THEORY OF THE RESOLVING POWER OF THE APN METHOD

A. G. Stromberg, V. F. Yankauskas

(Presented by Professor A. G. Stromberg)

Some questions in the theory of the resolving power of amalgam polarography with accumulation were considered in the work of V. E. Gorodovykh [1].

The aim of the present work is to derive mathematical relations for the dependence of resolving power on various factors in the method of amalgam polarography with accumulation (APN) on a stationary mercury drop at a linearly varying potential, on the basis of the parametric theory of APN.

For a sufficiently small difference between the peak potentials $\Delta \varphi_{12}$ of the anodic waves of two elements, the error $(g \times 100\%)$ in measuring the height of the second (more positive) element is equal to

\[ g = \frac{i}{I_2} = \frac{i}{I_1}\cdot \frac{I_1}{I_2}, \tag{1} \]

where $I_1$, $I_2$ are the heights of the waves of the interfering (more electronegative) and the determined (more electropositive) elements; $i$ is the current of the descending branch of the anodic wave of the interfering element at the peak potential of the determined element.

Equation (1) may be represented in the form:

\[ j = P \cdot g; \qquad j = \frac{I_1}{I_2}; \qquad P = \frac{I_1}{i}. \tag{2} \]

It is expedient to call the quantity $j$ the resolving power of the APN method. The resolving power $j$ is equal to the maximum ratio of the wave heights of the interfering and determined elements at a specified measurement error $g$ of the wave of the determined element and a potential difference $\Delta \varphi_{12}$ of the anodic waves, and under other specified experimental conditions; $P$ is the resolving-power coefficient, equal to the reciprocal of the measurement error of the wave of the second (determined) element at the value $j = 1$; or, in other words, the numerical value of $P$ is equal to the ratio of the height of the anodic wave of the first (interfering) element to the current of the descending branch of this element at the peak potential of the second (determined) element under the given experimental conditions.

The error \(g\) in the experiment is determined from the relation

\[ g=\frac{I_2'-I_2}{I_2}=\frac{I_2'}{I_2}-1;\qquad I_2'-I_2=i, \tag{3} \]

where \(I_2\) and \(I_2'\) are the depths of the anodic notch of the element being determined (the more electropositive one), when it is in the solution, respectively, in the absence and in the presence of the interfering element.

Let us obtain a theoretical expression for the resolving-power coefficient \(P\). It is obvious that, in the general case, the numerical value of \(P\) depends on many factors: the difference between the potentials of the anodic notches of both elements \((\Delta\varphi_{12})\), the ionic charge \((z)\), the diffusion coefficient of the metal in mercury \((D)\), the radius of the mercury drop \((r)\), the rate of change of the potential \((W,\ \text{volt/sec})\), the temperature \((T^\circ K)\), the charge-transfer coefficient \((\alpha)\) for irreversible processes, and, possibly, a number of other factors. In what follows we shall assume that the process considered is the anodic dissolution of a metal from a stationary mercury drop (amalgam) at a linearly varying potential and that the solution contains an excess of indifferent electrolyte.

The dependence of the current of the descending branch of the anodic notch, at a linearly varying potential after the limiting current has been reached, will obviously be described by the equation for the dependence of the limiting current on time at constant potential, since the limiting current does not depend on the electrode potential.

This dependence is given by the equation [1, 2]

\[ i=A\sum_{n=1}^{\infty} e^{-n^2Bt};\qquad A=z\cdot F\cdot C_a\cdot 8\pi rD;\qquad B=\frac{\pi^2D}{r^2} \tag{4} \]

(\(C_a\) is the concentration of metal atoms in mercury, \(z\)-atom/cm\(^3\)).

In this case the origin of the time scale \((t=0)\) does not coincide with the peak potential and, according to calculations by Gokhshtein [3], confirmed by experimental data [4], lies 0.013 volt more positive for a cathodic reversible peak in the presence of linear semi-infinite diffusion. For the anodic notch this quantity has not been considered theoretically and has not been determined experimentally.

Let us estimate the potential difference \(\Delta\varphi_{12}\) when in expression (1) for the current \(i\) all terms of the series except the first may be neglected with an error smaller than a specified value \(g'\) \((g'\times100\%)\):

\[ g'e^{-Bt}\geq e^{-4Bt};\qquad Wt\simeq \Delta\varphi_{12}\geq -\frac{2.3}{3B}\,W\lg g'. \tag{5} \]

It follows from formula (5) that for the usual values of the quantities used in APN practice \((W=0.01\ \text{V/sec};\ r=0.04\ \text{cm})\), and taking \(D=1.6\cdot10^{-5}\ \text{cm/sec};\ z=2;\ g'=0.05\), we obtain \(\Delta\varphi_{12}\geq 0.1\) volt. Consequently, for \(\Delta\varphi_{12}>0.1\) volt,

\[ i=A\cdot e^{-Bt};\qquad P=\frac{I}{A}e^{Bt}. \tag{6} \]

For the depth of the anodic notch, on the basis of the parametric theory, we have the expression [5, 6]:

\[ I=K_a\cdot S\cdot C_a;\qquad K_a=\frac{K_0}{f(\varepsilon)};\qquad \varepsilon=\frac{1}{r}\left(\frac{D}{z\cdot W}\right)^{1/2}, \tag{7} \]

\[ K_0=\mu\cdot 2.72\cdot10^5\cdot z\,(z\cdot D\cdot W)^{1/2} \begin{cases} \mu=1 — \text{rev.},\\[4pt] \mu=\dfrac{0.9}{\alpha^{1/2}} — \text{irrev.}, \end{cases} \tag{8} \]

where \(K_a\) is the coefficient of the anodic notch; \(K_0\) is the limiting value of the coefficient \(K_a\) for semi-infinite linear diffusion; \(\varepsilon\) is a parameter; \(S=4\pi r^2\) is the electrode surface. The function \(f(\varepsilon)\) is related to the function \(\psi(\varepsilon)\) in a more rigorous form [7] for a reversible anodic notch¹ by the relation \(f(\varepsilon)=0{,}142/\varepsilon\psi(\varepsilon)\). For the time \(t\) we have the expression

\[ Wt=\Delta\varphi_{ni}+\Delta\varphi_{\text{ini}};\qquad x=z\cdot W\cdot t, \tag{9} \]

where \(\Delta\varphi_{ni}=\Delta\varphi_{12}\); \(\Delta\varphi_{\text{ini}}\) is the difference between the potential at the time origin (for the curve of limiting current versus time) and the potential of the anodic notch.

Substituting (7), (8), and (9) into (6), we obtain:

\[ P(\varepsilon,x)=\frac{\exp\{(\pi\varepsilon)^2 x\}}{\mu\cdot 0{,}71\,\varepsilon f(\varepsilon)} \tag{10} \]

or

\[ \lg P\cdot\mu=-\lg 0{,}71\cdot \varepsilon f(\varepsilon)+\frac{(\pi\varepsilon)^2}{2{,}3}\,x . \tag{11} \]

Thus, the resolving-power coefficient depends only on three parameters: \(\varepsilon\), \(x\), and \(\mu\), while the product \(P\cdot\mu\) depends only on two parameters: \(\varepsilon\) and \(x\). From formulas (10) and (11) the important role of the parameter \(\varepsilon\) in the theory of diffusion currents is evident; special attention was already drawn to this earlier in the works of Gorodovykh [1, 7].

Knowing the form of the dependence of the function \(f(\varepsilon)\) on \(\varepsilon\), one can use formula (11) to calculate the dependence of \(\lg_{\mu} P\) on \(x\) at different \(\varepsilon\). From formula (11) it is seen that the graph in the coordinates \(\lg_{\mu} P, x\) has the form of a straight line. With the aid of this graph one can calculate the resolving-power coefficient for any values of the parameters \(x\) and \(\varepsilon\), i.e., for any experimental conditions \((\Delta\varphi_{12}, z, D, r, W, \alpha)\) within the range of parameter values \(\varepsilon\) and \(x\) used to construct the graph.

From formulas (10) and (11) it is seen that the logarithm of the resolving-power coefficient depends linearly on the difference between the potentials of the anodic notches of the two elements considered.

The dependence of the resolving-power coefficient \(P\) on the parameter \(\varepsilon\) (i.e., on the factors \(W, r, D\), and \(z\); see formula (7)) has a more complex character, since the parameter \(\varepsilon\) enters expressions (10) or (11) twice explicitly and, moreover, also in the form of the function \(f(\varepsilon)\). If for the function \(f(\varepsilon)\) one adopts the approximate value \(f(\varepsilon)=1+1{,}25\varepsilon\), established in work [6], then an estimate of the magnitude \(\lg_{\mu} P\) in formula (11) in the range of \(\varepsilon\) from \(0{,}5\) to \(2{,}0\) (which is usually encountered in the practical use of the APN method) shows that the first term in formula (11) changes only slightly (from 0 to \(-0{,}5\)), whereas the second term, in this case (at \(x=0{,}6\)), changes from 1 to 10. Thus, in this range of \(\varepsilon\) values, the quantity \(\lg_{\mu} P\) varies approximately inversely proportional to the radius of the mercury drop and to the square root of the rate of potential change, and practically does not depend on the ionic charge (since the product \(\varepsilon^2 x\) does not depend on \(z\)). With an increase in the diffusion coefficient of metal atoms in mercury, the quantity \(\lg_{\mu} P\) increases. With temperature, the quantity \(\lg_{\mu} P\) increases in the same way as the diffusion coefficient, i.e., by \(2{,}5\)--\(3\%\) per degree. For irreversible processes

the value \(P\) changes in proportion to the square root of the charge-transfer coefficient \(\alpha\).

The nature of the element being determined (the more electronegative one) does not affect the resolving-power coefficient, as is seen from formula (2).

When the stationary mercury drop is replaced by a mercury film electrode, the resolving power of the APN method increases substantially. However, the theoretical consideration of the resolving power of the APN method with a mercury film electrode is at present made difficult by the absence of a theoretical expression for the descending branch of the anodic tooth on this electrode.

Knowing the coefficient \(P\), one can calculate the resolving power of the APN method from formula (2), for any specified error due to the superposition of the tooth of the element being determined (the more positive one) on the descending branch of the interfering element (the more electronegative one).

Taking into account what was stated above about the character of the dependence of \(\lg P\) on \(x\), this dependence in the range of values of \(\varepsilon\) from 0.5 to 2.0 can be expressed by the approximate formula

\[ \lg P \simeq -0.4+\frac{(\pi\varepsilon)^2}{2.3}\,z\cdot\Delta\varphi_{ni}; \qquad P=0.4\cdot e^{(\pi\varepsilon)^2 z\Delta\varphi_{ni}} . \tag{12} \]

Hence, for the resolving power \(j\), we obtain the approximate expression (for \(\varepsilon\) from 0.5 to 2.0):

\[ \lg j=\lg 0.4g+\frac{(\pi\varepsilon)^2}{2.3}\,z\cdot\Delta\varphi_{12}; \qquad j=0.4\cdot g e^{(\pi\varepsilon)^2\Delta\varphi_{12}\cdot z}. \tag{13} \]

For calculating the experimental value of the function \(f(\varepsilon)\), it is convenient to use the formula

\[ f(\varepsilon)=a\cdot u; \qquad a=8.45\,\frac{(z\cdot D)^{1/2}}{r}; \qquad u=W^{1/2}\frac{q}{I}. \tag{14} \]

This formula is obtained from formulas (7) and (8), if one also takes into account the relations:

\[ C_a=\frac{q}{z\cdot F\cdot v}; \qquad \frac{v}{S}=\frac{r}{3}; \qquad K_a=\frac{z\cdot F\cdot r}{3}\cdot\frac{I}{q}. \tag{15} \]

Determining, at different rates of change of the potential \((W)\), the depth of the tooth \((I)\), the area under the tooth \((q,\ \text{coulomb})\), and then the variable \(u\) and the constant \(a\), we calculate from formula (12) the function \(f(\varepsilon)\) at different \(\varepsilon\) (see formula 7).

In order to estimate from experiment the numerical value of the potential difference \(\Delta\varphi_{np}\), entering into the expression for \(x\) (see formula 9), we rewrite formula (11) in the form

\[ \lg P=M+Nx', \tag{16} \]

\[ M=\left(-\lg \mu\,0.71\varepsilon f(\varepsilon)+\frac{(\pi\varepsilon)^2}{2.3}\,z\Delta\varphi_{np}\right); \qquad N=\frac{\pi\varepsilon^2}{2.3}. \tag{17} \]

Determining \(P\) and \(x'\) from experimental data for an anodic tooth:

\[ P=\frac{I}{i}; \qquad x'=z\cdot\Delta\varphi_{pi}, \tag{18} \]

we plot a graph in the coordinates \(\lg P,\ x'\) and, extrapolating the straight line obtained, find the value \(M\) \((M=\lg P\) at \(x'=0)\) and calculate

by formula (16) the sought quantity \(\Delta \varphi_{\mathrm{an}}\), since all the other quantities \((\varepsilon, f(\varepsilon), z, \mu)\) are known to us.

Conclusions

1. The concept of the “resolving power of the APN method” has been refined. It is proposed that the “resolving power of the APN method” be understood as the maximum ratio of the depths of the interfering and determined (more electropositive) elements, for a given difference in the potentials of their anodic peaks on the selected background and under other given conditions, at a preassigned value of the error from superposition of the peak of the more electronegative element on the current of the descending branch of the anodic peak of the more electropositive element.

2. The concept of the “resolving-power coefficient” has been introduced. By the resolving-power coefficient we mean the ratio of the depth of the anodic peak of the interfering element to the current of the descending branch of this element at the potential of the anodic peak of the determined (more electropositive) element. The resolving-power coefficient is numerically equal to the reciprocal of the error due to superposition of the anodic peak of the determined (more electropositive) element on the current of the descending branch of the anodic peak of the interfering element, at the same depth of both anodic peaks.

3. On the basis of the parametric theory of amalgam polarography with accumulation, a mathematical relation has been obtained that makes it possible theoretically to calculate the resolving-power coefficient in the method of amalgam polarography with accumulation on a stationary mercury drop at a linearly varying potential, under any specified conditions, if the dependence of the anodic-peak coefficient on the parameter \(\varepsilon\) (the function \(f(\varepsilon)\)) is known.

4. It has been shown that, of the three factors (the rate of change of potential, the radius of the mercury drop, and the coefficient of diffusion of the metal in mercury), the resolving-power coefficient depends through the parameter

\[ \varepsilon=\frac{1}{r}\left(\frac{D}{z \cdot W}\right)^{1/2}, \]

which once again confirms the importance of this parameter in the theory of APN. The nature of the influence of various factors on the resolving-power coefficient has been clarified.

1. A method has been proposed for the experimental determination of the potential difference between the potential of the anodic peak (obtained on a stationary mercury drop at a linearly varying potential) and the potential that corresponds to the origin of the time scale when considering the dependence of the limiting anodic current (on the descending branch of the anodic peak) on time.

References

1. V. E. Gorodovykh. Dissertation. Tomsk Polytechnic Institute. Tomsk, 1964.

2. N. G. Chovnyk, V. V. Vashchenko. Journal of Physical Chemistry, 37, 538, 1963.

3. Ya. P. Gokhshtein. Doctoral dissertation. GEOCHI, Academy of Sciences of the USSR. Moscow, 1963.

4. A. G. Stromberg, V. Z. Bashkatov. This collection.

5. A. G. Stromberg. Proceedings of the Siberian Branch of the Academy of Sciences of the USSR, no. 5, 76, 1962.

6. V. E. Gorodovykh, A. G. Stromberg, B. F. Nazarov. This collection.

7. V. E. Gorodovykh. Proceedings of TPI, 28, 3, 1965.

1. The expression for the depth of the anodic notch in work [7] is not entirely rigorous, since in its derivation boundary conditions were used that are strictly valid only for infinite diffusion. ↩