Estimation of uncertainty in pK a values determined by potentiometric titration

However, what seems to be almost missing from the literature is such approach whereby all uncertainty sources of a pKavalue are taken into account and propagated using the corresponding

Abstract A procedure is presented for estimation of

un-certainty in measurement of the pKaof a weak acid by

po-tentiometric titration The procedure is based on the ISO

GUM The core of the procedure is a mathematical model

that involves 40 input parameters A novel approach is

used for taking into account the purity of the acid, the

im-purities are not treated as inert compounds only, their

pos-sible acidic dissociation is also taken into account

Appli-cation to an example of practical pKadetermination is

pre-sented Altogether 67 different sources of uncertainty are

identified and quantified within the example The relative

importance of different uncertainty sources is discussed

The most important source of uncertainty (with the

exper-imental set-up of the example) is the uncertainty of pH

measurement followed by the accuracy of the burette and

the uncertainty of weighing The procedure gives

uncer-tainty separately for each point of the titration curve The

uncertainty depends on the amount of titrant added, being

lowest in the central part of the titration curve The

possi-bilities of reducing the uncertainty and interpreting the

drift of the pKavalues obtained from the same curve are

discussed

Electronic Supplementary Material Supplementary

ma-terial is available in the online version of this article at

http://dx.doi.org/10.1007/s00216-004-2586-1 A full

de-scription of derivation of the mathematical model and

quan-tification of the uncertainty components is available as

file pKa_u_ESM.pdf (portable document format) Full

de-tails of the uncertainty calculation are available in two

cal-culation files: MS Excel workbook pKa_u.xls (MS Excel 97

format) and GUM Workbench file pKa_u.smu GUM

Workbench software is not very widespread and we have

also included the report generated from the pKa_u.smu file

in PDF format (file pKa_u_GWB.pdf) That report contains

all the details of the calculation

Keywords Measurement uncertainty · Sources of

uncertainty · ISO · Eurachem · Dissociation constants ·

pKa· pH

Introduction

In recent years quality of results of chemical measurements – metrology in chemistry (traceability of results, measure-ment uncertainty, etc.) – has become an increasingly im-portant topic It is reflected by the growing number of publications, conferences, etc [1, 2, 3, 4] One of the main points that is now widely recognized is that every mea-surement result should be accompanied by an estimate of uncertainty – a property of the result characterizing the dispersion of the values that could reasonably be attrib-uted to the measurand [2, 3]

Dissociation constant Kaor the corresponding pKavalue

is one of the most important physicochemical characteris-tics of compounds having acidic (or basic) properties

Re-liable pKadata are indispensable in analytical chemistry, biochemistry, chemical technology, etc A huge amount of

pKadata has been reported in the literature and collected into several compilations [5, 6, 7]

Potentiometric titration methods for determination of

pKausing the glass electrode are the most widely used and

the art of such pKameasurement can be considered mature Numerous methods have been described, starting from those described in the classic book of Albert and Serjeant [8] and finishing with the modern computational ap-proaches (for example Miniquad [9], Minipot [9], Super-quad [9], Phconst [9], Pkpot [10], Miniglass [11] etc) for

calculation and refinement of pKa values from potentio-metric data

Efforts have also been devoted to investigating the

sources of uncertainty of pKa values The various com-puter programs mentioned above are very useful in this respect They can be used in the search of systematic er-rors, because many parameters are adjustable Standard errors of the parameters are obtained by weighted or un-weighted non-linear regression and curve-fitting [9, 10,

Eve Koort · Koit Herodes · Viljar Pihl · Ivo Leito

Estimation of uncertainty in pKa values determined

by potentiometric titration

Anal Bioanal Chem (2004) 379 : 720–729

DOI 10.1007/s00216-004-2586-1

Received: 4 January 2004 / Revised: 1 March 2004 / Accepted: 4 March 2004 / Published online: 22 April 2004

O R I G I N A L PA P E R

E Koort · K Herodes · V Pihl · I Leito (✉)

Institute of Chemical Physics, Department of Chemistry,

University of Tartu, Jakobi 2, 51014 Tartu, Estonia

e-mail: leito@ut.ee

© Springer-Verlag 2004

11, 12, 13, 14] The influence of various sources of

uncer-tainty in pH and titrant volume measurements on the

ac-curacy of acid–base titration has been studied using

loga-rithmic approximation functions by Kropotov [15] The

uncertainty of titration equivalence point (predict values

and detect systematic errors) was investigated by a

graph-ical method using spreadsheets by Schwartz [16] Gran

plots can also be used to determine titration equivalence

point [17] and they are useful for assessing the extent of

carbonate contamination of the alkaline titrant

The various sources of uncertainty have thus been

in-vestigated quite extensively However, what seems to be

almost missing from the literature is such approach

whereby all uncertainty sources of a pKavalue are taken

into account and propagated (using the corresponding

mathematical model) to give the combined uncertainty of

the pKavalue, which takes simultaneously into account the

uncertainty contributions from all the uncertainty sources

This combined uncertainty, which is obtained as a result,

is a range in which the true pKavalue remains with a stated

level of confidence In addition, the full uncertainty

bud-get gives a powerful tool for finding bottlenecks and for

optimizing the measurement procedure, because it shows

what the most important uncertainty sources are

In this paper we present a procedure of estimation of

uncertainty of pKa values determined by potentiometric

titration that takes into account as many uncertainty

sources as possible We also provide the realisation of the

procedure in two different software packages – MS Excel

and GUM Workbench – available in the electronic

sup-plementary material (ESM)

The procedure is based on a mathematical model of

pKameasurement and involves identification and

quantifi-cation of individual uncertainty sources according to the

ISO GUM/Eurachem approach [2, 3] This approach for

estimation of measurement uncertainty consists of the

fol-lowing steps:

1 specifying the measurand and definition of the

mathe-matical model;

2 identification of the sources of uncertainty;

3 modification of the model (if necessary);

4 quantification of the uncertainty components; and

5 calculating combined uncertainty

In this paper the section “Derivation of the uncertainty

es-timation procedure” includes steps 1–3 This is followed

by a detailed application example, which includes steps 4

and 5 To save space in the printed journal sections on

rivation of the uncertainty estimation procedure and

de-scription of the application example are only very briefly

outlined in the main paper Detailed description and

ex-planations are given in the file pKa_u_ESM.pdf in the

electronic supplementary material (ESM)

Derivation of the uncertainty estimation procedure

The dissociation of a Brønsted acid, HA, is expressed by

the (simplified) equation:

(1) and the dissociation constant is given by:

(2)

(3)

where a(H+), a(A–), and a(HA) are the activities of the

hy-drogen ion, the anion, and the undissociated acid

mole-cules, respectively The method of pKadetermination

con-sists in potentiometric titration of a given amount Va0(mL)

of a solution of an acid HA of known concentration Ca0 (mol L–1) with a solution of strong base MOH of known

concentration Ct0 (mol L–1) From the pH measurements

and the amounts and concentrations of the solutions a[H+]

and the ratio a[A–]/a[HA] can be calculated and a Ka(and

pKa) value can be calculated for every point of the

titra-tion curve In our approach the pKavalue corresponding

to an individual point “x” of the titration curve – denoted

pKax– is the measurand

The uncertainty estimation procedure derived below is

intended for the mainstream routine pKa measurement equipment An electrode system consisting of a glass elec-trode and reference elecelec-trode (or a combined elecelec-trode) with liquid junction, connected to a digital pH-meter with multi-point calibration This procedure is valid for mea-surements of acids that are neither too strong nor too weak The model equation and the full detailed list of

quanti-ties of pKameasurement of the acid HA corresponding to one point of the titration curve is presented in Table 1 De-tailed description of the derivation of the model equation and finding the sources of uncertainty is given in the file

pKa_u_ESM.pdf in the ESM The factors that are taken into account include all uncertainty sources related to weighing and volumetric operations, purities of the mea-sured acid HA, carbonate content of the titrant, and pH-re-lated uncertainty sources, such as accuracy of the calibra-tion buffer solucalibra-tions, repeatability uncertainty of the in-strument, residual liquid junction potential, temperature effects, etc

The equations given in Table 1 form the mathematical

model for pKameasurement The main equations are Eqs (4) and (5) together with Eqs (7), (21), (23), (24), and (25) in the ESM

(4)

(5)

where Exis the electromotive force (emf) of the electrode system in the measured solution at point “x” of the titra-tion curve, pHxis the pH of the measured solution, Eisand

pHisare the co-ordinates of the isopotential point of the electrode system (the intersection point of calibration lines

at different temperatures) [18, 19], s is the slope of the

cal-ibration line,αis the temperature coefficient of the slope

[ ] [ ]

D

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$

I

&

−

−

⋅

−

( [ LV )

PHDV FDO



( (

V α W W

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D D

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HA corresponding to one point of the titration curve

Table 1 (continued)

[19], and tmeas and tcal are the measurement temperature

and the calibration temperature, respectively The slope s

and the isopotential pHisare found by calibrating the sys-tem using standard solutions of known pH values pHi

hav-ing emf values Ei Cais the total concentration of the acid

HA in the titration cell, [A–] is the equilibrium concentra-tion of the anion A– and f1is the activity coefficient for singly charged ions (found from Debye–Hückel theory)

See comments in Table 1 and the file pKa_u_ESM.pdf in the ESM for detailed explanations

The model involves altogether 40 input parameters and

67 sources of uncertainty are taken into account

Application example

Experimental set-up

Detailed description of the experimental set-up is given in the ESM, only a brief outline is provided here The

uncer-tainty estimation procedure is applied to pKa determina-tion of benzoic acid Mainstream equipment was used for

pKameasurement – a pH meter with 0.001 pH unit reso-lution and a glass electrode with inner reference electrode and porous liquid junction were used The electrode was calibrated using five calibration solutions prepared accord-ing to the NIST procedure with pH values 1.679, 3.557, 4.008, 6.865, and 9.180 A piston burette with 5 mL capac-ity was used for titration Titration was carried out in a cell thermostatted to 25.0±0.1 °C, maintaining an atmosphere

of nitrogen over the solution and using a magnetic stirrer for stirring the solution The system was run under com-puter control providing fully automatic titration Main-stream volumetric glassware and analytical balance were used for preparation of solutions

Quantification of the uncertainty components and calculation of the uncertainty

The titration curve corresponding to the example is

avail-able in the ESM (file pKa_u.xls) The uncertainty calcula-tion was carried out using two different software pack-ages: MS Excel (Microsoft) and GUM Workbench (Metrodata) The MS Excel calculation workbook (the file

724

Table 1 (continued)

pKa_u.xls, in MS Excel 97 format) is available in the ESM The spreadsheet method for calculation of uncertainty has been used [3] Uncertainty calculation has been carried out for seven different titration points corresponding to 6,

12, 30, 50, 70, 90 and 95% of the overall titrant volume required to arrive at the equivalence point

Results

The detailed uncertainty budget for one single titrant

vol-ume (Vt=0.8 mL) is presented in Table 2 and Fig 1 It is

also available as GUM Workbench file pKa_u.smu in the ESM The uncertainty budgets of the pH values at the

dif-725

HA corresponding to one point of the titration curve (added titrant

volume: 0.8 ml) a

a The headings of the columns: standard uncertainty – uncertainty given at standard deviation level; distribution – probability distribu-tion funcdistribu-tion of the value; sensitivity coefficient – evaluated as

c i= ∆y/∆xi , describes how the value of y varies with changes in x i; un-certainty contribution – the square of a standard unun-certainty multiplied

by the square of the relevant sensitivity coefficient; index – ratio of the uncertainty contribution of an input quantity to the sum which is taken over all uncertainty contributions of input quantities, expressed

as percentages

ferent Vtvalues are presented in Table 3 The uncertainty

budgets, the resulting pKaxvalues and the resulting

com-bined standard uncertainties uc(pKax) and expanded

un-certainties U(pKax) are presented in Table 4 Figure 2

illus-trates the variation of uncertainty of pKavalues obtained

from different points of the titration curve

Discussion

The main sources of uncertainty in pKadetermination

The uncertainty budgets of the pKax values found from different points of the titration curve are presented in Table 4 As is expected, the uncertainty is the lowest in the middle of the titration curve The relationship is roughly symmetrical with respect to the half-neutralization point (see Figure 2) From Table 4 it follows that different sources of uncertainty dominate at the beginning of the curve and at the end

pH is clearly the key player in the uncertainty budgets corresponding to most of the titration curve points In turn, the uncertainty of pH is in all titration points almost en-tirely determined by the uncertainty of the EMF

measure-ment in the measured solution u(Ex): leaving out all other

uncertainty sources changes the uc(pHx) by only around

0.001 pH units The u(Ex), which consists of four

compo-nents (four rows next to the Exrow), is in turn determined mainly by the residual liquid junction potential uncertainty

It is interesting to note the different contributions of uncertainty of pHxto the u(pKax) in different parts of the titration curve, while the uncertainty of all the pH mea-surements is practically identical (see Table 1): the

influ-ence of u(pHx) is stronger in the beginning and in the mid-dle of the titration curve where it is clearly the dominating source of uncertainty At the end of the curve the

dominat-ing factors are the uncertainties of the concentrations Ca0 and Ct0and the titrant volume Vt This behaviour can be

726

Fig 1 Uncertainty contributions of the most important input

quan-tities of pKaxat the titration point Vt=0.8

Table 3 Uncertainty budgets

and combined uncertainties of

pH x corresponding to different

points on the titration curve

a The uncertainty contribution

percentages are given for the

uncertainty of the respective

pKax value (i.e the percentages

(excluding the row “Ex “ b ) sum

to give the uncertainty

contri-bution of the pHxvalue in

Table 4) The uncertainty

con-tributions have been found

ac-cording to Eq 58 in the ESM

(file pKa_u_ESM.pdf) The

full uncertainty budgets can be

found in the ESM (files

pKa_u.smu and pKa_u.xls)

b The separate uncertainty

con-tributions of components of Ex

– the most important input

quantity – are given in the next

four rows.

Titrant volume and pH

Uncertainty contributions of input quantities (%) a

Ex,rep 1.7 1.7 1.8 1.8 1.4 0.4 0.1

Ex,read 0.1 0.1 0.1 0.1 0.1 0.0 0.0

Ex,drift 15.6 16.0 17.1 16.9 13.4 4.1 0.9

Standard uncertainties of pH values

easily rationalised – in the region of the equivalence point

of the curve the relatively low concentration of neutral

[HA] is calculated as a difference between two relatively

high concentrations Caand [A–], which in turn are

depen-dent on the three parameters Ca0, Ct0and Vt At the

begin-ning and in the middle of the curve where the [HA] is low

this effect is not pronounced In contrast, at the beginning

of the titration curve there is pronounced self-dissociation

of the acid HA Thus, in addition to determining the a(H+)x

in Eq (2) pHxalso influences [A–]

The purity of the acid under investigation, P, is, in this

treatment, not related just to inert compounds but involves

also contaminants with acidic properties (see the

mathe-matical model section in the ESM file pKa_u_ESM.pdf

for a more detailed explanation) In the application

exam-ple it has been assumed that the acid contains in addition

to inert impurities also three different kinds of acidic

im-purity with different acidity (pKavalues around 2.5, 7, and

10) Concentrations and acidity of all those acidic

impuri-ties enter the measurement equations and are thus taken

into account As is seen from Table 4, impurities with

dif-ferent pKavalues have different influence on the final

re-sult The impurity with the lowest pKavalue has the highest influence The total uncertainty contribution of the four impurities is different in the different parts of the titration curve, ranging from 8.1% (in the middle of the curve) to

31% at Vt=1.55 mL The input quantities related to the pu-rity of the acid are the biggest source of uncertainty in the

initial acid concentration Ca0

The uncertainty of Vtis mainly determined by the ac-curacy of the mechanical burette The uncertainty of the concentration of the titrant depends on several sources of similar magnitude, the most important of these are again the weighing uncertainty, the purity of standard substance, and the accuracy of the burette The effect of contamina-tion of the titrant with carbonate becomes (at the level of carbonate, assumed in the example) visible only in the last

portion of the titration curve because the pKa value of

H2CO3is ca 6.3, which is well above the pHxvalues

Possibilities of optimizing the pKameasurement procedure

The uncertainty budget is a powerful tool for optimizing the measurement procedure From Tables 3 and 4 it can be concluded that the glassware used and the burette are in general appropriate for this work The stability of temper-ature in the laboratory is adequate There is no need to in-volve more calibration standards in the calibration of the

pH meter (it is also the recommendation of IUPAC to use

up to five buffers for multi-point calibration of pH meters [28]) The target uncertainty of pH measurement using multi-point calibration is estimated as 0.01–0.03 pH units

(expanded uncertainty, k=2), in agreement with our

re-sults The changes that could be introduced: instead of a

50 mL flask a 250 mL flask could be used, so that a larger amount of the acid could be weighed; a smaller piston

727

Table 4 Uncertainty budgets

and combined uncertainties of

pKax values calculated for

dif-ferent added titrant volumes Vt

a The uncertainty contributions

have been found according to

Eq 57 in the ESM (file

pKa_u_ESM.pdf) Those input

quantities that contribute

negli-gibly to the overall uncertainty

of pKax have been omitted The

full uncertainty budgets can be

found in the ESM (files

pKa_u.smu and pKa_u.xls)

Titrant volume and pH

Uncertainty contributions of input quantities (%) a

pKavalues and their uncertainties (standard and expanded)

curve

could be used for the piston burette (that can, in fact, be

difficult, because at least with this manufacturer 5 mL is

the smallest size) However these changes do not reduce

the uncertainty significantly The most significant decrease

of the overall uncertainty of pKawould be achieved if the

residual liquid junction potential could be estimated or

eliminated That is difficult, however, without introducing

significant changes to the experimental set-up [20, 23, 26]

Finding the overall pKavalue and its uncertainty

The procedure described here is intended for finding the

uncertainty of the pKaxdetermined from a single point of

the titration curve Obviously the best estimate of the pKa

value is the mean of the pKaxvalues that are in the region

of the lowest uncertainty (see the table and figure in the

ESM, file pKa_u.xls, sheet “final pKa”)

The overall uncertainty of pKashould consider all the

uncertainty sources in the method, including the variability

between the pKaxvalues found from different points of the

titration curve However, since the sources of variability

(the various repeatabilities) are already included in the

uncertainty estimates of the individual pKaxvalues, it is no

longer necessary to add any repeatability contribution

Based on this we take the average value of U(pKax) as the

estimate of U(pKa) It is unreasonable to divide the

uncer-tainty U(pKax) by the square root of n (the number of pKax

values used for calculating the overall pKavalue), because

the pKaxvalues are not statistically independent

On the basis of this reasoning we get, for our example

(using the pKax values corresponding to Vt 0.2, 0.4, 0.8

and 1.15 mL): pKa=4.219, uc(pKa)=0.017, U(pKa)=0.034

(k=2).

Interpretation of the drift of pKaxvalues

From Table 4 it is apparent that the pKa values increase

slightly with increasing Vt This drift is caused by various

effects of systematic nature Some of them influence the

first part of the curve, some the rear part For example,

some mismatch always exists between the four terms Ct,

Vt, Ca0, and Va0 That leads to an increasingly erroneous

concentration of the undissociated acid [HA] as the Vtgets

higher ([HA] is calculated from [HA]=Ca–[A–] (Eq (8) in

the ESM) and in the rear part of the curve the [HA] is

found as the small difference between two relatively large

quantities of similar magnitude) causing the pKax values

also to drift Because our uncertainty estimation procedure

takes into account all the uncertainty sources causing the

drift (including the uncertainties of the four terms of this

example), this drift is also automatically taken into account

by the uncertainty estimate Therefore, some drift of the

pKaxvalues is normal

The question remains, however, how much drift is

ac-ceptable We propose the following criterion: the drift of a

pKax value from the overall pKa value is acceptable as

long as the overall pK value lies within the limits of

ex-panded uncertainty pKax–U(pKax) pKax+U(pKax) Accord-ing to this approach the drift in Table 4 is acceptable

Comparison of the obtained uncertainty

of the pKavalue with literature data

The main problem with the literature is that very often no uncertainty estimate is given with the results For example,

there are 174 pKavalues for pKaof benzoic acid measured under different conditions given in Palm tables [7] Only for

24 of those values were uncertainty estimates reported The results of this work can be used to obtain rough estimates

of the uncertainty in such literature values if experimental details are available from the original publications The second aspect is the validity of the reported uncer-tainty values As can be seen from the results of this work,

“normal” expanded uncertainties (at k=2 level) for pKa values in the region of 3–5 pKaunits obtained from poten-tiometric titration with an electrode system containing

liquid junction, are in the range ±0.03–0.05 pKaunits It is doubtful whether with a similar experimental set-up it

would be possible to obtain expanded uncertainty (k=2) below 0.02 pKaunits It is outside the scope of this paper

to carry out an extensive review of literature data but we note that for carboxylic acids, for example, uncertainties

in the range 0.005 to 0.02 pKa units are more frequently found in Ref [7] than uncertainties in the range 0.03 to

0.05 pKaunits A situation encountered quite frequently is that values from different authors do not agree within the combined uncertainty limits This clearly indicates under-estimated uncertainties

Concerning the compound under study in this work, benzoic acid, acidic dissociation of benzoic acid has been

extensively studied (using all major methods for pKa mea-surement) and many different values have been found The values given in Ref [7] (at 25 °C) vary from 4.16 to 4.24, the values of higher quality (estimated by the limited information available on reliability of the values) are around 4.20 to 4.21 In the compilation of Kortüm et al [5] the values estimated by the compilers as the most

reli-able are pKa=4.20 Our result 4.219±0.034 agrees with the literature data well within the uncertainty limits

Acknowledgments This work was supported by the grant 5800

from the Estonian Science Foundation.

References (All references are included, also those that are cited only in ESM)

1 Bièvre PD, Taylor PDP (1997) Metrologia 34:67–75

2 BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, ISO (1993) Guide to the expression of uncertainty in measurement Geneva

3 Ellison SLR, Rösslein M, Williams A (2000) (eds) Quantifying uncertainty in analytical measurement, 2nd edn Eurachem/ CITAC

4 Kuselman I (2000) Rev Anal Chem 19:217–233

5 Kortüm G, Vogel W, Andrussow K (1961) Dissoziationkonstan-ten Organischer Säuren in Wässeriger Lösung Butterworths, London

728

6 Perrin DD (1965) Dissociation constants of organic bases in

aqueous solution Published as supplement to Pure and Applied

Chemistry Butterworths, London

7 Palm V (1975–1985) (eds) Tables of rate and equilibrium

con-stants of heterolytic organic reactions Viniti, Moscow–Tartu

8 Albert A, Serjeant EP (1984) Ionization constants of acids and

bases Cambridge University Press, UK

9 Baeza JJB, Ramos GR, Fernandez CM (1989) Anal Chim Acta

223:419–427

10 Barbosa J, Barron D, Beltran JL, Sanz-Nebot V (1995) Anal

Chim Acta 317:75–81

11 Izquierdo A, Beltran JL (1986) Anal Chim Acta 181:87–96

12 Avdeef A (1983) Anal Chim Acta 148:237–244

13 Kateman G, Smit HC, Meites L (1983) Anal Chim Acta 152:

61–72

14 (a) Barry DM, Meites L (1974) Anal Chim Acta 68:435–445.

(b) Barry DM, Meites L, Campbell B H (1974) Anal Chim

Acta 69:143–151

15 Kropotov VA (1999) J Anal Chem 54:134–137

16 Schwartz LM (1992) J Chem Educ 69:879–883

17 Gran G (1952) Analyst 77:661 in Gans P, O’Sullivan B (2000) Talanta 51:33–37

18 Bates RG (1973) Determination of pH Theory and Practice Wiley, New York

19 Galster H (1991) pH Measurement VCH, Weinheim

20 (a) Bagg J (1990) Electrochim Acta 35:361–365 (b) Bagg J (1990) Electrochim Acta 35:367–370

21 Schmitz G (1994) J Chem Educ 71:117–118

22 Heydorn K, Hansen EH (2001) Accred Qual Assur 6:75–77

23 Meinrath G, Spitzer P (2000) Microchim Acta 135:155–168

24 Bièvre P De, Peiser PHS (1997) Metrologia 34:49–59

25 Kragten J (1994) Analyst 119:2161–2165

26 Baucke FGK, Naumann R, Alexander-Weber C (1993) Anal Chem 65:3244–3251

27 Basic laboratory skills training guides: measurement of pH VAM Guide LGC, VAM, 2001

28 Buck RP et al (2002) (eds) Pure Appl Chem 74:2169–2200

29 Leito I, Strauss L, Koort E, Pihl V (2002) Accred Qual Assur 7:242–249