atomic spectra capstone - University of...
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PHY143 LAB 4: ATOMIC SPECTRA Introduction When an atom is excited it eventually falls back to its ground state, releasing the extra energy as photons. Since the energy of these photons directly corresponds to the gap between different energy levels in the atom, we can study the energy structure of the atom by measuring the wavelengths of these photons. Every atom emits a unique set of energies known as its emission spectrum. Once measured, these spectra allow scientists to identify atoms or molecules based purely on the light they emit: a technique known as spectroscopy. This technique allows us to investigate the material composition of objects ranging from very small samples to distant stars. In this lab you will use a diffraction based spectrometer to measure the emission spectrum of hydrogen and use the Rydberg formula to match each line in the spectrum with an atomic transition. You will then use the spectrometer to identify three different elements enclosed in electric discharge tubes.
Transcript of atomic spectra capstone - University of...
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?
