Accred Qual Assur (2005) 10: 430–438 DOI 10.1007/s00769-005-0009-4

A. R. Sousa M. A. Trancoso

Received: 28 February 2005 Accepted: 24 June 2005 Published online: 7 October 2005 C Springer-Verlag 2005


A. R. Sousa (
) · M. A. Trancoso Instituto Nacional de Engenharia, Tecnologia e Inovac¸a˜ o, I. P. - LAACQ, Estrada do Pac¸o do Lumiar, 1649-038 Lisboa, Portugal e-mail: [email protected]

GENERAL PAPER

Uncertainty of measurement for the determination of fluoride in water and wastewater by direct selective electrode potentiometry

Abstract A procedure to estimate the uncertainty of measurement applied to the fluoride determination of waters and wastewaters matrices by selective electrode potentiometry was implemented based on Eurachem Guide. The major sources of uncertainty were identified as the calibration standard solutions, fluoride concentration obtained by potential interpolation of the regression line and the precision. However the relative uncertainties depend on the anion concentration levels. The methodology proposed was presented to two fluoride concentration levels that are in the range of surface water samples (Csample =1.12 mg F l−1 ) and of wastewater matrices (Csample =101.4 mg F l−1 ). The expanded uncertainties calculated were 0.40 and 9.1 mg l−1 for low and high concentration levels, respectively, using the reproducibility uncertainty as precision evaluation. The relative expanded uncertainty was around ±10% for the highest concentration, which can be considered acceptable for the ion selective electrode potenciometric methods and ±36% for the lowest concentrations. In this case the sample fluoride content is very close to the limit of quantification which has a relative uncertainty of about ±30%. If the repeatability was used in spite of duplicate analysis the same conclusions were obtained (Csample =1.12 ± 0.39 mg F l−1 and Csample =101.4 ± 7.0 mg F l−1 ).

Although the calculated expanded uncertainties and consequently the combined uncertainty, do not vary significantly in the cases where it was used the repeatability or reproducibility for evaluating the precision, each relative variances uncertainty contributions do. When the repeatability is used to determine the combined uncertainty, the CSS and CF− uncertainties contributions are the most dominant ones. However, if reproducibility is used, relative uncertainty variance contributions are distributed among CSS, CF , and precision. In both cases, the rcF contribution increases and rCSS contribution decreases with the increasing of the concentration level. The precision variance contribution is only significant in the case where the reproducibility is used, and increases with the increasing of the concentration level. The uncertainty in the result calculated using the proposed methodology (Csample ± Usample = 2.17 ± 0.42 mg F l−1 ) is in satisfactory agreement with the estimated expanded uncertainty obtained using the relative reproducibility standard deviation obtained in interlaboratory studies (Csample ± UCsample = 2.17 ± 0.44 mg F l−1 ). Keywords Measurement uncertainty . Uncertainty component . Water and wastewater analysis . Ion selective electrodes . Fluoride

431

Symbols a aF− b cF CF CSS i CCSS

CPS Csample γF− Fdil E i L m MMF MMNaF n pur rec. R r Xi s t ua ub ubias ucal u CF u CPS i u CCSS uCSS u Cc sample uE u Fdil um u MMF u MMNaF uprecision

intercept of the calibration regression line fluoride activity (mol l−1 ) slope of the calibration regression line fluoride concentration (mol l−1 ) fluoride concentration (mg F l−1 ) fluoride calibration standard solutions (mg F l−1 ) concentration of each fluoride calibration standard solutions (mg F l−1 ) fluoride principal standard solution (mg F l−1 ) sample fluoride concentration (mg F l−1 ) activity coefficient dilution factor by which the sample was diluted electrode potential (mV) number of the calibration standard solutions constant term weight of the principal standard NaF (mg l−1 ) molar mass of fluoride (g mol−1 ) molar mass of NaF (g mol−1 ) sample size/values taken from a population purity of the NaF reagent given as mass fraction average recovery digital resolution of the balance relative uncertainty variance standard deviation of the mean of n values statistic test standard uncertainty of a standard uncertainty of b standard uncertainty of the bias (mg F l−1 ) linearity of the balance obtained from the balance calibration certificate standard uncertainty of CF (mg F l−1 ) standard uncertainty of CPS (mg F l−1 ) standard uncertainty of each calibration standard (mg F l−1 ) standard uncertainty associated to the calibration standard solutions (mg l−1 ) combined uncertainty of the fluoride analysis in the sample (mg l−1 ) standard uncertainty of E (mV) standard uncertainty of Fdil standard uncertainty of m (mg F l−1 ) standard uncertainty of MMF (g mol−1 ) standard uncertainty of MMNaF (g mol−1 ) standard uncertainty associated with precision (repeatability studies or duplicate analysis) (mg F l−1 )

upur u rec. uV(cal) u Vf u Vele i u Vpip u VPS u Vsam UCsample u Xi uY Vele Vf i Vpip

VPS Vsam X 1, . . . X i · X N ∂Y ∂ Xi

Y

uncertainty of purity of reagents uncertainty of rec. standard uncertainty to calibration of the volumetric equipment (ml) standard uncertainty of Vf (ml) standard uncertainty of Vele (ml) i standard uncertainty of Vpip (ml) standard uncertainty of VPS (ml) standard uncertainty of Vsam (ml) expanded uncertainty (mg F l−1 ) standard uncertainties of the input parameters standard uncertainty of the function Y volume of the electrolyte (ml) volume of the solution contained in the volumetric flask (ml) volume taken with a pipette of the fluoride principal standard solution (ml) volume of the volumetric flask used (ml) sample volume taken with a pipette (ml) N input variables estimates partial derivates of the function Y in rapport to the input variables Xi result of the measurement/function

Introduction The quality of receiving ecosystems is disturbed by residues disposal, agriculture practices, and residual wastewaters discharges resulting from industrial and urban activities. To minimize this impact, specific legislation for water and wastewater has been advised by the European Union and followed by the member states. Fluoride is one of the anions most usually analyzed in the field of water and wastewater analysis. Their monitoring becomes more important with the growth of the practice of fluoridation of water supplies as a public health measure [1]. Potentiometry using ion selective electrodes is a relatively easy and inexpensive method adequate to measure the fluoride concentration levels indicated by the legal requirements from potable waters to residual wastewaters discharges in the environment. The fluoride electrode selective membrane is made of a crystal of LaF3 doped with EuF3 to increase its conductivity. It is robust inside the temperature range of 0–80 ◦ C. The potential of the cell containing this selective electrode is related to the fluoride activity (mol l−1 ) by Eq. (1) [2]: E = L − b log aF−

(1)

where the slope b is about 59 mV at 25 ◦ C in ideal conditions. The slope value is indicative of the performance of the electrodes system. However, several factors including reference junction blockage, electrolyte loss, electrode interference, and the use of incorrect calibration solutions,

432

contribute to the variations on slope values. Equation (1) can be rewritten in terms of an activity coefficient as E = L − b log(γF− cF )

(2)

In the presence of an inert electrolyte, with a constant and high ionic strength, the electrode potential is a measure of CF according to Eq. (3): E = a − b log CF

(3)

In this work, the inert electrolyte used contains citrate, because it complexes preferentially with some cations like Al3+ that would otherwise bind F− and interfere with the potential measurements, and the pH value is close to 6 to minimize the interference of hydroxide. The same pH is maintained in standard and sample solutions. As shown in previous studies, the hydroxide does not interfere if the pH of the solutions were within the range 4–8 [3]. Whenever the result of an analysis is reported it is important to give a quantitative indication of its uncertainty because it gives an assessment of the quality of the analytical measures, enables comparisons of analytical results, and improves the guarantee of the conformity analysis. Therefore, the uncertainty estimation is a very important key in quality control and metrology. In this work, a procedure for estimating the uncertainty in the determination of fluoride in water and wastewater matrices by selective electrode potentiometry was implemented based on the Guide Eurachem of the Uncertainty Measurement [4]. This guide recommends the estimation of each component of uncertainty and then combines them into total uncertainty using the rule for propagation of errors. The methodology proposed was assessed to two fluoride concentration levels that are in the range of surface water samples (low concentration level) and of wastewater matrices (high concentration level). Experimental

Fig. 1 Analytical diagram

The calibration standard solutions were prepared through successive additions of the principal standard solution to 25 ml of the background electrolyte. Upon each standard solution addition, potential measurements were done after an equilibration period. The equilibrium was considered attained whenever two consecutive potential measurements separated by 30 s led to equal values within the experimental error (±0.2 mV). The principal standard additions and fluoride concentration of the calibration standard solutions are presented in Table 1. Of each sample 10 ml was diluted to a 25 ml volumetric flask, the pH adjusted to 6 and 12.5 ml of 1.0 mol l−1 C6 H5 Na3 O7 ·2H2 O were added. The potential measurement was done in the same conditions of the calibration standard solutions. Potential measurements were performed using an electrochemical cell constituted by an INGOLD selective fluoride electrode as indicator electrode and an INGOLD Ag|AgCl, KCl (sat.) as reference electrode connected to an ORION digital potentiometer model 420A with 0.1 mV resolution. The working temperature was 20 ± 2 ◦ C.

Chemicals and Solutions All solutions were prepared with chemicals of analyticalreagent grade and water purified by means of Millipore Milli-Q system. A fluoride principal standard solution containing 1000 mg F l−1 was prepared by dissolving 0.2232 g of NaF, previous dried at 105 ◦ C in water and then transferred to a 100 ml volumetric flask [1, 5]. The background electrolyte (or inert electrolyte) contains 0.5 mol l−1 sodium citrate at pH = 6.0 ± 0.2. Procedures The analytical procedure used in the determination of fluoride by direct potentiometry using a selective electrode is described in Fig. 1.

Table 1 Calibration standard solution concentrations and standard uncertainties Additionsa (ml)

i CCSS (mg F l−1 )

uCi

0.010 0.020 0.020 0.200 0.250 0.500 0.500

0.4000 1.198 1.996 9.899 19.604 38.455 56.593

0.0029 0.0092 0.016 0.080 0.16 0.32 0.47

CSS

(mg F l−1 )b

i = 1–7 a Successive principal standard additions into the inert electrolyte (25 ml) b Calculated by Eq. (14)

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The limit of quantification determined by ASTM D 41278 [6] was CF = 0.4 mg l−1 .

was calculated as Csample = CF Fdil

(6)

with Uncertainty estimation procedure The procedure used to evaluate the uncertainty associated to the fluoride determined by ion selective electrode potentiometry was divided into the following steps: Description of the method (see Experimental), specification of the measurand and identification of uncertainty sources, quantification of uncertainty components, and calculation of combined and expanded uncertainty. If a result of measurement is determined from other quantities the relationship between Y and the values of the input parameters can be expressed by a function, f [4]: Y = f (X 1 . . . X i . . . X N )

(4)

The uncertainty of the result depends on the uncertainty of the input parameters and can be described by Eq. (5) following the law of propagation of the errors [5, 7]. This equation is an approximation that is valid if there are no correlated input estimates [4]. u 2Y

=


 ∂Y 2 i

∂ Xi

u 2X i

(5)

Specification of the measurand and identification of uncertainties sources. The specification of the measurand requires a quantitative expression relating the value of the measurand to the parameters on which it depends. In this case, the fluoride concentration in the water or wastewater samples, Csample ,

Fdil =

Vf Vsam

In these conditions the CF is given by Eq. (6): CF = 10(−E+a)/b

(8)

In Fig. 2, uncertainty sources that affect the measurand were schematically presented. They can be identified as from the CF , Fdil , CCSS , precision, and bias. So, the measurand can be expressed by the model: Csample = CF Fdil f CSS f precision f bias

(9)

Standard uncertainty of fluoride concentration The uncertainty of fluoride concentration is described by Eq. (10). As can be seen u CF depends on the potential uncertainty and the calibration regression line (slope and intercept uncertainties).       2.303CF 2 2 2.303CF 2 2 u CF =  u + ua − E  b b     2.303CF (E − a) 2 2  + ub (10) b2

Fig. 2 Cause and effect diagram showing sources of uncertainty associated with the determination of fluoride in waters and wastewaters by selective electrode potentiometry

434

The uncertainty associated with the dilution factor

u CSS

The standard uncertainty associated with Fdil was determined by Eq. (11):   2   u Vf u Vsam 2 + (11) u Fdil = Fdil × Vf Vsam

Standard uncertainty associated with the CSS The calibration standard solutions were prepared through successive additions of the principal standard solution into a known electrolyte volume. Each standard solution concentration (mg l−1 ) was calculated using Eq. (12). n  i = CCSS

i=1

Vele +

i Vpip n  i=1

 CPS

(12)

i Vpip

 2  2 2  n 1 2 u CCSS = + u CCSS + . . . . u CCSS

(17)

Precision studies The precision (repeatability and reproducibility) of the measurements procedures may be matrix dependent or concentration dependent. In these conditions the repeatability of fluoride determinations was assessed for the two concentrations levels (Csample = 1.12 and 101.4 mg F l−1 ) using Eq. (18) [4]. s u precision = √ (Repeatability studies) n

(18)

The reproducibility of the fluoride determinations estimated from sample duplicate analysis seems to be more adequate because different concentration levels and matrices are considered. According to quality criteria all relative duplicate ranges must be lower than 10%. In these conditions the precision uncertainty can be calculated by Eq. (19):

with CPS

m.pur MMF = 106 VPS MMNaF

(13)

For each standard, their uncertainties were calculated using Eqs. (14)–(16).

i i u CCSS = CCSS

   2  n 
i u V  pip   i  V pip  i=1 2     n u     i Vele + Vpip    u CPS 2 i=1   +   +   n  CPS i Vpip Vele + i=1

u

Vele +

n 

i=1

u CPS

 i Vpip

  n  2
  2 i u Vpip = (u Vele ) +

(14)

0.1 Csample (Duplicate analysis) √ 2 3

(19)

Bias measurement—Recovery The matrix effects were investigated using a long term study of spiked samples of various types. A set of 10 results were collected with an average recovery of 98.5% and a standard deviation of 4.0%. The statistic test t (Eq. (20)) was used to check whether the average recovery is significantly different from 1. The calculated t-value (1.2) was compared with the two-tailed critical value, tcrit , for n−1 degrees of freedom at 95% confidence. In our conditions tcrit is 1.833. Since t < tcrit , rec. can be considered not significantly different from 1 [4]. t=

|1 − rec.| u rec.

(20)

Although the recovery is not significantly different from 100% the uncertainty concerning recovery must be included. Based on these results the laboratory establishes the range 100 ± 10% as the recovery limit within which the matrix effects can be considered minimal. So, the corresponding uncertainty can be calculated by dividing the √ limits ±0.1 by 3.

i=1

      2   um u pur 2 u VPS 2 = CPS  + +  m pur VPS  2     u MMNaF u MMF 2 + + MMNaF MMF

u precision =

(16)

The standard uncertainty associated with the calibration standard solutions was obtained from Eq. (17):

Quantification of uncertainty components Uncertainty sources can be divided into the following categories: purity of reagents, volumetric operations, molar

435

Table 2 Standard uncertainty components (u X i ) and combined uncertainty(u Cc sample ) for low concentration level

Csample = 1.12 mg F l−1 u Cc sample = 0.20 mg F l−1 using repeatability and reproducibility (Eq. 26) Table 3 Standard uncertainty components (u X i ) combined uncertainty(u Cc sample ) for high concentration level

Input quantities (Xi )

Symbol

Estimated value u X i

Remark

Calibration Standard solutions (0.4–20 mg F l−1 ) Fluoride concentration (mg F l−1 ) Dilution factor Repeatability (mg F l−1 ) Reproducibility (mg F l−1 ) Bias (recovery)

CSS

Table 1

0.179

Table 4

CF

0.446

0.020

Table 6

Fdil Precision Precision rec.

2.5 1 1 1

0.0043 0.021 0.032 0.058

Table 7 This work Eq. (18) This work Eq. (19) This work (100 ± 10%)

Input quantities (Xi )

Symbol

Estimated value u X i

Remark

CSS

Table 1

0.592

Table 4

CF

40.56

1.37

Table 6

Csample = 101.4 mg F l u Cc sample = 3.5 mg F l−1 using repeatability (Eq. 26) u Cc sample = 4.6 mg F l−1 using reproducibility (Eq. 26)

Calibration standard solutions (2–57 mg F l−1 ) Fluoride concentration (mg F l−1 ) Dilution factor Repeatability (mg F l−1 ) Reproducibility (mg F l−1 ) Bias (recovery)

Fdil Precision Precision rec.

2.5 1 1 1

0.0043 0.146 2.93 0.058

Table 7 This work (Eq. 18) This work (Eq. 19) This work (100 ± 10%)

Table 4 Uncertainty of calibration standard solutions

Input quantities (Xi )

Symbol

Estimated value u X i

Remark

Principal standard concentration (mg F l−1 ) Inert electrolyte volume (ml) Automatic pipette volume (ml) Calibration standard solutions (low level 0.4–20 mg F l−1 ) Calibration standard solutions (high level: 2–57 mg F l−1 )

CPS

999.8

5.9

Table 5

Vele i Vpip CSS

25 Table 1 Table 1

0.0276 Table 1 0.179

CSS

Table 1

0.592

This work (Eq. 22) This work (Eq. 22) This work (Eqs. (14)–(17)) This work (Eq. (14)–(17))

−1

Table 5 Uncertainty of principal standard concentration, CPS

Table 6 Uncertainty of the fluoride concentration, CF

∗

The quality control criterion to accept potential differences between successive measurements of the same solution was 1 mV. Then, assuming a rectangular distribution uE = 0.58 mV

Input quantities (Xi )

Symbol

Estimated value u X i

Remark

Purity of NaF Weight of NaF (g) Volume of the solution (ml) F− Molar mass (g mol−1 ) NaF Molar mass (g mol−1) Principal standard concentration (mg F l−1 )

pur m VPS MMF MMNaF CPS

0.99 0.2232 100 18.998403 41.98817 999.82

Supplier certification This work (Eq. (23)) This work (Eq. (22)) This work (Eq. (25)) This work (Eq. (24)) This work (Eq. (16))

0.00577 8.16×10−5 0.0837 2.89×10−7 1.16×10−5 5.90

Input quantities (Xi )

Symbol

Estimated value u X i

Remark

Low concentration level Electrode (mV)∗

E

224.3

0.577

b a

56.42 204.53

1.12 0.856

Own quality control criterion This work This work

E

112.1

0.577

b a

58.72 206.52

0.312 0.401

Slope (mV) Intercept (mV) CF = 0.446 mg F l−1 u CF = 0.0201 mg F l−1 (Eq. (10)) High concentration level Electrode (mV)∗ Slope (mV) Intercept CF = 40.56 mg F l−1 u CF = 1.37 mg F l−1 (Eq. (10))

Own quality control criterion This work This work

436

mass, first order coefficients regression line, precision, and bias of the method.

the calibration certificate [4, 9]. Because two successive weightings are used the uncertainties have to be counted twice:

Purity of reagents

   2   R u m = 2 u 2cal + √ 2 3

The uncertainty of purity of reagents was determined from the manufacturer’s indications. Since there were no data on distribution, a rectangular distribution was assumed and therefore upur is given by: u pur =

tolerance √ 3

(21)

Uncertainty of volumetric operations The uncertainty of volumetric operations is associated with the following uncertainty sources: the uncertainty associated with the calibration of the volumetric flask or pipette, the uncertainty associated with the use of glassware at a temperature which is different from the one in the calibration, and the repeatability of volumetric deliveries [4, 8]. The limits of accuracy of the glassware used (tolerance) are given by the manufacturer. However, no data on the distribution is reported by what a rectangular distribution was assumed because values are expected to be more likely in the centre than near bounds [9]. According to the manufacturer, the volumetric flasks used have been calibrated at a temperature of 20 ◦ C, although inside the laboratory the temperature varies around 20 ± 5 ◦ C. The uncertainty associated with the temperature can be calculated from the estimate of the temperature range and the volume expansion coefficient. The volume expansion of water is considerably larger than that of the glass, therefore, the water volume expansion coefficient is considered (2.1×10−4 ◦ C−1 ) [4]. As can be seen in Fig. 2, all contributions of repeatability, including the repeatability of the volume deliveries were combined in an individual source of uncertainty named precision. In these conditions, the uncertainty associated to the volume measurements is given by Eq. (22):  uV =

u 2V (cal)

 +

5 × 2.1 × 10−4 × V √ 3

(23)

Uncertainty of molar mass The molar mass standard uncertainty was √ found by dividing the IUPAC quoted uncertainty [7] by 3 assuming that it forms the bounds of a rectangular distribution [4, 10]. So the uncertainty of the NaF and F molar mass are calculated by the Eqs. (24) and (25), respectively.  u MMNaF =

u MMF =

tolerance(Na) √ 3

2

 +

tolerance(F) √ 3

tolerance(F) √ 3

2 (24)

(25)

Combined and expanded uncertainty The relation between the combined uncertainty for the determination of fluoride in water and wastewaters and the uncertainty of the independent input quantities is expressed by the model (Eq. 26):      2  2 2  u CF u Fdil   u Csample c =  Csample × +  C Fdil F  2 2 2 +u CSS + u precision + u bias (26) To obtain the expanded uncertainty at the 95% confidence level, the combined uncertainty was multiplied by the coverage factor k of 2 [4].

2 (22)

The uncertainty of weighting operations The usual identified uncertainties sources for gravimetric operations are readability (digital resolution) of the balance scale, the repeatability (included in the precision of the method), and the calibration function of the scale which includes the sensibility (neglected because the weight by difference is done on the same balance over a very narrow range) and the linearity of the balance obtained from

Results and discussions The values of the input quantities CSS, CF , Fdil , precision, and bias as well as their respective standard uncertainties are listen in Tables 2 and 3 for the low and high concentration levels used, respectively. Standard uncertainties of the model input quantities are also based on uncertainty budgets. The details for those quantities are presented in Tables 4–7. The expanded standard uncertainties were estimated as 0.40 and 9.1 mg F l−1 for low and high concentrations levels, respectively, using the duplicate analysis as precision. In these conditions, the results of the measurements should be reported as indicated in Eqs. (27) and (28). It can be seen

437

Table 7

Csample ± UCsample = 101.4 ± 9.1 mgF l−1

Uncertainty of dilution factor, Fdil

Input quantities (Xi ) Final volume (mL)

Symbol Vf

Sample volume (mL) Vsam

Estimated value

u Xi

25

0.028 This work (Eq. (22)) 0.013 This work (Eq. (22))

10

Fdil = 2.5 u Fdil = 0.0043 (Eq. (11))

that the relative expanded uncertainty is around ±10% the result for the highest concentration which can be considered acceptable for the selective electrode potentiometry methods [11] and ±36% for the lowest concentrations. In this case, the result is very close to the limit of quantification of the method that corresponds to a relative uncertainty of about ±30% [12]. Csample ± UCsample = 1.12 ± 0.40 mgF l−1

(28)

Remark

(27)

Fig. 3 Evolution of the relative contributions from CF , Fdil , CSS, precision and reproducibility and bias (recovery) to the combined standard uncertainty variance (u Cc sample ) for different sample fluoride concentrations using the repeatability (a) or the reproducibility (b) for precision. Csample (mg F l−1 ) = (
) 1.12 (2.eps) 6.92 (f3.eps) 22.9 (f4.eps)30.8 (f5.eps) 101

In the cases where the repeatability are used in spite of duplicate analysis, the same conclusions were obtained (Csample =1.12 ± 0.39 mg F l−1 and Csample =101.4 ± 7.0 mg F l−1 ). Relative uncertainty variance contributions calculated by Eq. (29) were used to illustrate the relative impact of different uncertainty components. The evolution of the relative uncertainty variance contributions from the single input variables u CF ,u Fdil , uCSS , uprecision , ubias , to the combined uncertainty for different concentration levels and sample matrix are shown in Fig. 3a using the repeatability and in Fig. 3b using the reproducibility as precision. r Xi =

( ∂∂YX i )2 u 2X i (u Cc sample )2

· 100

(29)

Although the calculated expanded uncertainties and consequently the combined uncertainty do not vary significantly in the cases where they were used the repeatability or reproducibility for evaluating the precision, each relative variances uncertainty contributions do. When the repeatability is used to determine the combined uncertainty, the CSS and CF uncertainty contributions are the most dominant ones. However, if reproducibility is used, relative uncertainty variance contributions are distributed among CSS, CF , and precision. In both cases, the rCF contribution increases and rCSS contribution decreases with the increasing of the concentration level. The precision variance contribution is only significant in the case where the reproducibility is used, and increases with the increasing of the concentration level. The relative reproducibility standard deviation obtained in interlaboratory study can be assumed as an estimate of the combined standard uncertainty of a laboratory [13]. The fluoride content obtained from the RELACRE (PORTUGAL) exercise on the determination of fluoride in mineral water carried out in 2004 were 2.18 mg F l−1 with a relative reproducibility standard deviation of 0.22 mg F l−1 . Using the recommended coverage factor k = 2 the UCsample value of 0.44 mg F l−1 was obtained. The expanded standard uncertainty estimated is 0.42 mg F l−1 using the methodology proposed in this work. In these conditions, the methodology presented to estimate uncertainty can be considered validated. The uncertainty in the results using the relative reproducibility standard deviation obtained in interlaboratory studies can only be considered to be a rough estimate useful for getting an idea about the order of the uncertainty, but it should not replace estimates of uncertainty from your measurements [13]. Conclusions In the present study, a procedure for evaluation of uncertainty components was proposed to determine fluoride by

438

ion selective electrode potentiometry in water and wastewaters matrices at different concentration levels based on the Eurachem Guide of Uncertainty Measurements. The uncertainty sources that affect the measurand were identified as due to CF , Fdil , CCSS , precision, and bias. The standard uncertainty of fluoride concentration is a function of the potential measurement uncertainty and the regression coefficient uncertainties of the calibration regression line also. The standard uncertainty associated with the dilution factor was calculated from the sample volumes diluted into a volumetric flask. Standard uncertainty associated with the fluoride calibration solutions depends on the fluoride principal standard uncertainty and the uncertainties associated with the procedure used to prepare each calibration solution. The standard precision uncertainty were assessed by the repeatability of the fluoride measurements and by the reproducibility estimated from sample duplicate analysis. In our conditions, this last situation seems the most adequate because matrix and concentration dependence were considered. The standard uncertainty due to bias was assessed by recovery tests using the range recoveries of 100 ± 10%.

The standard uncertainty components were organized in tables which allow an easy overview and it may facilitate the comparison of the uncertainty contributions. Relative uncertainty variance contributions were calculated to illustrate impact of each uncertainty components. For low fluoride concentrations levels, the results show that CSS and CF uncertainties contributions are the most dominant sources of combined uncertainty. Their evolutions with the increasing of fluoride concentration level do not vary significantly with the use of the repeatability or reproducibility for precision assessment. The precision contribution was only significant in those cases where there were good reproducibility and high fluoride concentration levels. The evaluation of the contribution from individual uncertainty components made it possible to improve the performance of the determination. The uncertainty in the result calculated using the proposed methodology (Csample ± UCsample = 2.17 ± 0.42 mg F l−1 ) is in satisfactory agreement with the estimated expanded uncertainty obtained using the relative reproducibility standard deviation obtained in interlaboratory studies (Csample ± UCsample = 2.18 ± 0.44 mg F l−1 ).

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