Development of a coulometry system at the Korea Research Institute of Standards and Science and evaluation of the measurement uncertainty originating from the system

Coulometry is used to realize the definition of the mole and enables the direct determination of the purity of high-purity substances that are used as standards for other relative measurements with very small measurement uncertainty. However, despite several studies reporting the development of accurate and precise coulometry systems, there is a lack of detailed description of the system and systematic evaluation of the measurement uncertainty arising from the system. In this study, coulometry systems constructed at the Korea Research Institute of Standards and Science (KRISS) are described in detail and the uncertainties associated with these systems are systematically evaluated using a bottom-up approach. The relative expanded uncertainty resulting from the measurement system for the amount-of-substance content of acid in high-purity potassium hydrogen phthalate is approximately 0.0005%. The performance of the KRISS coulometry systems has been verified through several international comparisons and comparative measurements using certified reference materials.


Introduction
Chemical electrolysis, more specifically coulometry, has been regarded as a primary measurement method that practically realizes the definition of the mole without relying on a measurement standard for the same quantity [1,2].In this method, the electric charge of electrons required for the chemical electrolysis of a sample, along with the sample mass, is measured.By utilizing the measured electric charge and mass, as well as the stoichiometric number of electrons involved in the electrolysis and Faraday constant, the amount of substance per unit mass of the sample (referred to as the amountof-substance content or amount content) can be accurately determined according to Faraday's law of electrolysis.
Coulometry as a primary method is predominantly implemented under the constant-current mode using a constantcurrent coulometry system because this mode of operation offers smaller uncertainties compared to others [3].The Electrochemical Analysis Working Group within the Consultative Committee for Amount of Substance: Metrology in Chemistry and Biology (CCQM) first introduced this method in the CCQM-P7 international comparison to measure the amount content of various materials, such as K 2 Cr 2 O 7 , KCl, and NaCl [4].Since then, national measurement institutes (NMIs) around the world have made efforts to establish national measurement standards for the amount contents of high-purity primary standards used in acid-base, redox, and precipitation titrations using this method, and numerous CCQM key comparisons (KCs) have been conducted to demonstrate the international equivalence between the established national measurement standards [5][6][7][8][9][10][11][12][13].As part of these efforts, the Korea Research Institute of Standards and Science (KRISS) built its first coulometry system in 2000 and has recently constructed a second coulometry system to support ongoing research in the field of coulometry.
A detailed description of the measuring system and a better understanding of the uncertainty of measurement results can improve the reliability of measurement results.Several studies have reported the development of accurate and precise coulometry systems [14][15][16].However, little attention has been given to providing detailed descriptions of the system and conducting a systematic evaluation of the measurement uncertainty associated with the system.The reasons why such descriptions are necessary are as follows: First, a deep understanding of the configuration and the principle of the system and of the uncertainty components coming from the system enables one to understand coulometry as the primary method.Second, a systematic evaluation of the uncertainty arising from the measurement system is crucial for estimating the uncertainties originating from chemical sources, which significantly affect the uncertainty of the measurement results but are difficult to estimate.This evaluation is also important as NMI's measurement capability gradually improves over time, resulting in lower measurement uncertainty.
In this study, two coulometry systems from the KRISS were discussed in detail and uncertainty arising from the systems was systematically evaluated using a bottom-up approach.A target uncertainty of 0.001% was set for the systems so that their contribution to the final measurement uncertainty is negligible.Note that all sources of uncertainties associated with the chemical analysis being performed were not considered, as they depend on the chemical nature of the target analytes.These chemical sources of uncertainty often greatly exceed the instrumentation-specific uncertainty reported in the study [17].The performance of the first coulometry system was verified by numerous CCQM KC results reported by the KRISS.The comparison results between the first and second systems indicate that the measurement results obtained by the two systems were equivalent and the instrumentation was reproducible.

General description of the coulometry system
Figure 1 shows a simplified circuit diagram of the constantcurrent coulometry system at the KRISS.Details of the circuit diagrams are provided in the supplementary material.A constant current is generated by the current source and flows through a dummy cell resistor or the electrolysis cell by the operation of a double-pole, double-throw (DPDT) electromechanical relay.When current flows through the electrolysis cell, its magnitude is determined by Ohm's law, which takes into account the resistance of the standard resistor (SR) as well as the voltage difference between the voltage drop across the SR and the output voltage of the voltage standard.The DPDT relay enables the measurement of the time interval of the current flow by initiating a negative-going signal that decreases from +6 V to 0 V (digital ground) and terminating with the reverse signal.The mass of a sample is measured after correction for the buoyancy effect.An electrometer and sourcemeter are employed to determine the point where the chemical electrolysis of a sample is completed using potentiometric and amperometric endpoint detection methods, respectively.Most instruments are connected to a computer through a general-purpose information bus or RS232C serial interface and are controlled using a LabVIEW-based control program for system control and data acquisition.

Current source
For the first coulometry system, a commercial sourcemeter and homemade current source are used to generate constant currents of 10 mA and 100 mA, respectively.Details of a homemade 100 mA current source can be found in the supplementary material.In contrast, the second coulometry system utilizes a commercial sourcemeter to supply various constant currents, namely 10 mA, 50 mA, and 100 mA. Figure 2 illustrates the variations in the measured currents generated by the 100 mA current sources of the two coulometry systems to monitor current stability over a typical measurement time of 2 h.As shown in figure 2(a), the measured current for the homemade 100 mA current source remained stable with no drift for 2 h, with a relative standard deviation (RSD) of 0.8 ppm.Although a slight drift was observed for the 100 mA  current generated by the commercial sourcemeter, the RSD of the measured current was comparable to that observed for the homemade current sourcemeter (figure 2(b)).This drift is likely due to the instability of the commercial current source, as it was not observed when using the homemade current source.Note that the stability of all devices used for performing the current measurement, namely the current source, SR, digital voltmeter, and voltage standard, affects the changes in the measured current.Therefore, the change in current caused solely by the stability of the current source is expected to be smaller than the overall change in the measured current.

SR
Homemade SR sets, comprising 10 Ω and 100 Ω SRs for the first system, and 10 Ω, 20 Ω, and 100 Ω SRs for the second system, were built to measure different current levels.To minimize the resistance change caused by Joule heating, the SRs were constructed by connecting several precision resistors (PRs) in parallel, each with a nearly zero temperature coefficient of resistance (TCR), and mounting them between two copper blocks that served as heat sinks (figure 3).As presented in table 1, the 10 Ω and 20 Ω SRs were composed of ten 100 Ω and 200 Ω PRs, respectively.In the case of the 100 Ω SR, five 500 Ω PRs were connected in parallel.Using this approach, the power dissipated by each individual PR was reduced proportionally to the number of PRs used in the connection.As a result, the power dissipation achieved was more than an order of magnitude lower than the power rating of the individual PR.As each of the PRs connected in parallel had the same nominal resistance and TCR, the equivalent TCR of the SRs was estimated to be equal to the TCR of an individual PR.Based on the estimated power dissipation and TCR of the SRs, Joule heating was expected to cause a negligible change in the resistance of the SRs.For the 10 Ω SR, where the Joule heating is expected to be the highest, the temperature increased by approximately 1 • C when a 100 mA current was applied for 2 h.Taking into account the TCR of the SR, the change in resistance was negligible, with a relative resistance change of 0.3 ppm.Therefore, in conclusion, although the temperature of the SR may increase owing to Joule heating, its resistance value does not significantly change.Thus, the homemade SRs were enclosed in an aluminum box in a temperature-controlled laboratory environment (24 • C ± 1 • C) without requiring a temperature-controlled oil bath.

Uncertainty evaluation
The measurement uncertainty obtained using the coulometry system was evaluated by separately evaluating and combining the uncertainty that contributed to the random dispersion of the measured values in multiple measurements (hereinafter referred to as the uncertainty due to random effect) and the uncertainty that did not contribute to the random dispersion of values under replication conditions (hereinafter referred to as the uncertainty due to the systematic effect).The rationale for this approach is described elsewhere [18][19][20].The uncertainty attributed to the random effect was assessed from the between-sample standard deviation for the repeated measurements, which included contributions from the measurement repeatability and sample inhomogeneity.The uncertainty due to the systematic effect was evaluated by considering the uncertainty arising from the coulometry system.To eliminate possible biases due to chemical sources, such as diffusion and migration losses of analytes, spray losses of analytes, and inert gas impurities, the following measurement steps were included in the measurement procedure.A horizontal coulometric cell with two intermediate compartments was used with appropriate rinsing because the diffusion loss was expected to be negligible in this cell type [21,22]; samples were applied at the appropriate time to avoid loss due to migration; the cell was shaken to wash its walls and lid during the final rinse; and ultrahigh-purity (99.9999%) nitrogen gas was used with specific gas traps, such as oxygen or carbon dioxide traps.
The uncertainty of the amount content arising from the coulometry system was evaluated using the model equation, as follows [3] where A is the amount content; Q is the total charge; m is the sample mass; z is the number of electrons involved in the electrochemical reaction; and F is the Faraday constant.Based on the bottom-up evaluation described in the Guide to the Expression of Uncertainty in Measurement, the uncertainty is expressed as follows Since u(z) is assumed to be zero due to neglecting chemical uncertainties, and u(F) is defined to be zero from the definition of F, the above equation can be simplified to: In the following sections, the contributions of each uncertainty component are discussed and they are combined to estimate u(A).Detailed information on the parameters used for the uncertainty evaluation is provided in the supplementary material.Note that herein, the parameters were obtained from actual measurement results of the amount content of highpurity potassium hydrogen phthalate using the first coulometry system.

Charge uncertainty
A coulometric measurement was conducted in three stages, namely initial titration, main titration, and final titration.The initial and final titrations used multiple low-current pulses of the same magnitude, whereas the main titration used a single high-current pulse.Thus, the total charge obtained from the measurement can be expressed as follows where i is the current; t is the time interval of the current flow; and the subscripts L and H represent low and high currents, respectively.Note that the reference point for t L,j is the endpoint of the initial titration, and the uncertainty associated with its determination is due to the chemical source and is not considered here.As the i and t are not correlated and correlation does not exist between i L and i H or between t L and t H , the u(Q) can be expressed as: As the values of each i L and t L are correlated, u(i L,j ) ≈ u(i L ) and u(t L,j ) ≈ u(t L ), and consequently, equation ( 5) can be simplified to: The current was calculated from the resistance of the SR and the voltage drop across the SR using Ohm's law.Thus, u(i) can be obtained as follows where i = V SR /R SR ; R SR is the resistance of the SR; and V SR is the voltage drop across the SR.Although this calculation is valid for a single current measurement, it can also be applied to the present approach, in which multiple measurements are averaged to obtain a single pulse current.In this case, the uncertainties of each current (i k ) are correlated, and the standard uncertainty of the average current (i) equals that of i k , as shown in the following equation [23] u where N represents the number of measurements.
As the resistance is proportional to the temperature change, R SR can be expressed as ), and its standard uncertainty is expressed as: where R SR cal is the calibrated resistance of the SR; α SR is the TCR of the SR; and T SR and T SR cal are the temperatures of the SR at the time of measurement and calibration, respectively.
The V SR was calculated using the voltage difference measured by the digital voltmeter (V DVM ) with respect to the output of the voltage standard (V VS ) to obtain an accurate measurement.However, as relays are used in voltage measurements, the voltage difference can be affected by the thermal EMFs of the relay contacts (V c ), even though the expected value of the V c is zero.To account for this, the V SR is expressed as V SR = V VS + V DVM + ΣV c , and its standard uncertainty can be expressed as follows The V VS is given by V VS = V VS cal + α VS (T VS − T VS cal ), where V VS cal is the calibrated output voltage of the voltage standard, α VS is the temperature coefficient of the voltage standard, and T VS and T VS cal are the internal temperatures of the voltage standard at the time of measurement and calibration, respectively.Thus, the above equation can be rewritten as: By substituting equations ( 9) and (11) into equation (7), the u(i) for currents of 100 mA and 10 mA can be estimated.The estimated uncertainties are presented in tables 2 and 3.
To evaluate the u(t), the effects of the time delay between poles in the DPDT relay were considered.Note that the two poles of the DPDT relay do not operate at the same time; this causes a difference (i.e. a time delay) between the time of the current flow and the measured time.If the time delay is considered as the time delay of the ON/OFF current signal relative to those of the time signal, then the time of the current flow can be expressed as t = t m − t on + t off , where t m is the time measured using the time interval counter, and t on and t off are the on-and off-time delays for a single relay operation.There might be a correlation between time delays for 100 mA and 10 mA current flows, as the same relay is used.However, since the u(t) has little effect on the u(Q) in comparison to the u(i), we assume that there are no correlations between time delays.With this assumption, the u(t) is expressed as follows The uncertainty budget for the time measurement during current flows of 100 mA and 10 mA is presented in tables 4 and 5, respectively.Note that the time delay associated with the DPDT relay can be eliminated by using a mechanical relay along with an inline, optically-isolated current sensor [14].
Table 6 presents the uncertainty evaluation for the charge, which is calculated by substituting u(i H ) and u(i L ) values taken from tables 2 and 3, along with the u(t H ) and u(t L ) values from tables 4 and 5 into equation (6).The u(Q) is determined primarily by the u(i H ) and u(t H ), whereas the contribution of the uncertainty associated with the 10 mA current is insignificant.Consequently, if a high current is used to apply most of the charge (>99%), the contribution of the charge resulting from low current pulses can be neglected.

Sample mass uncertainty
Considering that the analytical balance is calibrated using a calibrated standard weight with the calibration factor (f cal ) and that the buoyancy correction is made with the buoyancy correction factor(f b ), the relationship between the balance reading (m r ) and the mass of a sample object (m) is given by the following equation [24] where ρ a , ρ w , and ρ s are the densities of the air, standard weight, and sample, respectively; m w is the mass of the standard weight; and m r,w is the balance reading of the standard weight.Therefore, the u(m) can be expressed as:

Uncertainty components u(x
Uncertainty budget for the time measurement during a 100 mA current flow.
1.1 × 10 −5 s 12 Table 5. Uncertainty budget for the time measurement during a 10 mA current flow.
As the complete weighing operation includes tare weighing, the uncertainty contribution from non-linearity considered twice when evaluating the u(m r ) [25].
When equation ( 13) is considered, the u(f b ) is estimated using the following equation For the calculation of the ρ a , an International Organization of Legal Metrology formula for the air density is used, as follows [26] ρ a / ( kgm −3 ) = 0.34848p/hPa − 0.009h/% × e 0.061T/K (273.15+ T) /K (16) where p is the atmospheric pressure of air; h is the relative humidity; and T is the temperature.Hence, the u(ρ a ) can be calculated according to the following equations The u(f cal ) is evaluated as follows when considering the f cal term in equation ( 13) Table 7 presents the uncertainty budget for the mass measurement.The relative u(m) is approximately 0.0005% (for a sample mass of 0.5 g), with the main contribution coming from the u(f b ), which is due to the u(ρ s ) since a sample density range of ±1% is assumed.If the ρ s can be measured with an uncertainty much better than 1%, then the u(m r ) may become the main contributing factor.

Amount content uncertainty
Table 8 summarizes the uncertainty contributions resulting from the charge and mass.The relative u(A) arising from the measurement system is approximately 0.0005% for an amount content of 4.8 mol kg −1 (high-purity potassium hydrogen phthalate).However, it may be further reduced if a sample with a higher amount content is measured.Based on this study, the uncertainty resulting from the measurement system can be expected to be within 0.001% if a precise measurement system is used.

Performance of the coulometry systems
The performance of the first coulometry system was verified by the CCQM KC results [5][6][7][8][9][10]12] reported by the KRISS, as presented in table 9.The absolute values of the degrees of equivalences (d i ) divided by the expanded uncertainty of the d i for all results are less than 1, which indicates that all of the results obtained using the first coulometry system are consistent with the KC reference values within the associated expanded uncertainties.The performance of the system is sufficient to conduct coulometric analyses as the relative uncertainty arising from the system is one or two orders of magnitude lower than the RSD values observed in repeated measurements.
To evaluate the performance of the second coulometry system, coulometric analyses were conducted on the same reference materials using both the first and second systems, and their results were compared (figure 4).Among many candidate reference materials, K 2 Cr 2 O 7 (KRISS CRM 104-02-003) was selected because it exhibits relatively lower uncertainties.Thus, the equivalence of the results obtained from the two systems can be compared with a low level of uncertainty.The mean and expanded uncertainty of the six independent measurements obtained using the first system were 3.397 15 mol kg −1 and 0.000 06 mol kg −1 , respectively, whereas those for the second system were 3.397 13 mol kg −1 and 0.000 07 mol kg −1 .In addition, their RSDs were 0.0016% and 0.0018% and they were comparable to the RSD of the KRISS result observed in CCQM-K96.As the two results obtained from the two coulometry systems were equivalent within their expanded uncertainties, both systems can be used for the development of primary reference measurement  procedures and relevant certified reference materials with international equivalences.

Conclusions
In this study, constant-current coulometry systems developed at the KRISS were described and a systematic approach for the evaluation of the measurement uncertainties arising from the systems was introduced.The first and second systems were constructed based on the same working principle and performed two primary functions, namely charge and mass measurements.The charge measurement module comprised a current source, current path selection board, voltage standard, voltage standard selection board, voltmeter, SR, SR selection board, time interval counter, and endpoint system.The mass measurement module included a barometer, thermometer/hygrometer, and analytical balance.Although most of the devices were commercially available, the 100 mA current source in the first system and the SRs in both systems were homemade to ensure stable and accurate current measurement.
To assess the measurement uncertainty arising from the system, a bottom-up approach was employed.The uncertainty of the amount content was estimated from the uncertainties of the charge and sample mass.For the charge uncertainty, uncertainty sources related to the SR and voltage measurement system used to measure current, as well as sources related to the time measurement, were considered.Regarding mass uncertainty, the uncertainty sources that affected the balance reading, buoyancy correction, and balance calibration were considered.Note that the approach presented for evaluating uncertainty is generally valid.However, as the evaluation method described herein is specific to the KRISS coulometry system, details may vary when applied to other coulometry systems.
The performance of the first system was verified using the results obtained in several CCQM KCs, whereas that of the second system was examined by comparing it with the first system.The comparison was performed using the same reference material as a sample, and the results obtained from both systems were consistent with each other within the expanded measurement uncertainties.This indicated that both systems can be used equally.We believe that the detailed information provided in this study will be useful for researchers who wish to understand the measuring system that enables coulometry as a primary method and to conduct related research.
the measurement standards for amount of substance based on Avogadro's constant' (Grant No. 23011034).

Figure 1 .
Figure 1.Simplified circuit diagram of the constant-current coulometry system at the KRISS.

Figure 2 .
Figure 2. Changes in measured current generated by (a) a 100 mA-homemade current source in the first coulometry system and (b) the commercial sourcemeter in the second coulometry system.The mean and standard deviations of the measured values are shown in the figures.

Figure 3 .
Figure 3. Schematic of the standard resistor with front and side views.Current is supplied via a pair of current path connections and the resulting voltage drop between a pair of voltage sense connections is measured.

Figure 4 .
Figure 4. Summary of the comparison results for the assay of K 2 Cr 2 O 7 obtained using the first and second coulometry systems.The measurement procedure employed is the same as that used in CCQM-K96 and CCQM-K96.1.The individual results obtained from each system are represented by colored spheres, and their averages and corresponding expanded uncertainties are denoted by short horizontal and error bars, respectively.

Table 1 .
Configuration and specification of the standard resistors in the first and second coulometry systems.

Table 2 .
Uncertainty budget for the 100 mA current measurement.The top-level individual uncertainty components are left-aligned, while the lower-level components are indented further according to their respective levels.The combined standard uncertainty is placed at the bottom of the table.

Table 3 .
Uncertainty budget for the 10 mA current measurement.

Table 7 .
Uncertainty budget for the mass measurement.

Table 8 .
Uncertainty budget for the amount content measurement.

Table 9 .
Summary of results obtained from relevant key components (KC) in which the Korea Research Institute of Standards and Science participated.