Suary Goal A calibration standard is prepared fro a high purity etal (cadiu) with a concentration of ca.1000 g l -1. Measureent procedure The surface of the high purity etal is cleaned to reove any etal-oxide containation. Afterwards the etal is weighed and then dissolved in nitric acid in a voluetric flask. The stages in the procedure are show in the following flow chart. Clean Clean etal etal surface where :concentration of the calibration standard [g l -1 ] 1000 :conversion factor fro [l] to [l] P :ass of the high purity etal [g] :purity of the etal given as ass fraction :volue of the liquid of the calibration standard [l] Identification of the uncertainty sources: The relevant uncertainty sources are shown in the cause and effect diagra below: Teperature Weigh Weigh etal etal Dissolve and and dilute dilute RESULT Figure A1. 1: Preparation of cadiu standard Measurand!! P 1000 [g l -1 ] (tare) Linearity Sensitivity Readability Readability Linearity (gross) Sensitivity c(cd) Quantification of the uncertainty coponents The values and their uncertainties are shown in the Table below. Cobined Standard Uncertainty The cobined standard uncertainty for the preparation of a 100.7 g l -1 Cd calibration standard is 0.9 g l -1 The different contributions are shown diagraatically in Figure A1.. Table A1.1: alues and uncertainties Description alue Standard uncertainty Relative standard uncertainty u(x)/x P of the etal 0.9999 0.000058 0.000058 Mass of the etal 100.8 g 0.05 g 0.0005 olue of the flask 100.0 l 0.07 l 0.0007 concentration of the calibration standard 100.7 g l -1 0.9 g l -1 0.0009 QUAM:000.1 Page 34
Figure A1.: Uncertainty contributions in cadiu standard preparation c(cd) 0 0. 0.4 0.6 0.8 1 u(y,x i ) (g l -1 ) The values of u(y,x i )(!y/!x i ).u(x i ) are taken fro Table A1.3 QUAM:000.1 Page 35
Exaple A1: Preparation of a calibration standard. Detailed discussion A1.1 Introduction This first introductory exaple discusses the preparation of a calibration standard for atoic absorption spectroscopy (AAS) fro the corresponding high purity etal (in this exaple!1000 g l -1 Cd in dilute HNO 3 ). Even though the exaple does not represent an entire analytical easureent, the use of calibration standards is part of nearly every deterination, because odern routine analytical easureents are relative easureents, which need a reference standard to provide traceability to the SI. A1. Step 1: Specification The goal of this first step is to write down a clear stateent of what is being easured. This specification includes a description of the preparation of the calibration standard and the atheatical relationship between the easurand and the paraeters upon which it depends. Procedure The specific inforation on how to prepare a calibration standard is norally given in a Standard Operating Procedure (SOP). The preparation consists of the following stages Figure A1.3: Preparation of cadiu standard Clean Clean etal etal surface Weigh Weigh etal etal The separate stages are: i) The surface of the high purity etal is treated with an acid ixture to reove any etaloxide containation. The cleaning ethod is provided by the anufacturer of the etal and needs to be carried out to obtain the purity quoted on the certificate. ii) The voluetric flask (100 l) is weighed without and with the purified etal inside. The balance used has a resolution of 0.01 g. iii) 1 l of nitric acid (65% /) and 3 l of ionfree water are added to the flask to dissolve the cadiu (approxiately 100 g, weighed accurately). Afterwards the flask is filled with ion-free water up to the ark and ixed by inverting the flask at least thirty ties. Calculation: The easurand in this exaple is the concentration of the calibration standard solution, which depends upon the weighing of the high purity etal (Cd), its purity and the volue of the liquid in which it is dissolved. The concentration is given by where " " P 1000 g l -1 :concentration of the calibration standard [g l -1 ] 1000 :conversion factor fro [l] to [l] P :ass of the high purity etal [g] :purity of the etal given as ass fraction :volue of the liquid of the calibration standard [l] Dissolve and and dilute dilute RESULT A1.3 Step : Identifying and analysing uncertainty sources The ai of this second step is to list all the uncertainty sources for each of the paraeters which affect the value of the easurand. QUAM:000.1 Page 36
The purity of the etal (Cd) is quoted in the supplier's certificate as 99.99 ±0.01%. P is therefore 0.9999 ±0.0001. These values depend on the effectiveness of the surface cleaning of the high purity etal. If the anufacturer's procedure is strictly followed, no additional uncertainty due to the containation of the surface with etaloxide needs to be added to the value given in the certificate. There is no inforation available that 100% of the etal dissolves. Therefore one has to check with a repeated preparation experient that this contribution can be neglected. Mass The second stage of the preparation involves weighing the high purity etal. A 100 l quantity of a 1000 g l -1 cadiu solution is to be prepared. The relevant ass of cadiu is deterined by a tared weighing, giving 0.1008 g The anufacturer s literature identifies three uncertainty sources for the tared weighing: the repeatability; the readability (digital resolution) of the balance scale; and the contribution due to the uncertainty in the calibration function of the scale. This calibration function has two potential uncertainty sources, identified as the sensitivity of the balance and its linearity. The sensitivity can be neglected because the ass by difference is done on the sae balance over a very narrow range. NOTE: Buoyancy correction is not considered because all weighing results are quoted on the conventional basis for weighing in air [H.19]. The reaining uncertainties are too sall to consider. Note 1 in Appendix G refers. olue The volue of the solution contained in the voluetric flask is subject to three ajor sources of uncertainty: The uncertainty in the certified internal volue of the flask. ariation in filling the flask to the ark. The flask and solution teperatures differing fro the teperature at which the volue of the flask was calibrated. The different effects and their influences are shown as a cause and effect diagra in Figure A1.4 (see Appendix D for description). Figure A1.4: Uncertainties in Cd Standard preparation Teperature (tare) Linearity Sensitivity Readability Readability Linearity (gross) Sensitivity c(cd) A1.4 Step 3: Quantifying the uncertainty coponents In step 3 the size of each identified potential source of uncertainty is either directly easured, estiated using previous experiental results or derived fro theoretical analysis. The purity of the cadiu is given on the certificate as 0.9999 ±0.0001. Because there is no additional inforation about the uncertainty value, a rectangular distribution is assued. To obtain the standard uncertainty u(p) the value of 0.0001 has to be divided by 3 (see Appendix E1.1) Mass 0.0001 u( P) 0.000058 3 The uncertainty associated with the ass of the cadiu is estiated, using the data fro the calibration certificate and the anufacturer s recoendations on uncertainty estiation, as 0.05 g. This estiate takes into account the three contributions identified earlier (Section A1.3). NOTE: Detailed calculations for uncertainties in ass can be very intricate, and it is iportant to refer to anufacturer s literature where ass uncertainties are doinant. In this exaple, the calculations are oitted for clarity. QUAM:000.1 Page 37
olue The volue has three ajor influences; calibration, repeatability and teperature effects. i) : The anufacturer quotes a volue for the flask of 100 l ±0.1 l easured at a teperature of 0 C. The value of the uncertainty is given without a confidence level or distribution inforation, so an assuption is necessary. Here, the standard uncertainty is calculated assuing a triangular distribution. 0.1l 0.04 l 6 NOTE: A triangular distribution was chosen, because in an effective production process, the noinal value is ore likely than extrees. The resulting distribution is better represented by a triangular distribution than a rectangular one. ii) : The uncertainty due to variations in filling can be estiated fro a repeatability experient on a typical exaple of the flask used. A series of ten fill and weigh experients on a typical 100 l flask gave a standard deviation of 0.0 l. This can be used directly as a standard uncertainty. iii) Teperature: According to the anufacturer the flask has been calibrated at a teperature of 0 C, whereas the laboratory teperature varies between the liits of ±4 C. The uncertainty fro this effect can be calculated fro the estiate of the teperature range and the coefficient of the volue expansion. The volue expansion of the liquid is considerably larger than that of the flask, so only the forer needs to be considered. The coefficient of volue expansion for water is.1!10 4 C 1, Table A1.: alues and Uncertainties Description alue x u(x) u(x)/x of the etal P Mass of the etal (g) olue of the flask (l) 0.9999 0.000058 0.000058 100.8 0.05 g 0.0005 100.0 0.07 l 0.0007 which leads to a volue variation of ± (100! 4!.1! 10 " 4 ) ± 0.084 l The standard uncertainty is calculated using the assuption of a rectangular distribution for the teperature variation i.e. 0.084 l 0.05 l 3 The three contributions are cobined to give the standard uncertainty u() of the volue u( ) 0.04 + 0.0 + 0.05 0.07 l A1.5 Step 4: Calculating the cobined standard uncertainty is given by 1000 # # P -1 [g l ] The interediate values, their standard uncertainties and their relative standard uncertainties are suarised overleaf (Table A1.) Using those values, the concentration of the calibration standard is 1000! 100.8! 0.9999 " 100.7 g l 100.0 For this siple ultiplicative expression, the uncertainties associated with each coponent are cobined as follows. uc ( c c Cd Cd ) 0.000058 0.0009 u ( c c Cd ) c Cd & u( P) # $! % P " 0.9 g l + 0.0005 " 1 & u( ) # + $! % " + 0.0007 & u( ) # + $! % "! 0.0009 100.7 g l " 1 1! 0.0009 It is preferable to derive the cobined standard uncertainty (u c ( )) using the spreadsheet ethod given in Appendix E, since this can be utilised even for coplex expressions. The copleted spreadsheet is shown in Table A1.3. The contributions of the different paraeters are shown in Figure A1.5. The contribution of the uncertainty on the volue of the flask is the QUAM:000.1 Page 38
largest and that fro the weighing procedure is siilar. The uncertainty on the purity of the cadiu has virtually no influence on the overall uncertainty. The expanded uncertainty U( ) is obtained by ultiplying the cobined standard uncertainty with a coverage factor of, giving U ( ) " 0.9 g l Table A1.3: Spreadsheet calculation of uncertainty! 1 1.8g l A B C D E 1 P alue 0.9999 100.8 100.00 3 Uncertainty 0.000058 0.05 0.07 4 5 P 0.9999 0.999958 0.9999 0.9999 6 100.8 100.8 100.33 100.8 7 100.0 100.00 100.00 100.07 8 9 c(cd) 100.6997 100.75788 1003.19966 1001.9983 10 u(y,x i ) 0.05816 0.49995-0.70140 11 u(y), u(y,x i ) 0.7459 0.00338 0.4995 0.49196 1 13 u(c(cd)) 0.9 The values of the paraeters are entered in the second row fro C to E. Their standard uncertainties are in the row below (C3-E3). The spreadsheet copies the values fro C-E into the second colun fro B5 to B7. The result (c(cd)) using these values is given in B9. The C5 shows the value of P fro C plus its uncertainty given in C3. The result of the calculation using the values C5-C7 is given in C9. The coluns D and E follow a siilar procedure. The values shown in the row 10 (C10-E10) are the differences of the row (C9-E9) inus the value given in B9. In row 11 (C11-E11) the values of row 10 (C10-E10) are squared and sued to give the value shown in B11. B13 gives the cobined standard uncertainty, which is the square root of B11. Figure A1.5: Uncertainty contributions in cadiu standard preparation! 1 c(cd) 0 0. 0.4 0.6 0.8 1 u(y,x i ) (g l -1 ) The values of u(y,x i )(#y/#x i ).u(x i ) are taken fro Table A1.3 QUAM:000.1 Page 39
Appendix E Statistical Procedures Appendix E. Useful Statistical Procedures E.1 Distribution functions The following table shows how to calculate a standard uncertainty fro the paraeters of the two ost iportant distribution functions, and gives an indication of the circustances in which each should be used. EXAMPLE A cheist estiates a contributory factor as not less than 7 or ore than 10, but feels that the value could be anywhere in between, with no idea of whether any part of the range is ore likely than another. This is a description of a rectangular distribution function with a range a3 (sei range of a1.5). Using the function below for a rectangular distribution, an estiate of the standard uncertainty can be calculated. Using the above range, a1.5, results in a standard uncertainty of (1.5/!3) 0.87. Rectangular distribution For Use when: Uncertainty a ( ±a ) A certificate or other specification gives liits without specifying a level of confidence (e.g. 5l ± 0.05l) u ( x) a 3 1/a An estiate is ade in the for of a axiu range (±a) with no knowledge of the shape of the distribution. x Triangular distribution For Use when: Uncertainty 1/a a ( ± a ) The available inforation concerning x is less liited than for a rectangular distribution. alues close to x are ore likely than near the bounds. An estiate is ade in the for of a axiu range (±a) described by a syetric distribution. a u ( x) 6 x QUAM:000.1 Page 10
Appendix E Statistical Procedures Noral distribution For Use when: Uncertainty "! An estiate is ade fro repeated observations of a randoly varying process. An uncertainty is given in the for of a standard deviation s, a relative standard deviation s / x, or a coefficient of variance C% without specifying the distribution. u(x) s u(x) s u(x)x!( s/ x) u(x) C%! x 100 x An uncertainty is given in the for of a 95% (or other) confidence interval x±c without specifying the distribution. u(x) c / (for c at 95%) u(x) c/3 (for c at 99.7%) QUAM:000.1 Page 103