PHY143 LAB 4: ATOMIC SPECTRA
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PHY143LAB4:ATOMICSPECTRA Introduction Whenanatomisexcitediteventuallyfallsbacktoitsgroundstate,releasingtheextra energyasphotons.Sincetheenergyofthesephotonsdirectlycorrespondstothegapbetween differentenergylevelsintheatom,wecanstudytheenergystructureoftheatombymeasuringthe wavelengthsofthesephotons. Everyatomemitsauniquesetofenergiesknownasitsemissionspectrum.Oncemeasured, thesespectraallowscientiststoidentifyatomsormoleculesbasedpurelyonthelighttheyemit:a techniqueknownasspectroscopy.Thistechniqueallowsustoinvestigatethematerialcomposition ofobjectsrangingfromverysmallsamplestodistantstars. Inthislabyouwilluseadiffractionbasedspectrometertomeasuretheemissionspectrum ofhydrogenandusetheRydbergformulatomatcheachlineinthespectrumwithanatomic transition.Youwillthenusethespectrometertoidentifythreedifferentelementsenclosedin electricdischargetubes. THEORY TheBohrmodelcoupledwiththephotontheoryoflightaccuratelydescribesthespectrum ofhydrogen.WeonlyneedclassicalmechanicsandBohr’sassumptionthatangularmomentumis quantizedaccordingtoπΏ = πβ,whereβ = ! !! isthereducedPlanck’sconstantandnisaninteger. IntheBohrmodelofthehydrogenatom,theelectronorbitstheprotoninafixed,perfectly circularorbit.WestartwithCoulomb’slaw,whichgivesthemagnitudeoftheforceoftheprotonon theelectron: πΉ= whereπ = ! !!!! ππ! π! ππ ! = ! π! π isCoulomb’sconstant,risthedistanceoftheelectronfromtheproton,andeisthe chargeontheelectron.Inorderfortheelectrontomaintainacircularorbit,asassumed,theproton mustexertaforceofmagnitude π! π£ ! π whereπ! isthemassoftheelectronandvisthevelocityoftheelectron.Equatingtheseforces yields πΉ= ππ ! π! π£ ! ππ ! = ⇔ = π! π£ ! π! π π NowwecansubstituteinPlanck’squantizationassumption.Notethat πΏ = π! π = π! π£π = πβ ⇔ (π! π£)! = πβ π! ! ⇔ π! π£ ! = 1 π ! β! π! π ! Substitutingtoeliminatev, ππ ! = π ! β! 1 π ! β! ⇔π= π! π π π! π ! Nowwecanlookatthetotalenergyoftheelectron,givenbythedifferenceofthekineticand potentialenergy: π!! ππ ! 1 πβ πΈ! = − = 2π! π 2π! π ! ππ ! 1 − = πβ π 2π! = ! ππ! π ! π ! β! ! − ππ ! ππ! π ! π ! β! −π ! π! π ! 2π! β! NotethatallenergiesareconvenientlyproportionaltothegroundstateenergyπΈ! .Solvingforthis energy,wefind −π ! π! π ! πΈ! = = −13.6 ππ 2β! ThisenergyisknownastheRydbergenergyπ ! .Thenwecanexpressanyenergylevelofthe electronas πΈ! = 1 −π ! π! π ! 1 13.6ππ = ! πΈ! = − ! ! π 2β π π! Whentheelectronjumpstoalowerenergylevel,itemitsaphotonthatcarriesthereleasedenergy. Thisenergyisthedifferenceoftheinitialandfinalenergies. πΈ! = πΈ! − πΈ! = π ! 1 1 !− ! π! π! Whereπ! istheelectron’sinitialenergylevelandπ! istheelectron’sfinalenergylevel.Accordingto thephotonmodel,aphotonhasenergyπΈ = βπ/π,wherecisthespeedoflightandλisthe wavelengthofthephoton.Sincethewavelengthisobservableintheatomicspectra,wecanusethe wavelengthtoobservehydrogentransitions. πΈ! = Whereπ ! = − !".!!" !! βπ 1 1 1 1 1 = π ! ! − ! ⇔ = π ! ! − ! π π π! π! π! π! istheRydbergconstantforhydrogen. DiffractionGrating Thespectrophotometerusesadiffractiongratingtoseparatethecomponentwavelengthsofincident light.Aswesawinthediffractionlab,adiffractiongratingwithlineseparationdwillyieldapatternwith maximaatanglesθfromthenormalfor π ππππ = ππ, π = (0,1,2,3 … ) Measuringthepeakpositionsofaknownspectrumallowstheseparationdtobecalculatedprecisely. Thisrelationcanthenbeusedtofindthewavelengthsinunknownspectra. Setup 1)Placethespectrophotometerontheopticstrack. 2)Mountthetrackontwostands. 3)Placethecollimatingslitholderatoneendofthetrack.Placethedischargelampbehindit, proppingitupthewoodenplatformifnecessarytoalignthemiddleofthelamportheholewiththe slits.Coverthelampwiththeblackclothtoblocktheextralight. 4)Liftthetrackupuntilthecenterofthecollimatingslitsislevelwiththemiddleofthedischarge lamp.Fixthetrackinplaceatthisheight. 5)Attachthediffractiongrating(whichismagnetic)tothemountinthecenterofthe spectrophotometer.Avoidtouchingthesurfaceofthegrating! 6)AttachtheHighSensitivityLightSensorandtheApertureBrackettothearmofthe spectrophotometerusingtheblackrod.PlugtheHighSensitivityLightSensorintoAnalogChannel AontheScienceWorkshopinterface.Turnonthedischargelamp. 7)Placethecollimatinglensabout12cmfromthecollimatingslits.Havesomeonewith20/20 vision(correctedwithglassesisok)lookthroughthecollimatinglensattheslits.Adjustthe collimatinglensuntiltheslitsareinsharpfocus.Thecollimatinglensshouldbeabout10cmfrom thecollimatingslits.Movethespectrophotometerclosetothecollimatinglens. 8)Placethefocusinglensonthespectrophotometerarminbetweenthelightsensorandthe grating.YoushouldthenadjustthefocusinglenssothatthelightthatisshownontheAperture Bracketisfocused. ComputerSetup 1) OpentheRotarySensorcalibrationCapstonefile.The purposeofthisprogramistodeterminetherelationship betweentherotationofthespectrometerarmandthe rotationrecordedbytherotarysensor. 2) Click“Record”,thenrotatethespectrometerarm betweentwodegreemarks.Ifthereadinggoesnegative, reversetherotarysensor’sconnectiontothescience workshopinterface. 3) Writedownthenumberofradianstherotarymotion sensorrotates(shownonthescreen)foryourgiven rotation. • • Shouldyousweepthrough asmallorlargeangleto maketheproceduremore accurate? Howwillambientlight affectyour measurements?Whatare thesourcesofambient lightaroundyour experiment,andhowcan youminimizethem? 4) Takethenumberofdegreesthatyourotatedthespectrophotometerarmanddivideit bythenumberofradiansthatyougot.Thenumberyoushouldgetshouldbearound 0.95-0.96. 5) OpentheatomicspectraCapstonefile.ClickonCalculatorfoundontheleftsideofthe screen.Online3,replacethenumber.9569withthenumberthatyougotinthe previousstep.ClickAccept,thenclickCalculatoragain. 6) Calibrate the High Sensitivity Light Sensor. Click Calibration (the green circle on the left side of the screen). From the menu, select Light Intensity and click next. Select the dot that says One Standard (1 point offset) and click next. Cover the Light Sensor then click Set Current Value to Standard Value. Click Finish. Click Calibration again to close the calibration menu. THINGSTOTHINKABOUT Experiment Sweepingthedetectorarmthroughwillnowrecorda spectrumofthelightfromthedischargelamp.Trydifferent apertureandslitsizes,andadjustingthelenslocations,to recordasmanyofthespectralpeaksaspossible.Dim spectralpeakswillrequirecarefultuningoftheaperturesto observe. 1)OneoftheunlabeledtubesisHydrogen.Thevisiblepart ofthehydrogenisknownastheBalmerseries.Usethe Rydbergformulatodeterminethefinalandinitialenergy states(nfandni)foreachtransitionlineyouobserved: ! !!! →!! = π ( ! ! !! − ! !!! ),whereR=1.097x107m-1 -Howshouldyoudecidewhatslitapertures andsensorgaintouse? -Sincethisspectrometerusesadiffraction grating,youwillobservecopiesofthe spectrumoneithersideofthecenterposition (the1storderdiffractionpeaks).Youwillalso observe2ndorderdiffractionpeaks,which mayoverlapwiththe1storder.Howcanyou tellthemapart? -Howcouldyoucalibrateyourreadings, relatingthemeasuredanglestowavelengths? 2)Usethespectrophotometertomeasurethespectrumofeachunlabeleddischargetube. Determinethewavelengthofeachpeakandthencompareyourdatawithatableofspectradatato identifyeachsample.Agoodtableofspectralinescanbefoundhere:http://hyperphysics.phyastr.gsu.edu/hbase/quantum/atspect.html
