Experimental Verification of the Dependence of Sensitivity and Accuracy of Determination on the Solution Volume and Electrode Surface

A. A. Kaplin, A. G. Stromberg, A. N. Pokrovskaya

We have carried out experiments on the experimental evaluation of the effect of deviations of \(V\) and \(S\) from the optimal values, calculated from the relations [1], on the sensitivity and accuracy of the APN method. According to [2], for a film electrode,

\[ a_*=\frac{0.1\cdot \alpha_0\cdot \delta \cdot h_c}{W_k\cdot q_*\gamma_*\cdot \cos \beta_*}\cdot i_* . \tag{1} \]

The quantities \(\nu\) and \(\delta\), at constant \(w_*\) and film thickness \(l\), are constant. When \(V\) deviated from the optimum, the concentration of the analyzed solution was changed correspondingly so that \((C_1V)\) was constant, which corresponded to constancy of the sample mass \(g\).

Denoting parameter values differing from the optimal ones by a prime \((\prime)\), we obtain the theoretical value of the relative sensitivity at a specified accuracy:

\[ \rho_t=\frac{a_*}{a'}=\frac{\cos \beta'}{\cos \beta_*}\cdot \frac{\gamma'}{\gamma_*}\cdot \frac{i_*}{i'} ; \tag{2} \]

the relative change in peak depth is determined by the relation

\[ \rho_t'=\frac{\gamma'}{\gamma_*}\cdot \frac{i_*}{i'} . \tag{3} \]

Relation (3) is applicable, in accordance with [1], when the solution is stirred by nitrogen bubbling, at \(\beta' > 30^\circ\). Then \(\sigma_c^2=\sigma_B^2+\sigma_\beta^2\); \((h_\phi=0;\ \sigma_\phi=0)\). The error of determination is due to the irreproducibility of the results and to fluctuations of the residual current. For measuring a peak with a specified accuracy \((\beta'=\beta_*)\), the optimum concentration must be increased by a factor of \(\dfrac{\cos \beta'}{\cos \beta_*}\). Let us recall that

\[ \beta'=\operatorname{arc\,tg}\left(\frac{S_1}{S_*}\right);\quad \beta'=\operatorname{arc\,tg}\left(\frac{i_1}{i_*}\cdot \operatorname{tg}\beta_*\right). \tag{4} \]

If \(\beta'<30^\circ\), then \(\sigma_\beta^2\ll\sigma_B^2\); the accuracy of determination is not related to fluctuations of the residual current, and

\[ \rho_t=\frac{\gamma'}{\gamma_*}\cdot \frac{i_*}{i'},\quad 1-e^{-b}=\gamma;\quad b=\frac{K_{\mathrm{e}}}{zF}\,S\,\frac{\tau}{V}. \tag{5–6} \]

The expected tooth depth \(h_\tau\) upon changing the conditions can be calculated from the relation:

\[ h_\tau = h_* \cdot \frac{i_*}{i'} \cdot \frac{\gamma'}{\gamma_*}. \tag{7} \]

The experiments were carried out with bismuth in a hydrochloric-acid background \((0.25\,k\ \mathrm{HCl})\). In contrast to the theoretical calculations, the effect on the sensitivity and accuracy of a fivefold deviation of \(V\), \(S\), and \(i\) from the optimal values on the value of \(\rho\) was studied experimentally. Such a condition, without changing the final conclusions, is more acceptable for experimental implementation.

The value \(V_x\) was taken equal to \(2\ \mathrm{ml}\), which made it possible to use \(V' = 0.04\ \mathrm{ml}\) and \(10\ \mathrm{ml}\), without changing the stirring method (nitrogen bubbling). The value \(\gamma\) was calculated taking into account the dependence of \(K_{\mathrm{e}}\) on \(V\); \(i_* = 1 \cdot 10^{-8}\ \mathrm{A/mm}\). The optimal electrode surface \(S_*\) was calculated from relation (4) using the experimental value \(\beta'\) for \(S_1\) \((i = 1 \cdot 10^{-8}\ \mathrm{A/mm})\) and \(\beta_* = 25^\circ\). \(S_* = 0.14\ \mathrm{cm}^2\). At \(W = 7 \cdot 10^{-3}\ \mathrm{g/sec}\); \(Z = 3\); \(F = 96500\ \mathrm{C/eq}\); \(\delta = 0.04\ \mathrm{g}\); \(h_c = 6 + 3.5 \simeq 10\ \mathrm{mm}\); \(\nu = 1.2\); \(K_{\mathrm{e}} = 1000 Z\ \mathrm{A \cdot cm/g\text{-}ion^{-1}}\); \(V = 2\ \mathrm{ml}\); \(A = 207\); \(\tau = 1800\ \mathrm{sec}\), we have \(\gamma_* = 0.72\); and from relation (1), \(c_* = 2 \cdot 10^{-9}\ \mathrm{g\text{-}ion/l}\) \((m = 8 \cdot 10^{-10}\ \mathrm{g}\) in a volume of \(2\ \mathrm{ml})\).

The value \(h = 10\ \mathrm{mm}\) corresponds to the minimum concentration \(c_*\), determined with the prescribed accuracy or with the prescribed error. By the latter is meant the error caused only by nonuniform stirring of the solution and by other factors of the accumulation stage. The magnitude of the measurement error is determined by the interference \(\bar{\beta}_c\) and its fluctuations \(\sigma_\beta \ll \sigma_v\). This condition is maintained at \((25^\circ\text{--}30^\circ)\), and the value \(h_c = h_{\mathrm{meas}} + \Delta h_c\); \(\alpha = 0.997\). With an increase in \(\bar{\beta}\) and \(\sigma_v\), the accuracy of determination decreases; to preserve the former accuracy of determination it is necessary to increase \(h_c\), which leads to a deterioration of the sensitivity \(\rho_\tau\).

The experimental value of the relative sensitivity:

\[ \rho_{\mathrm{e}} = \frac{h'}{h_*} \cdot \frac{i_*}{i'} \tag{8} \]

where \(h'\) is the mean of the tooth heights obtained in the experiment \((n = 3;\ \alpha = 0.68)\).

Experimental data are presented in Table 1. It should be noted that \(\rho < 1\) means a decrease in the sensitivity of determination, and vice versa.

It follows from Table 1 that, in most of the variants considered, deviation of the solution volume and electrode surface from the optimal values leads to a decrease in the sensitivity or accuracy of determination. Only in 4 cases out of 14, as follows from the experimental data, does the sensitivity of determination remain the same or may be somewhat increased while preserving the prescribed accuracy of determination. At \(i = i_x\), \(S = S_*\), \(V = \frac{1}{S}V_*\), and also \(S = \frac{1}{S}S_*\), \(V = \frac{1}{S}V_*\),

\[ i = \frac{1}{S} i_x;\quad \rho = 1.3\ \text{and}\ 1.0 \]

owing to an increase in the degree of depletion of the solution when the solution volume is decreased. If at \(V_*\), \(\gamma_* = 0.95\), this possibility of increasing the sensitivity is excluded. An increase in \(\rho_{\mathrm{e}}\) through the use of a higher polarograph sensitivity [1] is possible if \(i_*\) is not limiting for the given type of polarograph and \(h_\phi = 0\) at \(i_x\) and \(i' = \frac{1}{n} i_*\). The sensitivity of the polarograph—

phase \(i'=\dfrac{1}{S} i_*\) is optimal for the parameters \(S'=\dfrac{1}{S}S_*;\ V=\dfrac{1}{S}V_*\) and the corresponding amount of substance in solution \(\left(m'=\dfrac{1}{S}m_*=1.6\cdot10^{-10}\ \mathrm{g}\right)\). In this case the amount of substance in the electrode \(m_0=m_x\gamma_*\) corresponds to the minimum tooth \((h_c)\), determined with the specified accuracy. For the indicated conditions, the variant examined experimentally by us as optimal \((S_*;\ V_*;\ i_*=1\cdot10^{-8}\ \mathrm{A/mm})\) is a deviation from the optimum that leads to a deterioration of sensitivity \((8\cdot10^{-10}\ \mathrm{g}\) instead of \(1.6\cdot10^{-10}\ \mathrm{g})\). By analogy, in the absence of fluctuation interferences of the second kind, it may be assumed that at

\[ i=4\cdot10^{-11}\ \mathrm{A/mm}=\frac{1}{250}i_*;\quad V=\frac{1}{250}V_*\simeq0.01\ \mathrm{ml}\quad S=\frac{1}{250}S_*=6\cdot10^{-4}\ \mathrm{cm^2} \]

one can determine the amount of substance

\[ m=\frac{1}{250}m_0=8\cdot\frac{10^{-10}}{250}=3\cdot10^{-12}\ \mathrm{g} \]

at an anodic-tooth depth \(h_c=10\ \mathrm{mm}\) and \(\gamma=0.7\). If, however, \(i=4\cdot10^{-10}\ \mathrm{A/mm};\ s'=6\cdot10^{-3}\ \mathrm{cm^2};\ V=0.1\ \mathrm{ml}\), then \(m=3\cdot10^{-11}\ \mathrm{g}\).

Table 1

Effect of deviations of the experimental conditions from the optimum on the sensitivity of the method

The experiments on determination of the indicated amounts were carried out by us in a rotating microcell and with an electrode consisting of a mercury hemisphere on a silver contact.

At \(i=3.6\cdot10^{-10}\ \mathrm{A/mm}\) \((LP\text{—}60);\ V=0.2\ \mathrm{ml};\ S=7\cdot10^{-3}\ \mathrm{cm^2};\ \tau=40\) min, i.e., under conditions close to those listed above, anodic bismuth teeth were obtained at its content in the solution of \(3\cdot10^{-11}\ \mathrm{g}\). From the area under the tooth, \(K_e=3000\ \mathrm{A\cdot cm/g\text{-}ion^{-1}},\ b=2.5;\ \gamma=0.92\) at \(\tau=40\) min. In experiments in which an electrode size \(S=7\cdot10^{-4}\ \mathrm{cm^2}\) was used (a hemisphere of radius \(0.011\ \mathrm{cm},\ V=0.02\ \mathrm{ml},\ \tau=40\) min, \(K_e=3000\ \mathrm{A\cdot cm/g\text{-}ion^{-1}};\ B=2.5;\ V=0.02\)), anodic bismuth teeth were obtained at its content in the solution of \(3\cdot10^{-12}\ \mathrm{g}\).

Thus, the experimental studies carried out showed the applicability of the derived relations for calculating the optimal parameters, and the presence of a decrease in sensitivity and accuracy of determination when the conditions deviate from the optimal ones. The possibility of increasing sensitivity by using higher sensitivities of the polarograph, with a corresponding decrease in \(S\) and \(V\), was experimentally confirmed. By selecting optimal conditions, a maximum sensitivity of determination of \(3 \cdot 10^{-12}\) g for bismuth was achieved, which corresponds to \(3 \cdot 10^{-10}\%\) at \(g = 1\) g. The schemes for selecting optimal conditions were used by us in developing accelerated and highly sensitive methods for determining impurities in high-purity indium.

The data in Table 1 make it possible to assess the nature of the independent influence of \(V\) and \(S\) on the sensitivity and accuracy of determination. Thus, the influence of \(V\) on the sensitivity of determination at a constant sample weight \((m—\mathrm{const})\) can be traced in the vertical columns at \(S—\mathrm{const}\). From the data in Table 1 it follows that the sensitivity of determination increases as the volume of solution decreases, which is in agreement with the theoretical conclusions [3]. The influence of the electrode surface on the sensitivity of the method can be assessed from the horizontal columns, at \(V—\mathrm{const}\).

REFERENCES

1. A. G. Stromberg, A. A. Kaplin, ZhAKh, 10, 1965.
2. A. A. Kaplin, A. G. Stromberg. Proceedings of the Tomsk Polytechnic Institute (in press).
3. A. G. Stromberg, A. A. Kaplin. Zav. labor., 4, 1964.