Theory of the Resolving Power of the APN Method. Symmetric Peaks
Abstract Generated abstract
The paper develops a quantitative estimate for the limiting resolving power of the APN method with a linearly varying potential when two closely spaced, symmetric anodic peaks overlap. Using a Gaussian approximation for peak shape and assuming equal half-width parameters, the authors formulate the total current as the sum of two peak functions and define the resolution limit by the condition that the tangent at the inflection point between peaks becomes horizontal. Solving the resulting derivative equations yields parametric expressions for the critical peak-position coordinate and limiting peak-height ratio as functions of normalized peak separation. The analysis shows how the limiting resolving power and required difference between peak potentials depend on peak half-width, providing a calculational basis for estimating resolution under specified experimental conditions.
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Theory of the Resolving Power of the APN Method. Symmetric Peaks
A. A. Kaplin, A. G. Stromberg, A. A. Zheltonozhko
(Presented by the scientific-methodological seminar of the Department of Physical Chemistry)
At present, there are comparatively few works in the literature on the theory of the resolving power of the APN method with a linearly varying potential [1—4]. The question of the limit of resolving power was not considered in these works.
The purpose of the present work is to consider the question of a quantitative estimate of the limit of resolving power on the basis of a mathematical analysis of the equation for the total current of two close peaks as a function of the difference between the peak potentials, the shape of the peaks, and the concentration ratio.
According to [2], the measure of the resolving power of the APN method is taken to be the maximum ratio of the peak heights of the interfering and determined elements at which it is still possible to determine the element under consideration with a prescribed accuracy, for a given difference between the peak potentials on the selected background and under other specified conditions.
Fig. 1. Diode peaks of elements in mutual position:
a) asymmetric and symmetric peak of the interfering element (the more electronegative one), b) mutual influence of two anodic asymmetric peaks
For definiteness, let us consider the process of electro-dissolution from a limited-volume amalgam of two metals with close values of peak potentials under a linearly varying potential.
In this case, the electro-dissolution currents of both metals add together, and the total polarogram has the form of a curve with two maxima and mi-
minimum between them, and the tangent \(MN\) at the point of inflection is located at an angle to the abscissa axis (Fig. 1, b).
As the limit of resolution we take the ratio of peak heights when the tangent at the point of inflection \(D\) is horizontal (Fig. 2).
We shall adopt the following initial and simplifying assumptions:
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Instead of considering an asymmetric peak with branches \(I_a\) and \(I\), we shall consider a symmetric peak with branches \(1\) and \(1b\) (Fig. 1, a).
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We assume that this symmetric peak can be described by a Gaussian-type function:
\[ i=I e^{-\alpha^2(\varphi-\varphi_n)^2}, \tag{1} \]
where \(I\) is the peak height, \(\varphi_n\) is the peak potential. The parameter \(\alpha\), as is easily seen, is related to the half-width of the symmetric peak \(\delta\) by the relation \((i=0.5I,\ \text{Fig. }1,a)\)
\[ \alpha=\frac{1.66}{\delta}. \tag{2} \]
Experience shows that function (1) satisfactorily describes the rising or falling branch of an anodic peak on a mercury drop or film electrode.
Fig. 2. Mutual influence of two symmetric peaks (limiting case)
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When two peaks are considered together, both peaks are regarded as symmetric; moreover, for definiteness we take the peak of the interfering element to be more electronegative, and the peak of the element being determined to be more electropositive. Consequently, for the interfering peak the descending (right-hand) branch is considered, and for the peak being determined the ascending (left-hand) branch. All the mathematical conclusions considered depend on the section of the curve \(AB\) between the maxima on the polarogram (Fig. 1, b). The shape of the curve to the left of \(A\) and to the right of \(B\) has practically no significance for conclusions about the resolving power and its limit.
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In the present work, for simplification, a particular case is considered in which the parameters \(\delta\) (or \(\alpha\)) of both peaks are identical.
In what follows we agree to denote the characteristics of the interfering peak (the more electronegative, left-hand one) by the index 1, and the characteristics of the peak being determined (the more electropositive, right-hand one) by the index 2.
Let us consider two closely spaced peaks. For the total current we have the equation
\[ i=i_1+i_2=I_1 e^{-\alpha_1^2(\varphi-\varphi_{n1})^2}+I_2 e^{-\alpha_2^2(\varphi-\varphi_{n2})^2}. \tag{3} \]
We make the change of variables in equation (3)
\[ y=\frac{i}{I_1}; \qquad x=\alpha_1(\varphi-\varphi_{n1}), \tag{4} \]
then instead of equation (3) we obtain
\[ y=e^{-x^2}+K e^{-n^2(x-b)^2}, \tag{5} \]
where
\[ n=\frac{\alpha_2}{\alpha_1}; \qquad b=\alpha_1(\varphi_{n2}-\varphi_{n1}); \tag{6} \]
\[ K=\frac{I_2}{I_1}. \tag{7} \]
For \(\alpha_1=\alpha_2\), \(n=1\).
For this particular case, equation (5) will take the form
\[ y=e^{-x^2}+Ke^{-(x-b)^2}. \tag{8} \]
The graph of equation (8) for \(K<1\) is shown in Fig. 1,b.
It follows from formula (4) that, on the \(x\)–\(y\) graph, the first peak (the interfering one) is located at the origin and its height is equal to unity. The second peak (the one being determined) is shifted to the right by a distance \(b\) along the \(x\)-axis, and its height is equal to \(K\).
The expression for the resolving power of the APN method for a linearly varying potential, according to [2], in our case has the form
\[ j=ge^{\frac{1.66\,\Delta\varphi_{n\,1,2}}{\delta}}, \tag{9} \]
where
\[ j=\frac{I_1}{I_2}=\frac{1}{K} \]
is the resolving power,
\[ P=\frac{I_1}{i}=e^{\frac{1.66\,\Delta\varphi_{n\,1,2}}{\delta}} \]
is the resolving-power coefficient,
\[ g=\frac{i}{I_2} \]
is the error in measuring the height of the peak of the second element being determined.
Proceeding from the definition given above, the limit of resolving power can be described by the following mathematical relations (Fig. 2):
\[ \frac{\partial y}{\partial x}=0;\qquad \frac{\partial^2 y}{\partial x^2}=0. \tag{10} \]
Taking the first and second derivatives of equation (8) and setting them equal to zero, we obtain the system of equations:
\[ \begin{cases} \dfrac{\partial y}{\partial x} = -x_{\mathrm{per}}e^{-x_{\mathrm{per}}^2} - K_{\mathrm{per}}(x_{\mathrm{per}}-b)e^{-(x_{\mathrm{per}}-b)^2} =0; \\[6pt] \dfrac{\partial^2 y}{\partial x^2} = -e^{-x_{\mathrm{per}}^2} + 2x_{\mathrm{per}}^2e^{-x_{\mathrm{per}}^2} - K_{\mathrm{per}}e^{-(x_{\mathrm{per}}-b)^2} + \\[4pt] \quad +2K_{\mathrm{per}}(x_{\mathrm{per}}-b)^2e^{-(x_{\mathrm{per}}-b)^2} =0. \end{cases} \tag{11} \]
We assume that the parameter \(b\) is specified by us, while \(x_{\mathrm{per}}\) and \(K_{\mathrm{per}}\) are unknowns. Solving the system of equations (11), we find
\[ x_{\mathrm{per}\,1,2}=\frac{b}{2}\pm\frac{1}{2}(b^2-2)^{1/2}, \tag{12} \]
\[ \frac{1}{j_{\mathrm{pred}}} = K_{\mathrm{per}} = -\frac{x_{\mathrm{per}}}{x_{\mathrm{per}}-b} e^{-(2bx_{\mathrm{per}}-b^2)}, \tag{13} \]
the sign \((+)\) in formula (12) corresponds to values \(K_{\mathrm{per}}<1\), and the sign \((-)\) to values \(K_{\mathrm{per}}>1\).
Thus, expression (13) makes it possible to calculate directly the limit of the resolving power of the APN method for a linearly varying potential.
The dependence of \(x_{\mathrm{trans}}, K_{\mathrm{trans}}, y_{\mathrm{trans}}\) on the parameter \(b\) is presented in Fig. 3. The theory under consideration is parametric, since the equation of the peak is described by the empirical interpolation formula (1), in which the numerical values of the parameters \(I, \varphi_n, \delta\) are found from experiment.
Fig. 3. Dependence of \(K_{\mathrm{trans}}, y_{\mathrm{trans}}, x_{\mathrm{trans}}\) on the parameter \(b\)
As is evident from equation (12) (Fig. 3), the value of \(b\) cannot be less than two.
It follows from (2) that the parameter \(\delta\) enters (8), (12–13) in implicit form and depends on a number of factors [2].
If it is assumed that the limit of resolving power, representing the length of the horizontal section \(\sigma = 2(b - x_{\mathrm{trans}})\), is some quantity that can be recorded with sufficient reliability on the polarogram, for example \(4\ \mathrm{mm}\), and the rate of change of the potential \((w\ \mathrm{V/sec})\) is \(0.066\ \mathrm{V/sec}\), the speed of motion of the recorder tape is \(0.66\ \mathrm{mm/sec}\), then for \(\delta = 0.14\ \mathrm{V}\) we have \(j_{\mathrm{lim}} = 1.2\), \(\Delta \varphi_{n\,1,2} = 0.16\ \mathrm{V}\); and for \(\delta = 0.073\ \mathrm{V}\), \(j_{\mathrm{lim}} = 200\), \(\Delta \varphi_{n\,1,2} = 0.88\ \mathrm{V}\).
Fig. 4. Dependence of the resolving-power limit and \(\Delta \varphi_{n\,1,2}\) on \(\delta\)
The dependence of the limit of resolving power \((j_{\mathrm{lim}})\) and of the limiting difference of the potentials \((\Delta \varphi_{n\,1,2})\) of two peaks on the half-peak width \((\delta)\) is presented in Fig. 4.
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