ON THE CALCULATION OF THE ELECTROLYSIS CONSTANT IN THE APN METHOD

A. G. Stromberg

(Presented by the Scientific-Methodological Seminar of the KhTF)

Our mathematical investigation [1, 2, 3] of ways to increase the sensitivity of the method of amalgam polarography with accumulation (APN) showed that one of the important reserves for increasing sensitivity is an increase in the electrolysis constant.

The aim of the present work is to systematize methods for calculating the electrolysis constant and to give their comparative characterization, as well as to develop several new methods of calculation.

The principal defining formula for the electrolysis constant \(K\) in the APN method in a stirred solution is the expression

\[ K=\frac{i}{sc}, \tag{1} \]

where \(i\) and \(c\) are the electrolysis current (A) and the concentration (g-ion/ml) of metal ions in the solution at the time \(t\) (sec); \(s\) is the electrode surface \((\mathrm{cm}^2)\). Thus, the dimension of the electrolysis constant is \(\mathrm{A}\cdot\mathrm{cm}\cdot\mathrm{g\text{-}ion}^{-1}\). This formula is valid both in the presence and in the absence of depletion of the solution during electrolysis. Therefore, calculation of the electrolysis constant presents no difficulty if one knows the electrolysis current, the concentration of metal ions in the solution, and the electrode surface at a definite moment of electrolysis time.

The simplest case is the carrying out of preliminary electrolysis in the APN method under conditions of practically no depletion of the solution (for example, less than 5%), when the dimensionless parameter

\[ b=\frac{K}{zF}\,s\,\frac{t}{V} \tag{2} \]

is less than 0.1 (\(V\) is the volume of the solution, ml; \(z\) is the number of electrons participating in the electrode process per ion; \(F\) is the Faraday constant). The practical absence of depletion is also indicated by constancy of the electrolysis current. In this case, instead of (1) we may write:

\[ K=\frac{i_0}{sc_0}, \tag{3} \]

where \(i_0\) and \(c_0\) are the electrolysis current and the concentration of the solution at the given moment of electrolysis, practically equal to the initial current \(i_0\) and the initial concentration of the solution \(c_0\), if depletion of the solution is absent.

Instead of the electrolysis current, one may measure the quantity of electricity that has passed through the electrode, which, in the absence of depletion of the solution, may be expressed by the formula

\[ q=i_0 t . \tag{4} \]

In this case, from (3) and (4) we obtain

\[ K=\frac{q}{s c_0 t}. \tag{5} \]

Instead of formula (4), the quantity of electricity may be determined from the area under the anodic wave, under the condition of 100% yield of the metal from the drop in the process of anodic dissolution of the metal from the amalgam when recording an anodic polarogram by means of the usual procedure employed in the APN method*. To obtain 100% yield of the metal from the drop it is expedient to apply the potential-stopping technique proposed by Igolinskii [4] and Zakharova [5].

This technique consists in the following: after the anodic limiting current has been reached (at several hundredths of a volt more positive than the anodic wave; more by \(34/z\) mV for a reversible anodic wave), the change in potential is stopped, and then the curve of the decrease of the limiting current practically to zero is recorded on the chart recorder tape. Stopping the potential affects the area under the anodic wave, since the limiting current does not depend on the electrolysis potential.

In the presence of depletion of the solution, the electrolysis constant can be calculated from the electrolysis current from relations (6), (7), and (2):

\[ \frac{i}{i_0}=\frac{c}{c_0}=e^{-b}. \tag{6} \]

From (6) and (2) it is easy to obtain the expression:

\[ \lg i=\lg i_0-\frac{K}{2.3 z F}\frac{s}{V}t, \tag{7} \]

From expression (7) it follows that on a plot in the coordinates \((\lg i, t)\) a straight line should be obtained, from whose slope \(\tg \alpha\) the electrolysis constant is determined:

\[ K=2.3 zF \frac{V}{s}\lg \alpha . \tag{8} \]

Let us also note that the intercept on the ordinate axis at \(t=0\) (obtained by extrapolating the straight line on the plot) gives, in accordance with formula (7), the value \(\lg i_0\), where \(i_0\) is the initial electrolysis current. Knowing \(i_0\) and \(c_0\), one can also calculate the electrolysis constant by formula (1).

The formulas described above for calculating the electrolysis constant from the electrolysis current are not very convenient in practice, since under practical working conditions in the APN method the concentrations of the ions under study in the solution are very small. Therefore the electrolysis currents are also very small, comparable with (or less than) the residual current, and usually cannot be measured with sufficient accuracy.

However, the electrolysis current required for calculating the electrolysis constant by formula (7) need not be measured directly, but may be—

* That is, with a potential varying linearly with time at a rate of 400–600 mV/min.

calculate it from the dependence of the area under the anodic tooth \(q\) or the depth of the anodic tooth \(l\) on time, from the relations

\[ i=\frac{dq}{dt};\quad i=\left(\frac{q}{l}\right)\cdot\frac{dl}{dt}. \tag{9} \]

From relations (9) it is seen that the electrolysis current can be calculated from the curve on a graph in the coordinates \(q, t\) by the tangent of the angle of the tangent at a given point of the curve, or from the curve on a graph in the coordinates \(l, t\) by the tangent of the angle of the tangent to this curve, multiplied by the constant coefficient \(q/l\), which does not depend on the time of electrolysis or on the concentration of ions in the solution and can be determined from one of the anodic teeth.

A convenient formula for calculating the electrolysis constant is expression (5), but it is applicable only in the absence of depletion of the solution, and, moreover, it is necessary to know the concentration of ions in the solution. Therefore, below we shall consider methods of calculating the electrolysis constant in the presence of depletion of the solution, which do not require measuring the electrolysis current and do not require knowledge of the concentration of the substance in the solution.

The starting formula in these calculations is the expression for the degree of depletion of the solution [6]:

\[ \gamma=1-e^{-b}. \tag{10} \]

Taking formula (2) into account, we obtain from this the basic expression for calculating the electrolysis constant in the presence of depletion of the solution:

\[ K=\frac{2.3zF}{t}\frac{V}{s}\lg\frac{1}{1-\gamma}. \tag{11} \]

For practical calculations it is convenient to rewrite this expression in the following form:

\[ \lg\frac{1}{1-\gamma}=\frac{K}{2.3zF}\frac{s}{V}t. \tag{12} \]

If the degree of depletion of the solution \(\gamma\) at the time \(t\) is known, then the electrolysis constant is easily calculated by formula (11). If the degrees of depletion at different time intervals are known, then on a graph in the coordinates \(\left(\lg \frac{1}{1-\gamma}, t\right)\) a straight line should be obtained (passing through the origin), from whose slope the electrolysis constant is determined by a formula identical with formula (9).

Let us also note that for a hanging mercury drop \(V/s=r/3\) (where \(r\) is the radius of the drop), and for a mercury film electrode \(V/s=L\) (where \(L\) is the thickness of the mercury film).

In what follows we shall consider methods of calculating the degree of depletion \(\gamma\) of the solution, the value of which must be known in order to calculate the electrolysis constant by formulas (11) and (12).

From formulas (6), (7), and (10) one obtains the expression

\[ \gamma=1-\frac{i}{i_0}, \tag{13} \]

which makes it possible to calculate the degree of depletion of the solution, knowing the electrolysis current at a given moment of electrolysis. But electrolysis currents under condi—

...in the conditions of the APN method is inconvenient to measure, and therefore we shall not dwell on this method of measuring \(\gamma\).

The amount of electricity expended on the deposition of the metal into the amalgam (at 100% current efficiency) is related to the degree of depletion of the solution by the obvious relation:

\[ q = zF\gamma c_0 V . \tag{14} \]

Hence

\[ \gamma = \frac{q}{zFc_0 V}. \tag{15} \]

In practice, to determine \(q\), the area under the anodic tooth is calculated.

If, in addition, the area under the anodic tooth \(q_\infty\) is known for complete depletion of the solution \((\gamma = 1)\), then the degree of depletion can be calculated from the formula

\[ \gamma = \frac{q}{q_\infty}, \tag{16} \]

which is a consequence of formula (15). Taking into account the proportionality between the area under the anodic tooth and the depth of the anodic tooth, one may write, instead of (16), the expression

\[ \gamma = \frac{I}{I_\infty}, \tag{17} \]

where \(I_\infty\) is the depth of the anodic tooth under the condition of complete depletion of the solution during preliminary electrolysis.

If the time \(t_{1/2}\) is known at which the depth of the anodic tooth is half its depth at complete depletion \(\left(I = \frac{1}{2} I_\infty \text{ and } \gamma = 0.5\right)\), then from (11), for calculating the electrolysis constant, we obtain the expression \(\left(\lg \frac{1}{1-\gamma} = 0.3,\right)\)

\[ K = \frac{0.69\, zF}{t_{1/2}} \frac{V}{s}. \tag{18} \]

A practical inconvenience of formulas (16), (17), and (18) is that carrying out electrolysis to complete depletion of the solution (necessary for determining \(I_\infty\) or \(q_\infty\)) usually takes a very long time.

Below we shall set forth a comparatively simple method for calculating the value \(I_\infty\) from the curve of the dependence of the depth of the anodic tooth on the time \(t\), which curve is very easily obtained experimentally.

Let us note that in formula (11) the right-hand side does not depend on the electrolysis time, that is:

\[ \frac{1}{t}\lg \frac{1}{1-\gamma} = \mathrm{const}. \tag{19} \]

Let us consider on the graph \((I, t)\) two points for the electrolysis times \(t_1\) and \(t_2\), with \(t_2 = 2t_1\). Then from (19) we obtain:

\[ (1-\gamma_1)^2 = 1-\gamma_2. \tag{20} \]

This expression is easily transformed to the form

\[ \gamma_1 = 2 - \frac{\gamma_2}{\gamma_1}. \tag{21} \]

Taking (17) into account, we obtain

\[ \gamma_1 = 2 - \frac{I_2}{I_1}. \tag{22} \]

From formula (22), the quantity \(\gamma\) in formula (11) is calculated directly.

From formula (22), taking (17) into account, one can calculate \(I_\infty\):

\[ I_\infty = \frac{I_1}{2 - \dfrac{I_2}{I_1}} . \tag{23} \]

More accurate results are obtained by graphical calculation if, on the graph \((I,t)\), several pairs of time values satisfying the condition \(t_2 = 2t_1\) are chosen, and for each pair the value of \(\gamma\) is calculated by formula (22), after which the graph is plotted in the coordinates \((\gamma, I)\). From formula (17) it is evident that this graph should be represented by a straight line passing through the origin, with slope coefficient \(\operatorname{ctg}\alpha = I_\infty\). Then, choosing some point on this straight line (with \(\gamma_1, I_1, t_1\)), we calculate \(K\) by formula (11).

In conclusion, we note that if, under conditions of depletion, the electrolysis constant is calculated by formula (5), then the erroneous value \(K_*\) obtained will differ from the true value by the amount

\[ \frac{K}{K_*} = \frac{2.3}{\gamma}\lg \frac{1}{1-\gamma}. \tag{24} \]

This expression is obtained if (5) is divided by (11) and (14) and (16) are taken into account.

For example, if \(\gamma = 0.5\), then calculation by formula (5) gives \(K_* = 0.7K\), i.e., the electrolysis constant calculated by formula (5) will be 30% lower than the true value.

Conclusions

1. Expressions have been considered for calculating the electrolysis constant from the electrolysis current and from the area under the anodic tooth in the absence and in the presence of depletion of the solution.

2. A new method is proposed for calculating the electrolysis constant under conditions of appreciable depletion of the solution, in which it is necessary to know only the experimental dependence of the depth of the anodic tooth on time and it is not necessary to know the concentration of the substance in the solution.

References

1. A. G. Stromberg. Izv. TPI, 128, 13, 1965.
2. A. G. Stromberg. Zavodskaya laboratoriya, 31, 1175, 1965.
3. A. G. Stromberg, A. A. Kaplin. Zh. anal. khim., 20, 1043, 1965.
4. V. A. Igolinsky. Dissertation. Tomsk Polytechnic Institute, Tomsk, 1963.
5. E. A. Zakharova. Dissertation. Tomsk State University, Tomsk, 1965. This collection, p. 196.
6. A. G. Stromberg. Izv. SO AN, No. 5, 76, 1962.