A novel method for measuring electrolytic conductivity with a polarization impedance control function

The article presents an idealized model of the differential method utilized in conductometry to measure electrolytic conductivity (EC). It identifies the main drawback of the widely used differential method: the inability to control the influence of polarization impedance on EC measurement results. A criterion for the absence of error caused by polarization effects is proposed, involving equality between the ratios of resistance differences and cell constant differences. A novel method for EC measurement, termed the double differential method, is introduced. This method relies on repeated differential impedance measurements and incorporates an additional third impedance measurement of a virtual liquid column. The new approach enables the identification of polarization imped - ance influence on EC measurements and the calculation of measurement errors for correction. A 3 times lower type A measurement uncertainty value can be achieved by averaging the corrected results.

Although each type of cell has electrodes with minimal polarization impedance, they can still exhibit variations due to several destabilizing factors.

These factors may include different measuring cur-
rent values for two lengths (and therefore resistance) of the cell, resulting from changes in the measurement ranges of the LCR-meter; changes in the porosity of electrode surfaces; alterations in the effective area of the electrodes due to contamination; or temperature differences during the use of the differential method.Thus, differing polarization impedance current distribution density ( A m -2 ⋅ ) inside the cells.
The mathematical models and conclusions of this section will also be applicable to cells with a removable central extension section and piston-type cells.
The complexity of studying the effectiveness of the differential method in suppressing polarization impedance for contact cells lies in the peculiarities of the electrode/electrolyte interface model, which is represented by two phenomena: the phenomenon at the boundary of two phases of the metal and the solution and the phenomenon in the bulk of the solution (Fig. 1c).The electrical model of the cell should be a series connection of two impedances: polarization and bulk (Fig. 1b).The problem is that the junction point between polarization and bulk impedances is virtual (point P 1 in Fig. 1c).Measuring the potential difference separately at each impedance is impossible in practice.
values during two cell resistance measurements lead to incomplete compensation, resulting in additional error in EC measurement.Therefore, this publication proposes a new method for measuring EC that allows for controlling the influence of polarization impedance, calculating the error caused by incomplete compensation, and using it as a correction for the measurement results.

Differential method and idealized model of a two-electrode cell
Let us consider the principle of operation of the known method using the example of a differential double cell (Fig. 1a and 1b), which consists of two two-electrode cells of different lengths but the same diameter.An important condition for the implementation of the method is the uniformity of the electric Vector signal analyzers allow us to obtain results separately for the real and imaginary parts of the cell impedance.Thus, we are justified in writing: The effectiveness of the differential method depends on the extent to which we can consider the term in the first parentheses to be zero.To achieve this minimum, several conditions must be met.The components of the polarization impedance of each cell, Z P1 (Ω) and Z P 2 (Ω), must have minimal magnitudes, be temperature-independent, and remain stable over the time interval of two measurements.Ensuring these conditions for the idealized model allows us to consider the first term of equation ( 2) in parentheses to be equal to zero.However, experimental verification is necessary for the final confirmation of this conclusion.
With these considerations in mind, we can state: In the future, to avoid complicating work with formulas, we will omit the indices P and B. Howe ver, it must be remembered that whenever measurement results are utilized, we are referring to the real (resistive) part of the bulk impedance in a series two-element equivalent circuit.
Structurally, the differential cell (see Fig. 1a) Given that the cross-sectional area A (m 2 ) of the long and short cells is the same ( ) This mathematical model is used to determine EС in almost all national measurement standards [1]- [10] of the leading countries worldwide.In terms of the conductivity cell constant K (m -1 ), where K L A = / , equation ( 5) is as follows: where K 1 (m -1 ) and K 2 (m -1 ) are the constants of the long and short cells, respectively.

Model of a differential two-electrode cell under real conditions
Although the differential cell (Fig. 1a and 1b) has electrodes with the lowest polarization impedance values, Z P1 (Ω) and Z P 2 (Ω), in real measurement mode, they may vary in magnitude due to the previously mentioned destabilizing factors.Consequently, the polarization impedance will not be completely compensated, and the term ( ) (2) will not equal zero.This will result in an additional error δ 12 in the difference of the measurement re- sults of electrical resistance ∆R m 12 (Ω), expressed as follows: where ∆R 12 (Ω) is the resistance difference that does not include the error δ 12 .

EC measurement method with the function of controlling polarization impedance
Under conditions where the ideal (without the influence of polarization impedance) liquid column resistance ∆R 12 (Ω) cannot be achieved in practice, obtaining additional information through extra measurements with different values of cell constants becomes necessary.To implement such a variational error correction method [23] , the existing design, consisting of two two-electrode cells, is supplemented with a third cell.This third cell should also have the same diameter as the first two but a different length, L 3 (m), Fig. 2a.
In the case of a piston-type cell, this necessitates performing numerous measurements with different positions of the piston L 1 , L 2 , and L 3 (m) rela tive to the lower electrode (Fig. 2b).This measurement approach enables the use of the differential method to obtain three independent EC results in the form of differences: the first and second liquid column k 12 ( S⋅m -1 ), the first and third k 13 (S⋅m -1 ), and the second and third k 23 (S⋅m -1 ): The measurement results will vary due to errors δ ij , caused by differences in polarization impedanc- es, which have different magnitudes.Let us apply the first variational procedure to the results (2).We will take the ratios of the results (8) and denote them with Latin letters.This yields a system of three equations for the three EC values: In an ideal model (where polarization impedances are identical), there are no errors caused by differences in polarization impedance.That is:

Charged electrode Charged electrode
Then all EC values in the system (8) will be equal, and the factors a , b , and c will be equal to one.Ac- cordingly, system (9) turns into the following: The ratios in the system (11) hold true for an ideal model and serve as the criterion for the absence of errors caused by differences in polarization impedances.This criterion can be formulated as follows: if the ratio of differences in the resistance of the liquid columns is equal to the ratio of differences in the calculated values of the constants or the differences in the lengths of the liquid columns, then the polarization impedance does not affect the result of the EC measurement.In the absence of the influence of polarization impedance, the measurement results in equation ( 8) will be the same.
Substituting ratios (11) into (9), we obtain: System (9) could serve as the foundation for solving the error δ ij , given its three unknowns and three equations.However, the equations in the system (12) are not independent.Unfortunately, each equation in the system is a combination of two other equations.To derive an independent equation, a twostep procedure is employed.The well-known differential method is utilized in the first stage to obtain three distinct resistance differences (7).Subsequently, the EC is determined according to (8).In the second stage, a different type of variation is employed.
For instance, the difference is taken from the differ-ences in resistances of the first and second cells and the first and third: It should be pointed out that an important property of equation ( 13) is that the error of the difference between the first two differences is determined by the error of the third difference.
Therefore, by substituting the result ( ) R R − into equation ( 13), we obtain from equation ( 14): From the last equation and system (12), it is easy to derive equations for the errors in EC measurement due to incomplete suppression of the polarization impedance: This way, equations are derived to calculate unknown errors in the system (8).If we use the results of calculating errors (16-18) as corrections to the measurement results (8), then three identical corrected results without errors will be obtained.Averaging the measurement results using this method, assuming a normal distribution for the quantity, will result in a reduction of type A measurement uncertainty by a factor of 3.
An essential feature of the novel method is a repeated differential impedance measurement, namely the basic equation (13).Therefore, the novel method for measuring EC is called the double differential method.
Briefly, the algorithm of the double differential method consists of the following operations: 1) Measuring the resistance of cells (4) at three values of constants (in the case of a piston-type cell, measuring resistance at different positions of the piston L 1 , L 2 , and L 3 (m)).
3) Checking for the influence of polarization impedance using equation (11).
4) In the absence of the influence of polarization impedance, calculating the EC using equation (8)   and averaging the value.
6) Correction of the EC results using equation (8)   and averaging of the EC value.

Conclusions
When performing EC measurements using the known differential method and primary conductivity cells of any design, it is impossible to assess the completeness of polarization impedance suppression.
The proposed method allows: 1) Formulating criteria for identifying the influence of polarization impedances on the result of EC measurement.
2) Calculating the error caused by incomplete suppression of polarization impedances and using it as a correction to the measurement results in order to reduce possible systematic errors.
3) After applying corrections and averaging the values, reducing the type A uncertainty of the measurement results by a factor of 3 .

Fig. 1 .
Fig. 1. a) physical model of a differential cell; b) equivalent circuit of a cell; c) electrode/electrolyte interface of two separate two-electrode cells, which are based on tubes having the same cross-sectional area A (m 2 ) but different lengths L 1 (m) and L 2 (m), at the ends of which platinized electrodes are fixed.System (4) represents a mathematical model for calculating the solution resistance inside each of the two cells.This model is idealized and valid, provided that the distribution of current density J

)Fig. 2 .
Fig. 2. Physical models of differential cells with the function of controlling the equivalence of polarization impedances Re( ) Z X