ad Rydbrg Costat by Dr. Jams E. Parks Dpartmt of Physics ad Astroomy 401 Nils Physics Buildig Th Uivrsity of Tss Koxvill, Tss 37996-100 Copyright March 00 by Jams Edgar Parks* *All rights ar rsrvd. No part of this publicatio may b rproducd or trasmittd i ay form or by ay mas, lctroic or mchaical, icludig photocopy, rcordig, or ay iformatio storag or rtrival systm, without prmissio i writig from th author. Objctivs Th objctivs of this xprimt ar: (1) to study th missio of light from a hydrog discharg sourc, () to lar th mpirical formulas to charactriz th pattr of spctral lis from hydrog, (3) to lar th postulats for dvlopig th Bohr modl of th hydrog atom, (4) to study ad dvlop th Bohr thory of th hydrog atom, (5) to masur th wavlgths of th Balmr sris of visibl missio lis from hydrog, ad (6) to lar to aalyz th wavlgth data to dtrmi th Rydbrg costat usig th Bohr modl formulatio. Thory Hydrog atoms i a discharg lamp mit a sris of lis i th visibl part of th spctrum. This sris is calld th Balmr Sris aftr th Swiss tachr Joha Balmr (185-1898) who, i 1885, foud by trial ad rror a formula to dscrib th wavlgths of ths lis. This formula is giv by 1 1 1 = R λ (1) whr ar itgrs, 3, 4, 5, up to ifiity ad R is a costat ow calld th Rydbrg costat. I th SI systm of uits, R=1.097 x 10 7 m -1. Balmr suggstd that his formula may b mor gral ad could dscrib spctra from othr lmts. Th i 1889, Johas Robrt Rydbrg foud svral sris of spctra that would fit a mor
gral rlatioship, similar to Balmr s mpirical formula. This gral rlatioship is kow as th Rydbrg formula ad is giv by 1 1 1 = R - i > f, () λ f i whr i ad f ar itgrs, 1,, 3, 4, 5, up to ifiity, with i > f. For th hydrog atom, i = corrspods to th Balmr sris. Thr ar othr sris i th hydrog atom that hav b masurd. Th Lyma sris is a st of ultraviolt lis that fit th rlatioship with i = 1. A sris i th ifrard rgio of th spctrum is th Pasch sris that corrspods to i = 3. Th Bracktt ad Pfud sris ar two mor i th ifrard rgio corrspodig to i = 4 ad i = 5. Th idividual lis i th Balmr sris ar giv th ams Alpha, Bta, Gamma, ad Dlta, ad ach corrspods to a i valu of 3, 4, 5, ad 6 rspctivly. Wavlgths of ths lis ar giv i Tabl 1. Tabl 1. Balmr Sris Som Wavlgths i th Visibl Spctrum. Nam of Li f i Symbol Wavlgth Balmr Alpha 3 H α 656.8 m Balmr Bta 4 H β 486.13 m Balmr Gamma 5 H γ 434.05m Balmr Dlta 6 H δ 410.17 m I 1913 th Daish physicist Nils Bohr was th first to postulat a thory dscribig th li spctra obsrvd i light maatig from a hydrog discharg lamp. With Albrt Eisti s thory for th photolctric ffct, whr a photo has rgy proportioal to its frqucy, Bohr postulatd th xistc of rgy lvls i th atom. H assumd that th rgy associatd with th photos of light wr th rsult of trasitios i th atom from o lvl to aothr, with th rgy of th photo big qual to th diffrc i th itral rgis spcific to rgy lvls ivolvd i th trasitio from a iitial stat to a fial stat, so that Ergy of Photo = - [ Ergy of Fial Stat - Ergy of Iitial Stat ] (3) or Δ E = [ E E ] = E E. (4) f i i f Th gativ sigs i frot of th brackts i Equatios (3) ad (4) ar bcaus th photos ar big mittd istad of big absorbd. Also, th lvls ar boud stats ad ar assigd gativ valus. I othr words, th atom is givig up rgy istad of havig rgy supplid to it.
I 1905 Albrt Eisti workd out th thory for th photolctric ffct usig a cocpt that Max Plack had usd to dscrib black body radiatio. I this thory, for which Eisti rcivd his Nobl priz, h postulatd that light cosists of packts of rgy calld photos or quata ad that ach quatum of light has rgy proportioal to its frqucy. H dtrmid that th rgy of a photo, E photo is giv by E photo = hυ (5) whr ν is th frqucy of th photo ad h is a costat of proportioality calld Plack s costat. Plack s costat is qual to 6.660755(40)x10-34 J s. From th lctromagtic wav thory of light, frqucy ν ad wavlgth λ ar rlatd to th c spd of light by ν =, so that λ c E photo = hυ = h (6) λ ad c h = Ei Ef =Δ E (7) λ which rlats th wavlgth of th mittd light to th diffrc i rgy lvls btw th fial ad iitial stats of th atom. Bohr postulatd that a lctro ca mov about th uclus of a atom i stabl orbits, without mittig radiatio ad losig rgy. Thus its rgy would b costat i ay sigl orbit ad its rgy would chag oly if it chagd orbits ad a trasitio occurrd by th lctro movig from o stabl orbit to aothr. This postulat was rvolutioary i that it cotradictd lctromagtic thory, which prdictd that th acclratig lctro would radiat rgy. I ordr to hav stabl orbits, Bohr furthr postulatd that th magitud of th orbitig lctro s agular momtum would b quatizd ad that it must b a itgral multipl of th quatity h/π. Sic th agular momtum L for a lctro of mass m movig i a orbit of radius r with a spd v is L = mvr, (8) his postulat th yilds mvr h = = 1,, 3, (9) π whr v is th spd of th lctro ad r is its radius as it orbits with a itgral umbr of th quatity h/π. Th umbr is rfrrd to as th Pricipal Quatum Numbr. 3
Excpt for th assumptio that th agular momtum is quatizd, Bohr s modl for th hydrog atom was dvlopd usig simpl classical cocpts. H assumd that th lctro was small i mass compard to th sigl proto i th uclus ad that it movd about this proto i a circular orbit. H said that th lctro was hld i a orbit by th lctrical forc F E btw th lctro ad proto, that is giv by Coulomb s Law, 1 FE = (10) 4πε0 r whr is th magitud of chargs of th lctro ad proto, r is th radius of th orbit, ad ε 0 is a costat kow as th prmittivity of fr spac. Th lctrical forc is th forc that supplis th ctriptal forc, F C, dd to kp th lctro i orbit ad is giv by F C mv = (11) r whr m is th mass of th lctro ad v is th spd of th lctro as movs i th circular orbit of radius r. Sttig Equatios (10) ad (11) qual givs th rlatioship 1 4πε r 0 m v =. (1) r By solvig Equatios (9) ad (1) simultaously for r ad v th Bohr modl fids xprssios for th radii ad spds of th orbitig lctro giv by r h = ε (13) πm 0 ad v 1 =. (14) ε h 0 Th miimum radius occurs wh =1 ad is rfrrd to as th Bohr radius, r 1, ad is giv by h r = ε. (15) πm 1 0 I Bohr s modl, th total rgy E of th lctro as it orbits th proto is th sum of th kitic rgy KE du to its motio ad th pottial rgy PE it has bcaus it is lctrically boud to th proto so that 4
E = KE + PE. (16) Th kitic rgy is giv by th classical quatio for kitic rgy, KE=½mv, ad yilds KE 1 1 m = m v = (17) ε 8 4 0 h wh th orbital spd foud i Equatio (14) is isrtd. Th lctrical pottial rgy is giv by which givs PE 1 = (18) 4πε r 0 PE 1 m = ε (19) 4 0 4h wh th Bohr radius giv by Equatio (13) is substitutd. Th gativ sig is a rsult of th lctro big boud to th proto, ad th pottial is tak to b zro wh th lctro is ifiitly rmovd from th proto. Th total rgy is th giv by which rducs simply to E = 1 m 1 m ε (0) E 4 4 o 8h ε0 4h 4 1 m = ε h. (1) o 8 Sic th pricipal quatum umbr charactrizs th orbit, a chag of rgy ΔE will occur wh it udrgos a trasitio from a iitial valu of i to a fial valu f so that th rgy chags from a iitial valu of E i to a fial valu of E f. Usig Equatio (1), ΔE is th giv by 4 4 Δ 1 m 1 m E = E i Ef = ε 8h + ε 8h () which rducs to Substitutig Equatio (7) givs o i o f 4 m 1 1 Δ E =. (3) 8ε oh f i 5
4 1 m 1 1 = 3 λ 8εohc f i (4) ad is rcogizd to b i th form of Equatio (), 1 1 1 = R λ, (5) f i with th Rydbrg costat R giv by m R =. (6) 8ε hc 4 3 o I this xprimt, Equatio (5) will b usd to dtrmi R from masurmts of λ. For th visibl, Balmr sris i is ad valus of f will b matchd to th obsrvd spctral pattr. Figur 1. Apparatus ad stup for th Balmr sris xprimt. Apparatus Th apparatus is show i Figur 1 ad cosists of: (1) a Pasco Modl SP-968 prcisio studt spctromtr, () a gratig with ithr 300 lis/mm or 600 lis/mm, (3) a hydrog discharg lamp ad powr supply, (4) a magifyig glass, (5) a small ight light, ad (6) a black cloth to block out stray light. Th spctromtr is a prcisio istrumt ad is show i Figur with th importat parts labld. 6
Figur. Pasco prcisio studt spctromtr. Figur 3. Schmatic diagram of studt spctromtr. A schmatic diagram of th studt spctromtr is show i Figur 3, which illustrats th pricipls of its opratio. Th studt spctromtr cosists of thr basic compots: a collimator, a diffractio gratig, ad a tlscop. Th light to b aalyzd trs th spctromtr through a adjustabl slit, which forms a arrow, wll-dfid li sourc of light. This slit is locatd at th focal poit of th collimator, which trasforms th light ito a paralll, collimatd bam of light. Th bam uiformly 7
illumiats th gratig so that all light rays strik th gratig at th sam agl of icidc. A paralll bam of light is cssary to illumiat th gratig so that a sharp imag of th slit ca b formd wh viwd with th tlscop. Th diffractio gratig disprss th light so that th diffractio agl, θ, dpds o th wavlgth of light, λ, ad th gratig spacig, d, accordig to th gratig quatio, whr is a itgr giv by dsiθ = mλ (7) m = 0,1,,3,. (8) Th itgr, m, is rfrrd to as th diffractio ordr. Light of a giv color will b diffractd at a spcific agl for a giv ordr. All light rays of that color will b paralll if th icidt light rays ar paralll. Light of a diffrt color is diffractd at a diffrt agl, but its light rays will also b paralll. Wh th tlscop is focusd at ifiity, paralll light of a spcific color trig th tlscop will form a imag of th slit of that color i th focal pla of th objctiv ls of th tlscop. Thus a gr imag of th slit will b formd at o agl ad a rd imag of th slit will b formd at aothr, largr agl. Th tlscop is attachd to a rotatig arm whos agl of rotatio ca b prcisly ad accuratly masurd. This dvic, oft rfrrd to as a goiomtr, masurs th diffractio agls of th spctra of colors of light. By rotatig th tlscop arm, imags of th slit of various colors ca b viwd ad magifid with th ypic ad thir rspctiv agls of diffractio masurd. Th ypic has a graticl with a orthogoal st of cross-hairs, which hlp alig ad rfrc th viwd imags of th slit. From th gratig spacig ad masurmts of th diffractio agls, th wavlgth of th obsrvd light ca b dtrmid from th gratig quatio, Equatio (7). Procdur 1. Th studt spctromtr will provid th bst ad most accurat rsults if it is proprly aligd ad focusd. Th light sourc should b locatd about 1 cm from th collimator slit. A block of wood is providd so that th discharg light sourc ca rst partially o th spctromtr platform. Th slit should b xamid ad adjustd so that it is just barly op. Not that oly o sid of th slit movs whil th othr sid rmais fixd as adjustmts ar mad.. Look through th ypic of th tlscop ad viw th cross-hairs of th graticul. (This should b do prfrably without your glasss if you war thm. Th ypic is adjustabl so that you ca corrct for your ysight.) Slid th ypic i ad out util th cross-hairs com ito sharp focus. By rotatig th graticul aligmt rig, 8
adjust th oritatio of th cross-hairs so that thy ar aligd vrtically ad horizotally. You may d to rfocus th cross-hairs. 3. Th tlscop should b focusd for a objct locatd at ifiity. Loos th tlscop rotatio lock-scrw (s Figur ) ad rotat th tlscop arm to viw a distat objct o a wall across th room, or a buildig outsid. Adjust th tlscop focus kob so that th viwd imag of th distat objct is clarly i focus. 4. Rotat th tlscop arm so that it is dirctly across ad aligd with th collimator. Th tlscop focus should rmai uchagd, kpt focusd at ifiity, durig this stp ad at all tims followig. 5. Tur th light sourc o ad viw th slit of th collimator. Th gratig should ot b i its mout. Th tlscop arm may d to b rotatd slightly to s th rd light ad outli of th slit. Th light sourc may d to b movd slightly to giv th bst, most its illumiatio of th slit. You may loos ad r-tight th tlscop rotatio lock-scrw as dd to facilitat fixig ad movig th arm positio. CAUTION: Th hydrog discharg lamp is powrd by high voltag ad th tub gts hot. DO NOT touch th tub aywhr spcially ar th ds whr th lctrical cotacts ar mad. Not: Simpl hydrog discharg tubs ca los thir hydrog by ractig with impuritis isid th tub ad by small laks from th outsid. A good tub will hav a bright rd color ar its ctr, ad a poor tub will b mor pikish ar th ctr. For bst rsults, it is importat to hav a good tub. 6. Us th focus kob to adjust th collimator focus for th clarst, sharpst imag of th slit. If dd, r-adjust th light sourc to provid th bst illumiatio of th slit ad to giv th sharpst imags. 7. Gratigs with ithr 300 lis/mm or 600 lis/mm ar providd for this xprimt, ad th gratig spacigs d ar ithr 1/300000 m or 1/600000 m. Th umbr of lis pr mm should b labld o th gratig, but if ot, ask your istructor for this valu. Th gratig is a rplica gratig approximatly 1 ich squar that is moutd i th ctr, o o sid of a 1½ x x5/3 glass plat. Th gratig itslf should ot b touchd ad th glass mout should b hadld oly by its dgs. Th diffractio gratig should ow b placd i th gratig mout with th gratig sid of th glass mout agaist th vrtical posts. Loos th spctromtr tabl lock-scrw ad alig th pla of th gratig so that it is prpdicular to th optical axis formd by th collimator ad tlscop wh thy ar dirctly opposit o aothr. Thr is a li iscribd o th spctromtr tabl to assist with th oritatio. This adjustmt dos t hav to b prcis, but i your bst judgmt, it should appar to b prpdicular. 9
8. Th imag you s with th tlscop dirctly i li with th collimator is th udiffractd, ctral imag. Now with th black cloth ovr th apparatus to shild out stray light, rotat th tlscop arm to th lft ad right to survy th diffrt lis ad colors of diffractd light. Movig away from th ctral imag, you should s th first ordr diffractio lis of violt, turquois, ad rd light followd by th scod ordr diffractio pattr of lis of th sam colors. Ths visibl lis ar thr of th Balmr lis corrspodig to f = 5, 4, ad 3. Thr is a fourth o corrspodig to f = 6, but it is wak, ad somtims so xtrmly wak that it caot b s. Th room may hav to b compltly dark to s this li, if at all possibl. Not: Thr ar bads of vry wak trasitios du to molcular hydrog that ca somtims b obsrvd btw th blu, gr, ad rd lis. Th missios from ths bads grow strogr as a tub ags, ad at th sam tim, th hydrog lis grow wakr. 9. Th followig stps ivolv aligig ad sttig th rfrc valu for makig th prcis agular masurmts of th diffractd light. It is importat to do ths stps carfully, sic all masurmts will b mad rlativ to this sttig. 10. Rotat th tlscop arm back to th positio whr it is dirctly opposit ad i li with th optical axis of th collimator ad whr th ctral, udiffractd imag ca b obsrvd. Adjust th arm so that th vrtical cross-hair is ar or o th ctral imag ad tight th tlscop rotatio lock-scrw. Us th fi adjust kob ar ad just to th right of th lock-scrw to alig th vrtical cross-hair xactly with th lft, statioary slit, th lft dg of th ctral li imag. S Figur. Not th xact positio of th vrtical cross-hair ad lft dg as this will b th rlativ positio of ths for all subsqut aligmts. 11. Agls ar masurd with th 0-360º scal gravd ito th rotatig tlscop bas ad ar rad rlativ to th zro fiducial rulig o th vrir scal that is fixd to th rotatig tabl bas. St th zro rfrc agl by loosig th tabl rotatio lockscrw ad rotatig th tabl bas util th zro fiducial mark o th vrir scal is closly aligd with th zro mark o th 0-360º mai scal. Tight th lock-scrw ad us th tabl rotatio fi adjust kob to positio th two zro marks so that thy ar xactly aligd with ach othr. Us th magifyig glass to bttr mak this adjustmt. As a chck, th 30 rulig o th vrir scal also should b aligd xactly with th 14.5º rulig o th mai scal. S Figur 4. 10
Figur 4. Tlscop viw of aligmt of slit with cross hairs ad aligmt of vrir scal with rotatig scal. 1. At this poit th followig coditios should xist: (1) th tlscop should b aligd with th optical axis of th collimator, () th vrtical cross-hair i th ypic should b aligd with th lft dg of th ctral imag,(3) th zro marks of th vrir ad mai scals should b aligd xactly, (4) th pla of th gratig should b prpdicular to th optical axis, ad (5) all adjustmts should b fixd with thir rspctiv lock-scrws. Th spctromtr should b rady for masurmts. 13. Th gral procdur for masurig th agls of th various diffractio lis is to u-tight th tlscop arm rotatio lock-scrw, rotat th tlscop to viw th diffractio li of itrst, ctr th imag of th li ar th vrtical cross-hair, tight th lock-scrw, us th tlscop rotatio fi adjust kob to alig th vrtical cross-hair with th lft dg of th slit imag, ad th rad th agular scal. 14. Th agls for ach of th diffractd lis ca b rad most simply by avoidig th vrir scal fatur ad stimatig th agls to th arst 0.1º o th scal o th rotatig tabl bas. Th prcisio ad accuracy providd by this stimat will b mor tha adquat for obtaiig xcllt rsults i this xprimt. Th agl ca b rad mor accuratly by usig th magifir. Th ight light will b dd i th darkd room. 15. Th zro o th vrir scal is th rfrc mark usd to idicat th xact agular positio o th mai scal. Rmmbr that th smallst divisio o th mai scal is 0.5º ad that th stimat oly has to ivolv 5 qual imagiary icrmts of 0.1º i ach of thos. 16. Op up a Excl spradsht ad labl th colums ad rows as show i Figur 5. 11
Figur 5. Exampl of Excl spradsht for hydrog Balmr sris xprimt. 17. I your spradsht, masur ad rcord th radigs o th mai scal for th agls θ R ad θ L i dgrs. This should b do for i = 5, 4, ad 3, both right ad lft diffractd Balmr lis, ad for both th first ad scod ordr pattrs, m = 1 ad m =. If possibl th masurmt should also iclud th i = 6 li. Th radigs for θ R should b i th rag of 0-30 ad th radigs for θ L should b i th rag 330-360. Of cours th valus of θ L should b i th rag of 0-30, but will b foud latr i th aalysis of th data by subtractig th radig valu from th iitial valu of 360. Aalysis 1. I th xtra colum for th corrctd valu for th agl θ L, subtract th radig for θ L from 360. Ths valus will b i th rag from 0 to 60, ad should b arly th sam valus as θ R. Exami your valus of θ L ad θ R to s if thy ar approximatly th sam o both sids. This will allow you to chck for ay mistaks i th agl masurmts. Th pairs of valus should b withi about 0.5 of ach othr, You should rpat th masurmt if thr appars to b a mistak.. I th colums for θ L (rad) ad θ R (rad) covrt th agls i dgrs to radia uits. Excl rquirs th agls for trigoomtry fuctios to b i radias. Th covrsio factor is 180 = π radias, thrfor multiply th agl i dgrs tims th valu for π ad th divid by 180. I Excl, th valu of π ca b obtaid with th Math & Trig fuctio, PI() with o argumt. 3. Us Equatio (7) to calculat λ from ach of th valus for θ L (rad) ad θ R (rad) ad rcord th valus i th colum labld for ths valus. Th valu for th si of th agl i cll H4 is writt as =SIN(H4). Rmmbr that th argumt must b i radias ad placd isid parthss. 1
4. I th colums for 1/λ ad (1/4-1/i ), comput ths valus. 5. Equatio (1) suggsts that a plot of 1/λ vrsus (1/4-1/i ) should giv a straight li passig through zro with a slop of R. Mak a graph of 1/λ vrsus (1/4-1/i ) ad labl th axs with propr titls ad uits. 6. Us th Add Trdli fatur of Excl to fid th slop of th straight li that bst fits your data. Forc th itrcpt to b zro. 7. Compar your valu for th slop ad R with th accptd valu of th Rydbrg costat. Calculat th prct diffrc ad tr your rsult i your spradsht. Accptd Valu - Masurd Valu % Diffrc = 100% Accptd Valu (9) 8. Us your masurd valu for th Rydbrg costat to calculat th wavlgths of th obsrvd missio lis. Compar ths valus with your masurmts of th wavlgth ad calculat th prct diffrc usig Equatio (9). 9. Prit your graph ad spradsht of data. Qustios 1. Show that th SI uits for Plack s costat h ar qual to th SI uits of agular momtum.. Calculat th magitud of th Bohr radius. 3. Calculat th spd of th lctro movig i a orbit whos radius is quivalt to th Bohr radius. 4. Usig your valu for th Rydbrg costat, calculat th wavlgth of a ultraviolt trasitio i th Lyma sris from th = lvl ad to th = 1 lvl. 5. Usig your valu for th Rydbrg costat, calculat th wavlgth of a ifrard trasitio i th Pasch sris from th = 4 lvl ad to th = 3 lvl. 6. How much rgy dos it tak to ioiz a hydrog atom i its groud stat? That is how much rgy has to b supplid to mov a lctro from th = 1 lvl to th = lvl? 13
7. I th aalysis of your rsults, suppos you had plottd 1/λ vrsus 1/ i istad of (¼ - 1/ i ). Would your data giv a straight li ad would its slop giv th sam rsults for R? How would th itrcpt chag? Prlab Qustios 1. What is th rgy of a rd photo of wavlgth 600 m? Of a blu photo of wavlgth 400 m? Exprss your aswrs i both Jouls ad lctro-volts. (1 V=1.60 x 10-19 J). How is th wavlgth of th light to b masurd i this xprimt? Dscrib ach of th trms i Equatio (7). 3. Of th followig thr trasitios: from = to = 1, from = 3 to =, from = 4 to = 3, which producs photos with th shortst wavlgth? Th logst? Which is i th visibl part of th spctrum? Which is i th ultraviolt? Which is i th ifrard? 14