COMPARISON OF TWO METHODS FOR ESTIMATING THE SENSITIVITY OF THE APN METHOD

A. G. Stromberg, M. P. Potapov

(Presented by Professor A. G. Stromberg)

In a paper published by one of us [1], a method was proposed for estimating the sensitivity of the method of amalgam polarography with accumulation (APN). In order to obtain an expression for the minimum determinable concentration of a substance, the concept of the effective half-width of a peak is introduced \((\delta_{\mathrm{eff}},\ \text{volt})\):

\[ \frac{\delta_{\mathrm{eff}}}{w}=\frac{q}{I}=\frac{\gamma \delta}{w}, \tag{1} \]

where \(w\) is the rate of change of potential, \(\text{volt/sec}\); \(q\) is the quantity of electricity expended in the process of anodic dissolution of the metal from the amalgam (equal to the area under the anodic peak), coulomb; \(I\) is the depth of the anodic peak, a; \(\delta\) is the half-width of the peak, volt; \(\gamma=\delta_{\mathrm{eff}}/\delta\) is a coefficient. For the quantity of electricity \(q\) we have the obvious relation:

\[ q=zF\gamma C_1V, \tag{2} \]

where \(z\) is the number of electrons participating in the anodic process per one metal atom; \(F\) is Faraday’s constant; \(C_1\) is the concentration of metal ions in the solution, \(\text{g-ion}/\text{cm}^3\); \(V\) is the volume of the solution, \(\text{cm}^3\); \(\gamma\) is the degree of depletion of the solution.

From relations (1) and (2) we obtain the desired expression for the minimum determinable concentration

\[ C_1'=\frac{\nu \sigma I}{zFw\gamma'}. \tag{3} \]

The minimum peak depth is determined from the relation:

\[ I_1=hi_*, \tag{4} \]

where \(i_*\) is the sensitivity of the polarograph, \(\text{a}/\text{mm}\); \(h\) is the minimum depth of the anodic peak that can be measured with a specified accuracy, or, for example, \(h=10\ \text{mm}\). The optimal electrode surface is determined from the relation

\[ hi_*=mS'j \quad \text{or} \quad S'=\frac{hi_*}{mj}, \tag{5} \]

where \(j\) is the residual-current density, \(\text{a}/\text{cm}^2\); \(m\) is a coefficient indicating how many times the depth of the anodic peak must be greater

residual current sufficient for the tooth to be measured with the specified accuracy (for example, 10%).

Other experimental conditions necessary for obtaining the required depth of the anodic tooth \(l'\) for the given optimum surface \(S'\) of the electrode are determined from relation (2):

\[ b'=\frac{K_1}{zF}\,S'\frac{t}{V}, \tag{6} \]

where \(K_1\) is the electrolysis constant, \(a\cdot cm/z\text{-ion}\); \(t\) is the time of preliminary electrolysis; \(b\) is a dimensionless parameter related to the degree of depletion \(\gamma\) by the relation [2]:

\[ \gamma=1-e^{-b} \tag{7} \]

In [1] it is assumed that \(\gamma=\gamma'=0.95\) (practically complete depletion of the solution) and, correspondingly, \(b=b'=3\). From formula (3), taking into account relations (4)—(7), one can estimate the minimally determinable concentration of ions in the solution and the conditions that must be fulfilled for this.

In earlier works [2,3] we derived an expression for the dependence of the depth of the anodic tooth on the radius of the mercury drop \((r)\) and other factors, starting from the expression

\[ I=K_2SC_2, \tag{8} \]

where \(K_2\) is the anodic-tooth constant, which in the cited work [2] is considered independent of the radius of the mercury drop\(^1\); \(C_2\) is the concentration of metal atoms in mercury, \(g\text{-ion}/cm^3\).

Taking into account the obvious relation

\[ \gamma C_1V=C_2v, \tag{9} \]

we obtain the desired expression for the current \((S/v=3/r)\):

\[ I=3K_2VC_1\frac{\gamma}{r}. \tag{10} \]

Solving the problem of the maximum of the function \(I\) in formula (10), taking into account the dependence of \(\gamma\) on \(r\) according to formulas (6)—(7), we obtain that the maximum value of the anodic-tooth depth is reached at the radius of the mercury drop \(r\) corresponding to the value of the dimensionless parameter \(b''=1.25\), i.e.,

\[ b''=\frac{K_1}{zF}\,S''\frac{t}{V}=1.25. \tag{11} \]

Hence the minimally determinable concentration from formula (10) is given by the expression:

\[ C_1''=\frac{I''}{3K_2V}\,\frac{r''}{\gamma''}, \tag{12} \]

where \(\gamma''=0.71\) at \(b''=1.25\) according to formula (7); the values \(I''\) and \(r''\) are determined from relations (4) and (5), and the other experimental conditions \((K_1,t_1,V)\) must be chosen so that condition (11) is satisfied.

It is of interest to compare the two estimates of the minimally determinable concentration given by formulas (3) and (12), and to determine how closely the results of these two estimates agree with one another.

First of all, let us show that formulas (3) and (12) are essentially identical. To do this, express the half-width of the tooth \(\delta\) in the formu-

\(^1\) Later we showed [3] that this assumption is approximate.

where (3) is expressed through the other quantities. The concentration of metal atoms in the mercury drop is given by the obvious relation:

\[ C_2=\frac{q}{zFv}. \tag{13} \]

From formulas (1), (8), and (13) we obtain:

\[ \delta=\left(\frac{zF}{3K_2}\frac{w}{V}\right)r . \tag{14} \]

Substitution of formula (14) into (3) gives for \(C'_1\) the expression

\[ C'_1=\frac{I}{3K_2V}\frac{r'}{\gamma'}, \tag{15} \]

which is identical with formula (12). Let us compare the values of \(C'_1\) and \(C''_1\) at the same depth \((h,\ \text{mm})\) of the anodic notch \((l'=l'')\) and with the constancy of the other conditions \((K_1,\ t_1,\ V)\), except for \(r,\ S,\ v\) and \(\gamma,\ b\). From (12) and (15) we obtain

\[ \frac{C'_1}{C''_1}=\frac{r'}{r''}\frac{\gamma''}{\gamma'}=1.16. \tag{16} \]

From formulas (6) \((b'=3)\) and (11) \((b''=1.25)\), with \(K,\ t,\ V\) constant, it follows:

\[ \frac{r'}{r''}=\left(\frac{S'}{S''}\right)^{1/2} =\left(\frac{b'}{b''}\right)^{1/2}=1.55. \tag{17} \]

The ratio \(\gamma''/\gamma'=0.71/0.95=0.75\). Hence we obtain for \(C'_1/C''_1\) the value 1.16 indicated above. Thus, we arrive at an interesting conclusion. When electrolysis is carried out until incomplete depletion of the solution \((b''=1.25;\ \gamma''=0.71)\), it is possible to measure somewhat smaller concentrations (by 16%) than with complete depletion of the solution, with the other conditions \((l,\ K_1,\ t,\ V)\) constant, except for the size of the mercury drop. Although this difference is small and has no great practical significance, from theoretical considerations it is interesting to clarify in more detail the cause of this phenomenon.

From relation (9) we obtain, for the two degrees of depletion considered (0.71 and 0.95) and \(V=\text{const}\):

\[ \frac{C'_1}{C''_1}=\frac{C'_2}{C''_2}\cdot \frac{v'}{v''}\cdot \frac{\gamma''}{\gamma'}. \tag{18} \]

Thus, the change in the minimum determinable content of the element in the solution with a change in the radius of the mercury drop is associated with a change in the degree of concentration of the metal in the drop, with a change in the volume of the drop \((v'/v''=(b'/b'')^{3/2}=3.73)\), and with a change in the degree of depletion of the solution \((\gamma''/\gamma'=0.75)\).

From formula (8) we obtain (for \(l'=l''\)) \(C'_2/C''_2=S''/S'=b''/b'=0.42\). Thus, the numerical estimate of relation (18) gives:

\[ C'_1/C''_1=0.417\cdot 3.72\cdot 0.75=1.16 \]

in agreement with relation (16).

From the expressions given above one can obtain a number of useful relations for calculations by the APN method. From formula (6) we obtain the value of the ratio of the volume of solution to the volume of the mercury drop, which—

... should take place in order, under the given conditions \((K_1, t, r)\), to reach the required value of the degree of depletion \((b=-2.3\lg(1-\gamma))\):

\[ \frac{V}{v}=\frac{3K_1t}{zFr}\cdot\frac{1}{b}. \tag{19} \]

From formulas (9) and (19) we obtain an expression for the degree of concentration:

\[ \frac{C_2}{C_1}=\gamma\left(\frac{V}{v}\right)=\frac{3K_1t}{zFr}\frac{\gamma}{b}. \tag{20} \]

From formula (10) we obtain an expression for the angular coefficient of the calibration curve, which characterizes the relative sensitivity of the determination of elements:

\[ \frac{I}{C_1}=K_2S\left(\frac{V}{v}\right)_\gamma=\frac{3K_2V\gamma}{r}. \tag{21} \]

In conclusion, let us make a numerical estimate of the sensitivity of the APN method when operating with a hanging mercury drop under conditions of the maximum value of the depth of the anodic tooth \((\gamma'=0.71;\ b'=1.25)\). Let us take the radius of the mercury drop as \(r=0.05\ \text{cm}\). We shall estimate to what extent such an electrode surface \((S=4\pi r^2=3.14\cdot10^{-2}\ \text{cm}^2)\) corresponds to the optimum. Let the residual-current density exceed the capacitive current by a factor of 3, i.e. \(j=3\cdot1.4\cdot10^{-7}=4.2\cdot10^{-7}\ \text{A}/\text{cm}^2\) (with the capacitance of the double layer at the electrode \(20\cdot10^{-6}\ \text{farad}/\text{cm}^2\) and a rate of change of potential \(7\cdot10^{-3}\ \text{V}/\text{s}\), the capacitive current is \(20\cdot10^{-6}\cdot7\cdot10^{-3}\ \text{A}/\text{cm}^2=1.4\cdot10^{-7}\ \text{A}/\text{cm}^2\)).

Let us estimate the numerical value of the coefficient \(m\) in formula (5), taking the sensitivity of the polarograph as \(i_*=10^{-9}\ \text{A}/\text{mm}\) and the minimum depth of the teeth as \(h=10\ \text{mm}\):

\[ m=\frac{hi_*}{Sj}\simeq0.8. \]

Our experiments show that even at smaller values of the coefficient \(m\), with \(h=10\ \text{mm}\), sufficiently reproducible teeth can be obtained. Therefore the adopted electrode size \((r=0.05\ \text{cm})\) is approximately optimal under the conditions under consideration \((h, i_*, j)\). Let us take \(K_1=5\cdot10^3\ \text{A}\cdot\text{cm}/\text{g-ion};\ z=2;\ t=30=1800\ \text{s}\) and estimate the optimal value of the volume \(V\) from formula (11) at \(b=1.25\):

\[ V=\frac{K_1St}{1.25zF}=1.46\ \text{ml}. \]

Now we have all the necessary data for estimating, by formula (12), the minimum determinable concentration of metal ions in solution \((K_2\simeq100\ \text{A}\cdot\text{cm}/\text{g-atom})\) under the conditions considered \((\gamma=0.71)\)

\[ C_1=\frac{I_2r}{3K_2V\gamma}=1.6\cdot10^{-12}\ \text{g-ion}/\text{ml}=1.6\cdot10^{-9}\ \text{g-ion}/\text{l}, \]

which corresponds to a content of \(10^{-4}\ \mu\text{g}/\text{ml}\) (for atomic weight 100) and about \(2\cdot10^{-8}\%\) impurity in the material (for a 1 g sample).

If, by increasing the size of the mercury drop while keeping the other conditions constant, we wished to obtain a degree of depletion \(\gamma=0.95\), it would be necessary to use a mercury drop of size \(r''=r'(b''/b')=0.078\ \text{cm}\), while the minimum determinable concentration would be greater by a factor of 1.16, i.e. \(C_1=1.85\cdot10^{-9}\ \text{g-ion}/\text{l}\).

Let us next estimate the ratios in formulas (19)—(21) (for $\gamma = 0.71$): $V/v = 5600$; $C_2/C_1 = 4000$; $I/C_1 = 6.3\ a\cdot h/2\text{-ion}$. Thus, under the conditions considered, on an ordinary hanging mercury drop it is possible, from a volume of $1.5$ ml with an electrolysis time of $30$ min, to achieve a 4000-fold concentration of the metal and to determine $10^{-4}$ $\mu\text{g}/\text{ml}$ ($10^{-8}\%$) of a substance in solution. Such sensitivity for a large number of elements is possessed by none of the physicochemical methods known to us, except neutron radioactivation and mass-spectroscopic methods.

Conclusions

1. It has been shown that estimating the sensitivity of the APN method under conditions of practically complete depletion of the solution and under depletion of the solution corresponding to the maximum depth of the anodic peak, as a function of the radius of the mercury drop, gives practically identical results in both cases.

2. The numerical calculations performed show that a determination sensitivity of $10^{-4}$ $\mu\text{g}/\text{ml}$ ($0.1$ nanogram/ml) is quite attainable under ordinary operating conditions for the APN method, if optimal values of the electrode surface area and solution volume are chosen for the work.

LITERATURE

1. A. G. Stromberg. Zavod. laboratoriya, XX, 10, 1175, 1965.
2. A. G. Stromberg. Izv. SO AN SSSR, No. 5, 76, 1962.
3. A. G. Stromberg. Zavod. laboratoriya, 29, 387, 1963.