atomic spectra capstone - university of...
TRANSCRIPT
PHY143LAB4 : ATOMIC SPECTRA
Introduction
Whenanatomisexcitediteventuallyfallsbacktoitsgroundstate,releasingtheextraenergyasphotons.Sincetheenergyofthesephotonsdirectlycorrespondstothegapbetweendifferentenergylevelsintheatom,wecanstudytheenergystructureoftheatombymeasuringthewavelengthsofthesephotons.
Everyatomemitsauniquesetofenergiesknownasitsemissionspectrum.Oncemeasured,thesespectraallowscientiststoidentifyatomsormoleculesbasedpurelyonthelighttheyemit:atechniqueknownasspectroscopy.Thistechniqueallowsustoinvestigatethematerialcompositionofobjectsrangingfromverysmallsamplestodistantstars.
InthislabyouwilluseadiffractionbasedspectrometertomeasuretheemissionspectrumofhydrogenandusetheRydbergformulatomatcheachlineinthespectrumwithanatomictransition.Youwillthenusethespectrometertoidentifythreedifferentelementsenclosedinelectricdischargetubes.
THEORY
TheBohrmodelcoupledwiththephotontheoryoflightaccuratelydescribesthespectrumofhydrogen.WeonlyneedclassicalmechanicsandBohr’sassumptionthatangularmomentumisquantizedaccordingto𝐿 = 𝑛ℏ,whereℏ = !
!!isthereducedPlanck’sconstantandnisaninteger.
IntheBohrmodelofthehydrogenatom,theelectronorbitstheprotoninafixed,perfectlycircularorbit.WestartwithCoulomb’slaw,whichgivesthemagnitudeoftheforceoftheprotonontheelectron:
𝐹 =𝑘𝑞!𝑞!𝑟!
=𝑘𝑒!
𝑟!
where𝑘 = !!!!!
isCoulomb’sconstant,risthedistanceoftheelectronfromtheproton,andeisthe
chargeontheelectron.Inorderfortheelectrontomaintainacircularorbit,asassumed,theprotonmustexertaforceofmagnitude
𝑚!𝑣!
𝑟
where𝑚! isthemassoftheelectronandvisthevelocityoftheelectron.Equatingtheseforcesyields
𝐹 =𝑘𝑒!
𝑟!=𝑚!𝑣!
𝑟⇔
𝑘𝑒!
𝑟= 𝑚!𝑣!
NowwecansubstituteinPlanck’squantizationassumption.Notethat
𝐿 = 𝑝!𝑟 = 𝑚!𝑣𝑟 = 𝑛ℏ⇔ (𝑚!𝑣)! =𝑛ℏ !
𝑟!⇔ 𝑚!𝑣! =
1𝑚!
𝑛!ℏ!
𝑟!
Substitutingtoeliminatev,
𝑘𝑒! =𝑛!ℏ!
𝑚!𝑟⇔ 𝑟 =
1𝑘𝑛!ℏ!
𝑚!𝑒!
Nowwecanlookatthetotalenergyoftheelectron,givenbythedifferenceofthekineticandpotentialenergy:
𝐸! =𝑝!!
2𝑚!−𝑘𝑒!
𝑟=
12𝑚!
𝑛ℏ𝑟
!
−𝑘𝑒!
𝑟=
12𝑚!
𝑛ℏ ! 𝑘𝑚!𝑒!
𝑛!ℏ!
!
− 𝑘𝑒!𝑘𝑚!𝑒!
𝑛!ℏ!
=−𝑘!𝑚!𝑒!
2𝑛!ℏ!
Notethatallenergiesareconvenientlyproportionaltothegroundstateenergy𝐸!.Solvingforthisenergy,wefind
𝐸! =−𝑘!𝑚!𝑒!
2ℏ!= −13.6 𝑒𝑉
ThisenergyisknownastheRydbergenergy𝑅! .Thenwecanexpressanyenergyleveloftheelectronas
𝐸! =1𝑛!−𝑘!𝑚!𝑒!
2ℏ!=1𝑛!𝐸! = −
13.6𝑒𝑉𝑛!
Whentheelectronjumpstoalowerenergylevel,itemitsaphotonthatcarriesthereleasedenergy.Thisenergyisthedifferenceoftheinitialandfinalenergies.
𝐸! = 𝐸! − 𝐸! = 𝑅!1𝑛!!−1𝑛!!
Where𝑛! istheelectron’sinitialenergyleveland𝑛! istheelectron’sfinalenergylevel.Accordingtothephotonmodel,aphotonhasenergy𝐸 = ℎ𝑐/𝜆,wherecisthespeedoflightandλisthewavelengthofthephoton.Sincethewavelengthisobservableintheatomicspectra,wecanusethewavelengthtoobservehydrogentransitions.
𝐸! =ℎ𝑐𝜆= 𝑅!
1𝑛!!−1𝑛!!
⇔1𝜆= 𝑅!
1𝑛!!−1𝑛!!
Where𝑅! = − !".!!"!!
istheRydbergconstantforhydrogen.
DiffractionGratingThespectrophotometerusesadiffractiongratingtoseparatethecomponentwavelengthsofincidentlight.Aswesawinthediffractionlab,adiffractiongratingwithlineseparationdwillyieldapatternwithmaximaatanglesθfromthenormalfor
𝑑 𝑆𝑖𝑛𝜃 = 𝑚𝜆,𝑚 = (0,1,2,3… )
Measuringthepeakpositionsofaknownspectrumallowstheseparationdtobecalculatedprecisely.Thisrelationcanthenbeusedtofindthewavelengthsinunknownspectra.
Setup1)Placethespectrophotometerontheopticstrack.
2)Mountthetrackontwostands.
3)Placethecollimatingslitholderatoneendofthetrack.Placethedischargelampbehindit,proppingitupthewoodenplatformifnecessarytoalignthemiddleofthelamportheholewiththeslits.Coverthelampwiththeblackclothtoblocktheextralight.
4)Liftthetrackupuntilthecenterofthecollimatingslitsislevelwiththemiddleofthedischargelamp.Fixthetrackinplaceatthisheight.
5)Attachthediffractiongrating(whichismagnetic)tothemountinthecenterofthespectrophotometer.Avoidtouchingthesurfaceofthegrating!
6)AttachtheHighSensitivityLightSensorandtheApertureBrackettothearmofthespectrophotometerusingtheblackrod.PlugtheHighSensitivityLightSensorintoAnalogChannelAontheScienceWorkshopinterface.Turnonthedischargelamp.
7)Placethecollimatinglensabout12cmfromthecollimatingslits.Havesomeonewith20/20vision(correctedwithglassesisok)lookthroughthecollimatinglensattheslits.Adjustthecollimatinglensuntiltheslitsareinsharpfocus.Thecollimatinglensshouldbeabout10cmfromthecollimatingslits.Movethespectrophotometerclosetothecollimatinglens.
8)Placethefocusinglensonthespectrophotometerarminbetweenthelightsensorandthegrating.YoushouldthenadjustthefocusinglenssothatthelightthatisshownontheApertureBracketisfocused.
• Shouldyousweepthroughasmallorlargeangletomaketheproceduremoreaccurate?
• Howwillambientlightaffectyourmeasurements?Whatarethesourcesofambientlightaroundyourexperiment,andhowcanyouminimizethem?
ComputerSetup
1) OpentheRotarySensorcalibrationCapstonefile.Thepurposeofthisprogramistodeterminetherelationshipbetweentherotationofthespectrometerarmandtherotationrecordedbytherotarysensor.
2) Click“Record”,thenrotatethespectrometerarmbetweentwodegreemarks.Ifthereadinggoesnegative,reversetherotarysensor’sconnectiontothescienceworkshopinterface.
3) Writedownthenumberofradianstherotarymotionsensorrotates(shownonthescreen)foryourgivenrotation.
4) Takethenumberofdegreesthatyourotatedthespectrophotometerarmanddivideitbythenumberofradiansthatyougot.Thenumberyoushouldgetshouldbearound0.95-0.96.
5) OpentheatomicspectraCapstonefile.ClickonCalculatorfoundontheleftsideofthescreen.Online3,replacethenumber.9569withthenumberthatyougotinthepreviousstep.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.
Experiment
Sweepingthedetectorarmthroughwillnowrecordaspectrumofthelightfromthedischargelamp.Trydifferentapertureandslitsizes,andadjustingthelenslocations,torecordasmanyofthespectralpeaksaspossible.Dimspectralpeakswillrequirecarefultuningoftheaperturestoobserve.
1)OneoftheunlabeledtubesisHydrogen.ThevisiblepartofthehydrogenisknownastheBalmerseries.UsetheRydbergformulatodeterminethefinalandinitialenergystates(nfandni)foreachtransitionlineyouobserved:
! !!!→!!
= 𝑅( !!!! −
!!!!),whereR=1.097x107m-1
2)Usethespectrophotometertomeasurethespectrumofeachunlabeleddischargetube.Determinethewavelengthofeachpeakandthencompareyourdatawithatableofspectradatatoidentifyeachsample.Agoodtableofspectralinescanbefoundhere:http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/atspect.html
THINGS TO THINKABOUT
-Howshouldyoudecidewhatslitaperturesandsensorgaintouse?
-Sincethisspectrometerusesadiffractiongrating,youwillobservecopiesofthespectrumoneithersideofthecenterposition(the1storderdiffractionpeaks).Youwillalsoobserve2ndorderdiffractionpeaks,whichmayoverlapwiththe1storder.Howcanyoutellthemapart?
-Howcouldyoucalibrateyourreadings,relatingthemeasuredanglestowavelengths?