DFM-2015-R03 Transferring the unit of mass between weights ... · DFM‐2015‐R03 Page 2 of 16 O...
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DFM‐2015‐R03 Page 1 of 16 Transferring the unit of mass between weights kept in air and in vacuum Lars Nielsen, DFM Abstract In the current watt balance and crystal density measurements used to assign a value to the Planck constant h in terms of the international prototype of the kilogram , the unit of mass has to be transferred from air to vacuum. In the future, when the kilogram has been redefined in terms of a fixed value of the Planck constant, the unit of mass has to be transferred from vacuum to air. This report describes how this mass transfer may be done in a way that enables evaluation of the standard uncertainty associated with the mass transfer. 1. Introduction The unit of mass, kilogram is currently defined as the mass of the international prototype of the kilogram immediately after cleaning and washing using a specified procedure involving solvents and steamሾ1ሿሾ2ሿሾ3ሿ. As the prototype is kept and used in humid air, a natural layer of adsorbed water on the surface of the artefact is contributing to the unit of mass. When measuring the Planck constant in the current watt balance experiments or the Avogadro constant in the current crystal density experiment, mass standards ሺweightsሻ that are traceable to the prototype but used in vacuum are required. Once a new definition of the kilogram in terms of a fixed value of the Planck constant is in place, the kilogram will be realized in vacuum and needs to be transferred to air before being disseminated. A gravimetric method is often used to measure the mass change due to water sorption/desorption associated with the transfer of weights between air and vacuum. This method uses at least two mass standards ሺknown as sorption artefactsሻ having the same nominal values of mass and volume, the same surface finish, but a large difference in their surface areas. Based on the assumption that the sorption per unit area is the same for the two sorption artefacts, the change in sorption can be calculated from a measured change in mass difference between the two sorption artefacts when transferred between air and vacuum. Experiments have shown that the sorption is not always reversible. The modelling described in this report accounts for that. 2. Modelling Assume that sorption artefact S is cycled repeatedly between the two media air ሺAሻ and vacuum ሺBሻ. The cycling is described by a media sequence ܯൌ ሺA, B, A, B, … ሻ, so that ܯଵ ൌ A, ܯଶ ൌ B, ܯଷ ൌ A, etc. When the sorption artefact is transferred from medium ܯ to medium ܯାଵ the mass of the artefact is assumed to change according to the model: ୗ,ାଵ ൌ ୗ, ୗ ܣୗ ݏ , ሺ1ሻ where ௌ, is the mass of the artefact S in medium ܯ , ௌ,ାଵ is the mass of the artefact S in medium ܯାଵ , ܣୗ is the geometrical surface area of the artefact S,
Transcript of DFM-2015-R03 Transferring the unit of mass between weights ... · DFM‐2015‐R03 Page 2 of 16 O...
DFM‐2015‐R03 Page1of16
TransferringtheunitofmassbetweenweightskeptinairandinvacuumLarsNielsen,DFM
Abstract
InthecurrentwattbalanceandcrystaldensitymeasurementsusedtoassignavaluetothePlanckconstanthintermsoftheinternationalprototypeofthekilogram ,theunitofmasshastobetransferredfromairtovacuum.Inthefuture,whenthekilogramhasbeenredefinedintermsofafixedvalueofthePlanckconstant,theunitofmasshastobetransferredfromvacuumtoair.Thisreportdescribeshowthismasstransfermaybedoneinawaythatenablesevaluationofthestandarduncertaintyassociatedwiththemasstransfer.
1. IntroductionTheunitofmass,kilogramiscurrentlydefinedasthemassoftheinternationalprototypeofthekilogram immediatelyaftercleaningandwashingusingaspecifiedprocedureinvolvingsolventsandsteam 1 2 3 .Astheprototype iskeptandusedinhumidair,anaturallayerofadsorbedwateronthesurfaceoftheartefactiscontributingtotheunitofmass.WhenmeasuringthePlanckconstant inthecurrentwattbalanceexperimentsortheAvogadroconstant inthecurrentcrystaldensityexperiment,massstandards weights thataretraceabletotheprototype butusedinvacuumarerequired.OnceanewdefinitionofthekilogramintermsofafixedvalueofthePlanckconstantisinplace,thekilogramwillberealizedinvacuumandneedstobetransferredtoairbeforebeingdisseminated.
Agravimetricmethodisoftenusedtomeasurethemasschangeduetowatersorption/desorptionassociatedwiththetransferofweightsbetweenairandvacuum.Thismethodusesatleasttwomassstandards knownassorptionartefacts havingthesamenominalvaluesofmassandvolume,thesamesurfacefinish,butalargedifferenceintheirsurfaceareas.Basedontheassumptionthatthesorptionperunitareaisthesameforthetwosorptionartefacts,thechangeinsorptioncanbecalculatedfromameasuredchangeinmassdifferencebetweenthetwosorptionartefactswhentransferredbetweenairandvacuum.
Experimentshaveshownthatthesorptionisnotalwaysreversible.Themodellingdescribedinthisreportaccountsforthat.
2. ModellingAssumethatsorptionartefactSiscycledrepeatedlybetweenthetwomediaair A andvacuum B .Thecyclingisdescribedbyamediasequence A, B, A, B, … ,sothat A, B, A,etc.Whenthesorptionartefactistransferredfrommedium tomedium themassoftheartefactisassumedtochangeaccordingtothemodel:
, , , 1
where
, isthemassoftheartefactSinmedium ,
, isthemassoftheartefactSinmedium ,
isthegeometricalsurfaceareaoftheartefactS,
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isthechangeinsorptionperunitsurfaceareaoftheartefactwhentransferredfrommedium tomedium ,
isafactorforthesorptionartefactSconvertingitsgeometricsurfaceareatoaneffectivesurfaceareaforthesorption.
Notethatthequantity maybenegativeorpositivedependingonthemediabetweenwhichtheartefactistransferred.
Ifasetof sorptionartefactsS , S , . . . , S arecycledbetweenthetwomedia,equation 1 isreplacedby
, ,
, ,
⋮, ,
2
Thesuccessfuluseofsorptionartefactsrelyontheassumptionsthatthechangeinsorption isthesameforallsorptionartefactsinvolvedinthesametransferbetweenmediaAandB.Inordertotestthatassumption,atleastthreesorptionartefactsneedtobeused.Uptonowithasbeenusualpracticetoassumethatthefactors , … , areidentical andequalto1exactly .Thisimpliesthatthe
sorption isdefinedintermsofgeometricarearatherthanintermsofeffectivearea,andthattheratioofeffectiveareatogeometricalareaisthesameforallsorptionartefactsinaset.Thefactors
, … , canbeinterpretedastheratiosofeffectiveareatogeometricareasofthesorption
artefacts.Ifthesurfaceroughnessesoftheartefactshavebeenmeasured,someinformationabouttheseratiosisavailable.Inthatcase,bestestimates largerthan1 andassociatedstandarduncertaintiesmaybeassignedtothefactors , … , .Ifnoinformationoftheeffectiveareasofthe
artefactsisavailable,variationinthesurfacefinishofthesorptionartefactsinasetmaybetakenintoaccountbyassigningapriorvalue1toallfactors , … , butwithnon‐zerostandarduncertainties.
AssumethatoneweightM ispermanentlystoredinair mediumA ,andanotherweightM ispermanentlystoredinvacuum mediumB ;bothweightshavethesamenominalmassvaluesasthesorptionartefactsS , S , . . . , S .Thegoalistomeasurethemass oftheweightM storedinvacuumintermsofthemass oftheweightM storedinair,orviceversa.ThevolumesoftheweightsM andM aredenoted respectively ,andthevolumesofthesorptionartefactsS , S , . . . , S aredenoted , , . . . , .
Atstage inthemediasequencethesorptionartefactsS , S , . . . , S arecomparedwitheachotherandwitheithertheweightstoredinairortheweightstoredinvacuum.Thiscomparisonismodelledby
∆ 3
where
,, , , , … , , isavectorofthemassvaluesoftheweightsandthesorption
artefactsatstage inthemediasequence, , , , … , isavectorofthevolumevaluesofthemassandsorptionartefacts
involvedinthecomparisons, isthedesignmatrixaccordingtowhichthemassandsorptionartefactshavebeencomparedat
stage inthemediasequence, isthevectorofairdensitiesmeasuredforeachmasscomparisonatstage inthemedia
sequence,
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∆ isthevectorofindicationdifferencesmeasuredforeachmasscomparisonperformedatstage ofthemediasequence,
isthescalefactorofthemasscomparatoratstage ofthemediasequence.
Whenweighingisdoneinair A ,theweightM storedinvacuumisnotinvolvedinthemasscomparison,soif A,thentheelementsofthesecondcolumnof areallzero.Similarly,whenweighingaredoneinvacuum B ,theweightM isnotinvolvedinthecomparisons,soif B,thentheelementsofthefirstcolumnof areallzero.Themasses and areassumedtoconstant;thisassumptionisplausibleonlyiftheweightM iskeptpermanentlyinair,andtheweightM iskeptinpermanentvacuum.Whenweighingisdoneinvacuum B ,theairdensityisnegligible,soif B,then .
Theequations 2 and 3 formthenecessarybasisfortransferringtheunitofmassbetweenairandvacuum.TheseequationsareeasilysolvedbythegeneralmethodofleastsquaresdevelopedatDFM 4 5 ,knownasDFM‐LSQ.Toillustratethis,considerthatthemassoftheweightM keptinairisknown,andthatwewanttomeasurethemassoftheweightM keptinvacuumusingtwosorptionartefactsS andS .Iftheshortestpossiblemediasequence A, B isapplied,threemassdifferencesmaybemeasuredinair,
Δ ,
Δ ,
Δ ,
1 1 01 0 10 1 1
,
,
, , 0
, 0 ,
0 , ,
,
,
, 4
andthreemassdifferencesmaybemeasuredinvacuum:
Δ ,
Δ ,
Δ ,
1 1 01 0 10 1 1
,
,
. 5
Themassesofthesorptionartefactsinairandinvacuumarerelatedtoasingle negative sorptioncoefficient throughtheequations
,
,
,
,6
Inthismeasurementthereare 6quantities,forwhichnopriorinformationisavailable,
,,,
,,
,,
,, , 7
and 16quantities
, , Δ , , . . , Δ , , , ,,,
,, , , , 8
forwhichwehavebestestimates andanassociatedcovariancematrix .Amongthequantities andthereare 8constraintsontheform
, , 9
whicharederivedfromequations 4 , 5 ,and 6 bysubtractiontherighthandsidesfromthelefthandsidesoftheequations.Followingref. 5 ,bestestimates and ofthequantities and ,aswellastheassociatedcovariancematrix,isfoundbyminimisingthechi‐squarefunction
10
subjecttotheconstraint , .Theconsistencyofmodelanddatashouldbecheckedbycomparingtheobservedchi‐squarevalue
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11
withitsexpectationvalue,whichisequaltothedegreesoffreedom oftheleastsquaresadjustment:
, 12
where isthenumberofconstraintsand numberofquantities,forwhichnopriorinformationisavailable.Assumingthat istheoutcomeofachisquaredistribution ,consistencybetweendataandmodelmaybequestionedif
Pr , 13
where istheprobabilitythat bychance theoutcomeofachi‐squaredistribution islargerthanthevalueactuallyobserved,and isachosenlevelofsignificance.Itisrecommendedtousealevelofsignificanceintherange0.1 0.2.
Thelargerthenumberofdegreesoffreedomare,thestrongeristheconsistencytestexpressedbyequation 13 .Inthecasedescribedabove,wehave 8 6 2degreesoffreedomfortheleastsquaresadjustment;thesetwodegreesoffreedomwouldreducetozero,ifthetwo redundant masscomparisonofthesorptionartefactsinequations 4 5 wereomitted.Evenifthetworedundantmasscomparisonswereincluded,therewouldbenoredundancyinthedeterminationofthesorptioncoefficient.Inordertoobtainsuchredundancy,athirdsorptionartefactwouldhavetobeincluded.Assumingthatthemassvalues and areconstant,furtherredundancywouldbeobtainedbyexpandingthemediacyclefromjusttwostagestothreeormorestages.
Inadditiontothegeneraltestforconsistencydescribedabove,itisrecommendedtotestforconsistencybetweeneachmeasuredvalue inthearray withthecorrespondingadjustedvalue inthearray .Thisisdonebycalculatingthenormaliseddeviations
, 1, … , . 14
Measuredvalues forwhich| | 2arepotentialoutliers,althoughthereisaprobabilityofabout0.05thatthiscouldhappenbychance,assumingthat followsanormaldistributionN 0,1 .
3. AnalysisofsorptiondataUnfortunatelytherearenodataavailableforamasstransferbetweenmassandvacuum,whichhasbeencarriedoutusingtherecommendedprocedure.However,inasorptioncomparisoncarriedoutinthetaskgroupCCMWGMTG1,datawerereportedthathavebeenusedtodemonstratehowthemethodworksandhowaccurateitis.
IntheTG1sorptioncomparisonthreesorptionartefactS , S ,andS werecirculatedamonganumberofparticipants.ThesorptionartefactS wasanintegralweightofstainlesssteelidentifiedas‘71DD’,S wasastackoftwostainlesssteeldiscsidentifiedas‘Stack1’,andS wasastackoffourstainlesssteeldiscsidentifiedas‘Stack2’.Themeasuredvolumesandareasarelistedintable1.
Table1.Measuredvolumesandsurfaceareasofsorptionartefacts
S1 S2 S3V /cm3 126.7730 126.7737 126.7700
u (V )/cm3 0.0011 0.0014 0.0010
A /cm2 140.0 188.3 285.4
u (A )/cm22.3 2.6 3.3
K48(m1 kg)/mg 0.160
u (m )/mg 0.008
V /cm346.4879
u (V )/cm30.0005
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Table2.MasscorrectionandvolumeofDanish nationalprototypeK48Eachparticipatinglaboratorydidmeasurementsinair mediumA andvacuum mediumB usingamediacycle A, B, … , withatleastonemeasurementinvacuumandtwomeasurementsinair.Thelaboratoriescomparedthethreesorptionartefactswitheachotherinairandinvacuum,andwithanunspecifiedreferencestandardkeptinair.Thesorptionartefactswerenotcomparedtoaweightkeptincontinuousvacuum.Ateachstageinthemediasequence,thelaboratoriesreportedthemeasuredmassvalueof andthethreemassdifferencesmeasuredbycomparingS , S ,andS .Theaverageairdensitiesassociatedwiththemasscomparisonsinairwerealsoreported.
Inordertoassesstheuncertaintyatwhichthemassofanationalprototypekeptinaircouldbetransferredfromairtovacuum,theindicationdifferences,whichwouldhavebeenobservedbycomparingtheintegralsorptionartefactS withtheDanishprototypeK48inair,werecalculatedusingtheairdensitiesandmassvaluesofS reportedbythelaboratories,thevolumeofS listedintable1andthemass andvolume oftheprototypelistedintable2.Thecalculationwasdoneusingequation 3 with 1 exactly .Similarly,allreportedmassdifferencesmeasuredinairwereconvertedintoindicationdifferences.
Measurementdatawerereconstructedthiswayforfiveparticipatinglaboratoriesthatprovidedthenecessaryinformation:BIPM,LNE,PTB,METAS,andINRIM.ThereconstructedmeasurementdatawereanalysedusingDFM‐LSQ.Acommonstandarduncertainty Δ 0.001mgwereassignedtothereconstructedindicationdifferencesΔ ,andforthefactors , , usedtodescribevariations
inthesorptionefficienciesofthesurfacesofthesorptionartefacts,apriorvalueequalto1withstandarduncertainty 0.1wasassignedtoeachsorptionartefact.
Theresultsoftheanalysisofthereconstructedmeasurementdataareshowninfigure1inthecaseofINRIM.Thefigureindicatesthatthesorptionisnotreversible;whatisremovedfromthesurfaceinthefirsttransitionfromairtovacuumislessthanthatremovedinthesubsequenttransitionsfromairtovacuum.Furthermore,moreisaddedtothesurfacebythetransferfromvacuumtoairthanthatremovedbytheprecedingtransitionfromairtovacuum.Asaresult,themassinairofthesorptionartefactsincreasesfromonestageinairtothenext,andthesameistrueforthemassinvacuum.Thisiswhyitisrecommendedtocomparethesorptionartefactswithaweightpermanentlykeptinvacuumaswellaswithaweightpermanentlykeptinair.Bydoingsoitcanbetestedifthemassdifferencebetweentheweightkeptinairandtheweightkeptinvacuumisconstantwithinthemeasurementuncertainty.
InthecaseofINRIM,thedegreesoffreedomoftheleastsquaresanalysisis 21,andtheobservedchi‐squarevalueis 2.5.Inthiscase,thereisnoindicationofinconsistenciesbetweendataandmodel.Thisisconfirmedbythenormaliseddeviationsshowninfigure2,whichallfallswellwithintherange| | 2.Iftheprioruncertaintiesofthefactors , , arereducedbyafactorof10 from
0.1to 0.01 ,theobservedchi‐squarevalueincreasesfrom 2.5to 11.1,whichalsonotlargerthanexpected.However,thenormaliseddeviationsassociatedthesurfaceareas
, , andthefactors , , failtomeetthecriteria| | 2byasignificantmargin
2.4 | | 3.1 .Thisindicatesthatthereisasignificantdifferenceinthesorptionefficienciesofthesurfacesofthethreesorptionartefacts.Hadonlytwosorptionartefactsbeenused,suchadifferencecouldnothavebeendetected.
Thestandarduncertaintiesofthemassvaluesfoundforthesorptionartefactsinairisabout0.010mg,whereasthestandarduncertaintiesofthemassvaluesfoundinvacuumareabout0.008mg.This
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meansthattheuncertaintyassociatedwiththemasstransferfromairtovacuum orfromvacuumtoair isabout0.006mg.
FormtheanalysisofthedataprovidedbyBIPM,LNE,PTB,METAS,andINRIMintheCCMWGMTG1sorptioncomparison,fivesetsoffactors , , werefoundforthecirculatedsorptionartefacts
S , S ,andS .Thesefivesetsareshowninfigure3.Itisnotedthatthefivesetsofvaluesareconsistenttakingintoaccounttheassociatedstandarduncertainties,andthatsorptionartefactS seemstohaveasorptionefficiencythatisabout10%higherthanS andS .
Figure1.Thesorptioncoefficients andmasses , , , , , ofthesorptionartefactsS , S ,andS measuredbyINRIMattheninestagesofthemediasequence A, B, A, B, A, B, A, B, A ,wheremediumAisair oddvaluesofstageindex andBisvacuum evenvaluesofstageindex .Theerrorbarsindicatestandarduncertainties.
Figure2.Thenormaliseddeviationsassociatedwiththe48measuredquantitiesinvolvedintheleastsquaresadjustment.
1.50
1.52
1.54
1.56
1.58
1.60
1 2 3 4 5 6 7 8 9(m
i1 kg)/m
gStage in media sequence, i
Mass of S1
‐0.0003
‐0.0002
‐0.0001
0.0000
0.0001
0.0002
0.0003
1 2 3 4 5 6 7 8
s i/(mg/cm
2)
Stage in media sequence, i
Sorption coefficient
1.64
1.66
1.68
1.70
1.72
1.74
1 2 3 4 5 6 7 8 9
(mi1 kg)/m
g
Stage in media sequence, i
Mass of S2
1.40
1.42
1.44
1.46
1.48
1.50
1 2 3 4 5 6 7 8 9
(mi1 kg)/m
g
Stage in media sequence, i
Mass of S3
‐2.00
‐1.00
0.00
1.00
2.00
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47
d
Measured quantity no.
Normalised deviations
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Figure3.Thefactors , , measuredbyfivelaboratoriesforthesamesetofsorptionartefactsS , S ,andS .Theerrorbarsindicatestandarduncertainties.
4. SimulationstudyInordertovalidatethemasstransferfromairtovacuumusingthemodeldescribedinsection2,asimulationstudywascarriedout.ThestudyinvolvedoneweightM permanentlystoredinairmediumA ,oneweightM permanentlystoredinvacuum mediumB ,and 3threesorptionartefactsS , S ,andS cycledbetweenairandvacuumusingthemediasequence A, B, A, B, A, B, A, B, A .Ateachstageinthemediasequence,allsixpossiblemassdifferencesbetweenthethreesorptionartefactsandtheweightkeptpermanentlyinthemediawereassumedtobemeasured.
4.1. SimulationofdataInthefirststep,truevalueswereassignedtothefollowingquantities:
Themass oftheweightkeptpermanentlyinvacuum. Themass andvolume oftheweightkeptpermanentlyinair. Thevolumes , , ,geometricalareas , , ,conversionfactors , , andinitial
masses,,
,,
,ofthesorptionartefactsS , S ,andS .
Theairdensities , , , , duringmasscomparisonsin assumedtobeconstantduringmasscomparisonsatagivenstageofthemediasequence .
Thebalancescalefactors . Theeightsorptioncoefficients , … , .
Usingequation 2 ,truemassvaluesofthethreesorptionartefactsatstage2–9inthemediasequencewerecalculated,andusingequation 3 atotalof9 6 54truedifferences∆ , , … , ∆ , werecalculated.
Inasecondsteparealisticstandarddeviation wasassignedtoeachquantity exceptforthefactors , , ,andameasuredvalue wasdrawfromanormaldistributionN , ,where
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
BIPM LNE METAS CEM INRIM
Factors p of sorption artefacts
S1
S2
S3
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isthetruevalueofthequantity assignedorcalculatedinthefirststep.Forthefactors , ,
measuredvaluesequalto1wereassigned.
Atotaloftwentysimulationssimulationwascarriedoutinfourcases:
Case1:Reversiblesorption/desorptionofthesameamountofwater 0.0001mg/cm2 ateachmediatransfer.
Case2:Reversiblesorption/desorptionasinCase1,butacontaminationof0.010mgwasaddedtoallthreesorptionartefactsinthetransferfrommediastage4 vacuum tostage5 air andnotremovedagain.
Case3:Irreversiblesorption/desorptionsimilartothatobservedbyINRIMinthesorptioncomparisondescribedinsection3,seefigure1.
Case4:Irreversiblesorption/desorptionasinCase3,butacontaminationof0.010mgwasaddedtoallthreesorptionartefactsinthetransferfrommediastage4 vacuum tostage5 air asinCase2.
Ineachofthefourcases,thesameseedoftherandomnumbergeneratorwasusedforthesimulations.Thismeansthesequenceofrandomcomponentsisthesameineachofthefourcases;onlythetruevaluesofthequantitiesaredifferent.
Forthemassoftheweightkeptinvacuum,atruevalue 1kgwasassigned,andforthemasskeptinair,thebestestimatesgivenintable2wereusedastruevaluesofthemass andvolume oftheweightkeptinair,whereasthestandarduncertaintiesintable2wereusedasstandarddeviationsforthesimulation.Forthesorptionartefacts,thebestestimatesgivenintable1wereusedastruevaluesofthevolumes , , andthegeometricalareas , , ,whereasthestandarduncertainties
intable1wereusedasstandarddeviationsforthesimulation;twentysetsofconversionfactors, , randomlyselectedfromanormaldistributionN 1, with 0.1wereusedastrue
values,onesetforeachofthetwentysimulations.Asinitialmassvalues,,
,,
,ofthe
sorptionartefacts,thebestestimatesfoundbyINRIMinthesorptioncomparisondescribedinsection3 seefigure1 wereusedastruevalues.ThefiveairdensitiesreportedbyINRIMinthesorptioncomparisonwereusedastruevaluesoftheairdensities , , , , ;thestandarddeviationusedforthesimulationofairdensitieswas 0.00005kg/m .Forsimplicity,thevalues 1 exactly wereselectedforthebalancescalefactors.Forthesimulationoftheindicationdifferences∆ ,thestandarddeviation 0.001mgwasused.
4.2. AnalysisofsimulateddataToeachsimulatedvalue ofaquantity exceptthefactors , , ,astandarduncertainty
wasassigned.Forthefactors , , ,thebestestimate 1andfivedifferentbut
commonstandarduncertainties wereassigned:
1,
, ∈ 0.2, 0.1, 0.05, 0.02, 0.01 .
Eachsimulatedsetofdatawasanalysedfivetimes,onetimeforeachstandarduncertainty assignedtothefactors , , .Onlyoneofthestandarduncertainties isequaltothe
standarddeviation 0.1usedtosimulatevaluesofthefactors , , .Analyseswithlargerand
smallervaluesof wereperformedinordertoassesstheimportanceofselectingarealisticvalueof .Asummaryoftheresultsoftheanalysesperformedareshowninfigure4‐7.
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Incaseofcompletelyreversiblesorption/desorption Case1 ,theadjustedvalueofthemass oftheweightkeptinairisfairlyindependentofthevalueoftheuncertainty assignedtothefactors , , intheleastsquaresanalysis,seefigure4.Onlyinthecase 0.2,wherethe
uncertaintyisoverestimatedbyafactoroftwo,thereareafewcases simulationno.14and19 ,wheretheadjustedvaluesofthemass aresignificantlydifferentcomparedtothecases,where
isequaltoorsmallerthanthestandarddeviation 0.1usedforthesimulationofthefactors , , .Thestandarduncertainty ,thevalueofwhichisindicatedwitherrorbarsinfigure
4,decreasesas decreases,butasthereareothersignificantuncertaintycontributionsto ,e.g.fromthestandarduncertainty 0.0080mgassignedtothemass ofthereferenceweightkeptinair,thedecreasein ispredominantwhen isreducedfrom 0.2,where0.8mg 1.4mg,to 0.1,where0.0096mg 0.0103mg.Inthecase
0.2,thevariationintheuncertainty oftheadjustedvalueofthemass isverylarge,especiallyforthoseadjustedvaluesof thatdiffersmostlyfromthetruevalue.Whathappensinthesecasesisthattheadjustedvaluesof , , areratherextremeinsuchawaythatthe
differencesoftheeffectivesorptionareas , , becomesrathersmall,orevengetthe
wrongsign.Itshouldbenotedthatforallfivestandarduncertainties ,theadjustedvalueofthemass isconsistentwiththetruevaluewithin 2 .Thesameisnottrueforthefactors , , ;forthesefactorsthetruevaluesareonlyexpectedtoberecovered,ifthevalueof is
equaltoorlargerthanthestandarddeviation 0.1usedforthesimulationofthetruevaluesof , , .Thegraphsintherightcolumnoffigure4confirmthat.Thegraphsinthemiddleoffigure
4showthatthetestofconsistenceofdatawithmodelisalmostindependentofthevalueoftheuncertainty assignedtothefactors , , .Atfirstthismightbeseenasaparadox,sincefor
smallvaluesof theadjustedvaluesof , , differsignificantlyfromthetruevalues.The
explanationis,however,thatduetothesignificantuncertaintyintheairbuoyancycorrection,theinformationaboutdifferencesintheeffectivesorptionareas , , cannotbeextracted
fromthemeasuredmassdifferenceswhensorption/desorptionisreversible.
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Case1:Reversiblesorption/desorption
Figure4.ResultsoftheleastsquaresanalysisoftwentysimulationsinCase1withstandarduncertaintya 0.2,b 0.1,c 0.05,d 0.02ande 0.01assignedtothefactors , , .Thegraphtotheleftshowstheadjustedvaluesfoundforthemass ofweightkeptinvacuumcomparedtoitstruevalue.Thegraphinthemiddleshowsobservedchi‐squarevalue comparedtotheexpectationvalue fullline andthecriticalvaluecorrespondingtosignificancelevel 10% brokenline .Thegraphtotherightshowstheadjustedvalues opendots ofthefactors , , comparedtotheirtruevalues closeddots .Errorbarsindicatestandarduncertainties.
a
b
c
d
e
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Case2:Reversiblesorption/desorption+0.01mgcontamination
Figure5.ResultsoftheleastsquaresanalysisoftwentysimulationsinCase2withstandarduncertaintya 0.2,b 0.1,c 0.05,d 0.02ande 0.01assignedtothefactors , , .
a
b
c
d
e
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Case3:Irreversiblesorption/desorption
Figure6.ResultsoftheleastsquaresanalysisoftwentysimulationsinCase3withstandarduncertaintya 0.2,b 0.1,c 0.05,d 0.02ande 0.01assignedtothefactors , , .
a
b
c
d
e
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Case4:Irreversiblesorption/desorption+0.01mgcontamination
Figure7.ResultsoftheleastsquaresanalysisoftwentysimulationsinCase4withstandarduncertaintya 0.2,b 0.1,c 0.05,d 0.02ande 0.01assignedtothefactors , , .
a
b
c
d
e
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Incase2,wherethereversiblesorption/desorptionisdisruptedbyadding0.010mgcontaminationtoallthreesorptionartefactsbetweenstage4and5inthemediacycle,itisonlypossibletoobtainconsistencybetweendataandmodeliftheadjustedvaluesofthefactors , , aresothatthe
differencesineffectivesorptionareas , , areclosetozero.Thisisachievedif
≅ 1.2, ≅ 1and ≅ 0.6asinthegraphtotherightoffigure5a .Thatis,toachieve
consistencyalargevalueof isrequired,suchas 0.2.However,asthedifferencesintheeffectivesorptionareas , , areclosetozero,theuncertainty becomes
extremelylargeasseeninthegraphintheleftinfigure5a .For 0.05,theuncertainty reachesausefullevel,buttheadjustedvaluesof aresignificantlybiasedcomparedtothetruevalue,seefigure5c .Forsmallervaluesof ,thisbiasdisappearsasseeninfigure5d ‐e .However,theresultswouldnot andshouldnot betrustedbytheexperimenter,astheconsistencetestshowsasignificantinconsistencybetweendataandmodel,whichisduetothefactthatacontaminationof0.010mgwasaddedbetweenstage4and5inthemediacyclewithoutaccountingforitinthemodelusedtoanalysethedata.
Contaminationofthesorptionartefactsisaproblemonlyifitisnotuniformlydistributedoverthesurfacesofthesorptionartefacts.Infact,uniformlydistributedcontaminationallowstheratiosoftheeffectivesorptionareastobemeasuredfairlyaccurate.ThisisillustratedinCase3,whereuniformirreversiblesorption/desorptionhasbeenassumed.Asseeninthegraphsintheleftcolumnoffigure6,theadjustedvaluesofthemass oftheweightkeptinvacuumarevirtuallyindependentofthevalueassignedto .Theassociatedstandarduncertainties alsohaveonlylittledependenceofthevalueassignedto intheleastsquaresanalysis;ontheaverageitdecreasesfrom
0.0096mgfor 0.2to 0.0087mgfor 0.01.Incontrasttocase1,thetestofconsistencydependsonthevalueassignedto ,asseeninthegraphsinthemiddlecolumnoffigure6.However,evenfor 0.01theobservedchi‐squarevaluefallsbelowthecriticalvalueforatestofconsistenceata10%levelofsignificanceinelevenoutoftwentysimulations.Asshowninthegraphsintherightcolumnoffigure6,theadjustedvaluesofthefactors , , areconsistent
withthetruevaluesaslongastheuncertainty isequaltoorlargerthanthestandarddeviation0.1usedforthesimulation.For 0.1,thestandarduncertaintiesassociatedwiththefactors
, , areabout30%smallerinCase3thaninCase1.Thereasonisthatthegaininmassofthe
sorptionartefactsfromonestageingivenmediatothenextstageinthesamemediaismeasuredbycomparingthesorptionartefactswiththeweightstoredinthatmedia,whichisassumedtohavethesamemassatallstagesinthemediacycle.Thelargerthesemassgainsare,themoreinformationtherewouldberegardingthevaluesofthefactors , , .
Incase4,theuniformirreversiblesorption/desorptionwasdisruptedbyaddinga0.010mgcontaminationtoallsorptionartefactsbetweenstage4andstage5inthemediacycle.Asseeninthegraphsintheleftcolumnoffigure7,thisleadtoasignificantbiasintheadjustedvaluesofthemass oftheweightkeptinvacuum,whichislargerthelargerthevalueassignedto is.For 0.2,theaveragebiasis0.0300mg,andfor 0.01,theaveragebiasis0.0061mg.Forcomparison,theaveragebiasesfoundinCase3were0.0034mgfor 0.2and0.0028mgfor 0.01.Asseeninthegraphsintherightcolumnoffigure7,theadjustedvaluesofthefactors , ,
seektoreducethedifferencesintheeffectivesorptionareas , , ,justasincase2.It
islesspronouncedinCase4,however,duetotheadditionalamountofinformationabouttheeffectivesorptionareas.Forthemajorityofthesimulations,thechi‐squaretestindicatestheinconsistency
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betweendataandmodelcreatedbyaddingthe0.010mgcontaminationtothedatawithoutaccountingforitinthemodel;seethegraphsintherightcolumnoffigure7.
5. DiscussionWhentransferringtheunitofmassfromairtovacuum,atraditionalassumptionhasbeenthatonlywaterisadsorbedanddesorbed,andthatthattheadsorption/desorptionisreversible.ThesorptioncomparisoncarriedoutinCMMWGMTG1shows,thatadsorption/desorptionisirreversibleingeneral,sothatthesorptionartefactsgainmassesastheyarerepeatedlycycledbetweenairandvacuum.Theirreversibilityseemstobemostpronouncedifthesorptionartefactsarecleanedjustbeforethemediacyclingexperiment.Aftersomecyclesbetweenairandvacuum,thesorption/desorptionseemstoapproachreversibility,andthemassesofthesorptionartefactstendtostabilizeinairaswellasinvacuum.
Thesimulationexperimentshowsthatirreversibilityofthesorption/desorptionisinfactusefulasitenablesameasurementoftherelativeabsorptionefficiencies , … , ofthesurfacesofasetof
sorptionartefactsS , S , . . . , S .Thequantities , … , maythereforebedeterminedbyperforming
anair‐vacuumcyclingexperimentonfreshlycleanedsorptionartefacts.Oncethesequantitieshavebeenmeasured,theirvaluesandassociatedcovariancematrixcouldbeusedasinputtotheanalysisofsubsequentair‐vacuumcyclingexperimentswithoutcleaningofthesorptionartefacts,wheretheirreversibilityofthesorption/desorptionprocessmightbelesspronounced.
Incaseofreversiblesorption/desorption,atleastthreesorptionartefactsareneededinordertotestthehypothesisthatthemassofthesorptionisproportionaltothegeometricalareaofthesorptionartefacts.Ifasorptionartefactismadeupof discs withspacers havingthesamenominalmass,theuniformityofthesorptionefficienciesamongthediscscouldbetestedinaseparateair‐vacuumcyclingexperimentinwhichthemassdifferencesamongthediscs withspacers aremeasuredinairandinvacuum.
Thesimulationswereperformedonanair‐vacuumcyclingexperiment,inwhichtheweightstoredinairwasaplatinum‐iridiumstandard,whereastheweightstoredinvacuumandthesorptionartefactsweremadeofstainlesssteel.ThiswasdoneinordertoevaluatethestandarduncertaintywithwhichthemassofaweightkeptinvacuumcouldbemeasuredintermsofanationalprototypekeptinairusingthesorptionstandardscirculatedintheCCMWGMTG1sorptioncomparison.Thisuncertaintyturnedouttobeontheorderof0.010mgandwasactuallydominatedbythestandarduncertainty0.008mgassignedtotheprototypeitself.Thevalidityofthisuncertaintycanonlybeprovenbyperformingaratherlargenumberofair‐vacuumcyclingexperiments similartotheperformedsimulation thatleadstoconsistentmassvaluesoftheweightstoredinvacuum,takingintoaccountthemeasurementuncertainty.Insuchanexperiment,theweightkeptinairshouldhavethesamenominalvolumeasthesorptionartefactsinordertoreducetheuncertaintyduetobuoyancyeffectswhenmeasuringinair.Forthesamereasonshouldthevolumes orvolumedifferences ofthesorptionartefactsandtheweightkeptinairbemeasuredwiththesmallestpossiblestandarduncertainty.
6. ConclusionAprocedurefortransferringtheunitofmassbetweenaweightkeptinairandaweightkeptinvacuumhasbeenpresented.Theprocedureinvolvestheuseofsorptionartefactsthatarerepeatedlycomparedtotheweightkeptinairandtotheweightkeptinvacuuminanair‐vacuumcycle.Amodelofthemeasurementbasedonhomogeneousabsorption/desorptionhasbeendescribed.Themodel
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hasbeentestedonadaptedmeasurementdataprovidedbyparticipantsinasorptioncomparisonandonsimulateddata.Thesensitivitytomodellingerrorshasbeentestedbysimulationaswell.
Ithasbeenshownthatamassvalueinairhavingastandarduncertaintyof0.008mgcanbetransferredtoamassvalueinvacuumwithastandarduncertaintyof0.010mg,whichmeansthatanuncertaintycontributionof0.006mgfromtheair‐vacuumtransfercanbeachieved,atleastinprinciple.Themodelisbasedonthecrucialassumptionthatsorption/desorptionishomogenousoverthesurfacesofthesorptionartefacts.Ifthisassumptionfailsinreality,themassvaluecalculatedfortheweightkeptinvacuummightbesignificantlybiased.Althoughtheproposedmethodforanalysingthedatafromanair‐vacuumcyclingexperimentincludesatoolfortestingtheconsistencyamongdataandmodel,thereisaratherlargeprobability,thataninvalidassumptionmightnotbedetectedinasingleair‐vacuumcyclingexperiment.
7. References1 BIPM2006TheInternationalSystemofUnits SI 8thedn,Sévres,BIPM2 DavisR2003TheSIunitofmassMetrologia,40 2003 299–3053 GirardG1990ThewashingandcleaningofkilogramprototypesattheBIPM,Sévres,BIPM4 NielsenL1998LeastsquaresestimationusingLagrangemultipliersMetrologia35,115‐18
NielsenL2000Metrologia47,183 erratum 5 NielsenL2002EvaluationofmeasurementsbythemethodofleastsquaresAlgorithmsfor
ApproximationIVedJLevesleyetal UniversityofHuddersfield pp170‐86RecommendationG1 2010 oftheCCM,Considerationofanewdefinitionofthekilogram.
