Determination of Uncertainty for Volume …

1 This application note provides the practical evaluation of Uncertainty forvolume measurements according to the Guide to the Expression ofUncertainty in measurement (GUM).1 3 ISO 8655-74covers nongravimetricmethods of Volume measurement . This methodology can serve as a rapid,automated test in lieu of traditional burette certification procedures recom-mended by manufacturers of automated titration titration of sodium chlorideFor the following example, a 20-mL cylinder built into a titrator was data were generated using a Titroline alpha plus and TW alpha samplechanger (Schott Instruments,Mainz, Germany). The requirements for sys-tematic error of a 20-mL cylinder are preparationThe sample, NaCl analytical degree (article no. 106404, Merck KGaA,Darmstadt, Germany), had a molecular weight of g/mol. Because of thevery small weight of about 11 mg for the 10% Volume of the cylinder, a knownsolution of NaCl was made.

2 Portions of this solution were used to get a higherweight instead of direct weight of solid NaCl. The test solution (sodium chloride)was as follows: NaCl weight g; water weight g. Becauseevery gram of the solution contained mg NaCl, about 1 g for 10% of thevolume of the cylinder had to be weighed instead of 11 solutions used were polyvinylalcohol (Merck114266) , nitric acid (Merck100452) diluted 1:4 with deionized (DI) water,and chloride-free DI titration sample was prepared in the following manner. The test solu-tion was weighed (see experimental results) and filled to 80 mL with DIwater. Five milliliters of polyvinylalcohol solution and 2 mL of nitric acidwere procedureAll titrations were done with the addition of dynamic titrant and precisedrift control with a sample changer. Three blank samples were followed by10 titrations (approx. 10% up to about 100% of the cylinder Volume of 20mL).

3 After each titration, three rinsing positions were used to clean theelectrode and titration comparison purposes, another procedure was employed as well. Thiscomprised 10 titrations for every Volume of about 10%, 50%, and 100% ofthe cylinder Volume , as typically outlined in traditional certification proce-dures: electrode silver chloride with Ag/AgCl reference electrode(NaNO3electrolyte); reagent mol/L (Merck109081).Cause and effectEach step in sample preparation analysis has an associated uncer-tainty. Cause-and-effect diagrams very clearly show the influence ofthese in NaClThe weight of the NaCl depends on the real molecular weight of the actualNaCl used; the error of the balance; and the purity of the salt, as specifiedby the producer. The main factors are purity and weighing error. Assumingprofessional laboratory personnel are performing the analyses, it can be sur-mised that only the specified error of the balance is used.

4 In the presentcase, a differential weight was not used but, rather, a well-dried the air moisture is usually about 50%, the weighing processshould be fast enough to avoid any moisture adsorption in this short periodof with waterThe weight of the water follows the same procedure. While the NaCl isweighed on an analytical balance with four decimal places, the water isweighed on a balance with two decimal places. Water has significant steampressure, which must be taken into the test solutionThe test solution contains the Uncertainty of the NaCl and the water. Inaddition, the single samples have some uncertainties. The weighing is doneon a differential basis. Every weight has an Uncertainty , the evaporation ofthe water important among them. Use of a syringe with small openings andnearly no evaporation is the preferred weighing diagramThe total diagram includes all of the single uncertainties.

5 The Volume ofthe burette and the reproducibility of the titration must be taken of singleuncertaintiesMolecular weight of NaClThe molecular weight of NaCl has an unexpected influence due to the highuncertainty of the chloride ion. The Uncertainty of all elements is publishedannually by the International Union of Pure and Applied Chemistry(IUPAC) and on the Internet. IUPAC recommends a rectangular distribu-tion. Therefore, every value ( , Cl) must be calculated as follows (seeTable 1also):uCl= The standard Uncertainty of NaCl is calculated by:uNaCl= (u Cl )2 + ( u N a) 2 where uClis the Uncertainty of Cl and uNaClis the Uncertainty of 3 14/OCTOBER 2004 AMERICAN LABORATORYAPPLICATION NOTED etermination of Uncertainty forVolume Measurements Made Using theTitration Methodby J rgen 01:00 AM Page 1416/OCTOBER 2004 AMERICAN LABORATORYW eighing error NaClThe NaCl is weighed on an analytical balance.

6 The linear error is definedby the manufacturer as mg in the range used. In this case, a rectangu-lar distribution is recommended as where umKClis the Uncertainty of weighted mass back weight was used. The salt was weighed onto weighing paper, whichhad no influence on the weight. The Uncertainty is thereforeumNaCl= mgwhere umNaClis the Uncertainty of weighted mass of NaClThe purity of the NaCl is specified by the manufacturer as P= Since a rectangular distribution is assumed, the Uncertainty is:uPNaCl= uPNaCl= uPNaClis the Uncertainty purity of weightThe combined Uncertainty for the NaCl weight is thereforeuwNaCl= where uwNaClis the Uncertainty weight of value is calculated asuNaCl= + + = mg 9 Evaporating waterSome tests gave a rate of evaporation of mg/min. The procedure neededa maximum of 1 min. Of course, the rate depends on the type of vessel. Inthe present tests, a 1-L glass bottle with a GL 45 opening was distribution was assumed to be:upwater= upwater= gwhere upwateris the Uncertainty evaporation of error waterFor this procedure, a two-decimal balance was used with a linearity of and a random error of g.

7 Each error is based on a rectangular distri-bution. Therefore, two uncertainties give a triangular 3 (umNaCl+ uPNaCl+ uNaCl) 9 3 3 APPLICATION NOTEum1water= um1water= gum2water= um2water= gwhere um1wateris the Uncertainty of the water weight based on the linearityerror of the balance, um2wateris the Uncertainty of water based on the ran-dom error of the balance. umwateris the combined Uncertainty of the waterbased on the errors of the combined Uncertainty is:umwater=umwater= + = g 6 Purity of waterFor acid base titrations, the risk of contaminated water is much higherthan for chloride determinations. In the present experiments, 21 blanksamples were run. No chloride contamination was found in any of thesetitrations. One producer specifies mg/L as possible on a rectangular distribution:upuwater= upuwater= mg/Lwhere upwateris the Uncertainty purity of weightThe combined Uncertainty for the water weight is:uwater=where uwateris the Uncertainty of water weight, upwateris the Uncertainty ofwater based on evaporation, upuwateris the Uncertainty purity of water, andumwateris the Uncertainty weight of combined value for the water is:uNaCl= + + = g 9 Weight test sampleThe solid NaCl was weighed directly.

8 The single portions are weighed in asyringe. In the first step the syringe was weighed with NaCl solution inwater; in the second step the syringe was weighed back empty. The differ-ence is the NaCl solution weight. Thus, the Uncertainty of the linearity ofthe balance has to be calculated 2 *( 0 .0 9 )2 = mgwhere utsis the Uncertainty weight of the of waterSee above 1000upwater+ upuwater+ umwater 9 3 10000um1water+ um2water 6 3 3 Table 1 Determination of Uncertainty for Volume measurementsElementAtomic mol wtUncertaintyStandard 01:00 AM Page 1618/OCTOBER 2004 AMERICAN LABORATORY upwater= upwater= gTest solution of NaClThe combined Uncertainty for the test solution in the case of a 1-g sampleis:uTest= + + + uTest= 9 = gwhere uTestis the Uncertainty of test solution NaCl in water used for titra-tion, upwateris the Uncertainty evaporation of water test sample, utsis theuncertainty weight of test sample, uNaClis the Uncertainty of NaCl weight,and uwateris the Uncertainty buretteThe Volume of the burette of the titrator is defined by three 1upwater+ uts+ uNaCl+ uwater 9 3 APPLICATION NOTE Uncertainty of systematic error.

9 The manufacturer s specifications of the cylinder Volume , , with a triangular distribution ofuVs= = mLwhere uVsis the Uncertainty of the cylinder Volume burette. Uncertainty of random error, which is following the same assump-tion:uVs= = mL Uncertainty of influence of temperature; for a 19-mL Volume and a waterexpansion coefficient of C, a temperature change of 3 C anda 95% confidence level is achieved:uVt= = mLThe Uncertainty of the Volume can be calculated by:uvol= (0 .0 0 8 2 )2 + ( 0 .0 0 5 7 )2 + ( 0 .0 0 6 )2 = mLwhere uvolis the Uncertainty of the reproducibility of the titration, remaining after these points, dependsmainly on such parameters as electrode behavior, correct titration parameters,equivalence point evaluation, and signal-to-noise ratios. Since it is preferableto avoid these problems than to correct them, the researchers employed pre-cise parameters and were able to calculate error-free equivalence signal-to-noise ratio for a chloride titration in the case of low concen-trations is 1/300; it can reach 1/1000 in other cases.

10 In some instances, theuncertainty of the titration has been a practical point of view, an Uncertainty would be defined as:utitration= mLTitration resultThe Uncertainty of the burette is the sought-after parameter. The titrationresult is calculated without the Volume Uncertainty of the Volume by:uTi t (0 .0 0 0 1 1 )2 + ( 0 .0 0 1 )2 = mLwhere uTi tis the Uncertainty of resultsThe titer of the AgNO3solution was determined independently via manytitrations and was found to be The calculation of the NaCl byweight is:NaCl [mg] = weight [g] * [mg/g]The NaCl found by the titration is calculated as follows (see Table 2also):NaCl [mg] = EQ [mL] * [mg/mL] * RSD is sufficient for this method (see Table 3). The linearity of themethod is shown in Figure volumes for 10%, 50%, and 100% (see Table 2) were marked and com-pared using a procedure of 10 titrations for every Volume . The results werecomparable (see Tables 4 6).