determining \(a\) are applicable only at \(a\geqslant 1\); smaller values of \(a\) may be found from the form of the adsorption isotherm (10).

We express our sincere gratitude to Academician A. N. Frumkin for discussion of the material of the present article.

Received
18 V 1963

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