Description
Inthislab,weuseddatacollectedfromacalorimetryexperimenttodeterminethespecific heat of an unknown sample. We also used error propagation techniques to find the error in thecalculatedspecificheat. Wethenusedourcalculatedspecificheattoidentifytheunknown sampleusingagiventableofcandidatematerialsandtheirproperties. Icalculatedthespecific heatoftheunknownsampletobe0.248±0.000268[J/kg*K].Thecalculatedvalueofspecific heat most closely matches the specific heat of Tellurium Copper Alloy, with a discrepancy of 0.013 [J/kg*K]. Thus, it was determined that the known sample was most likely Tellurium Copper(Alloy145).
I.Nomenclature
mc = massofcalorimeter ms = massofsample cc = specificheatcapacityofcalorimeter cs = specificheatcapacityofsample T2 = equilibriumtemperatureofcalorimeterandsample T1 = initialtemperatureofsample T0 = initialtemperatureofcalorimeter ∆U = changeininternalenergyofsystem U2 = initialinternalenergyofsystem U1 = finalinternalenergyofsystem
II.Introduction Thisprojectusesconceptsfromcalorimetrytofindthespecificheatofanunknownsample. Calorimetryisdefinedasa basictechniqueformeasuringthermodynamicproperties. Inatypicalcalorimetryexperiment,anunknownsampleis placedinsideawell-insulatedcontainerfilledwithaliquidwithknownproperties. Asheatistransferredfromthesample totheknownliquid,itbecomespossibletodeterminethepropertiesofthesample. Withregardstothisspecificexperiment,waterwaschosentobetheknownliquid,andastandardpurchasedcalorimeterwasusedastheinsulatedcontainer.
Theformulausedtocalculatethespecificheatofthesampleisasfollows:
cs =
mccc(T2−T0) ms(T1−T2)
(1)
Withregardstoeachtermintheaboveequation: mc and ms canbemeasuredbysometypeofmassbalance cc shouldbegivenbythemanufacturerofthecalorimeter T1 canbeobtainedfromthetemperatureofthewaterinwhichthesampleisheated T0 canbeobtainedfromthetemperatureprofileofthecalorimeter T2 canbeobtainedfromthetemperatureprofileofthecalorimeter
Notethatthebecausethecontainerusedforcalorimetryisn’tperfectlyinsulated,itisnecessarytouseaprocedure provided by the manufacturer of the calorimeter that uses the method of least squares fit and extrapolation to better approximate T2, T1, and T0. More specifically, T0 is approximated by fitting a line to the pre-sample temperature ∗UndergraduateStudent,AerospaceEngineering,5515NorthernLightsDrive
collectedduringthefirst10minutesoftheexperiment. Thenthelineisextrapolatedforwardtothetimethesamplewas added: 11minutesintotheexperiment. Asecondlineisthenfittedfromthemaximumtemperaturereadingtothelast temperaturereading. Thislineisthenextrapolatedbacktothetimewhenthesamplewasadded: 11minutesintothe experiment. Thesetwoextrapolatedtemperaturevaluesarethenaveragedandthesecondlineofbestfitisextrapolated backwardstothetimecorrespondingtothisaveragetemperaturevalue. Theresultingextrapolatedtemperaturevalueis T2,thefinaltemperatureofthecalorimeterandsampleatequilibrium. ThesenewextrapolatedapproximatesforT0 and T2 arethenpluggedintoformuladerivedaboveforcalculatingthespecificheatoftheunknownsample,togetherwith mc, ms, cc,andT1.
Inaddition,itisimportanttonotethateachoftheofthetermspresentinequation1hasuncertainty,andthatthese uncertaintiesmustbepropagatedusingthetechniquescoveredinclassinordertofindtheuncertaintyinthecalculated specificheatofthesample.
The general error propagation formula was used to calculate the error in the specific heat of the sample. This formula,alongwithallofthepartialderivativesusedinit,arelistedbelow. σcs =s( ∂cs ∂mc ∗σmc)2 +(∂cs ∂T2 ∗σT2)2 +(∂cs ∂T1 ∗σT1)2 +( ∂cs ∂ms ∗σms)2 +(∂cs ∂T0 ∗σT0)2 (2) ∂cs ∂mc = cc(−T0 +T2) ms(T1−T2) (3) ∂cs ∂T2 = ccmc ms(T1−T2) + ccmc(−T0 +T2) ms(T1−T2)2 (4) ∂cs ∂T0 = − ccmc ms(T1−T2) (5) ∂cs ∂T1 = − ccmc(−T0 +T2) ms(T1−T2)2 (6) ∂cs ∂ms = − ccmc(−T0 +T2) m2 s(T1−T2) (7)
III.ExperimentalMethod 1) Examinethecalorimeter. Notethatthecalorimeterismadeofaluminumwithanexactspecificheatof0.214 cal/(g*C). 2) Thesampleisweighedmultipletimestodetermineanaveragemass. 3) Athermocouplewithsoftwarecold-junctioncompensationandITLLLabStationswasusedtotaketemperature readingsofthealuminumcalorimeter. 4) Thethermocoupleisplacedintotheholdprovidedandsecuredwithhightemperaturecottonbeforetheinsulation capisreplaced. 5) Thesampleisimmersedinboilingwaterforabout10minutesuntilisinequilibriumwiththeboilingwater. 6) Atthesametime,thesampleisimmersedintheboilingwater,thetemperaturemeasuringsoftwareisinitiated. It takessampleseverysecond. 7) Thesampleisthenremovedfromtheboilingwaterusingtongs. Itisshakenandthenquicklyplacedinsidethe calorimeter. Thecalorimeteristhensealed. 8) Thetemperaturemeasuringsoftwarerunsforabout10minutesbeforetheexperimentends. 9) Theprogramisterminatedandthedataissaved
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Fig.1 TemperatureProfileofCalorimeterwithLeastSquareBestFitLines.
Fig.2 TemperatureProfileofBoilingWaterwithLeastSquareBestFitLines.
IV.Results Thespecificheatofthesamplewasfoundtobeapproximately0.248J/gKwithanerrorofapproximately0.000172 J/gK.Comparingthisvaluewiththethoseintheprovidedtableofcandidatematerialspropertiesshowsthatwhilethe specificheatofthesampledoesn’texactlymatchupwithanyofthe4providedmaterials,itmostcloselymatchesthe specificheatofTelluriumCopper(Alloy145),whichhasanspecificheatof0.261J/gK.Foramoreexplicitcomparison, considerthatthedifferencebetweenthespecificheatofthesampleandtheTelluriumCopperisamere0.013J/gK.On
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theotherhand,thedifferencebetweenthespecificheatofthesampleandZn(0.402J/gK),Pb(0.129J/gK),andAl(0.9 J/gK)is0.154J/gK,0.119J/gK,and0.652J/gK,respectively.
V.Discussion In this case, the deviation between the measured specific heat of the sample and the specific heat of Tellurium Copperissignificant,asthespecificheatofTelluriumCopperdoesnotfallwithintheerrorboundsofthemeasured specificheatofthesample. Thisindicatesthattheerrorsoftheparametersusedinmycalculationofthespecificheat ofthesampleneedtobelargerthantheyarenow. Lookingateachoftheparametersinequation1,theerrorsinthe temperaturemeasurementsT0,T1,T2wereallcalculatedusingthemethodstaughtinclass. Assumingthesemethods wereusedproperly,theonlytoobtainalargererrorinthesevaluesistoincreasetherangeoftemperaturemeasurements usedtoproducethelinesofbestfit,therebyintroducinggreatervariationbetweenthevalues. Perhapsamoresimple methodtoincreasetheerrorinthecalculatedspecificheatofthesampleistoincreasetheerrorassignedtothemassof thecalorimeterandthesample. Thisincreaseinerrormaybejustifiediftherewerehumanerrorsmadewhenmeasuring the mass of the calorimeter or sample. For example, it is possible that the scale used wasn’t calibrated properly, or not enough mass measurements were taken before calculating the average mass. Finally, perhaps a more practical justificationforthediscrepancybetweenthemeasuredandacceptedspecificheatisthatthemethodusedtoapproximate thevaluesofT0,T1,T2isflawed. Forexample,themeasuredspecificheatwouldbegreaterifthedifferencebetweenT2 andT0orT1andT2increased. ThatmeansthatiftheapproximatedvalueofT0werelowerandT1werehigher,the discrepancybetweenthemeasuredandacceptedspecificheatvaluescouldbereduced. Assuch,ifthisexperimentwere conductedusingamoreaccuratecalorimeterwithbetterinsulation,wewouldlikelyobtainbetterresults.
VI.Conclusion TheidentityofthesampleismostlikelyTelluriumCopper(Alloy145). Whilethecalculatedspecificheatofthe sampledoesdeviatesignificantlyfromtheacceptedspecificheat,itwasstillpossibletoidentifythethesamplewitha greatdegreeofcertainty. ThisisbecausethediscrepancybetweenthespecificheatofthesampleandTelluriumCopper isfarlowerthanthediscrepancybetweenthespecificheatofthesampleandZn-Cu-Ti,Pb,orthediscrepancybetween thespecificheatofthesampleand6063-T1Al. Thus,itisreasonabletodeemthisexperimentasuccess.
References Jackson,J.,“ASEN2012Project1Calorimetry,”Oct. 2017. Jackson,J.,“CalorimetryExperimentalProcedure,”Oct. 2017. Jackson,J.,“CalorimetryAdditionalData,”Oct. 2017. Jackson,J.,“CandidateMaterialsProperties,”Oct. 2017. Jackson,J.,“ASEN2012Project1Calorimetry,”Oct. 2017.
Appendix Thecompletederivationoftheformulausedtocalculatethespecificofthesampleisasfollows:
∆U = 0 (8)
U2−U1 = 0 (9) (mcuc2 + msus2)−(mcuc1 + msus1) = 0 (10) mscs(T1−T2) = mccc(T2−T0) (11)
cs =
mccc(T2−T0) ms(T1−T2)
(12)
ExplicitUseofEngineeringMethodforAlgorithmDevelopment
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• Problem: Ineedusethemethodofleastsquaresandtheerrorpropagationformulaslearnedinclasstoidentifyan unknownsamplebycalculatingitsspecificheat. • Knowns: Data that contain measurements of the temperature at different times of the calorimeter and boiling waterbath. Themassofthesampleandofthecalorimeterandtheiruncertainties. • Find: Thespecificheatofthesampleanditsuncertaintyusingthedataprovided. • Assumptions: Themassanduncertaintiesprovidedarecorrect. Thesampleisaddedtothecalorimeter10minutes afterthedatastartedtobecollected. Itisvalidtouseallerrorpropagationandothertechniquescoveredinclass. • Sketch: Seealgorithmdiagramsbelow. • Fundamentals: Seeequationsandotherconceptspresentedintheintroduction. • Alternatives: Coulduseabettercalorimeter,eliminatingtheneedtoextrapolateandapproximate. Couldusea massspectrometerorotherdevicestousethemetalscompositiontodetermineitsidentity. Couldalsocarryout chemistryexperimentsorvisuallycompareitwithothermaterialstotestitspropertiesandhelpidentifyit. • Steps: 1) Parsedata 2) FindT0 3) FinderrorinT0 4) FindT2 5) FinderrorinT2 6) FindT1 7) FinderrorinT1 8) Findthespecificheatofthesample 9) Findtheerrorinthespecificheatofthesample 10) Plotthetemperatureprofileandleastsquaresbestfitlinesforthecalorimeterandboilingwater 11) Printfindingstooutputfile • Check: Ihaveverifiedthatmyvalueforthespecificheatofthesampleandtheerrorinthespecificheatofthe samplematchesthatofmypeers. • MakesSense?: Ibelieveso. Therewasn’tverymuchvariationinthepointsusedtogenerateall3linesofbestfit. Inaddition,theerrorinthemasseswassmall. Thus,itmakessensefortheerrorinthespecificheattobesmall. ThecalculatedvalueofthespecificheatissimilartothatofTelluriumCopper,whichseemsreasonable.
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Fig.3 MainProgramAlgorithmFlowChart
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Fig.4 Subroutine1: readinputAlgorithmFlowChart
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Fig.5 Subroutine2: findT0AlgorithmFlowChart
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Fig.6 Subroutine3: findsigT0AlgorithmFlowChart
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Fig.7 Subroutine4: findT2AlgorithmFlowChart
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Fig.8 Subroutine5: findsigT2AlgorithmFlowChart
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Fig.9 Subroutine6: findT2AlgorithmFlowChart
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Fig.10 Subroutine7: findsigT2AlgorithmFlowChart
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Fig.11 Subroutine8: findcsAlgorithmFlowChart
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Fig.12 Subroutine9: findsigcsAlgorithmFlowChart
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Fig.13 Subroutine10: createplotsAlgorithmFlowChart
Fig.14 Subroutine11: writeoutputAlgorithmFlowChart
