Figure A1.5: Uncertainty contributions in cadmium standard preparation u( y, xi) ( mg l 1) 0 0.2 0.4 0.6 0.8 1 c( Cd) Purity m V The values of u( y, xi)=( ¶ y/ ¶ xi). u( xi) are taken from Table A1.3 Quantifying Uncertainty Example A2 QUAM: 2000. P1 Page 40 Example A2: Standardising a Sodium Hydroxide Solution Summary Goal A solution of sodium hydroxide ( NaOH) is standardised against the titrimetric standard potassium hydrogen phthalate ( KHP).

Measurement procedure The titrimetric standard ( KHP) is dried and weighed. After the preparation of the NaOH solution the sample of the titrimetric standard ( KHP) is dissolved and then titrated using the NaOH solution. The stages in the procedure are shown in the flow chart Figure A2.1.

Measurand:

] l mol [ 1000 1 - × × × = T KHP KHP KHP NaOH V M P m c where c NaOH : concentration of the NaOH solution [ mol l   1]

1000 : conversion factor [ ml] to [ l]

m KHP : mass of the titrimetric standard KHP [ g]

P KHP : purity of the titrimetric standard given as : mass fraction M KHP : molar mass of KHP [ g mol   1]

V T : titration volume of NaOH solution [ ml] Weighing KHP Weighing KHP Preparing NaOH Preparing NaOH Titration Titration RESULT RESULT Figure A2.1: Standardising NaOH Identification of the uncertainty sources:

The relevant uncertainty sources are shown as a cause and effect diagram in Figure A2.2.

Quantification of the uncertainty components The different uncertainty contributions are given in Table A2.1, and shown diagrammatically in Figure A2.3. The combined standard uncertainty for the 0.10214 mol l 1 NaOH solution is 0.00010 mol l 1 Table A2.1: Values and uncertainties in NaOH standardisation Description Value x Standard uncertainty u Relative standard uncertainty u( x)/ x rep Repeatability 1.0 0.0005 0.0005 m KHP Mass of KHP 0.3888 g 0.00013 g 0.00033 P KHP Purity of KHP 1.0 0.00029 0.00029 M KHP Molar mass of KHP 204.2212 g mol 1 0.0038 g mol 1 0.000019 V T Volume of NaOH for KHP titration 18.64 ml 0.013 ml 0.0007 c NaOH NaOH solution 0.10214 mol l 1 0.00010 mol l 1 0.00097 Quantifying Uncertainty Example A2 QUAM: 2000. P1 Page 41 Figure A2.2: Cause and effect diagram for titration c( NaOH) P( KHP)

M( KHP) V( T) calibration temperature end point bias m( KHP)

m( gross) m( tare) calibration linearity sensitivity calibration linearity sensitivity repeatability V( T)

end point m( KHP)

Figure A2.3: Contributions to Titration uncertainty u( y, xi) ( mmol l 1) 0 0.02 0.04 0.06 0.08 0.1 0.12 c( NaOH) Repeatability m( KHP) P( KHP) M( KHP) V( T)

The values of u( y, xi)=( ¶ y/ ¶ xi). u( xi) are taken from Table A2.3 Quantifying Uncertainty Example A2 QUAM: 2000. P1 Page 42 Example A2: Standardising a sodium hydroxide solution. Detailed discussion A2.1 Introduction This second introductory example discusses an experiment to determine the concentration of a solution of sodium hydroxide ( NaOH). The NaOH is titrated against the titrimetric standard potassium hydrogen phthalate ( KHP). It is assumed that the NaOH concentration is known to be of the order of 0.1 mol l   1. The end point of the titration is determined by an automatic titration system using a combined pH electrode to measure the shape of the pH curve. The functional composition of the titrimetric standard potassium hydrogen phthalate ( KHP), which is the number of free protons in relation to the overall number of molecules, provides traceability of the concentration of the NaOH solution to the SI system.

A2.2 Step 1: Specification The aim of the first step is to describe the measurement procedure. This description consists of a listing of the measurement steps and a mathematical statement of the measurand and the parameters upon which it depends.

Procedure:

The measurement sequence to standardise the NaOH solution has the following stages.

Figure A2.4: Standardisation of a solution of sodium hydroxide Weighing KHP Weighing KHP Preparing NaOH Preparing NaOH Titration Titration RESULT RESULT The separate stages are:

i) The primary standard potassium hydrogen phthalate ( KHP) is dried according to the supplier's instructions. The instructions are given in the supplier's catalogue, which also states the purity of the titrimetric standard and its uncertainty. A titration volume of approximately 19 ml of 0.1 mol l 1 solution of NaOH entails weighing out an amount as close as possible to g 388 . 0 0 . 1 1000 19 1 . 0 2212 . 204 = ´ ´ ´ The weighing is carried out on a balance with a last digit of 0.1 mg.

ii) A 0.1 mol l 1 solution of sodium hydroxide is prepared. In order to prepare 1 l of solution, it is necessary to weigh out » 4 g NaOH. However, since the concentration of the NaOH solution is to be determined by assay against the primary standard KHP and not by direct calculation, no information on the uncertainty sources connected with the molecular weight or the mass of NaOH taken is required.

iii) The weighed quantity of the titrimetric standard KHP is dissolved with » 50 ml of ion free water and then titrated using the NaOH solution. An automatic titration system controls the addition of NaOH and records the pH curve. It also determines the end point of the titration from the shape of the recorded curve.

Calculation:

The measurand is the concentration of the NaOH solution, which depends on the mass of KHP, its purity, its molecular weight and the volume of NaOH at the end point of the titration ] l mol [ 1000 1 - × × × = T KHP KHP KHP NaOH V M P m c where c NaOH : concentration of the NaOH solution [ mol l   1]

1000 : conversion factor [ ml] to [ l]

m KHP : mass of the titrimetric standard KHP [ g] Quantifying Uncertainty Example A2 QUAM: 2000. P1 Page 43 P KHP : purity of the titrimetric standard given as mass fraction M KHP : molar mass of KHP [ g mol   1]

V T : titration volume of NaOH solution [ ml]

A2.3 Step 2: Identifying and analysing uncertainty sources The aim of this step is to identify all major uncertainty sources and to understand their effect on the measurand and its uncertainty. This has been shown to be one of the most difficult step in evaluating the uncertainty of analytical measurements, because there is a risk of neglecting uncertainty sources on the one hand and an the other of double counting them. The use of a cause and effect diagram ( Appendix D) is one possible way to help prevent this happening. The first step in preparing the diagram is to draw the four parameters of the equation of the measurand as the main branches. Afterwards, each step of the method is considered and any further influence quantity is added as a factor to the diagram working outwards from the main effect. This is carried out for each branch until effects become sufficiently remote, that is, until effects on the result are negligible.

Mass m KHP Approximately 388 mg of KHP are weighed to standardise the NaOH solution. The weighing procedure is a weight by difference. This means that a branch for the determination of the tare ( m tare) and another branch for the gross weight ( m gross) have to be drawn in the cause and effect diagram. Each of the two weighings is subject to run to run variability and the uncertainty of the calibration of the balance. The calibration itself has two possible uncertainty sources: the sensitivity and the linearity of the calibration function. If the weighing is done on the same scale and over a small range of weight then the sensitivity contribution can be neglected.

All these uncertainty sources are added into the cause and effect diagram ( see Figure A2.6).

Purity P KHP The purity of KHP is quoted in the supplier's catalogue to be within the limits of 99.95% and 100.05%. P KHP is therefore 1.0000 ± 0.0005. There is no other uncertainty source if the drying procedure was performed according to the suppliers specification. c( NaOH) m( KHP) P( KHP)

M( KHP) V( T)

Figure A2.5: First step in setting up a cause and effect diagram c( NaOH) m( KHP) P( KHP)

M( KHP) V( T) m( gross) m( tare) repeatability repeatability calibration linearity sensitivity calibration linearity sensitivity Figure A2.6: Cause and effect diagram with added uncertainty sources for the weighing procedure Quantifying Uncertainty Example A2 QUAM: 2000. P1 Page 44 Molar mass M KHP Potassium hydrogen phthalate ( KHP) has the empirical formula C8H5O4K. The uncertainty in the molar mass of the compound can be determined by combining the uncertainty in the atomic weights of its constituent elements. A table of atomic weights including uncertainty estimates is published biennially by IUPAC in the Journal of Pure and Applied Chemistry. The molar mass can be calculated directly from these; the cause and effect diagram ( Figure A2.7) omits the individual atomic masses for clarity Volume V T The titration is accomplished using a 20 ml piston burette. The delivered volume of NaOH from the piston burette is subject to the same three uncertainty sources as the filling of the volumetric flask in the previous example. These uncertainty sources are the repeatability of the delivered volume, the uncertainty of the calibration of that volume and the uncertainty resulting from the difference between the temperature in the laboratory and that of the calibration of the piston burette. In addition there is the contribution of the end point detection, which has two uncertainty sources.

1. The repeatability of the end point detection, which is independent of the repeatability of the volume delivery.

2. The possibility of a systematic difference between the determined end point and the equivalence point ( bias), due to carbonate absorption during the titration and inaccuracy in the mathematical evaluation of the endpoint from the titration curve.

These items are included in the cause and effect diagram shown in Figure A2.7.

A2.4 Step 3: Quantifying uncertainty components In step 3, the uncertainty from each source identified in step 2 has to be quantified and then converted to a standard uncertainty. All experiments always include at least the repeatability of the volume delivery of the piston burette and the repeatability of the weighing operation. Therefore it is reasonable to combine all the repeatability contributions into one contribution for the overall experiment and to use the values from the method validation to quantify its size, leading to the revised cause and effect diagram in Figure A2.8.

The method validation shows a repeatability for the titration experiment of 0.05%. This value can be directly used for the calculation of the combined standard uncertainty.

Mass m KHP The relevant weighings are:

container and KHP: 60.5450 g( observed)

container less KHP: 60.1562 g( observed)

KHP 0.3888 g( calculated)

Because of the combined repeatability term identified above, there is no need to take into account the weighing repeatability. Any systematic offset across the scale will also cancel. c( NaOH) P( KHP)

M( KHP) V( T) repeatability calibration temperature end point bias repeatability m( KHP)

m( gross) m( tare) repeatability repeatability calibration linearity sensitivity calibration linearity sensitivity Figure A2.7: Cause and effect diagram ( all sources) Quantifying Uncertainty Example A2 QUAM: 2000. P1 Page 45 The uncertainty therefore arises solely from the balance linearity uncertainty.

Linearity: The calibration certificate of the balance quotes ± 0.15 mg for the linearity. This value is the maximum difference between the actual mass on the pan and the reading of the scale. The balance manufacture's own uncertainty evaluation recommends the use of a rectangular distribution to convert the linearity contribution to a standard uncertainty.

The balance linearity contribution is accordingly mg 09 . 0 3 mg 15 . 0 = This contribution has to be counted twice, once for the tare and once for the gross weight, because each is an independent observation and the linearity effects are not correlated.

This gives for the standard uncertainty u( m KHP) of the mass m KHP, a value of mg 13 . 0 ) ( ) 09 . 0 ( 2 ) ( 2 = Þ ´ = KHP KHP m u m u NOTE 1: Buoyancy correction is not considered because all weighing results are quoted on the conventional basis for weighing in air [ H. 19]. The remaining uncertainties are too small to consider. Note 1 in Appendix G refers.

NOTE 2: There are other difficulties when weighing a titrimetric standard. A temperature difference of only 1 ° C between the standard and the balance causes a drift in the same order of magnitude as the repeatability contribution. The titrimetric standard has been completely dried, but the weighing procedure is carried out at a humidity of around 50 % relative humidity, so adsorption of some moisture is expected.

Purity P KHP P KHP is 1.0000 ± 0.0005. The supplier gives no further information concerning the uncertainty in the catalogue. Therefore this uncertainty is taken as having a rectangular distribution, so the standard uncertainty u( P KHP) is 00029 . 0 3 0005 . 0 = .

Molar mass M KHP From the latest IUPAC table, the atomic weights and listed uncertainties for the constituent elements of KHP ( C8H5O4K) are:

Element Atomic weight Quoted uncertainty Standard uncertainty C 12.0107 ± 0.0008 0.00046 H 1.00794 ± 0.00007 0.000040 O 15.9994 ± 0.0003 0.00017 K 39.0983 ± 0.0001 0.000058 For each element, the standard uncertainty is found by treating the IUPAC quoted uncertainty as forming the bounds of a rectangular distribution. The corresponding standard uncertainty is therefore obtained by dividing those values by 3 .

The separate element contributions to the molar mass, together with the uncertainty contribution for each, are: c( NaOH) P( KHP)

M( KHP) V( T) calibration temperature end point bias m( KHP)

m( gross) m( tare) calibration linearity sensitivity calibration linearity sensitivity repeatability V( T)

end point m( KHP)

Figure A2.8: Cause and effect diagram ( Repeatabilities combined) Quantifying Uncertainty Example A2 QUAM: 2000. P1 Page 46 Calculation Result Standard uncertainty C8 8 ´ 12.0107 96.0856 0.0037 H5 5 ´ 1.00794 5.0397 0.00020 O4 4 ´ 15.9994 63.9976 0.00068 K 1 ´ 39.0983 39.0983 0.000058 The uncertainty in each of these values is calculated by multiplying the standard uncertainty in the previous table by the number of atoms.

This gives a molar mass for KHP of 1 mol g 2212 . 204 0983 . 39 9976 . 63 0397 . 5 0856 . 96 - = + + + = KHP M As this expression is a sum of independent values, the standard uncertainty u( M KHP) is a simple square root of the sum of the squares of the contributions:

1 2 2 2 2 mol g 0038 . 0 ) ( 000058 . 0 00068 . 0 0002 . 0 0037 . 0 ) (

- = Þ + + + = KHP KHP M u M u NOTE: Since the element contributions to M KHP are simply the sum of the single atom contributions, it might be expected from the general rule for combing uncertainty contributions that the uncertainty for each element contribution would be calculated from the sum of squares of the single atom contributions, that is, for carbon,

001 . 0 00037 . 0 8 ) ( 2 = ´ = C M u . Recall,

however, that this rule applies only to independent contributions, that is, contributions from separate determinations of the value. In this case, the total is obtained by multiplying the a single value by 8. Notice that the contributions from different elements are independent, and will therefore combine in the usual way.

Volume V T 1. Repeatability of the volume delivery: As before, the repeatability has already been taken into account via the combined repeatability term for the experiment.

2. Calibration: The limits of accuracy of the delivered volume are indicated by the manufacturer as a ± figure. For a 20 ml piston burette this number is typically ± 0.03 ml. Assuming a triangular distribution gives a standard uncertainty of ml 012 . 0 6 03 . 0 = .

Note: The ISO Guide ( F. 2.3.3) recommends adoption of a triangular distribution if there are reasons to expect values in the centre of the range being more likely than those near the bounds. For the glassware in examples A1 and A2, a triangular distribution has been assumed ( see the discussion under Volume uncertainties in example A1).

3. Temperature: The uncertainty due to the lack of temperature control is calculated in the same way as in the previous example, but this time taking a possible temperature variation of ± 3 ° C ( with a 95% confidence). Again using the coefficient of volume expansion for water as 2.1 ´ 10   4 ° C   1 gives a value of ml 006 . 0 96 . 1 3 10 1 . 2 19 4 = ´ ´ ´ - Thus the standard uncertainty due to incomplete temperature control is 0.006 ml.

NOTE: When dealing with uncertainties arising from incomplete control of environmental factors such as temperature, it is essential to take account of any correlation in the effects on different intermediate values. In this example, the dominant effect on the solution temperature is taken as the differential heating effects of different solutes, that is, the solutions are not equilibrated to ambient temperature. Temperature effects on each solution concentration at STP are therefore uncorrelated in this example, and are consequently treated as independent uncertainty contributions.

4. Bias of the end point detection: The titration is performed under a layer of Argon to exclude any bias due to the absorption of CO2 in the titration solution. This approach follows the principle that it is better to prevent any bias than to correct for it. There are no other indications that the end point determined from the shape of the pH curve does not correspond to the equivalence point, because a strong acid is titrated with a strong base. Therefore it is assumed that the bias of the end point detection and its uncertainty are negligible.

V T is found to be 18.64 ml and combining the remaining contributions to the uncertainty u( V T) of the volume V T gives a value of Quantifying Uncertainty Example A2 QUAM: 2000. P1 Page 47 ml 013 . 0 ) ( 006 . 0 012 . 0 ) ( 2 2 = Þ + = T T V u V u A2.5 Step 4: Calculating the combined standard uncertainty c NaOH is given by ] l mol [ 1000 1 - × × × = T KHP KHP KHP NaOH V M P m c The values of the parameters in this equation, their standard uncertainties and their relative standard uncertainties are collected in Table A2.2 Using the values given above:

1 l mol 10214 . 0 64 . 18 2212 . 204 0 . 1 3888 . 0 1000 - = ´ ´ ´ = NaOH c For a multiplicative expression ( as above) the standard uncertainties are used as follows:

2 2 2 2 2 ) ( ) ( ) ( ) ( ) (

) (

÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ = T T KHP KHP KHP KHP KHP KHP NaOH NaOH c V V u M M u P P u m m u rep rep u c c u 00097 . 0 00070 . 0 000019 . 0 00029 . 0 00033 . 0 0005 . 0 ) (

2 2 2 2 2 = + + + + = Þ NaOH NaOH c c c u ( ) 1 l mol 00010 . 0 00097 . 0 - = ´ = Þ NaOH NaOH c c uc Spreadsheet software is used to simplify the above calculation of the combined standard uncertainty ( see Appendix E. 2). The spreadsheet filled in with the appropriate values is shown as Table A2.3, which appears with additional explanation.

It is instructive to examine the relative contributions of the different parameters. The contributions can easily be visualised using a histogram. Figure A2.9 shows the calculated values | u( y, xi)| from Table A2.3.

The contribution of the uncertainty of the titration volume V T is by far the largest followed by the repeatability. The weighing procedure and the purity of the titrimetric standard show the same order of magnitude, whereas the uncertainty in the molar mass is again nearly an order of magnitude smaller.

A2.6 Step 5: Re evaluate the significant components The contribution of V( T) is the largest one. The volume of NaOH for titration of KHP ( V( T)) itself is affected by four influence quantities: the repeatability of the volume delivery, the calibration of the piston burette, the difference between the operation and calibration temperature Table A2.2: Values and uncertainties for titration Description Value x Standard uncertainty u( x) Relative standard uncertainty u( x)/ x rep Repeatability 1.0 0.0005 0.0005 m KHP Weight of KHP 0.3888 g 0.00013 g 0.00033 P KHP Purity of KHP 1.0 0.00029 0.00029 M KHP Molar mass of KHP 204.2212 g mol 1 0.0038 g mol 1 0.000019 V T Volume of NaOH for KHP titration 18.64 ml 0.013 ml 0.0007 Figure A2.9: Uncertainty contributions in NaOH standardisation u( y, xi) ( mmol l 1) 0 0.02 0.04 0.06 0.08 0.1 0.12 c( NaOH) Repeatability m( KHP) P( KHP) M( KHP) V( T) Quantifying Uncertainty Example A2 QUAM: 2000. P1 Page 48 repeatability of the volume delivery, the calibration of the piston burette, the difference between the operation and calibration temperature of the burette and the repeatability of the endpoint detection. Checking the size of each contribution, the calibration is by far the largest. Therefore this contribution needs to be investigated more thoroughly.

The standard uncertainty of the calibration of V( T) was calculated from the data given by the manufacturer assuming a triangular distribution. The influence of the choice of the shape of the distribution is shown in Table A2.4.

According to the ISO Guide 4.3.9 Note 1:

" For a normal distribution with expectation m and standard deviation s , the interval m ± 3 s encompasses approximately 99.73 percent of the distribution. Thus, if the upper and lower bounds a+ and a define 99.73 percent limits rather than 100 percent limits, Xi can be assumed to be approximately normally distributed rather than there being no specific knowledge about Xi [ between the bounds], then u2( xi) = a2/ 9. By comparison, the variance of a symmetric rectangular distribution of the halfwidth a is a2/ 3 ... and that of a symmetric triangular distribution of the half width a is a2/ 6 ... The magnitudes of the variances of the three distributions are surprisingly similar in view of the differences in the assumptions upon which they are based."

Thus the choice of the distribution function of this influence quantity has little effect on the value of the combined standard uncertainty ( uc( c NaOH)) and it is adequate to assume that it is triangular.

The expanded uncertainty U( c NaOH) is obtained by multiplying the combined standard uncertainty by a coverage factor of 2.

1 l mol 0002 . 0 2 00010 . 0 ) ( - = ´ = NaOH c U Thus the concentration of the NaOH solution is ( 0.1021 ± 0.0002) mol l   1. Table A2.3: Spreadsheet calculation of titration uncertainty A B C D E F G 1 Rep m( KHP) P( KHP) M( KHP) V( T)

2 Value 1.0 0.3888 1.0 204.2212 18.64 3 Uncertainty 0.0005 0.00013 0.00029 0.0038 0.013 4 5 rep 1.0 1.0005 1.0 1.0 1.0 1.0 6 m( KHP) 0.3888 0.3888 0.38893 0.3888 0.3888 0.3888 7 P( KHP) 1.0 1.0 1.0 1.00029 1.0 1.0 8 M( KHP) 204.2212 204.2212 204.2212 204.2212 204.2250 204.2212 9 V( T) 18.64 18.64 18.64 18.64 18.64 18.653 10 11 c( NaOH) 0.102136 0.102187 0.102170 0.102166 0.102134 0.102065 12 u( y, xi) 0.000051 0.000034 0.000030  0.000002  0.000071 13 u( y) 2, u( y, xi) 2 9.72E 9 2.62E 9 1.16E 9 9E 10 4E 12 5.041E 9 14 15 u( c( NaOH)) 0.000099 The values of the parameters are given in the second row from C2 to G2. Their standard uncertainties are entered in the row below ( C3 G3). The spreadsheet copies the values from C2 G2 into the second column from B5 to B9. The result ( c( NaOH)) using these values is given in B11. C5 shows the value of the repeatability from C2 plus its uncertainty given in C3. The result of the calculation using the values C5 C9 is given in C11. The columns D and G follow a similar procedure. The values shown in the row 12 ( C12 G12) are the differences of the row ( C11 G11) minus the value given in B11. In row 13 ( C13 G13) the values of row 12 ( C12 G12) are squared and summed to give the value shown in B13. B15 gives the combined standard uncertainty, which is the square root of B13. Quantifying Uncertainty Example A2 QUAM: 2000. P1 Page 49 Table A2.4: Effect of different distribution assumptions Distribution factor u( V( T; cal)) ( ml) u( V( T)) ( ml) uc( c NaOH)

Rectangular 3 0.017 0.019 0.00011 mol l 1 Triangular 6 0.012 0.015 0.00009 mol l 1 NormalNote 1 9 0.010 0.013 0.000085 mol l 1 Note 1: The factor of 9 arises from the factor of 3 in Note 1 of ISO Guide 4.3.9 ( see page 48 for details). Quantifying Uncertainty Example A3 QUAM: 2000. P1 Page 50 Example A3: An Acid/ Base Titration Summary Goal A solution of hydrochloric acid ( HCl) is standardised against a solution of sodium hydroxide ( NaOH) with known content.

Measurement procedure A solution of hydrochloric acid ( HCl) is titrated against a solution of sodium hydroxide ( NaOH), which has been standardised against the titrimetric standard potassium hydrogen phthalate ( KHP), to determine its concentration. The stages of the procedure are shown in Figure A3.1.

Measurand:

HCl KHP T T KHP KHP HCl V M V V P m c × × × × × = 1 2 1000 [ mol l 1]

where the symbols are as given in Table A3.1 and the value of 1000 is a conversion factor from ml to litres.

Identification of the uncertainty sources:

The relevant uncertainty sources are shown in Figure A3.2. Quantification of the uncertainty components The final uncertainty is estimated as 0.00016 mol l 1. Table A3.1 summarises the values and their uncertainties; Figure A3.3 shows the values diagrammatically.

Figure A3.1: Titration procedure Weighing KHP Weighing KHP Titrate KHP with NaOH Titrate KHP with NaOH Take aliquot of HCl Take aliquot of HCl Titrate HCl with NaOH Titrate HCl with NaOH RESULT RESULT Figure A3.2: Cause and Effect diagram for acid base titration V( T2)

Calibration Temperature m( KHP)

V( T2)

end point V( T2)

V( T1)

end point V( T1)

V( HCl) End point Temperature Calibration Bias Calibration Calibration sensitivity linearity sensitivity linearity same balance m( gross)

c( HCl) m( KHP) P( KHP)

M( KHP) V( HCl) V( T1) Bias End point Temperature Calibration m( tare)

Repeatability Quantifying Uncertainty Example A3 QUAM: 2000. P1 Page 51 Table A3.1: Acid base Titration values and uncertainties Description Value x Standard uncertainty u( x) Relative standard uncertainty u( x)/ x rep Repeatability 1 0.001 0.001 m KHP Weight of KHP 0.3888 g 0.00013 g 0.00033 P KHP Purity of KHP 1.0 0.00029 0.00029 V T2 Volume of NaOH for HCl titration 14.89 ml 0.015 ml 0.0010 V T1 Volume of NaOH for KHP titration 18.64 ml 0.016 ml 0.00086 M KHP Molar mass of KHP 204.2212 g mol 1 0.0038 g mol 1 0.000019 V HCl HCl aliquot for NaOH titration 15 ml 0.011 ml 0.00073 c HCl HCl solution concentration 0.10139 mol l 1 0.00016 mol l 1 0.0016 Figure A3.3: Uncertainty contributions in acid base titration u( y, xi) ( mmol l 1) 0 0.05 0.1 0.15 0.2 c( HCl) Repeatability m( KHP) P( KHP) V( T2) V( T1) M( KHP) V( HCl)

The values of u( y, xi)=( ¶ y/ ¶ xi). u( xi) are taken from Table A3.3. Quantifying Uncertainty Example A3 QUAM: 2000. P1 Page 52 Example A3: An acid/ base titration. Detailed discussion A3.1 Introduction This example discusses a sequence of experiments to determine the concentration of a solution of hydrochloric acid ( HCl). In addition, a number of special aspects of the titration technique are highlighted. The HCl is titrated against solution of sodium hydroxide ( NaOH), which was freshly standardised with potassium hydrogen phthalate ( KHP). As in the previous example ( A2) it is assumed that the HCl concentration is known to be of the order of 0.1 mol l   1 and that the end point of the titration is determined by an automatic titration system using the shape of the pH curve. This evaluation gives the measurement uncertainty in terms of the SI units of measurement.

A3.2 Step 1: Specification A detailed description of the measurement procedure is given in the first step. It compromises a listing of the measurement steps and a mathematical statement of the measurand.

Procedure The determination of the concentration of the HCl solution consists of the following stages ( See also Figure A3.4):

i) The titrimetric standard potassium hydrogen phthalate ( KHP) is dried to ensure the purity quoted in the supplier's certificate. Approximately 0.388 g of the dried standard is then weighed to achieve a titration volume of 19 ml NaOH.

ii) The KHP titrimetric standard is dissolved with » 50 ml of ion free water and then titrated using the NaOH solution. A titration system controls automatically the addition of NaOH and samples the pH curve. The endpoint is evaluated from the shape of the recorded curve.

iii) 15 ml of the HCl solution is transferred by means of a volumetric pipette. The HCl solution is diluted with de ionised water to give » 50 ml solution in the titration vessel.

iv) The same automatic titrator performs the measurement of HCl solution. Weighing KHP Weighing KHP Titrate KHP with NaOH Titrate KHP with NaOH Take aliquot of HCl Take aliquot of HCl Titrate HCl with NaOH Titrate HCl with NaOH RESULT RESULT Figure A3.4: Determination of the concentration of a HCl solution Calculation:

The measurand is the concentration of the HCl solution, c HCl. It depends on the mass of KHP, its purity, its molecular weight, the volumes of NaOH at the end point of the two titrations and the aliquot of HCl.:

] l mol [ 1000 1 1 2 - × × × × × = HCl KHP T T KHP KHP HCl V M V V P m c where c HCl : concentration of the HCl solution [ mol l 1]

1000 : conversion factor [ ml] to [ l]

m KHP : mass of KHP taken [ g]

P KHP : purity of KHP given as mass fraction V T2 : volume of NaOH solution to titrate HCl [ ml]

V T1 : volume of NaOH solution to titrate KHP [ ml]

M KHP: molar mass of KHP [ g mol   1]

V HCl : volume of HCl titrated with NaOH solution [ ml] Quantifying Uncertainty Example A3 QUAM: 2000. P1 Page 53 A3.3 Step 2: Identifying and analysing uncertainty sources The different uncertainty sources and their influence on the measurand are best analysed by visualising them first in a cause and effect diagram ( Figure A3.5).

Because a repeatability estimate is available from validation studies for the procedure as a whole, there is no need to consider all the repeatability contributions individually. They are therefore grouped into one contribution ( shown in the revised cause and effect diagram in Figure A3.5).

The influences on the parameters V T2, V T1, m KHP, P KHP and M KHP have been discussed extensively in the previous example, therefore only the new influence quantities of V HCl will be dealt with in more detail in this section.

Volume V HCl 15 ml of the investigated HCl solution is to be transferred by means of a volumetric pipette. The delivered volume of the HCl from the pipette is subject to the same three sources of uncertainty as all the volumetric measuring devices.

1. The variability or repeatability of the delivered volume 2. The uncertainty in the stated volume of the pipette 3. The solution temperature differing from the calibration temperature of the pipette.

A3.4 Step 3: Quantifying uncertainty components The goal of this step is to quantify each uncertainty source analysed in step 2. The quantification of the branches or rather of the different components was described in detail in the previous two examples. Therefore only a summary for each of the different contributions will be given.

repeatability The method validation shows a repeatability for the determination of 0.1% ( as % rsd). This value can be used directly for the calculation of the combined standard uncertainty associated with the different repeatability terms.

Mass m KHP Calibration/ linearity: The balance manufacturer quotes ± 0.15 mg for the linearity contribution. This value represents the maximum difference between the actual mass on the pan and the reading of the scale. The linearity contribution is assumed to show a rectangular distribution and is converted to a standard uncertainty:

mg 087 . 0 3 15 . 0 = The contribution for the linearity has to be accounted for twice, once for the tare and once for the gross mass, leading to an uncertainty u( m KHP) of mg 12 . 0 ) ( ) 087 . 0 ( 2 ) ( 2 = Þ ´ = KHP KHP m u m u NOTE 1: The contribution is applied twice because no assumptions are made about the form of the non linearity. The non linearity is accordingly treated as a systematic effect on each weighing, which varies randomly in magnitude across the measurement range. Figure A3.5: Final cause and effect diagram V( T2)

Calibration Temperature m( KHP)

V( T2)

end point V( T2)

V( T1)

end point V( T1)

V( HCl) End point Temperature Calibration Bias Calibration Calibration sensitivity linearity sensitivity linearity same balance m( gross)

c( HCl) m( KHP) P( KHP)

M( KHP) V( HCl) V( T1) Bias End point Temperature Calibration m( tare)

Repeatability Quantifying Uncertainty Example A3 QUAM: 2000. P1 Page 54 NOTE 2: Buoyancy correction is not considered because all weighing results are quoted on the conventional basis for weighing in air [ H. 19]. The remaining uncertainties are too small to consider. Note 1 in Appendix G refers.

P( KHP)

P( KHP) is given in the supplier's certificate as 100% ± 0.05%. The quoted uncertainty is taken as a rectangular distribution, so the standard uncertainty u( P KHP) is 00029 . 0 3 0005 . 0 ) ( = = KHP P u .

V( T2)

i) Calibration: Figure given by the manufacturer ( ± 0.03 ml) and approximated to a triangular distribution ml 012 . 0 6 03 . 0 = .

ii) Temperature: The possible temperature variation is within the limits of ± 4 ° C and approximated to a rectangular distribution ml 007 . 0 3 4 10 1 . 2 15 4 = ´ ´ ´ - .

iii) Bias of the end point detection: A bias between the determined end point and the equivalence point due to atmospheric CO2 can be prevented by performing the titration under Argon. No uncertainty allowance is made.

V T2 is found to be 14.89 ml and combining the two contributions to the uncertainty u( V T2) of the volume V T2 gives a value of ml 014 0 ) ( 007 0 012 0 ) (

2 2 2 2 . V u . . V u T T = Þ + = Volume V T1 All contributions except the one for the temperature are the same as for V T2 i) Calibration: ml 012 . 0 6 03 . 0 = ii) Temperature: The approximate volume for the titration of 0.3888 g KHP is 19 ml NaOH, therefore its uncertainty contribution is ml 009 . 0 3 4 10 1 . 2 19 4 = ´ ´ ´ - .

iii) Bias: Negligible V T1 is found to be 18.64 ml with a standard uncertainty u( V T1) of ml 015 . 0 ) ( 009 . 0 012 . 0 ) (

1 2 2 1 = Þ + = T T V u V u Molar mass M KHP Atomic weights and listed uncertainties ( from current IUPAC tables) for the constituent elements of KHP ( C8H5O4K) are:

Element Atomic weight Quoted uncertainty Standard uncertainty C 12.0107 ± 0.0008 0.00046 H 1.00794 ± 0.00007 0.000040 O 15.9994 ± 0.0003 0.00017 K 39.0983 ± 0.0001 0.000058 For each element, the standard uncertainty is found by treating the IUPAC quoted uncertainty as forming the bounds of a rectangular distribution. The corresponding standard uncertainty is therefore obtained by dividing those values by 3 .

The molar mass M KHP for KHP and its uncertainty u( M KHP) are, respectively:

1 mol g 2212 . 204 0983 . 39 9994 . 15 4 00794 . 1 5 0107 . 12 8 - = + ´ + ´ + ´ = KHP M 1 2 2 2 2 mol g 0038 . 0 ) ( 000058 . 0 ) 00017 . 0 4 ( ) 00004 . 0 5 ( ) 00046 . 0 8 ( ) (

- = Þ + ´ + ´ + ´ = KHP KHP F u M u NOTE: The single atom contributions are not independent. The uncertainty for the atom contribution is therefore calculated by multiplying the standard uncertainty of the atomic weight by the number of atoms.

Volume V HCl i) Calibration: Uncertainty stated by the manufacturer for a 15 ml pipette as ± 0.02 ml and approximated with a triangular distribution: 6 02 . 0 = 0.008 ml.

ii) Temperature: The temperature of the laboratory is within the limits of ± 4 ° C. Using a rectangular temperature distribution gives a standard uncertainty of 3 4 10 1 . 2 15 4 ´ ´ ´ - = 0.007 ml.

Combining these contributions gives ml 011 . 0 ) ( 007 . 0 008 . 0 0037 0 ) ( 2 2 2 = Þ + + = HCl HCl V u . V u Quantifying Uncertainty Example A3 QUAM: 2000. P1 Page 55 A3.5 Step 4: Calculating the combined standard uncertainty c HCl is given by HCl KHP T T KHP KHP HCl V M V V P m c × × × × × = 1 2 1000 NOTE: The repeatability estimate is, in this example, treated as a relative effect; the complete model equation is therefore rep V M V V P m c HCl KHP T T KHP KHP HCl ´ × × × × × = 1 2 1000 All the intermediate values of the two step experiment and their standard uncertainties are collected in Table A3.2. Using these values:

1  l mol 10139 . 0 1 15 2212 . 204 64 . 18 89 . 14 0 . 1 3888 . 0 1000 = ´ ´ ´ ´ ´ ´ = HCl c The uncertainties associated with each component are combined accordingly:

0018 . 0 001 . 0 00073 . 0 000019 . 0 00080 . 0 00094 . 0 00029 . 0 00031 . 0 ) ( ) ( ) ( ) ( ) ( ) ( ) (

) (

2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 = + + + + + + = + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ = rep u V V u M M u V V u V V u P P u m m u c c u HCl HCl KHP KHP T T T T KHP KHP KHP KHP HCl HCl c 1 l mol 00018 . 0 0018 . 0 ) ( - = ´ = Þ HCl HCl c c c u A spreadsheet method ( see Appendix E) can be used to simplify the above calculation of the combined standard uncertainty. The spreadsheet filled in with the appropriate values is shown in Table A3.3, with an explanation.

The sizes of the different contributions can be compared using a histogram. Figure A3.6 shows the values of the contributions | u( y, xi)| from Table A3.3.

Figure A3.6: Uncertainties in acid base titration u( y, xi) ( mmol l 1) 0 0.05 0.1 0.15 0.2 c( HCl) Repeatability m( KHP) P( KHP) V( T2) V( T1) M( KHP) V( HCl)

The expanded uncertainty U( c HCl) is calculated by multiplying the combined standard uncertainty by a coverage factor of 2:

 1 l mol 0004 . 0 2 00018 . 0 ) ( = ´ = HCl c U The concentration of the HCl solution is ( 0.1014 ± 0.0004) mol l   1 Table A3.2: Acid base Titration values and uncertainties ( 2 step procedure)

Description Value x Standard Uncertainty u( x) Relative standard uncertainty u( x)/ x rep Repeatability 1 0.001 0.001 m KHP Mass of KHP 0.3888 g 0.00012 g 0.00031 P KHP Purity of KHP 1.0 0.00029 0.00029 V T2 Volume of NaOH for HCl titration 14.89 ml 0.014 ml 0.00094 V T1 Volume of NaOH for KHP titration 18.64 ml 0.015 ml 0.00080 M KHP Molar mass of KHP 204.2212 g mol 1 0.0038 g mol 1 0.000019 V HCl HCl aliquot for NaOH titration 15 ml 0.011 ml 0.00073 Quantifying Uncertainty Example A3 QUAM: 2000. P1 Page 56 A3.6 Special aspects of the titration example Three special aspects of the titration experiment will be dealt with in this second part of the example. It is interesting to see what effect changes in the experimental set up or in the implementation of the titration would have on the final result and its combined standard uncertainty.

Influence of a mean room temperature of 25 ° C For routine analysis, analytical chemists rarely correct for the systematic effect of the temperature in the laboratory on the volume. This question considers the uncertainty introduced by the corrections required.

The volumetric measuring devices are calibrated at a temperature of 20 ° C. But rarely does any analytical laboratory have a temperature controller to keep the room temperature that level. For illustration, consider correction for a mean room temperature of 25 ° C. The final analytical result is calculated using the corrected volumes and not the calibrated volumes at 20 ° C. A volume is corrected for the temperature effect according to )] 20 ( 1 [ ' - a - = T V V where V' : actual volume at the mean temperature T V : volume calibrated at 20 ° C a : expansion coefficient of an aqueous solution [ ° C   1]

T : observed temperature in the laboratory [ ° C]

The equation of the measurand has to be rewritten:

HCl T T KHP KHP KHP HCl V V V M P m c ' ' ' 1000 1 2 × × × × = Table A3.3: Acid base Titration   spreadsheet calculation of uncertainty A B C D E F G H I 1 rep m( KHP) P( KHP) V( T2) V( T1) M( KHP) V( HCl)

2 value 1.0 0.3888 1.0 14.89 18.64 204.2212 15 3 uncertainty 0.001 0.00012 0.00029 0.014 0.015 0.0038 0.011 4 5 rep 1.0 1.001 1.0 1.0 1.0 1.0 1.0 1.0 6 m( KHP) 0.3888 0.3888 0.38892 0.3888 0.3888 0.3888 0.3888 0.3888 7 P( KHP) 1.0 1.0 1.0 1.00029 1.0 1.0 1.0 1.0 8 V( T2) 14.89 14.89 14.89 14.89 14.904 14.89 14.89 14.89 9 V( T1) 18.64 18.64 18.64 18.64 18.64 18.655 18.64 18.64 10 M( KHP) 204.2212 204.2212 204.2212 204.2212 204.2212 204.2212 204.2250 204.2212 11 V( HCl) 15 15 15 15 15 15 15 15.011 12 13 c( HCl) 0.101387 0.101489 0.101418 0.101417 0.101482 0.101306 0.101385 0.101313 14 u( y, xi) 0.000101 0.000031 0.000029 0.000095  0.000082  0.0000019  0.000074 15 u( y) 2, u( y, xi) 2 3.34E 8 1.03E 8 9.79E 10 8.64E 10 9.09E 9 6.65E 9 3.56E 12 5.52E 9 16 17 u( c( HCl)) 0.00018 The values of the parameters are given in the second row from C2 to I2. Their standard uncertainties are entered in the row below ( C3 I3). The spreadsheet copies the values from C2 I2 into the second column from B5 to B11. The result ( c( HCl)) using these values is given in B13. The C5 shows the value of the repeatability from C2 plus its uncertainty given in C3. The result of the calculation using the values C5 C11 is given in C13. The columns D to I follow a similar procedure. The values shown in the row 14 ( C14 I14) are the differences of the row ( C13 H13) minus the value given in B13. In row 15 ( C15 I15) the values of row 14 ( C14 I14) are squared and summed to give the value shown in B15. B17 gives the combined standard uncertainty, which is the square root of B15. Quantifying Uncertainty Example A3 QUAM: 2000. P1 Page 57 Including the temperature correction terms gives:

÷ ÷ ø ö ç ç è æ - a - × - a - - a - ´ ÷ ÷ ø ö ç ç è æ × × = × × × × = ) ] 20 ( 1 [ )] 20 ( 1 [ )] 20 ( 1 [ 1000 ' ' ' 1000 1 2 1 2 T V T V T V M P m V V V M P m c HCl T T KHP KHP KHP HCl T T KHP KHP KHP HCl This expression can be simplified by assuming that the mean temperature T and the expansion coefficient of an aqueous solution a are the same for all three volumes ÷ ÷ ø ö ç ç è æ - a - × × ´ ÷ ÷ ø ö ç ç è æ × × = ) ] 20 ( 1 [ 1000 1 2 T V V V M P m c HCl T T KHP KHP KHP HCl This gives a slightly different result for the HCl concentration at 20 ° C:

1 4 l mol 10149 . 0 )] 20 25 ( 10 1 . 2 1 [ 15 64 . 18 2236 . 204 89 . 14 0 . 1 3888 . 0 1000 - - = - ´ - ´ ´ ´ ´ ´ ´ = HCl c The figure is still within the range given by the combined standard uncertainty of the result at a mean temperature of 20 ° C, so the result is not significantly affected. Nor does the change affect the evaluation of the combined standard uncertainty, because a temperature variation of ± 4 ° C at the mean room temperature of 25 ° C is still assumed.

Visual end point detection A bias is introduced if the indicator phenolphthalein is used for visual end point detection, instead of an automatic titration system extracting the equivalence point from the pH curve. The change of colour from transparent to red/ purple occurs between pH 8.2 and 9.8 leading to an excess volume, introducing a bias compared to the end point detection employing a pH meter. Investigations have shown that the excess volume is around 0.05 ml with a standard uncertainty for the visual detection of the end point of approximately 0.03 ml. The bias arising from the excess volume has to be considered in the calculation of the final result. The actual volume for the visual end point detection is given by Excess T Ind T V V V + = 1 ; 1 where Ind T V ; 1 : volume from a visual end point detection 1 T V : volume at the equivalence point V Excess: excess volume needed to change the colour of phenolphthalein The volume correction quoted above leads to the following changes in the equation of the measurand HCl Excess Ind T KHP Excess Ind T KHP KHP HCl V V V M V V P m c × - × - × × × = ) ( ) ( 1000 ; 1 ; 2 The standard uncertainties u( V T2) and u( V T1) have to be recalculated using the standard uncertainty of the visual end point detection as the uncertainty component of the repeatability of the end point detection.

ml 034 . 0 03 . 0 009 . 0 012 . 0 004 . 0 ) ( ) (

2 2 2 2 ; 1 1 = + + + = - = Excess Ind T T V V u V u ml 033 . 0 03 . 0 007 . 0 012 . 0 004 . 0 ) ( ) (

2 2 2 2 ; 2 2 = + + + = - = Excess Ind T T V V u V u The combined standard uncertainty 1 l mol 0003 . 0 ) ( - = HCl c c u is considerable larger than before.

Triple determination to obtain the final result The two step experiment is performed three times to obtain the final result. The triple determination is expected to reduce the contribution from repeatability, and hence reduce the overall uncertainty.

As shown in the first part of this example, all the run to run variations are combined to one single component, which represents the overall experimental repeatability as shown in the in the cause and effect diagram ( Figure A3.5).

The uncertainty components are quantified in the following way:

Mass m KHP Linearity: mg 087 . 0 3 15 . 0 = mg 12 . 0 87 . 0 2 ) ( 2 = ´ = Þ KHP m u Purity P KHP Purity: 00029 . 0 3 0005 . 0 = Quantifying Uncertainty Example A3 QUAM: 2000. P1 Page 58 Volume V T2 calibration: ml 012 . 0 6 03 . 0 = temperature:

ml 007 . 0 3 4 10 1 . 2 15 4 = ´ ´ ´ - ( ) ml 014 . 0 007 . 0 012 . 0 2 2 2 = + = Þ T V u Repeatability The quality log of the triple determination shows a mean long term standard deviation of the experiment of 0.001 ( as RSD). It is not recommended to use the actual standard deviation obtained from the three determinations because this value has itself an uncertainty of 52%. The standard deviation of 0.001 is divided by the square root of 3 to obtain the standard uncertainty of the triple determination ( three independent measurements)

00058 . 0 3 001 . 0 = = Rep ( as RSD)

Volume V HCl calibration: ml 008 . 0 6 02 . 0 = temperature: ml 007 . 0 3 4 10 1 . 2 15 4 = ´ ´ ´ - ( ) ml 01 . 0 007 . 0 008 . 0 2 2 = + = Þ HCl V u Molar mass M KHP ( ) 1 mol g 0038 . 0 - = KHP M u Volume V T1 calibration: ml 02 . 0 6 03 . 0 = temperature: ml 009 . 0 3 4 10 1 . 2 19 4 = ´ ´ ´ - ( ) ml 015 . 0 009 . 0 012 . 0 2 2 1 = + = Þ T V u All the values of the uncertainty components are summarised in Table A3.4. The combined standard uncertainty is 0.00016 mol l   1, which is a very modest reduction due to the triple determination. The comparison of the uncertainty contributions in the histogram, shown in Figure A3.7, highlights some of the reasons for that result. Though the repeatability contribution is much reduced, the volumetric uncertainty contributions remain, limiting the improvement.

Figure A3.7: Replicated Acid base Titration values and uncertainties u( y, xi) ( mmol l 1) 0 0.05 0.1 0.15 0.2 c( HCl) Repeatability m( KHP) P( KHP) V( T2) V( T1) M( KHP) V( HCl) Replicated Single run Table A3.4: Replicated Acid base Titration values and uncertainties Description Value x Standard Uncertainty u( x) Relative Standard Uncertainty u( x)/ x Rep Repeatability of the determination 1.0 0.00058 0.00058 m KHP Mass of KHP 0.3888 g 0.00013 g 0.00033 P KHP Purity of KHP 1.0 0.00029 0.00029 V T2 Volume of NaOH for HCl titration 14.90 ml 0.014 ml 0.00094 V T1 Volume of NaOH for KHP titration 18.65 ml 0.015 ml 0.0008 M KHP Molar mass of KHP 204.2212 g mol 1 0.0038 g mol 1 0.000019 V HCl HCl aliquot for NaOH titration 15 ml 0.01 ml 0.00067 Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 59 Example A4: Uncertainty Estimation from In House Validation Studies. Determination of Organophosphorus Pesticides in Bread.

Summary Goal The amount of an organophosphorus pesticide residue in bread is determined employing an extraction and a GC procedure.

Measurement procedure The stages needed to determine the amount of organophosphorus pesticide residue are shown in Figure A4.1 Measurand:

6 10 × × × × × × = hom sample ref op ref op op F m Rec I V c I P mg kg 1 where Pop : Level of pesticide in the sample [ mg kg 1]

Iop : Peak intensity of the sample extract cref : Mass concentration of the reference standard [ m g ml 1]

Vop : Final volume of the extract [ ml]

106 : Conversion factor from [ g g 1] to [ mg kg 1]

Iref : Peak intensity of the reference standard Rec : Recovery msample : Mass of the investigated sub sample [ g]

Fhom : Correction factor for sample inhomogeneity Identification of the uncertainty sources:

The relevant uncertainty sources are shown in the cause and effect diagram in Figure A4.2.

Quantification of the uncertainty components:

Based on in house validation data, the three major contributions are listed in Table A4.1 and shown diagrammatically in Figure A4.3 ( values are from Table A4.5). Figure A4.1: Organophosphorus pesticides analysis RESULT RESULT Homogenise Homogenise Extraction Extraction Clean up Clean up ` Bulk up' ` Bulk up'

GC Determination GC Determination Prepare calibration standard Prepare calibration standard GC Calibration GC Calibration Table A4.1: Uncertainties in pesticide analysis Description Value x Standard uncertainty u( x) Relative standard uncertainty u( x)/ x Comments Repeatability( 1) 1.0 0.27 0.27 Based on duplicate tests of different types of samples Bias ( Rec) ( 2) 0.9 0.043 0.048 Spiked samples Other sources ( 3)

( Homogeneity) 1.0 0.2 0.2 Estimation based on model assumptions u( Pop)/ Pop     0.34 Relative standard uncertainty Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 60 Figure A4.2: Uncertainty sources in pesticide analysis P( op) I( op) c( ref) V( op)

m( sample) I( ref) V( op) Calibration Temperature dilution dilution Calibration V( ref) V( ref) Calibration Temperature m( ref) Calibration m( ref) Purity ( ref) I( op)

Calibration Recovery m( gross) I( ref)

Calibration Calibration Linearity m( tare) m( sample)

Calibration Linearity F( hom) Repeatability Linearity Figure A4.3: Uncertainties in pesticide analysis u( y, xi) ( mg kg 1) 0 0.1 0.2 0.3 0.4 P( op) Repeatability Bias Homogeneity The values of u( y, xi)=( ¶ y/ ¶ xi). u( xi) are taken from Table A4.5 Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 61 Example A4: Determination of organophosphorus pesticides in bread. Detailed discussion.

A4.1 Introduction This example illustrates the way in which inhouse validation data can be used to quantify the measurement uncertainty. The aim of the measurement is to determine the amount of an organophosphorus pesticides residue in bread. The validation scheme and experiments establish traceability by measurements on spiked samples. It is assumed the uncertainty due to any difference in response of the measurement to the spike and the analyte in the sample is small compared with the total uncertainty on the result.

A4.2 Step 1: Specification The specification of the measurand for more extensive analytical methods is best done by a comprehensive description of the different stages of the analytical method and by providing the equation of the measurand.

Procedure The measurement procedure is illustrated schematically in Figure A4.4. The separate stages are:

i) Homogenisation: The complete sample is divided into small ( approx. 2 cm) fragments, a random selection is made of about 15 of these, and the sub sample homogenised. Where extreme inhomogeneity is suspected proportional sampling is used before blending.

ii) Weighing of sub sampling for analysis gives mass msample iii) Extraction: Quantitative extraction of the analyte with organic solvent, decanting and drying through a sodium sulphate columns, and concentration of the extract using a Kuderna Danish apparatus.

iv) Liquid liquid extraction:

v) Acetonitrile/ hexane liquid partition, washing the acetonitrile extract with hexane, drying the hexane layer through sodium sulphate column. vi) Concentration of the washed extract by gas blown down of extract to near dryness.

vii) Dilution to standard volume Vop ( approx. 2 ml) in a 10 ml graduated tube.

viii) Measurement: Injection and GC measurement of 5 m l of sample extract to give the peak intensity Iop.

ix) Preparation of an approximately 5 m g ml 1 standard ( actual mass concentration cref).

x) GC calibration using the prepared standard and injection and GC measurement of 5 m l of the standard to give a reference peak intensity Iref.

Figure A4.4: Organophosphorus pesticides analysis RESULT RESULT Homogenise Homogenise Extraction Extraction Clean up Clean up ` Bulk up' ` Bulk up'

GC Determination GC Determination Prepare calibration standard Prepare calibration standard GC Calibration GC Calibration Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 62 Calculation The mass concentration cop in the final sample is given by 1 ml g - m × = ref op ref op I I c c and the estimate Pop of the level of pesticide in the bulk sample ( in mg kg 1) is given by 1 6 kg mg 10 - × × × = sample op op op m Rec V c P or, substituting for cop,

1  6 kg mg 10 × × × × × = sample ref op ref op op m Rec I V c I P where Pop : Level of pesticide in the sample [ mg kg 1]

Iop : Peak intensity of the sample extract cref : Mass concentration of the reference standard [ m g ml 1]

Vop : Final volume of the extract [ ml]

106 : Conversion factor from [ g g 1] to [ mg kg 1]

Iref : Peak intensity of the reference standard Rec : Recovery msample: Mass of the investigated sub sample [ g] Scope The analytical method is applicable to a small range of chemically similar pesticides at levels between 0.01 and 2 mg kg 1 with different kinds of bread as matrix.

A4.3 Step 2: Identifying and analysing uncertainty sources The identification of all relevant uncertainty sources for such a complex analytical procedure is best done by drafting a cause and effect diagram. The parameters in the equation of the measurand are represented by the main branches of the diagram. Further factors are added to the diagram, considering each step in the analytical procedure ( A4.2), until the contributory factors become sufficiently remote.

The sample inhomogeneity is not a parameter in the original equation of the measurand, but it appears to be a significant effect in the analytical procedure. A new branch, F( hom), representing the sample inhomogeneity is accordingly added to the cause and effect diagram ( Figure A4.5).

Finally, the uncertainty branch due to the inhomogeneity of the sample has to be included in the calculation of the measurand. To show the effect of uncertainties arising from that source clearly, it is useful to write Figure A4.5: Cause and effect diagram with added main branch for sample inhomogeneity P( op) I( op) c( ref) V( op)

m( sample) I( ref) Precision Calibration Temperature dilution Precision Calibration V( ref) Precision Calibration Temperature Precision Calibration m( ref) Purity( ref)

Precision Calibration Recovery m( gross) Precision Calibration Precision Calibration Linearity Sensitivity m( tare)

Precision Calibration Linearity Sensitivity F( hom) Linearity Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 63 ] kg mg [ 10 1 6 - × × × × × × = sample ref op ref op hom op m Rec I V c I F P where Fhom is a correction factor assumed to be unity in the original calculation. This makes it clear that the uncertainties in the correction factor must be included in the estimation of the overall uncertainty. The final expression also shows how the uncertainty will apply.

NOTE: Correction factors: This approach is quite general, and may be very valuable in highlighting hidden assumptions. In principle, every measurement has associated with it such correction factors, which are normally assumed unity. For example, the uncertainty in cop can be expressed as a standard uncertainty for cop, or as the standard uncertainty which represents the uncertainty in a correction factor. In the latter case, the value is identically the uncertainty for cop expressed as a relative standard deviation.

A4.4 Step 3: Quantifying uncertainty components In accordance with section 7.7., the quantification of the different uncertainty components utilises data from the in house development and validation studies:

· The best available estimate of the overall run to run variation of the analytical process.

· The best possible estimation of the overall bias ( Rec) and its uncertainty.

· Quantification of any uncertainties associated with effects incompletely accounted for the overall performance studies.

Some rearrangement the cause and effect diagram is useful to make the relationship and coverage of these input data clearer ( Figure A4.6).

NOTE: In normal use, samples are run in small batches, each batch including a calibration set, a recovery check sample to control bias and random duplicate to check precision. Corrective action is taken if these checks show significant departures from the performance found during validation. This basic QC fulfils the main requirements for use of the validation data in uncertainty estimation for routine testing.

Having inserted the extra effect ` Repeatability' into the cause and effect diagram, the implied model for calculating Pop becomes 1 6 kg mg 10 - × × × × × × × = Rep sample ref op ref op hom op F m Rec I V c I F P Eq. A4.1 That is, the repeatability is treated as a multiplicative factor FRep like the homogeneity. This form is chosen for convenience in Figure A4.6: Cause and effect diagram after rearrangement to accommodate the data of the validation study P( op) I( op) c( ref) V( op)

m( sample) I( ref) V( op) Calibration Temperature dilution dilution Calibration V( ref) V( ref) Calibration Temperature m( ref) Calibration m( ref) Purity ( ref) I( op)

Calibration Recovery m( gross) I( ref)

Calibration Calibration Linearity m( tare) m( sample)

Calibration Linearity F( hom) Repeatability Linearity Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 64 calculation, as will be seen below.

The evaluation of the different effects is now considered.

1. Precision study The overall run to run variation ( precision) of the analytical procedure was performed with a number of duplicate tests ( same homogenised sample, complete extraction/ determination procedure) for typical organophosphorus pesticides found in different bread samples. The results are collected in Table A4.2.

The normalised difference data ( the difference divided by the mean) provides a measure of the overall run to run variability. To obtain the estimated relative standard uncertainty for single determinations, the standard deviation of the normalised differences is taken and divided by 2 to correct from a standard deviation for pairwise differences to the standard uncertainty for the single values. This gives a value for the standard uncertainty due to run to run variation of the overall analytical process, including run to run recovery variation but excluding homogeneity effects, of 27 . 0 2 382 . 0 = NOTE: At first sight, it may seem that duplicate tests provide insufficient degrees of freedom. But it is not the goal to obtain very accurate numbers for the precision of the analytical process for one specific pesticide in one special kind of bread. It is more important in this study to test a wide variety of different materials and sample levels, giving a representative selection of typical organophosphorus pesticides. This is done in the most efficient way by duplicate tests on many materials, providing ( for the repeatability estimate) approximately one degree of freedom for each material studied in duplicate.

2. Bias study The bias of the analytical procedure was investigated during the in house validation study using spiked samples ( homogenised samples were split and one portion spiked). Table A4.3 collects the results of a long term study of spiked samples of various types.

The relevant line ( marked with grey colour) is the " bread" entry line, which shows a mean recovery for forty two samples of 90%, with a standard deviation ( s) of 28%. The standard uncertainty was calculated as the standard deviation of the mean 0432 . 0 42 28 . 0 ) ( = = Rec u . Table A4.2: Results of duplicate pesticide analysis Residue D1 [ mg kg 1] D2 [ mg kg 1] Mean [ mg kg 1] Difference D1 D2 Difference/

mean Malathion 1.30 1.30 1.30 0.00 0.000 Malathion 1.30 0.90 1.10 0.40 0.364 Malathion 0.57 0.53 0.55 0.04 0.073 Malathion 0.16 0.26 0.21  0.10  0.476 Malathion 0.65 0.58 0.62 0.07 0.114 Pirimiphos Methyl 0.04 0.04 0.04 0.00 0.000 Chlorpyrifos Methyl 0.08 0.09 0.085  0.01  0.118 Pirimiphos Methyl 0.02 0.02 0.02 0.00 0.000 Chlorpyrifos Methyl 0.01 0.02 0.015  0.01  0.667 Pirimiphos Methyl 0.02 0.01 0.015 0.01 0.667 Chlorpyrifos Methyl 0.03 0.02 0.025 0.01 0.400 Chlorpyrifos Methyl 0.04 0.06 0.05  0.02  0.400 Pirimiphos Methyl 0.07 0.08 0.75  0.10  0.133 Chlorpyrifos Methyl 0.01 0.01 0.10 0.00 0.000 Pirimiphos Methyl 0.06 0.03 0.045 0.03 0.667 Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 65 A significance test is used to determine whether the mean recovery is significantly different from 1.0. The test statistic t is calculated using the following equation ( ) 315 . 2 0432 . 0 9 . 0 1 ) ( 1 = - = - = Rec u Rec t This value is compared with the 2 tailed critical value tcrit, for n   1 degrees of freedom at 95% confidence ( where n is the number of results used to estimate Rec ). If t is greater or equal than the critical value tcrit than Rec is significantly different from 1.

021 . 2 31 . 2 41 ; @ ³ = crit t t In this example a correction factor ( 1/ Rec ) is being applied and therefore Rec is explicitly included in the calculation of the result.

3. Other sources of uncertainty The cause and effect diagram in Figure A4.7 shows which other sources of uncertainty are ( 1) adequately covered by the precision data, ( 2) covered by the recovery data or ( 3) have to be further examined and eventually considered in the calculation of the measurement uncertainty. All balances and the important volumetric measuring devices are under regular control. Precision and recovery studies take into account the influence of the calibration of the different volumetric measuring devices because during the investigation various volumetric flasks and pipettes have been used. The extensive variability studies, which lasted for more than half a year, also cover influences of the environmental temperature on the result. This leaves only the reference material purity, possible nonlinearity in GC response ( represented by the ` calibration' terms for Iref and Iop in the diagram), and the sample homogeneity as additional components requiring study.

The purity of the reference standard is given by the manufacturer as 99.53% ± 0.06%. The purity is potential an additional uncertainty source with a standard uncertainty of 00035 . 0 3 0006 . 0 = ( Rectangular distribution). But the contribution is so small ( compared, for example, to the precision estimate) that it is clearly safe to neglect this contribution.

Linearity of response to the relevant organophosphorus pesticides within the given concentration range is established during validation studies. In addition, with multi level studies of the kind indicated in Table A4.2and Table A4.3: Results of pesticide recovery studies Substrate Residue Type Conc. [ mg kg   1] N1) Mean 2) [%] s 2)[%]

Waste Oil PCB 10.0 8 84 9 Butter OC 0.65 33 109 12 Compound Animal Feed I OC 0.325 100 90 9 Animal & Vegetable Fats I OC 0.33 34 102 24 Brassicas 1987 OC 0.32 32 104 18 Bread OP 0.13 42 90 28 Rusks OP 0.13 30 84 27 Meat & Bone Feeds OC 0.325 8 95 12 Maize Gluten Feeds OC 0.325 9 92 9 Rape Feed I OC 0.325 11 89 13 Wheat Feed I OC 0.325 25 88 9 Soya Feed I OC 0.325 13 85 19 Barley Feed I OC 0.325 9 84 22 ( 1) The number of experiments carried out ( 2) The mean and sample standard deviation s are given as percentage recoveries. Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 66 Table A4.3, nonlinearity would contribute to the observed precision. No additional allowance is required. The in house validation study has proven that this is not the case.

The homogeneity of the bread sub sample is the last remaining other uncertainty source. No literature data were available on the distribution of trace organic components in bread products, despite an extensive literature search ( at first sight this is surprising, but most food analysts attempt homogenisation rather than evaluate inhomogeneity separately). Nor was it practical to measure homogeneity directly. The contribution has therefore been estimated on the basis of the sampling method used.

To aid the estimation, a number of feasible pesticide residue distribution scenarios were considered, and a simple binomial statistical distribution used to calculate the standard uncertainty for the total included in the analysed sample ( see section A4.6). The scenarios, and the calculated relative standard uncertainties in the amount of pesticide in the final sample, were:

§ Residue distributed on the top surface only: 0.58. § Residue distributed evenly over the surface only: 0.20.

§ Residue distributed evenly through the sample, but reduced in concentration by evaporative loss or decomposition close to the surface: 0.05 0.10 ( depending on the " surface layer" thickness).

Scenario ( a) is specifically catered for by proportional sampling or complete homogenisation: It would arise in the case of decorative additions ( whole grains) added to one surface. Scenario ( b) is therefore considered the likely worst case. Scenario ( c) is considered the most probable, but cannot be readily distinguished from ( b). On this basis, the value of 0.20 was chosen.

NOTE: For more details on modelling inhomogeneity see the last section of this example.

A4.5 Step 4: Calculating the combined standard uncertainty During the in house validation study of the analytical procedure the repeatability, the bias and all other feasible uncertainty sources had been thoroughly investigated. Their values and uncertainties are collected in Table A4.4. Figure A4.7: Evaluation of other sources of uncertainty P( op) I( op) c( ref) V( op)

m( sample) I( ref) V( op) Calibration( 2) Temperature( 2)

dilution dilution Calibration( 2) V( ref) V( ref) Calibration( 2) Temperature( 2) m( ref) Calibration ( 2)

m( ref) Purity( ref) I( op)

Calibration( 3)

Recovery( 2) m( gross) I( ref)

Calibration( 3) Calibration( 2) Linearity m( tare) m( sample)

Calibration( 2) Linearity F( hom)( 3) Repeatability( 1)

Linearity ( 1) Repeatabilty ( FRep in equation A4.1) considered during the variability investigation of the analytical procedure.

( 2) Considered during the bias study of the analytical procedure.

( 3) To be considered during the evaluation of the other sources of uncertainty. Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 67 The relative values are combined because the model ( equation A4.1) is entirely multiplicative:

op op c op op c P P u P P u ´ = Þ = + + = 34 . 0 ) ( 34 . 0 2 . 0 048 . 0 27 . 0 ) ( 2 2 2 The spreadsheet for this case ( Table A4.5) takes the form shown in Table A4.5. Note that the spreadsheet calculates an absolute value uncertainty ( 0.373) for a nominal corrected result of 1.1111, giving a value of 0.373/ 1.11= 0.34.

The relative sizes of the three different contributions can be compared by employing a histogram. Figure A4.8 shows the values | u( y, xi)| taken from Table A4.5. The repeatability is the largest contribution to the measurement uncertainty. Since this component is derived from the overall variability in the method, further experiments would be needed to show where improvements could be made. For example, the uncertainty could be reduced significantly by homogenising the whole loaf before taking a sample.

The expanded uncertainty U( Pop) is calculated by multiplying the combined standard uncertainty with a coverage factor of 2 to give:

op op op P P P U ´ = ´ ´ = 68 . 0 2 34 . 0 ) ( Table A4.4: Uncertainties in pesticide analysis Description Value x Standard uncertainty u( x) Relative standard uncertainty u( x) Remark Repeatability( 1) 1.0 0.27 0.27 Duplicate tests of different types of samples Bias ( Rec) ( 2) 0.9 0.043 0.048 Spiked samples Other sources ( 3)

( Homogeneity) 1.0 0.2 0.2 Estimations founded on model assumptions u( Pop)/ Pop     0.34 Relative standard uncertainty Figure A4.8: Uncertainties in pesticide analysis u( y, xi) ( mg kg 1) 0 0.1 0.2 0.3 0.4 P( op) Repeatability Bias Homogeneity The values of u( y, xi)=( ¶ y/ ¶ xi). u( xi) are taken from Table A4.5 Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 68 A4.6 Special aspect: Modelling inhomogeneity for organophosphorus pesticide uncertainty Assuming that all of the material of interest in a sample can be extracted for analysis irrespective of its state, the worst case for inhomogeneity is the situation where some part or parts of a sample contain all of the substance of interest. A more general, but closely related, case is that in which two levels, say L1 and L2 of the material are present in different parts of the whole sample. The effect of such inhomogeneity in the case of random sub sampling can be estimated using binomial statistics. The values required are the mean m and the standard deviation s of the amount of material in n equal portions selected randomly after separation.

These values are given by ( ) Þ + × = m 2 2 1 1 l p l p n ( ) 2 2 1 1 nl l l np + - × = m [ 1]

( ) ( ) 2 2 1 1 1 2 1 l l p np - × - × = s [ 2] where l1 and l2 are the amount of substance in portions from regions in the sample containing total fraction L1 and L2 respectively, of the total amount X, and p1 and p2 are the probabilities of selecting portions from those regions ( n must be small compared to the total number of portions from which the selection is made).

The figures shown above were calculated as follows, assuming that a typical sample loaf is approximately 24 12 12 ´ ´ cm, using a portion size of 2 2 2 ´ ´ cm ( total of 432 portions) and assuming 15 such portions are selected at random and homogenised.

Scenario ( a)

The material is confined to a single large face ( the top) of the sample. L2 is therefore zero as is l2; and L1= 1. Each portion including part of the top surface will contain an amount l1 of the material. For the dimensions given, clearly one in six ( 2/ 12) of the portions meets this criterion, p1 is Table A4.5: Uncertainties in pesticide analysis A B C D E 1 Repeatability Bias Homogeneity 2 value 1.0 0.9 1.0 3 uncertainty 0.27 0.043 0.2 4 5 Repeatability 1.0 1.27 1.0 1.0 6 Bias 0.9 0.9 0.943 0.9 7 Homogeneity 1.0 1.0 1.0 1.2 8 9 Pop 1.1111 1.4111 1.0604 1.333 10 u( y, xi) 0.30  0.0507 0.222 11 u( y) 2, u( y, xi) 2 0.1420 0.09 0.00257 0.04938 12 13 u( Pop) 0.377 ( 0.377/ 1.111 = 0.34 as a relative standard uncertainty)

The values of the parameters are entered in the second row from C2 to E2. Their standard uncertainties are in the row below ( C3: E3). The spreadsheet copies the values from C2 E2 into the second column from B5 to B7. The result using these values is given in B9 (= B5 ´ B7/ B6, based on equation A4.1). C5 shows the value of the repeatability from C2 plus its uncertainty given in C3. The result of the calculation using the values C5: C7 is given in C9. The columns D and E follow a similar procedure. The values shown in the row 10 ( C10: E10) are the differences of the row ( C9: E9) minus the value given in B9. In row 11 ( C11: E11) the values of row 10 ( C10: E10) are squared and summed to give the value shown in B11. B13 gives the combined standard uncertainty, which is the square root of B11. Quantifying Uncertainty Example A4 QUAM: 2000. P1 Page 69 therefore 1/ 6, or 0.167, and l1 is X/ 72 ( i. e. there are 72 " top" portions).

This gives 1 1 5 . 2 167 . 0 15 l l = ´ ´ = m 2 1 2 1 2 08 . 2 ) 17 . 0 1 ( 167 . 0 15 l l = ´ - ´ ´ = s 1 2 1 44 . 1 08 . 2 l l = = s Þ 58 . 0 = m s = Þ RSD NOTE: To calculate the level X in the entire sample, m is multiplied back up by 432/ 15, giving a mean estimate of X of X X l X = ´ = ´ ´ = 72 72 5 . 2 15 432 1 This result is typical of random sampling; the expectation value of the mean is exactly the mean value of the population. For random sampling, there is thus no contribution to overall uncertainty other that the run to run variability, expressed as s or RSD here.

Scenario ( b)

The material is distributed evenly over the whole surface. Following similar arguments and assuming that all surface portions contain the same amount l1 of material, l2 is again zero, and p1 is, using the dimensions above, given by 63 . 0 ) 24 12 12 ( ) 20 8 8 ( ) 24 12 12 ( 1 = ´ ´ ´ ´ - ´ ´ = p i. e. p1 is that fraction of sample in the " outer" 2 cm. Using the same assumptions then 272 1 X l = .

NOTE: The change in value from scenario ( a)

This gives:

1 1 5 . 9 63 . 0 15 l l = ´ ´ = m 2 1 2 1 2 5 . 3 ) 63 . 0 1 ( 63 . 0 15 l l = ´ - ´ ´ = s 1 2 1 87 . 1 5 . 3 l l = = s Þ 2 . 0 = m s = Þ RSD Scenario ( c)

The amount of material near the surface is reduced to zero by evaporative or other loss. This case can be examined most simply by considering it as the inverse of scenario ( b), with p1= 0.37 and l1 equal to X/ 160. This gives 1 1 6 . 5 37 . 0 15 l l = ´ ´ = m 2 1 2 1 2 5 . 3 ) 37 . 0 1 ( 37 . 0 15 l l = ´ - ´ ´ = s 1 2 1 87 . 1 5 . 3 l l = ´ = s Þ 33 . 0 = m s = Þ RSD However, if the loss extends to a depth less than the size of the portion removed, as would be expected, each portion contains some material l1 and l2 would therefore both be non zero. Taking the case where all outer portions contain 50% " centre" and 50% " outer" parts of the sample 296 2 1 2 1 X l l l = Þ ´ = ( ) 2 2 2 2 2 1 6 . 20 15 37 . 0 15 15 37 . 0 15 l l l l l l = ´ + ´ ´ = ´ + - ´ ´ = m 2 2 2 2 1 2 5 . 3 ) ( ) 37 . 0 1 ( 37 . 0 15 l l l = - ´ - ´ ´ = s giving an RSD of 09 . 0 6 . 20 87 . 1 = In the current model, this corresponds to a depth of 1 cm through which material is lost. Examination of typical bread samples shows crust thickness typically of 1 cm or less, and taking this to be the depth to which the material of interest is lost ( crust formation itself inhibits lost below this depth), it follows that realistic variants on scenario ( c) will give values of m s not above 0.09.

NOTE: In this case, the reduction in uncertainty arises because the inhomogeneity is on a smaller scale than the portion taken for homogenisation. In general, this will lead to a reduced contribution to uncertainty. It follows that no additional modelling need be done for cases where larger numbers of small inclusions ( such as grains incorporated in the bulk of a loaf) contain disproportionate amounts of the material of interest. Provided that the probability of such an inclusion being incorporated into the portions taken for homogenisation is large enough, the contribution to uncertainty will not exceed any already calculated in the scenarios above. Quantifying Uncertainty Example A5 QUAM: 2000. P1 Page 70 Example A5: Determination of Cadmium Release from Ceramic Ware by Atomic Absorption Spectrometry Summary Goal The amount of released cadmium from ceramic ware is determined using atomic absorption spectrometry. The procedure employed is the empirical method BS 6748.

Measurement procedure The different stages in determining the amount of cadmium released from ceramic ware are given in the flow chart ( Figure A5.1).

Measurand:

2 0 dm mg - × × × × × = temp time acid V L f f f d a V c r The variables are described in Table A5.1.

Identification of the uncertainty sources:

The relevant uncertainty sources are shown in the cause and effect diagram at Figure A5.2.

Quantification of the uncertainty sources:

The sizes of the different contributions are given in Table A5.1 and shown diagrammatically in Figure A5.2 Figure A5.1: Extractable metal procedure RESULT RESULT Surface conditioning Surface conditioning Fill with 4% v/ v acetic acid Fill with 4% v/ v acetic acid Leaching Leaching Homogenise leachate Homogenise leachate AAS Determination AAS Determination Prepare calibration standards Prepare calibration standards AAS Calibration AAS Calibration Preparation Preparation Table A5.1: Uncertainties in extractable cadmium determination Description Value x Standard uncertainty u( x) Relative standard uncertainty u( x)/ x co Content of cadmium in the extraction solution 0.26 mg l 1 0.018 mg l 1 0.069 d Dilution factor ( if used) 1.0 Note 1 0Note 1 0 Note 1 VL Volume of the leachate 0.332 l 0.0018 l 0.0054 aV Surface area of the vessel 2.37 dm2 0.06 dm2 0.025 facid Influence of the acid concentration 1.0 0.0008 0.0008 ftime Influence of the duration 1.0 0.001 0.001 ftemp Influence of temperature 1.0 0.06 0.06 r Mass of cadmium leached per unit area 0.036 mg dm 2 0.0033 mg dm 2 0.09 Note 1: No dilution was applied in the present example; d is accordingly exactly 1.0 Quantifying Uncertainty Example A5 QUAM: 2000. P1 Page 71 Figure A5.2: Uncertainty sources in leachable cadmium determination Result r c( 0) V( L)

d a( V) Reading Calibration Temperature 1 length 2 length area Filling calibration curve f( temperature) f( time) f( acid)

Figure A5.3: Uncertainties in leachable Cd determination u( y, xi) ( mg dm 2) x1000 0 1 2 3 4 r c( 0) V( L) a( V) f( acid) f( time) f( temp)

The values of u( y, xi)=( ¶ y/ ¶ xi). u( xi) are taken from Table A5.4 Quantifying Uncertainty Example A5 QUAM: 2000. P1 Page 72 Example A5: Determination of cadmium release from ceramic ware by atomic absorption spectrometry. Detailed discussion.

A5.1 Introduction This example demonstrates the uncertainty evaluation of an empirical method; in this case ( BS 6748), the determination of metal release from ceramic ware, glassware, glass ceramic ware and vitreous enamel ware. The test is used to determine by atomic absorption spectroscopy ( AAS) the amount of lead or cadmium leached from the surface of ceramic ware by a 4% ( v/ v) aqueous solution of acetic acid. The results obtained with this analytical method are only expected to be comparable with other results obtained by the same method.

A5.2 Step 1: Specification The complete procedure is given in British Standard BS 6748: 1986 " Limits of metal release from ceramic ware, glass ware, glass ceramic ware and vitreous enamel ware" and this forms the specification for the measurand. Only a general description is given here ( right).

A5.2.1 Apparatus and Reagent specifications The reagent specifications affecting the uncertainty study are:

· A freshly prepared solution of 4% v/ v glacial acetic acid in water, made up by dilution of 40 ml glacial acetic to 1 l.

· A ( 1000 ± 1) mg l 1 standard lead solution in 4% ( v/ v) acetic acid.

· A ( 500 ± 0.5) mg l 1 standard cadmium solution in 4% ( v/ v) acetic acid.

Laboratory glassware is required to be of at least class B and incapable of releasing detectable levels of lead or cadmium in 4% acetic acid during the test procedure. The atomic absorption spectrophotometer is required to have detection limits of at most 0.2 mg l 1 for lead and 0.02 mg l 1 for cadmium.

A5.2.2 Procedure The general procedure is illustrated schematically in Figure A5.4. The specifications affecting the uncertainty estimation are:

i) The sample is conditioned to ( 22 ± 2) ° C. Where appropriate (` category 1' articles), the surface area of the article is determined. For this example, a surface area of 2.37 dm2 was obtained ( Table A5.1 and Table A5.3 include the experimental values for the example).

ii) The conditioned sample is filled with 4% v/ v acid solution at ( 22 ± 2) ° C to within 1 mm from the overflow point, measured from the upper rim of the sample, or to within 6 mm from the extreme edge of a sample with a flat or sloping rim.

iii) The quantity of 4% v/ v acetic acid required or used is recorded to an accuracy of ± 2% ( in this example, 332 ml acetic acid was used).

iv) The sample is allowed to stand at ( 22 ± 2) ° C for 24 hours ( in darkness if cadmium is determined) with due precaution to prevent evaporation loss.

v) After standing, the solution is stirred sufficiently for homogenisation, and a test portion removed, diluted by a factor d if necessary, and analysed by AA, using Figure A5.4: Extractable metal procedure RESULT RESULT Surface conditioning Surface conditioning Fill with 4% v/ v acetic acid Fill with 4% v/ v acetic acid Leaching Leaching Homogenise leachate Homogenise leachate AAS Determination AAS Determination Prepare calibration standards Prepare calibration standards AAS Calibration AAS Calibration Preparation Preparation Quantifying Uncertainty Example A5 QUAM: 2000. P1 Page 73 appropriate wavelengths and, in this example, a least squares calibration curve.

vi) The result is calculated ( see below) and reported as the amount of lead and/ or cadmium in the total volume of the extracting solution, expressed in milligrams of lead or cadmium per square decimetre of surface area for category 1 articles or milligrams of lead or cadmium per litre of the volume for category 2 and 3 articles.

NOTE: Complete copies of BS 6748: 1986 can be obtained by post from BSI customer services, 389 Chiswick High Road, London W4 4AL England ( + 44 ( 0) 208 996 9001.

A5.3 Step 2: Identity and analysing uncertainty sources Step 1 describes an ` empirical method'. If such a method is used within its defined field of application, the bias of the method is defined as zero. Therefore bias estimation relates to the laboratory performance and not to the bias intrinsic to the method. Because no reference material certified for this standardised method is available, overall control of bias is related to the control of method parameters influencing the result. Such influence quantities are time, temperature, mass and volumes, etc.

The concentration c0 of lead or cadmium in the acetic acid after dilution is determined by atomic absorption spectrometry and calculated using 1

1 0 0 0 l mg ) (

B B A c - = where c0 : concentration of lead or cadmium in the extraction solution [ mg l 1 ] A0 : absorbance of the metal in the sample extract B0 : intercept of the calibration curve B1 : slope of the calibration curve For the ` category 1' item considered in the present example, the empirical method calls for the result to be expressed as mass r of lead or cadmium leached per unit area. r is given by 2

1 0 0 0 dm mg ) ( d B a B A V d a V c r V L V L × × - × = × × = where the additional parameters are r : mass of Cd or Pb leached per unit area [ mg dm 2 ] VL : the volume of the leachate [ l ] aV : the surface area of the vessel [ dm2 ] d : factor by which the sample was diluted The first part of the above equation of the measurand is used to draft the basic cause and effect diagram ( Figure A5.5).

Figure A5.5: Initial cause and effect diagram Result r c( 0) V( L)

d a( V) Reading Calibration Temperature 1 length 2 length area Filling calibration curve There is no reference material certified for this empirical method with which to assess the laboratory performance. All the feasible influence quantities, such as temperature, time of the leaching process and acid concentration therefore have to be considered. To accommodate the additional influence quantities the equation is expanded by the respective correction factors leading to temp time acid V L f f f d a V c r × × × × × = 0 These additional factors are also included in the revised cause and effect diagram ( Figure A5.6). They are shown there as effects on c0.

NOTE: The latitude in temperature permitted by the standard is a case of an uncertainty arising as a result of incomplete specification of the measurand. Taking the effect of temperature into account allows estimation of the range of results which could be reported whilst complying with the empirical method as well as is practically possible. Note particularly that variations in the result caused by different operating temperatures within the range Quantifying Uncertainty Example A5 QUAM: 2000. P1 Page 74 cannot reasonably described as bias as they represent results obtained in accordance with the specification.

A5.4 Step 3: Quantifying uncertainty sources The aim of this step is to quantify the uncertainty arising from each of the previously identified sources. This can be done either by using experimental data or from well based assumptions.

Dilution factor d For the current example, no dilution of the leaching solution is necessary, therefore no uncertainty contribution has to be accounted for.

Volume VL Filling: The empirical method requires the vessel to be filled ` to within 1 mm from the brim'. For a typical drinking or kitchen utensil, 1 mm will represent about 1% of the height of the vessel. The vessel will therefore be 99.5 ± 0.5% filled ( i. e. VL will be approximately 0.995 ± 0.005 of the vessel's volume).

Temperature: The temperature of the acetic acid has to be 22 ± 2 º C. This temperature range leads to an uncertainty in the determined volume, due to a considerable larger volume expansion of the liquid compared with the vessel. The standard uncertainty of a volume of 332 ml, assuming a rectangular temperature distribution, is ml 08 . 0 3 2 332 10 1 . 2 4 = ´ ´ ´ - Reading: The volume VL used is to be recorded to within 2%, in practice, use of a measuring cylinder allows an inaccuracy of about 1% ( i. e. 0.01VL). The standard uncertainty is calculated assuming a triangular distribution.

Calibration: The volume is calibrated according to the manufacturer's specification within the range of ± 2.5 ml for a 500 ml measuring cylinder. The standard uncertainty is obtained assuming a triangular distribution.

For this example a volume of 332 ml is used and the four uncertainty components are combined accordingly ( ) ml 83 . 1 6 5 . 2 6 332 01 . 0 ) 08 . 0 ( 6 332 005 . 0 2 2 2 2 = ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ ´ + + ÷ ÷ ø ö ç ç è æ ´ = L V u Cadmium concentration c0 The amount of leached cadmium is calculated using a manually prepared calibration curve. For this purpose five calibration standards, with a concentration 0.1 mg l 1, 0.3 mg l 1, 0.5 mg l 1, 0.7 mg l 1 and 0.9 mg l 1, were prepared from a 500 ± 0.5 mg l 1 cadmium reference standard. The Figure A5.6: Cause and effect diagram with added hidden assumptions ( correction factors)

Result r c( 0) V( L)

d a( V) Reading Calibration Temperature 1 length 2 length area Filling calibration curve f( temperature) f( time) f( acid) Quantifying Uncertainty Example A5 QUAM: 2000. P1 Page 75 linear least squares fitting procedure used assumes that the uncertainties of the values of the abscissa are considerably smaller than the uncertainty on the values of the ordinate. Therefore the usual uncertainty calculation procedures for c0 only reflect the uncertainty in the absorbance and not the uncertainty of the calibration standards, nor the inevitable correlations induced by successive dilution from the same stock. In this case, however, the uncertainty of the calibration standards is sufficiently small to be neglected.

The five calibration standards were measured three times each, providing the results in Table A5.2.

The calibration curve is given by 0 1 B B c A i j + × = where Aj : jth measurement of the absorbance of the ith calibration standard ci : concentration of the ith calibration standard B1 : slope B0 : intercept and the results of the linear least square fit are Value Standard deviation B1 0.2410 0.0050 B0 0.0087 0.0029 with a correlation coefficient r of 0.997. The fitted line is shown in Figure A5.7. The residual standard deviation S is 0.005486.

The actual leach solution was measured twice, leading to a concentration c0 of 0.26 mg l 1. The calculation of the uncertainty u( c0) associated with the linear least square fitting procedure is described in detail in Appendix E3. Therefore only a short description of the different calculation steps is given here.

u( c0) is given by 1 0 2 2 0 1 0 l mg 018 . 0 ) ( 2 . 1 ) 5 . 0 26 . 0 (

15 1 2 1 241 . 0 005486 . 0 ) ( 1 1 ) (

- = Þ - + + = - + + = c u S c c n p B S c u xx with the residual standard deviation S given by Table A5.2: Calibration results Concentration [ mg l 1] 1 2 3 0.1 0.028 0.029 0.029 0.3 0.084 0.083 0.081 0.5 0.135 0.131 0.133 0.7 0.180 0.181 0.183 0.9 0.215 0.230 0.216 Figure A5.7: Linear least square fit and uncertainty interval for duplicate determinations Concentration of Cadmium [ mg/ l] Absorption 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.05 0.10 0.15 0.20 0.25 Quantifying Uncertainty Example A5 QUAM: 2000. P1 Page 76 005486 . 0 2 )] ( [

1 2 1 0 = - × + - = å = n c B B A S n j j j and 2 . 1 ) (

1 2 = - = å = n j j xx c c S where B1 : slope p : number of measurements to determine c0 n : number of measurements for the calibration c0 : determined cadmium concentration of the leached solution c : mean value of the different calibration standards ( n number of measurements)

i : index for the number of calibration standards j : index for the number of measurements to obtain the calibration curve Area aV Length measurement: The total surface area of the sample vessel was calculated, from measured dimensions, to be 2.37 dm2. Since the item is approximately cylindrical but not perfectly regular, measurements are estimated to be within 2 mm at 95% confidence. Typical dimensions are between 1.0 dm and 2.0 dm leading to an estimated dimensional measurement uncertainty of 1 mm ( after dividing the 95% figure by 1.96). Area measurements typically require two length measurements, height and width respectively ( i. e. 1.45 dm and 1.64 dm)

Area: Since the item has not a perfect geometric shape, there is also an uncertainty in any area calculation; in this example, this is estimated to contribute an additional 5% at 95% confidence.

The uncertainty contribution of the length measurement and area itself are combined in the usual way.

2 2 2 2 dm 06 . 0 ) ( 96 . 1 37 . 2 05 . 0 01 . 0 01 . 0 ) (

= Þ ÷ ø ö ç è æ ´ + + = V V a u a u Temperature effect ftemp A number of studies of the effect of temperature on metal release from ceramic ware have been undertaken( 1 5). In general, the temperature effect is substantial and a near exponential increase in metal release with temperature is observed until limiting values are reached. Only one study1 has given an indication of effects in the range of 20 25 ° C. From the graphical information presented the change in metal release with temperature near 25 ° C is approximately linear, with a gradient of approximately 5% ° C 1. For the ± 2 ° C range allowed by the empirical method this leads to a factor ftemp of 1 ± 0.1. Converting this to a standard uncertainty gives, assuming a rectangular distribution:

u( ftemp)= 06 . 0 3 1 . 0 = Time effect ftime For a relatively slow process such as leaching, the amount leached will be approximately proportional to time for small changes in the time. Krinitz and Franco1 found a mean change in concentration over the last six hours of leaching of approximately 1.8 mg l 1 in 86 mg l 1, that is, about 0.3%/ h. For a time of ( 24 ± 0.5) h c0 will therefore need correction by a factor ftime of 1 ± ( 0.5 ´ 0.003) = 1 ± 0.0015. This is a rectangular distribution leading to the standard uncertainty 001 . 0 3 0015 . 0 ) ( @ = time f u .

Acid concentration facid One study of the effect of acid concentration on lead release showed that changing concentration from 4 to 5% v/ v increased the lead released from a particular ceramic batch from 92.9 to 101.9 mg l 1, i. e. a change in facid of 097 . 0 9 . 92 ) 9 . 92 9 . 101 ( = - or close to 0.1. Another study, using a hot leach method, showed a comparable change ( 50% change in lead extracted on a change of from 2 to 6% v/ v) 3. Assuming this effect as approximately linear with acid concentration gives an estimated change in facid of approximately 0.1 per % v/ v change in acid concentration. In a separate experiment the concentration and its standard uncertainty have been established using titration with a standardised NaOH titre ( 3.996% v/ v u = 0.008% v/ v). Taking the uncertainty of 0.008% v/ v on the acid concentration suggests an uncertainty for facid of 0.008 ´ 0.1 = 0.0008. As the uncertainty on the acid concentration is already expressed as a standard uncertainty, this value can be used directly as the uncertainty associated with facid. Quantifying Uncertainty Example A5 QUAM: 2000. P1 Page 77 NOTE: In principle, the uncertainty value would need correcting for the assumption that the single study above is sufficiently representative of all ceramics. The present value does, however, give a reasonable estimate of the magnitude of the uncertainty.

A5.5 Step 4: Calculating the combined standard uncertainty The amount of leached cadmium per unit area, assuming no dilution, is given by 2  0 dm mg temp time acid V L f f f a V c r × × × × = The intermediate values and their standard uncertainties are collected in Table A5.3. Employing those values 2 dm mg 036 . 0 0 . 1 0 . 1 0 . 1 37 . 2 332 . 0 26 . 0 - = ´ ´ ´ ´ = r In order to calculate the combined standard uncertainty of a multiplicative expression ( as above) the standard uncertainties of each component are used as follows:

( ) ( ) ( ) ( ) ( ) ( ) ( ) 095 . 0 06 . 0 001 . 0 0008 . 0 025 . 0 0054 . 0 069 . 0 2 2 2 2 2 2 2 2 2 2 2 2 0 0 = + + + + + = ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ = temp temp time time acid acid V V L L c f f u f f u f f u a a u V V u c c u r r u  2 dm mg 0034 . 0 095 . 0 ) ( = = Þ r r u c The simpler spreadsheet approach to calculate the combined standard uncertainty is shown in Table A5.4. A description of the method is given in Appendix E.

The contributions of the different parameters and influence quantities to the measurement uncertainty are illustrated in Figure A5.8, comparing the size of each of the contributions ( C13: H13 in Table A5.4) with the combined uncertainty ( B16).

The expanded uncertainty U( r) is obtained by applying a coverage factor of 2 Ur= 0.0034 ´ 2 = 0.007 mg dm 2 Thus the amount of released cadmium measured according to BS 6748: 1986 ( 0.036 ± 0.007) mg dm 2 where the stated uncertainty is calculated using a coverage factor of 2.

A5.6 References for Example 5 1. B. Krinitz, V. Franco, J. AOAC 56 869 875 ( 1973)

2. B. Krinitz, J. AOAC 61, 1124 1129 ( 1978)

3. J. H. Gould, S. W. Butler, K. W. Boyer, E. A. Stelle, J. AOAC 66, 610 619 ( 1983)

4. T. D. Seht, S. Sircar, M. Z. Hasan, Bull. Environ. Contam. Toxicol. 10, 51 56 ( 1973)

5. J. H. Gould, S. W. Butler, E. A. Steele, J. AOAC 66, 1112 1116 ( 1983)

Table A5.3: Intermediate values and uncertainties for leachable cadmium analysis Description Value Standard uncertainty u( x) Relative standard uncertainty u( x)/ x c0 Content of cadmium in the extraction solution 0.26 mg l 1 0.018 mg l 1 0.069 VL Volume of the leachate 0.332 l 0.0018 l 0.0054 aV Surface area of the vessel 2.37 dm2 0.06 dm2 0.025 facid Influence of the acid concentration 1.0 0.0008 0.0008 ftime Influence of the duration 1.0 0.001 0.001 ftemp Influence of temperature 1.0 0.06 0.06 Quantifying Uncertainty Example A5 QUAM: 2000. P1 Page 78 Table A5.4: Spreadsheet calculation of uncertainty for leachable cadmium analysis A B C D E F G H 1 c0 VL aV facid ftime ftemp 2 value 0.26 0.332 2.37 1.0 1.0 1.0 3 uncertainty 0.018 0.0018 0.06 0.0008 0.001 0.06 4 5 c0 0.26 0.278 0.26 0.26 0.26 0.26 0.26 6 VL 0.332 0.332 0.3338 0.332 0.332 0.332 0.332 7 aV 2.37 2.37 2.37 2.43 2.37 2.37 2.37 8 facid 1.0 1.0 1.0 1.0 1.0008 1.0 1.0 9 ftime 1.0 1.0 1.0 1.0 1.0 1.001 1.0 10 ftemp 1.0 1.0 1.0 1.0 1.0 1.0 1.06 11 12 r 0.036422 0.038943 0.036619 0.035523 0.036451 0.036458 0.038607 13 u( y, xi) 0.002521 0.000197  0.000899 0.000029 0.000036 0.002185 14 u( y) 2, u( y, xi) 2 1.199 E 5 6.36 E 6 3.90 E 8 8.09 E 7 8.49 E 10 1.33 E 9 4.78 E 6 15 16 uc( r) 0.0034 The values of the parameters are entered in the second row from C2 to H2, and their standard uncertainties in the row below ( C3: H3). The spreadsheet copies the values from C2: H2 into the second column ( B5: B10). The result ( r) using these values is given in B12. C5 shows the value of c0 from C2 plus its uncertainty given in C3. The result of the calculation using the values C5: C10 is given in C12. The columns D and H follow a similar procedure. Row 13 ( C13: H13) shows the differences of the row ( C12: H12) minus the value given in B12. In row 14 ( C14: H14) the values of row 13 ( C13: H13) are squared and summed to give the value shown in B14. B16 gives the combined standard uncertainty, which is the square root of B14.

Figure A5.8: Uncertainties in leachable Cd determination u( y, xi) ( mg dm 2) x1000 0 1 2 3 4 r c( 0) V( L) a( V) f( acid) f( time) f( temp)

The values of u( y, xi)=( ¶ y/ ¶ xi). u( xi) are taken from Table A5.4 Quantifying Uncertainty Example A6 QUAM: 2000. P1 Page 79 Example A6: The Determination of Crude Fibre in Animal Feeding Stuffs Summary Goal The determination of crude fibre by a regulatory standard method.

Measurement procedure The measurement procedure is a standardised procedure involving the general steps outlined in Figure A6.1. These are repeated for a blank sample to obtain a blank correction.

Measurand The fibre content as a percentage of the sample by weight, Cfibre, is given by:

Cfibre = ( ) a c b 100 ´ - Where:

a is the mass ( g) of the sample. ( Approximately 1 g)

b is the loss of mass ( g) after ashing during the determination;

c is the loss of mass ( g) after ashing during the blank test.

Identification of uncertainty sources A full cause and effect diagram is provided as Figure A6.9.

Quantification of uncertainty components Laboratory experiments showed that the method was performing in house in a manner that fully justified adoption of collaborative study reproducibility data. No other contributions were significant in general. At low levels it was necessary to add an allowance for the specific drying procedure used. Typical resulting uncertainty estimates are tabulated below ( as standard uncertainties) ( Table A6.1). Figure A6.1: Fibre determination.

Alkaline digestion Alkaline digestion Dry and weigh residue Dry and weigh residue Ash and weigh residue Ash and weigh residue RESULT RESULT Acid digestion Acid digestion Grind and weigh sample Grind and weigh sample Table A6.1: Combined standard uncertainties Fibre content (% w/ w) Standard uncertainty u( Cfibre) (% w/ w) Relative Standard uncertainty u( Cfibre) / Cfibre 2.5 0 29 0 115 0 31 2 2 . . . + = 0.12 5 0.4 0.08 10 0.6 0.06 Quantifying Uncertainty Example A6 QUAM: 2000. P1 Page 80 Example A6: The determination of crude fibre in animal feeding stuffs. Detailed discussion A6.1 Introduction Crude fibre is defined in the method scope as the amount of fat free organic substances which are insoluble in acid and alkaline media. The procedure is standardised and its results used directly. Changes in the procedure change the measurand; this is accordingly an example of an empirical method.

Collaborative trial data ( repeatability and reproducibility) were available for this statutory method. The precision experiments described were planned as part of the in house evaluation of the method performance. There is no suitable reference material ( i. e. certified by the same method) available for this method.

A6.2 Step 1: Specification The specification of the measurand for more extensive analytical methods is best done by a comprehensive description of the different stages of the analytical method and by providing the equation of the measurand.

Procedure The procedure, a complex digestion, filtration, drying, ashing and weighing procedure, which is also repeated for a blank crucible, is summarised in Figure A6.2. The aim is to digest most components, leaving behind all the undigested material. The organic material is ashed, leaving an inorganic residue. The difference between the dry organic/ inorganic residue weight and the ashed residue weight is the " fibre content". The main stages are:

i) Grind the sample to pass through a 1mm sieve ii) Weigh 1g of the sample into a weighed crucible iii) Add a set of acid digestion reagents at stated concentrations and volumes. Boil for a stated, standardised time, filter and wash the residue.

iv) Add standard alkali digestion reagents and boil for the required time, filter, wash and rinse with acetone. v) Dry to constant weight at a standardised temperature (" constant weight" is not defined within the published method; nor are other drying conditions such as air circulation or dispersion of the residue).

vi) Record the dry residue weight.

vii) Ash at a stated temperature to " constant weight" ( in practice realised by ashing for a set time decided after in house studies).

viii) Weigh the ashed residue and calculate the fibre content by difference, after subtracting the residue weight found for the blank crucible.

Measurand The fibre content as a percentage of the sample by weight, Cfibre, is given by:

Cfibre = ( ) a c b 100 ´ - Where:

a is the mass ( g) of the sample. Approximately 1 g of sample is taken for analysis.

b is the loss of mass ( g) after ashing during the determination.

c is the loss of mass ( g) after ashing during the blank test.

A6.3 Step 2: Identifying and analysing uncertainty sources A range of sources of uncertainty was identified. These are shown in the cause and effect diagram for the method ( see Figure A6.9). This diagram was simplified to remove duplication following the procedures in Appendix D; this, together with removal of insignificant components, leads to the simplified cause and effect diagram in Figure A6.10.

Since prior collaborative and in house study data were available for the method, the use of these data is closely related to the evaluation of different contributions to uncertainty and is accordingly discussed further below. Quantifying Uncertainty Example A6 QUAM: 2000. P1 Page 81 A6.4 Step 3: Quantifying uncertainty components Collaborative trial results The method has been the subject of a collaborative trial. Five different feeding stuffs representing typical fibre and fat concentrations were analysed in the trial. Participants in the trial carried out all stages of the method, including grinding of the samples. The repeatability and reproducibility estimates obtained from the trial are presented in Table A6.2.

As part of the in house evaluation of the method, experiments were planned to evaluate the repeatability ( within batch precision) for feeding stuffs with fibre concentrations similar to those of the samples analysed in the collaborative trial. The results are summarised in Table A6.2. Each estimate of in house repeatability is based on 5 replicates. Figure A6.2: Flow diagram illustrating the stages in the regulatory method for the determination of fibre in animal feeding stuffs Grind sample to pass through 1 mm sieve Weigh 1 g of sample into crucible Add filter aid, anti foaming agent followed by 150 ml boiling H2SO4 Add anti foaming agent followed by 150 ml boiling KOH Boil vigorously for 30 mins Filter and wash with 3x30 ml boiling water Boil vigorously for 30 mins Filter and wash with 3x30 ml boiling water Apply vacuum, wash with 3x25 ml acetone Dry to constant weight at 130 ° C Ash to constant weight at 475 500 ° C Calculate the % crude fibre content Weigh crucible for blank test Quantifying Uncertainty Example A6 QUAM: 2000. P1 Page 82 The estimates of repeatability obtained in house were comparable to those obtained from the collaborative trial. This indicates that the method precision in this particular laboratory is similar to that of the laboratories which took part in the collaborative trial. It is therefore acceptable to use the reproducibility standard deviation from the collaborative trial in the uncertainty budget for the method. To complete the uncertainty budget we need to consider whether there are any other effects not covered by the collaborative trial which need to be addressed. The collaborative trial covered different sample matrices and the pre treatment of samples, as the participants were supplied with samples which required grinding prior to analysis. The uncertainties associated with matrix effects and sample pre treatment do not therefore require any additional consideration. Other parameters which affect the result relate to the extraction and drying conditions used in the method. These were investigated separately to ensure the laboratory bias was under control ( i. e., small compared to the reproducibility standard deviation). The parameters considered are discussed below.

Loss of mass on ashing As there is no appropriate reference material for this method, in house bias has to be assessed by considering the uncertainties associated with individual stages of the method. Several factors will contribute to the uncertainty associated with the loss of mass after ashing:

§ acid concentration; § alkali concentration;

§ acid digestion time;

§ alkali digestion time;

§ drying temperature and time;

§ ashing temperature and time.

Reagent concentrations and digestion times The effects of acid concentration, alkali concentration, acid digestion time and alkali digestion time have been studied in previously published papers. In these studies, the effect of changes in the parameter on the result of the analysis was evaluated. For each parameter the sensitivity coefficient ( i. e., the rate of change in the final result with changes in the parameter) and the uncertainty in the parameter were calculated.

The uncertainties given in Table A6.3 are small compared to the reproducibility figures presented in Table A6.2. For example, the reproducibility standard deviation for a sample containing 2.3 % w/ w fibre is 0.293 % w/ w. The uncertainty associated with variations in the acid digestion time is estimated as 0.021 % w/ w ( i. e., 2.3 ´ 0.009). We can therefore safely neglect the uncertainties associated with variations in these method parameters.

Drying temperature and time No prior data were available. The method states that the sample should be dried at 130 ° C to " constant weight". In this case the sample is dried for 3 hours at 130 ° C and then weighed. It is then dried for a further hour and re weighed. Constant Table A6.2: Summary of results from collaborative trial of the method and in house repeatability check Fibre content (% w/ w)

Collaborative trial results Sample Mean Reproducibility standard deviation ( sR) Repeatability standard deviation ( sr) In house repeatability standard deviation A 2. 3 0.293 0.198 0.193 B 12.1 0.563 0.358 0.312 C 5.4 0.390 0.264 0.259 D 3.4 0.347 0.232 0.213 E 10.1 0.575 0.391 0.327 Quantifying Uncertainty Example A6 QUAM: 2000. P1 Page 83 weight is defined in this laboratory as a change of less than 2 mg between successive weighings. In an in house study, replicate samples of four feeding stuffs were dried at 110, 130 and 150 ° C and weighed after 3 and 4 hours drying time. In the majority of cases, the weight change between 3 and 4 hours was less than 2 mg. This was therefore taken as the worst case estimate of the uncertainty in the weight change on drying. The range ± 2 mg describes a rectangular distribution, which is converted to a standard uncertainty by dividing by Ö 3. The uncertainty in the weight recorded after drying to constant weight is therefore 0.00115 g. The method specifies a sample weight of 1 g. For a 1 g sample, the uncertainty in drying to constant weight corresponds to a standard uncertainty of 0.115 % w/ w in the fibre content. This source of uncertainty is independent of the fibre content of the sample. There will therefore be a fixed contribution of 0.115 % w/ w to the uncertainty budget for each sample, regardless of the concentration of fibre in the sample. At all fibre concentrations, this uncertainty is smaller than the reproducibility standard deviation, and for all but the lowest fibre concentrations is less than 1/ 3 of the sR value. Again, this source of uncertainty can usually be neglected. However for low fibre concentrations, this uncertainty is more than 1/ 3 of the sR value so an additional term should be included in the uncertainty budget ( see Table A6.4).

Ashing temperature and time The method requires the sample to be ashed at 475 to 500 ° C for at least 30 mins. A published study on the effect of ashing conditions involved determining fibre content at a number of different ashing temperature/ time combinations, ranging from 450 ° C for 30 minutes to 650 ° C for 3 hours. No significant difference was observed between the fibre contents obtained under the different conditions. The effect on the final result of small variations in ashing temperature and time can therefore be assumed to be negligible.

Loss of mass after blank ashing No experimental data were available for this parameter. However, as discussed above, the effects of variations in this parameter are likely to be small.

A6.5 Step 4: Calculating the combined standard uncertainty This is an example of an empirical method for which collaborative trial data were available. The in house repeatability was evaluated and found to be comparable to that predicted by the collaborative trial. It is therefore appropriate to use the sR values from the collaborative trial. The discussion presented in Step 3 leads to the Table A6.3: Uncertainties associated with method parameters Parameter Sensitivity coefficientNote 1 Uncertainty in parameter Uncertainty in final result as RSD Note 4 acid concentration 0.23 ( mol l 1) 1 0.0013 mol l 1 Note 2 0.00030 alkali concentration 0.21 ( mol l 1) 1 0.0023 mol l 1 Note 2 0.00048 acid digestion time 0.0031 min 1 2.89 mins Note 3 0.0090 alkali digestion time 0.0025 min 1 2.89 mins Note 3 0.0072 Note 1. The sensitivity coefficients were estimated by plotting the normalised change in fibre content against reagent strength or digestion time. Linear regression was then used to calculate the rate of change of the result of the analysis with changes in the parameter.

Note 2. The standard uncertainties in the concentrations of the acid and alkali solutions were calculated from estimates of the precision and trueness of the volumetric glassware used in their preparation, temperature effects etc. See examples A1 A3 for further examples of calculating uncertainties for the concentrations of solutions.

Note 3. The method specifies a digestion time of 30 minutes. The digestion time is controlled to within ± 5 minutes. This is a rectangular distribution which is converted to a standard uncertainty by dividing by Ö 3.

Note 4. The uncertainty in the final result, as a relative standard deviation, is calculated by multiplying the sensitivity coefficient by the uncertainty in the parameter. Quantifying Uncertainty Example A6 QUAM: 2000. P1 Page 84 conclusion that, with the exception of the effect of drying conditions at low fibre concentrations, the other sources of uncertainty identified are all small in comparison to sR. In cases such as this, the uncertainty estimate can be based on the reproducibility standard deviation, sR, obtained from the collaborative trial. For samples with a fibre content of 2.5 % w/ w, an additional term has been included to take account of the uncertainty associated with the drying conditions. Standard uncertainty Typical standard uncertainties for a range of fibre concentrations are given in the Table A6.4 below.

Expanded uncertainty Typical expanded uncertainties are given in Table A6.5 below. These were calculated using a coverage factor k of 2, which gives a level of confidence of approximately 95%.

Table A6.4: Combined standard uncertainties Fibre content (% w/ w) Standard uncertainty u( Cfibre) (% w/ w) Relative standard uncertainty u( Cfibre)/ Cfibre 2.5 0 29 0 115 0 31 2 2 . . . + = 0.12 5 0.4 0.08 10 0.6 0.06 Table A6.5: Expanded uncertainties Fibre content (% w/ w) Expanded uncertainty U( Cfibre) (% w/ w) Expanded uncertainty as CV (%)

2.5 0.62 25 5 0.8 16 10 0.12 12 Figure A6.9: Cause and effect diagram for the determination of fibre in animal feeding stuffs Crude Fibre (%)

Loss of mass after ashing ( b) Loss of mass after blank ashing ( c) Mass sample ( a) Precision sample weight precision weighing precision weight of crucible before ashing drying temp weight of sample and crucible after ashing ashing temp ashing time weighing of crucible balance linearity balance calibration extraction precision weight of crucible after ashing drying time balance calibration balance linearity ashing temp ashing time balance calibration balance linearity weighing of crucible weighing of crucible acid digest alkali digest alkali volume alkali conc digest conditions boiling rate extraction time acid vol acid conc digest conditions boiling rate extraction time balance linearity balance calibration drying temp drying time weighing of crucible ashing precision balance calibration balance linearity weight of sample and crucible before ashing acid digest* alkali digest*

* The branches feeding into these " acid digest" and " alkali digest" branches have been omitted for clarity. The same factors affect them as for the sample ( i. e., digest conditions, acid conc etc.). QUAM:

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P1 Page 85 Quantifying Uncertainty Example A6 Figure A6.10: Simplified cause and effect diagram alkali vol Crude Fibre (%)

Loss of mass after ashing ( b) Loss of mass after blank ashing ( c) Precision sample weight precision weighing precision weight of crucible before ashing drying temp ashing temp ashing time extraction weight of crucible after ashing drying time ashing temp ashing time acid digest alkali digest alkali conc digest conditions boiling rate extraction time acid vol acid conc digest conditions boiling rate extraction time drying temp drying time ashing precision weight of sample and crucible after ashing weight of sample and crucible before ashing acid digest* alkali digest*

* The branches feeding into these " acid digest" and " alkali digest" branches have been omitted for clarity. The same factors affect them as for the sample ( i. e., digest conditions, acid conc etc.). QUAM:

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P1 Page 86 Quantifying Uncertainty Example A6 Quantifying Uncertainty Example A7 QUAM: 2000. P1 Page 87 Example A7: Determination of the Amount of Lead in Water Using Double Isotope Dilution and Inductively Coupled Plasma Mass Spectrometry A7.1 Introduction This example illustrates how the uncertainty concept can be applied to a measurement of the amount content of lead in a water sample using Isotope Dilution Mass Spectrometry ( IDMS) and Inductively Coupled Plasma Mass Spectrometry ( ICP MS).

General introduction to Double IDMS IDMS is one of the techniques that is recognised by the Comité consultatif pour la quantité de matière ( CCQM) to have the potential to be a primary method of measurement, and therefore a well defined expression which describes how the measurand is calculated is available. In the simplest case of isotope dilution using a certified spike, which is an enriched isotopic reference material, isotope ratios in the spike, the sample and a blend b of known masses of sample and spike are measured. The element amount content cx in the sample is given by:

( ) ( ) å å × × × × - × × - × × × = i i R K R K R K R K R K R K m m c c yi yi xi xi x1 x1 b b b b y1 y1 x y y x ( 1)

where cx and cy are element amount content in the sample and the spike respectively ( the symbol c is used here instead of k for amount content1 to avoid confusion with K factors and coverage factors k). mx and my are mass of sample and spike respectively. Rx, Ry and Rb are the isotope amount ratios. The indexes x, y and b represent the sample, the spike and the blend respectively. One isotope, usually the most abundant in the sample, is selected and all isotope amount ratios are expressed relative to it. A particular pair of isotopes, the reference isotope and preferably the most abundant isotope in the spike, is then selected as monitor ratio, e. g. n( 208Pb)/ n( 206Pb). Rxi and Ryi are all the possible isotope amount ratios in the sample and the spike respectively. For the reference isotope, this ratio is unity. Kxi, Kyi and Kb are the correction factors for mass discrimination, for a particular isotope amount ratio, in sample, spike and blend respectively. The K factors are measured using a certified isotopic reference material according to equation ( 2). observed certified 0 bias 0 where ; R R K K K K = + = ( 2)

where K0 is the mass discrimination correction factor at time 0, Kbias is a bias factor coming into effect as soon as the K factor is applied to correct a ratio measured at a different time during the measurement. The Kbias also includes other possible sources of bias such as multiplier dead time correction, matrix effects etc. Rcertified is the certified isotope amount ratio taken from the certificate of an isotopic reference material and Robserved is the observed value of this isotopic reference material. In IDMS experiments, using Inductively Coupled Plasma Mass Spectrometry ( ICP MS), mass fractionation will vary with time which requires that all isotope amount ratios in equation ( 1) need to be individually corrected for mass discrimination.

Certified material enriched in a specific isotope is often unavailable. To overcome this problem, ` double' IDMS is frequently used. The procedure uses a less well characterised, isotopically enriched spiking material in conjunction with a certified material ( denoted z) of natural isotopic composition. The certified, natural composition material acts as the primary assay standard. Two blends are used; blend b is a blend between sample and enriched spike, as in equation ( 1). To perform double IDMS a second blend, b' is prepared from the primary assay standard with amount content cz, and the enriched material y. This gives a similar expression to equation ( 1):

( ) ( ) å å × × × × - × × - × × × = i i z z y y R K R K R K R K R K R K m m c c yi yi zi zi 1 1 b b b b 1 1 z y y z ' ' ' ' '

( 3)

where cz is the element amount content of the primary assay standard solution and mz the mass of the primary assay standard when preparing the new blend. m'y is the mass of the enriched spike solution, K'b, R'b, Kz1 and Rz1 are the K factor and the ratio for the new blend and the assay standard respectively. The index z represents the assay Quantifying Uncertainty Example A7 QUAM: 2000. P1 Page 88 standard. Dividing equation ( 1) with equation ( 3) gives ( ) ( ) ( ) ( ) å å å å × × × × - × × - × × × × × × × - × × - × × × = i i i i R K R K R K R K R K R K m m c R K R K R K R K R K R K m m c c c yi yi zi zi z1 z1 b b b b y1 y1 z y y yi yi xi xi x1 x1 b b b b y1 y1 x y y z x ' ' ' ' '

( 4)

Simplifying this equation and introducing a procedure blank, cblank, we get:

( ) ( ) blank zi zi xi xi b b y1 y1 z1 z1 b b x1 x1 b b b b y1 y1 y z x y z x ' ' ' ' '

c R K R K R K R K R K R K R K R K R K R K m m m m c c i i - × × × × - × × - × ´ × - × × - × × × × = å å ( 5)

This is the final equation, from which cy has been eliminated. In this measurement the number index on the amount ratios, R, represents the following actual isotope amount ratios: R1= n( 208Pb)/ n( 206Pb) R2= n( 206Pb)/ n( 206Pb)

R3= n( 207Pb)/ n( 206Pb) R4= n( 204Pb)/ n( 206Pb)

For reference, the parameters are summarised in Table A7.1.

A7.2 Step 1: Specification The general procedure for the measurements is shown in Table A7.2. The calculations and measurements involved are described below.

Calculation procedure for the amount content cx For this determination of lead in water, four blends each of b', ( assay + spike), and b, ( sample + spike), were prepared. This gives a total of 4 values for cx. One of these determinations will be described in detail following Table A7.2, steps 1 to 4. The reported value for cx will be the average of the four replicates. Table A7.1. Summary of IDMS parameters Parameter Description Parameter Description mx mass of sample in blend b [ g] my mass of enriched spike in blend b [ g]

m'y mass of enriched spike in blend b' [ g] mz mass of primary assay standard in blend b' [ g]

cx amount content of the sample x [ mol g 1 or m mol g 1] Note 1 cz amount content of the primary assay standard z [ mol g 1 or m mol g 1] Note 1 cy amount content of the spike y [ mol g 1 or m mol g 1] Note 1 cblank observed amount content in procedure blank [ mol g 1 or m mol g 1] Note 1 Rb measured ratio of blend b, n( 208Pb)/ n( 206Pb) Kb mass bias correction of Rb R'b measured ratio of blend b', n( 208Pb)/ n( 206Pb) K'b mass bias correction of R'b Ry1 measured ratio of enriched isotope to reference isotope in the enriched spike Ky1 mass bias correction of Ry1 Rzi all ratios in the primary assay standard, Rz1, Rz2 etc. Kzi mass bias correction factors for Rzi Rxi all ratios in the sample Kxi mass bias correction factors for Rxi Rx1 measured ratio of enriched isotope to reference isotope in the sample x Rz1 as Rx1 but in the primary assay standard Note 1: Units for amount content are always specified in the text. Quantifying Uncertainty Example A7 QUAM: 2000. P1 Page 89 Table A7.2. General procedure Step Description 1 Preparing the primary assay standard 2 Preparation of blends: b' and b 3 Measurement of isotope ratios 4 Calculation of the amount content of Pb in the sample, cx 5 Estimating the uncertainty in cx Calculation of the Molar Mass Due to natural variations in the isotopic composition of certain elements, e. g. Pb, the molar mass, M, for the primary assay standard has to be determined since this will affect the amount content cz. Note that this is not the case when cz is expressed in mol g 1. The molar mass, M( E), for an element E, is numerically equal to the atomic weight of element E, Ar( E). The atomic weight can be calculated according to the general expression:

( ) å å = = × = p i p i R M R A 1 i 1 i i r E ) E ( ( 6)

where the values Ri are all true isotope amount ratios for the element E and M( iE) are the tabulated nuclide masses.

Note that the isotope amount ratios in equation ( 6) have to be absolute ratios, that is, they have to be corrected for mass discrimination. With the use of proper indexes, this gives equation ( 7). For the calculation, nuclide masses, M( iE), were taken from literature values2, while Ratios, Rzi, and K0 factors, K0( zi), were measured ( see Table A7.8). These values give ( ) 1 1 i z i z 1 i z i z i z mol g 21034 . 207 E ) 1 Assay , Pb (

- = = = × × × = å å p i p i R K M R K M ( 7)

Measurement of K factors and isotope amount ratios To correct for mass discrimination, a correction factor, K, is used as specified in equation ( 2). The K0 factor can be calculated using a reference material certified for isotopic composition. In this case, the isotopically certified reference material NIST SRM 981 was used to monitor a possible change in the K0 factor. The K0 factor is measured before and after the ratio it will correct. A typical sample sequence is: 1. ( blank), 2. ( NIST SRM 981), 3. ( blank), 4. ( blend 1), 5. ( blank), 6. ( NIST SRM 981), 7. ( blank), 8. ( sample), etc.

The blank measurements are not only used for blank correction, they are also used for monitoring the number of counts for the blank. No new measurement run was started until the blank count rate was stable and back to a normal level. Note that sample, blends, spike and assay standard were diluted to an appropriate amount content prior to the measurements. The results of ratio measurements, calculated K0 factors and Kbias are summarised in Table A7.8.

Preparing the primary assay standard and calculating the amount content, cz.

Two primary assay standards were produced, each from a different piece of metallic lead with a chemical purity of w= 99.999 %. The two pieces came from the same batch of high purity lead. The pieces were dissolved in about 10 ml of 1: 3 w/ w HNO3: water under gentle heating and then further diluted. Two blends were prepared from each of these two assay standards. The values from one of the assays is described hereafter.

0.36544 g lead, m1, was dissolved and diluted in aqueous HNO3 ( 0.5 mol l 1) to a total of d1= 196.14 g. This solution is named Assay 1. A more diluted solution was needed and m2= 1.0292 g of Assay 1, was diluted in aqueous HNO3 ( 0.5 mol l 1) to a total mass of d2= 99.931g. This solution is named Assay 2. The amount content of Pb in Assay 2, cz, is then calculated according to equation ( 8)

( ) 1 1 8 1 1 2 2 g mol 092605 . 0 g mol 10 2605 . 9 1 , Pb 1 - - - m = ´ = × × × = Assay M d w m d m c z ( 8)

Preparation of the blends The mass fraction of the spike is known to be roughly 20 µ g Pb per g solution and the mass fraction of Pb in the sample is also known to be in this range. Table A7.3 shows the weighing data for the two blends used in this example. Quantifying Uncertainty Example A7 QUAM: 2000. P1 Page 90 Measurement of the procedure blank cBlank In this case, the procedure blank was measured using external calibration. A more exhaustive procedure would be to add an enriched spike to a blank and process it in the same way as the samples. In this example, only high purity reagents were used, which would lead to extreme ratios in the blends and consequent poor reliability for the enriched spiking procedure. The externally calibrated procedure blank was measured four times, and cBlank found to be 4.5 ´ 10 7 µ mol g 1, with standard uncertainty 4.0 ´ 10 7 µ mol g 1 evaluated as type A.

Calculation of the unknown amount content cx Inserting the measured and calculated data ( Table A7.8) into equation ( 5) gives cx= 0.053738 µ mol g 1. The results from all four replicates are given in Table A7.4.

A7.3 Steps 2 and 3: Identifying and quantifying uncertainty sources Strategy for the uncertainty calculation If equations ( 2), ( 7) and ( 8) were to be included in the final IDMS equation ( 5), the sheer number of parameters would make the equation almost impossible to handle. To keep it simpler, K0 factors and amount content of the standard assay solution and their associated uncertainties are treated separately and then introduced into the IDMS equation ( 5). In this case it will not affect the final combined uncertainty of cx, and it is advisable to simplify for practical reasons.

For calculating the combined standard uncertainty, uc( cx), the values from one of the measurements, as described in A7.2, will be used. The combined uncertainty of cx will be calculated using the spreadsheet method described in Appendix E. Uncertainty on the K factors i) Uncertainty on K0 K is calculated according to equation ( 2) and using the values of Kx1 as an example gives for K0:

9992 . 0 1699 . 2 1681 . 2 ) 1 (

observed certified 0 = = = R R x K ( 9)

To calculate the uncertainty on K0 we first look at the certificate where the certified ratio, 2.1681, has a stated uncertainty of 0.0008 based on a 95% confidence interval. To convert an uncertainty based on a 95% confidence interval to standard uncertainty we divide by 2. This gives a standard uncertainty of u( Rcertified)= 0.0004. The observed amount ratio, Robserved= n( 208Pb)/ n( 206Pb), has a standard uncertainty of 0.0025 ( as rsd). For the Kfactor the combined uncertainty can be calculated as:

( ) 002507 . 0 0025 . 0 1681 . 2 0004 . 0 ) 1 ( )) 1 ( ( 2 2 0 0 c = + ÷ ø ö ç è æ = x K x K u ( 10)

This clearly points out that the uncertainty contributions from the certified ratios are negligible. Henceforth, the uncertainties on the measured ratios, Robserved, will be used for the uncertainties on K0.