Some Issues in Estimating the Accuracy of Determination and Selecting Optimal Conditions in the APN Method

A. A. Kaplin

(Presented by the scientific seminar of the Faculty of Chemical Technology)

The method of amalgam polarography with accumulation [1, 2] is one of the most promising for determining traces of impurities in high-purity materials. In works [3–6] it has been shown that, by the APN method, impurities can be determined down to \(10^{-10}\)—\(10^{-12}\%\) and lower when using the corresponding sensitivities of the polarograph \((10^{-9}\)—\(10^{-12}\ \text{A/mm}\) and lower) and when selecting optimal conditions at the stages of accumulation, anodic oxidation, and recording of the peak. The minimum impurity content \((a,\ \%)\) determinable with a prescribed accuracy when a film electrode is used [6] is

\[ a\%= \frac{0.1\,\alpha_{0}\,\delta\,h_{c}\,\nu}{w\cdot g\,\gamma_{*}\cos\beta_{c}}\,i_{*}, \tag{1} \]

where \(h_c\) is the minimum height of the anodic peak, mm; \(\alpha_0\) is the electrochemical equivalent, \(g\cdot A^{-1}\text{sec}^{-1}\); \(\delta\) is the half-peak width, volts; \(w\) is the rate of change of potential, V/sec; \(g\) is the sample weight, g; \(\nu\) is the peak-shape coefficient; \(\nu=1.2\); \(i_*\) is the sensitivity of the polarograph, \(A\cdot \text{mm}^{-1}\). The degree of depletion of the solution is

\[ \gamma = 1-e^{-B}, \tag{2} \]

\[ B=\frac{K_{1}\cdot S_{*}\tau_{*}}{zFV_{*}}, \tag{3} \]

where \(K_1\) is the electrolysis constant, \(A\cdot cm\cdot g\text{-ion}^{-1}\); \(S_*\) is the electrode surface area, \(cm^2\); \(\tau_*\) is the electrolysis time, sec; \(V_*\) is the volume of solution, ml.

Let us consider several questions connected with estimating the accuracy of determination in the APN method. The true error of an experimental determination can be estimated either by comparing the results with the true content of the impurity in the sample, or by comparing the anodic current with a value calculated from the exact equation of the peak. For the time being, however, the determination data are the result of processing several parallel data and comparing them with peaks from additions of standard solutions. It is assumed that the conditions in determining the latter are analogous to the conditions in determining the impurities. However, the parameters of the recorded signal—the height or area of the peak—are determined by the combined action of the useful signal (the change in anodic current during oxidation of the amalgam of the element being analyzed) and interferences. Therefore the magnitude of the peak height has a random character associated with fluctuation-

by changes in the parameters at the stages of accumulation, oxidation, and tooth registration. The accuracy of determination must be characterized by the statistical characteristics of the parameters of all stages. The criterion that makes it possible to estimate the accuracy of concentration determination is the root-mean-square error \(\sigma_c\). To estimate the errors we shall use relation (1), which establishes the connection between \((a)\%\) and the parameters of all stages; then:

\[ \sigma_c=\sqrt{\sigma_{\mathrm{p}}^2+\sigma_{\mathrm{n}}^2+\sigma_{\mathrm{ok}}^2+\sigma_{\mathrm{r}}^2}. \tag{4} \]

1. The stage of sample preparation, including chemical separation of the base. This stage, although not directly related to APN determination, affects the total accuracy of the analysis. The total error \(\sigma_{\mathrm{p}}\) is due to: nonuniform distribution of the impurity in the sample, loss of impurity during sample treatment or separation of the base, adsorption, desorption, or hydrolysis of the impurity in the supporting electrolyte. Errors associated with these parameters, \(B\), may be characterized by the random error in measuring the sample weight \(g\).

2. The accumulation stage. The error of determination is associated with fluctuations in the degree of exhaustion (2–3). For \(B<0.05\),

\[ \gamma \cong B=\frac{K_1}{zF} S_* \frac{\tau_*}{V_*}. \]

Consequently, under the condition of mutual independence of the parameters,

\[ \sigma_{\mathrm{n}}=\sigma_\gamma=\sqrt{\sigma_k^2+\sigma_s^2+\sigma_V^2+\sigma_\tau^2} \tag{5} \]

The constant \(K\) is a function of the method of stirring, the shape of the electrodes and electrolyzer, and the diffusion coefficient of ions in solution (\(D_i^0,\ \mathrm{cm^2/sec}\)). Under thermostatting of the cell and a constant position of the electrodes, the influence of the latter factors is insignificant. The main error is introduced by the nonuniform nature of the stirring of the solution; as a systematic error it can so far be taken into account only for a rotating disk electrode. Fluctuation of the magnitude of the active surface of the electrode may be associated with adsorption and desorption of surfactants, evolution and adhesion of hydrogen bubbles, and the formation of sparingly soluble films on the electrode surface. A change in the volume of solution is possible because of splashing and evaporation. If \(B\geq 3\), \(\gamma=1\), and \(\sigma_\gamma\) does not depend on changes in the parameters of the accumulation stage, provided that \((B+\Delta B)\geq 3\) \((\alpha=0.997)\). At values \(0.05<B<3\), the quantity \(\sigma_\gamma\) depends in a complicated way on the parameter \(B\). This dependence becomes more complicated in the presence of a correlation between individual parameters. For example, when the solution is stirred by nitrogen bubbling,

\[ K_1=K_1^0+\frac{A}{V}\ [6]. \]

For \(K_1>\frac{A}{V}\) \((V>(10\div 15\ \mathrm{ml}))\), \(K_1\) does not depend on the solution volume; for \(V<(0.5\div 1\ \mathrm{ml})\), \(\sigma_k\) depends on \(V\), which must be taken into account by correlation coefficients.

3. The stages of anodic oxidation and tooth registration. Fluctuations of the parameters of these stages \((\delta;\ \beta;\ i_k)\) determine the accuracy of measuring the quantity \(h_c\) and its component \(h_{\mathrm{meas}}\). By \(h_{\mathrm{meas}}\) is meant such a minimum tooth height as can be measured with the aid of a measuring device with a systematic error not exceeding the error of the latter (for example, a ruler or microscope). Let us consider two cases:

a) When measuring the tooth height there is no fixed reference point. The depth of the tooth is taken as \(h_{\mathrm{meas}}\), the angle at the base of which is equal to \(90^\circ\). Then

\[ h_{\mathrm{meas}}=\sqrt{\delta_1\delta_2}, \]

where \(\delta_1\) and \(\delta_2\) are the widths of the base of the kinetic and diffusion branches of the tooth. At \(h'<h_{\mathrm{meas}}\), the fixed reference point for the height disappears. Usually \(h_{\mathrm{meas}}=5\div 10\ \mathrm{mm}\),

b) The peak potential of the wave is fixed, for example, by stopping the drum on the polarographs LP-55, LP-60. \(h'\) may be considerably smaller than \(h_{\mathrm{meas}} = 1/\hat{\delta}_1\hat{\delta}_2\). It must be taken into account that, for small wave magnitudes—for example 1 mm, measured with an MIM-7 microscope—the inertia of the instrument makes itself felt and there arises the probability of a random instrumental error \(\sigma_i\).

The magnitude \(h_{\mathrm{meas}}\) is affected by the half-wave width \(\hat{\delta}\). Fluctuations of \(\hat{\delta}\) may be caused by changes in the degree of reversibility, the radius of the drop, the diffusion coefficient of atoms in the amalgam, and the formation of a film on the electrode surface. An increase in \(\hat{\delta}\) leads to a decrease in \(h_{\mathrm{meas}}\). To eliminate the error associated with \(\sigma_{\hat{\delta}}\), one may use the integrating attachment of Yu. A. Ivanov [7].

The error of determination increases in the presence of interference during the stage of recording the wave. Taking the accuracy of determination in the absence of interference as given, let us consider the influence of interference on the sensitivity of determination.

In the APN method, one may distinguish interference of the first and second kinds. The first includes such interference whose amplitude, in the region of potentials equal to the width of the base of the wave, varies according to a linear or nearly linear law. The fluctuation period of such interference considerably exceeds the time required to record the wave. The magnitude of interference of the first kind may be characterized by the derivative of the residual current with respect to potential, or by the angle of inclination \(\beta\) of the residual-current line to the horizontal. The magnitude \(\beta\) and its fluctuations are caused by the discharge of oxygen ions, impurities of metals reduced at the electrode without amalgam formation, capacitive current, etc. During the stage of wave recording, an increase in \(\beta\) leads to a reduction in the accuracy of measuring \(h_{\mathrm{meas}}\), owing to the increasing influence of the inertia of the instrument as \(\dfrac{di_{\mathrm{res}}}{d\varphi}\) increases and the fixed reference point disappears. The measurement accuracy remains the same if \(h_{\mathrm{meas}}\) is increased by the amount \(1/\cos\beta_c\), where \(\beta_c(0.997)=\beta_0+\Delta\beta\). Interference of the first kind is in principle removable. Interference of the second kind consists of pulses whose frequency-spectrum width is smaller than the width of the wave spectrum. Unlike interference of the first kind, during the time of wave recording there occurs a single or multiple increase in the interference and its decrease to zero. The following groups of interference of the second kind may be distinguished.

a) Interference caused by the oxidation stage: evolution of hydrogen, and change in the active surface during wave recording. This also includes waves due to contamination of the supporting electrolyte and electrolyzer by the element being determined or by an element adjacent to it. Interference of the second kind includes the currents of one of the waves when it partially overlaps another.

b) Instrumental interference: vibration of the apparatus, electrical interference from neighboring operating instruments and from the power supply, interference from poor contacts, disruption of contact between the slider of the setting potentiometer and the rheochord during oxidation of the latter, especially in chemical laboratories, interference from nonuniform wear of the rheochord, etc. The interference listed in (a, b) is in principle removable: purification of the solution from contaminants, increase in resolving power, grounding of the instrument.

c) Completely irremovable fluctuation interference: thermal noise, shot-effect noise, magnetic fluctuations. The limit of such noise lies at \(10^{-15}\)—\(10^{-16}\) a, which is 3–4 orders of magnitude below the currents used in the APN method. Interference of the second kind may be characterized by the peak values of the amplitudes \(h_n\). Let us consider variants for estimating the influence of interference of the second kind.

1. In the region of potentials equal to the tooth base, a single interference pulse with peak amplitude \(h_{\mathrm{p}}\) is recorded, coinciding with the potential of the anodic tooth: teeth of the analyzed element in the blank sample, partial superposition of currents of neighboring elements, a combination of teeth of the analyzed impurity and of the standard solution:

a) \(\sigma_{\mathrm{p}}=0;\ \sigma_{\mathrm{c}}=0.\)

The condition for measuring the minimum tooth with a specified accuracy:

\[ h_{\mathrm{c}}=h_{\mathrm{c+p}}-h_{\mathrm{p}}=h_{\mathrm{meas}}, \tag{8} \]

where \(h_{\mathrm{c+p}}\) is the total tooth height from the signal and the interference.

b) \(\sigma_{\mathrm{p}}=0;\ \sigma_{\mathrm{c}}\ne 0.\)

\[ h_{i,\mathrm{c}}(0.997)=h_{\mathrm{meas}}+\Delta h_{\mathrm{meas}};\qquad h_{\mathrm{c}}=h_{\mathrm{c+p}}-h_{\mathrm{p}}; \]

\[ h_{\mathrm{c}}=h_{\mathrm{meas}}+\Delta h_{\mathrm{meas}} \tag{9}; \qquad \pm\Delta h_{\mathrm{c}}=\frac{\sigma_{\mathrm{c}}\cdot t_{\mathrm{k}}}{\sqrt{n}}\ [8], \]

where \(n\) is the number of measurements; \(t_{\mathrm{k}}\) is Student’s coefficient.

c) \(\sigma_{\mathrm{p}}\ne 0;\ \sigma_{\mathrm{c}}\ne 0.\)

Fluctuations of the signal and of the interference are of opposite sign:

\[ h_{i,\mathrm{c}}(0.997)=h_{\mathrm{meas}}\pm\Delta h_{\mathrm{meas}};\qquad h_{i,\mathrm{p}}(0.997)=h_{i,\mathrm{p}}\pm\Delta h_{\mathrm{p}}; \tag{10} \]

\[ h_{\mathrm{c}}(0.997)=h_{\mathrm{meas}}+\Delta h_{\mathrm{meas}}-\Delta h_{\mathrm{p}}. \tag{11} \]

d) \(\sigma_{\mathrm{c}}\ne 0;\ \sigma_{\mathrm{p}}\ne 0.\)

Fluctuations of the signal and of the interference are of the same sign:

\[ h_{\mathrm{c}}=h_{\mathrm{meas}}+\Delta h_{\mathrm{meas}}+\Delta h_{\mathrm{p}}. \tag{12} \]

Relations (8; 9; 11; 12) correspond to the limiting values of the tooth height measured with a specified accuracy, the error \(\Delta h_{\mathrm{meas}}\) being determined by the accuracy of the meter. The corresponding concentration \(a,\%\), for \(h_{\mathrm{c}}\), is the limit of analytical detection with the specified accuracy. The quantity \(h_{\mathrm{c}}\) enters into expression (1).

\[ h_{\mathrm{c}}=\frac{h_{\mathrm{meas}}}{1\pm k_0}, \]

where \(k_0\) is the magnitude of the maximum error \((\alpha=0.997)\) for the given method of mixing.

e) Interference of the second kind within the base width, with arbitrary potential location of the peak amplitude value \((h_{\mathrm{p}}=\mathrm{const})\). The error of determination is random in character; \(h_{\mathrm{c+p}}\) varies so that \(h_{\mathrm{c}}\le h_{\mathrm{c+p}}\le h_{\mathrm{c}}+h_{\mathrm{p}}\). To measure the tooth height with a specified accuracy, the following condition must be fulfilled: \(h_{\mathrm{p}}=n_0\cdot h_m\cdot m\), whence \(m=\dfrac{1}{n_0}\) (14). Here \(n_0\) is understood as the specified error relative to the value \(\Delta h_{\mathrm{meas}}\). For example, \(h_{\mathrm{p}}=10\ \mathrm{mm};\ h_{\mathrm{meas}}=10\pm2\%\). To measure \(h_{\mathrm{c}}\) with an error of \(\pm2.5\%\) \((n_0=0.005)\), the threshold ratio is \(m=200\). Consequently, at the given sensitivity of the polarograph field, the signal must exceed the interference by a factor of 200 in order to measure the tooth with the specified accuracy. The sensitivity of determination (1) is correspondingly worsened. For the detection limit under interferences of this kind, one may use relation [9]:

\[ m=\operatorname{arc}\Phi\left(1-\frac{1-\alpha}{n-1}\right), \tag{15} \]

where \(\alpha\) is the confidence probability; \(n\) is the number of measurements. The most complicated case is the recording of a fluctuating anodic tooth against a background of fluctuational interferences of random amplitude and phase.

Along with those described above, a considerable random error is introduced into the result by the different tooth depth for the same additions of standard solution. To eliminate these errors, a method is proposed for calculating the concentration in the solution without introducing standard additions. In the solution being analyzed, the anodic teeth \(h_1\) and \(h_2\) are recorded at different electrolysis times \(\tau_1\) and \(\tau_2\), and solution volumes \((V_1\) and \(V_2)\) [10]. In the first case, from the relation

\[ K_1=\frac{ZF\lg(h_1/h_2)}{0.43\cdot S(\tau_2-\tau_1)} \tag{16} \]

\(K_1\) is estimated, and \(\gamma=1-e^{-B}\) is calculated. The impurity content is

\[ a\%=\frac{0.1\alpha q}{\gamma g}, \tag{17} \]

where \(q\) is the area under the anodic tooth, coulombs. At \(B>3\), with an accuracy to within 5%, in relation (17) \(\gamma=0.95\). The calculation is simplified when \(K_1\) is known for the given method of stirring.

Let us consider the systematic errors in APN determinations that enter into the assigned total error of the determination. In accordance with relation (1),

\[ \frac{\Delta a}{a} = \frac{\Delta h_c}{h_c} + \frac{\Delta g}{g} + \frac{\Delta i}{i} + \frac{\Delta \gamma}{\gamma} + \frac{\Delta \beta}{\beta}. \tag{18} \]

1. The quantity \(\dfrac{\Delta h_c}{h_c}\) is determined by the quantity \(\dfrac{\Delta h_{\text{meas}}}{h_{\text{meas}}}\). At \(h_{\text{meas}}=10\) mm and measurement with a ruler, the error is \(\pm 2\%\).

2. The error in determining the weight of the sample, with analytical-balance accuracy down to \(10^{-4}\)–\(10^{-5}\) g and sample weights of 0.1–1 g, is negligible.

3. The systematic error of the recording instrument is determined by the accuracy class of the instrument; for the polarograph 7-77-4B it is 2.5. Consequently, \(\dfrac{\Delta i}{i}\) is \(\pm 2.5\%\).

4. The quantity \(\dfrac{\Delta\gamma}{\gamma}\) is determined by the value of the parameter \(B\). At \(\gamma\leq 0.05\),

\[ \frac{\Delta\gamma}{\gamma} \simeq \frac{\Delta B}{B} = \frac{\Delta V}{V} + \frac{\Delta\tau}{\tau} + \frac{\Delta S}{S}. \tag{19} \]

If \(K_1=1000Z\ \text{A}\cdot\text{cm}^{-2}\cdot\text{ion}^{-1}\); \(S=2\cdot10^{-2}\ \text{cm}^2\), then \(V\gg1\) ml; \(\tau\gg250\) sec.

At \(V\gg1\) ml, \(\dfrac{\Delta V}{V}=\pm1\%\); at \(\tau\gg250\) sec, \(\dfrac{\Delta\tau}{\tau}=\pm0.01\%\). For replaceable drops the error may be due to an error in reading the current \(\dfrac{\Delta i_p}{i_p}\) and the time of drop formation \((t)\): \(\dfrac{\Delta S}{S}=\dfrac{\Delta i_p}{i_p}+\dfrac{\Delta t}{t}\). The accuracy class of the rectifier for building up the electrode is 1.5. \(t=30\div60\) sec. \(\dfrac{\Delta S}{S}=1.5\%\). If the mercury current efficiency is incomplete, the quantity \(\dfrac{\Delta S}{S}\) can be taken into account by measuring the size of the electrode under a microscope. If repeated measurements are carried out in one and the same solution and on the same drop, \(\dfrac{\Delta S}{S}\) and \(\dfrac{\Delta V}{V}\) are equal to zero.

1. The error \(\dfrac{\Delta \gamma}{\gamma}\) does not depend on the accumulation parameters, provided that \((B+\Delta B)\leqslant 3\). At intermediate values \(0.05 < B < 3\), the total error \(\dfrac{\Delta \gamma}{\gamma}\) depends on \(B\) in a complicated way. The error in measuring \(\beta\) and \(h_n\) is determined by the accuracy of the measuring instruments (ruler and protractor). Thus, when working in one and the same solution on one and the same drop, with complete depletion of the solution, the total systematic error is \(\pm(4\div 5)\%\) and is due mainly to the accuracy of measuring the tooth and to the accuracy class of the polarograph. No systematic study has yet been carried out in the APN method of the influence of random and systematic errors depending on the parameters of the stages. Usually the total error (reproducibility \(\sigma_v\)) of all stages is estimated without allowance for the effect of first- and second-kind interferences. The quantity \(\sigma_v\) is known for the given mixing method and the given operating procedure. Consequently,

\[ \sigma_c=\sqrt{\sigma_v^2+\sigma_\beta^2+\sigma_{\mathrm{ir}}^2}. \tag{20} \]

Below are given variants of analytical problems and possible schemes, proposed by us, for selecting optimum conditions.

1. Determination of the concentration \(a\%\) of the given element with a determination error of \(\pm \Delta a\) \((\alpha=0.997)\) in \(n\)-parallel measurements. From the value of \(\Delta a\) (after subtracting the systematic error), \(n\), and \(t_k\), we find \(\sigma_c\) (10) and, taking \(h_p=0\), with known \(\sigma_v\), we estimate \(\sigma_\beta\). From relation (1), knowing \(h_c\) (8—9, 11—12) for \(1/\cos_\beta=1\), we calculate \(i_x=A\cdot \gamma_x\) (21). Taking \(\gamma_x=0.05\), we find \(i_{x\max}\) on an arbitrary surface \(S_1\); at the polarograph sensitivities \(i_x\), \(1/2\,i_x\), and \(2i_x\), we estimate \(\sigma_{i,\beta}\) in \(n\)-measurements and \(\beta_i\). From the dependence \(\sigma_{\beta,i}-\beta\) and the known \(\sigma_\beta\), we determine \(\beta\). From the relation

\[ S_x=S_1\cdot \frac{\tg \beta_x}{\tg \beta_i} \]

we determine \(S_x\), and then \(\tau_x\) and \(V_x\) for the given mixing method. If \(i_x\) is below the sensitivity limit of the polarograph, we increase \(i_x\) and, correspondingly, \(\gamma_x\). In the presence of second-kind interferences, the tooth magnitude \(h_c\) is calculated from the threshold ratio \(m\) (14); the deterioration of sensitivity can be compensated by increasing \(\gamma\) and decreasing \(S_x\), if \(h_p=f(S)\).

1. Estimation of the maximum sensitivity of determination with a prescribed accuracy. The calculation is analogous to item 1. The value \(\gamma_x=0.95\).

2. A lower limit is specified for the content of an impurity in a sample \((a,\%)\). Develop conditions for product control with a determination error of \(\pm \Delta a_x(0.997)\). The calculation of the sensitivity of the procedure \(a_x\) is determined by the relation

\[ a_x+\Delta a_x=a \tag{22} \]

and

\[ a_{x,i}(0.997)=a_x\pm\Delta a_x;\qquad n_1=\pm\frac{\Delta a_x}{a_x}. \]

Consequently,

\[ a_x=\frac{a}{1+n_1}. \tag{23} \]

The subsequent calculation is carried out in accordance with item 1.

Variants of the calculation of optimum parameters were used by us in developing procedures for the determination of \(10^{-8}\)—\(10^{-6}\%\) impurities in high-purity indium.

References

1. A. G. Stromberg, E. A. Stromberg, Zavod. labor., 27, 3, 1961.
2. A. G. Stromberg, E. A. Zakharova, Zavod. labor., 30, 261, 1964.
3. A. G. Stromberg, Izv. TPI, vol. 128, 1965.
4. A. G. Stromberg, Zavod. labor., vol. XXXI, 10, 1965.
5. A. G. Stromberg, A. A. Kaplin, ZhAKh, 10, 1965.
6. A. A. Kaplin, A. G. Stromberg, Izv. TPI (in press).
7. Yu. A. Ivanov, Zavod. labor. (in press).
8. A. N. Zaidel, Elementary Estimates of Measurement Errors. Nauka, Moscow, 1965.
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10. A. G. Stromberg, V. I. Kuleshov, ZhAKh (in press).