II.6. THE DETERMINATION OF THE RYDBERG CONSTANT


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1 II.6. TE DETERMINATION OF TE RYDBERG CONSTANT. Work purpos Th dtrination of th constant involvd in th spctral sris of th hydrognoid atos / ions.. Thory Th atos of ach lnt it, whn thy ar xcitd (for instanc in an lctrical discharg in a gas), an optical discrt spctru on th basis of which th lnt can b idntifid. Th spctra of chical lnts ar as coplicatd as thir atoic nubr Z is gratr. Th optical spctra of th atos ar du to th optical lctrons, that is th lctrons situatd on th priphral orbitals. Th spctroscopists stablishd that all th lins of diffrnt spctral lins of th hydrognoid ato / ion (a hidrognoid ion is ford by a nuclus and a singl lctron) could b dscribd by a gnral rlation, which givs us th wavlngth of th spctral lins: ~ R R νn, = T ( ) T ( n) = = R. () λ n n n, In th abov rlation n and ar intgr nubrs, T() and T(n) ar spctral trs, and whr = Z R R Z R + M is th lctron ass, M is th nuclus ass and, () R is Rydbrg constant. For hydrogn, this constant bcos th hydrogn Rydbrg constant: 9
2 R R = + M + R. (3) Th xplanation of th spctral sris of th hydrogn ato rprsntd a succssful vrification of th hydrogn ato thory givn by Nils Bohr. Bohr sustains that thr ar only crtain orbits allowd for th lctron, corrsponding to crtain stationary stats. forulatd th following postulats: I. Th ato can xist in a discrt sris of stationary stats dtrind by th discrt sris E, E,..., E,... of th total nrgy. In n ths stats th ato dos not it or absorb nrgy. II. Th nrgy of th ato can chang discontinuously, by th transition fro on stationary stat of total nrgy stat of total nrgy E to anothr stationary E n, with th absorption or ission of a photon. Th frquncy of th absorbd or ittd photon is givn by th rlation: En E ν n, = (4) h (h bing th Planck constant), th procss of absorption taking plac in th cas whn th lctron passs fro a closr orbit to nuclus to a or distant on, and th ission taking plac in th opposit cas. Th spcification of th allowd orbits is obtaind by introducing th quantification condition statd by Bohr, who found that th angular ontu of th lctron on th allowd orbits ust b an intgr ultipl of ћ: L = v r = n h, (5) 93
3 h whr h = is th rationalisd Planck constant and n =,, 3,... is π calld th principal quantu nubr. Th total nrgy E, on th nth orbit is quantizd by th xprssion: whr E is th rducd ass and n µ = 8ε h n h n µ = + M + 0 n, (6) (7) 0 = (8) 4π ε is th rationalisd lntary charg ( ε 0 bing th vacuu pritivity). In th quantu chanics, th lctron nrgy for th hydrogn ato (Eq. (6)) is found by intgrating th Schrödingr quation. Th total lctron nrgy on diffrnt stationary orbits is ngativ, which ans that th lctron is bound by th lctroagntic fild of th nuclus. Using th rlations (4) and (6), on can obtain: 4 µ 0 = λ 3 n, 4π h n c,, n =,, 3,..., < n (9) (c bing th spd of light in vacuu) and, coparing this with (), (3) and (7), it rsults that: 4 0 R =. (0) 3 4π h c 94
4 Using Eq. (4), vry wavlngth of th lins fro diffrnt spctral sris for hydrogn can b found. A spctral sris rprsnts th su of spctral lins that hav a coon dsxcitd nrgtic lvl (s Figur ). Figur. So, thr is th Lyan sris, which has th coon nrgtic lvl corrsponding to = (in Eq. (9)), and n and is situatd in th ultraviolt rang; Balr sris, for which = and n 3, in th visibl rang; Paschn sris, for which = 3 and n 4 and th following sris, all in th infrard rang. In this xprint w will study Balr spctral sris and dtrin th wavlngths for th lins α, β, γ, δ, and prsntd in Figur. In th cas of Balr sris, th rlation () bcos: ε ~ ν n, = = R, n = 3, 4, 5, 6, 7, () λ n, n 95
5 fro which th Rydbrg constant rsults: R 4n =. () ( n 4) λ n, Figur. 3. Exprintal stup Th hydrogn spctru in th visibl band is rcordd on a photographic plat (spctrogra), which is st btwn two plxiglas plats. On th sa photographic plat, it appars, paralll to th hydrogn spctru, th rcury spctru, usd as a coparison spctru, bing rcordd at th sa spctrograph and in idntical conditions. Th rcury spctru prsnts a nubr of lins, thir configuration apparing in Figur, on which thr ar also indicatd th wavlngths corrsponding to ach spctral lin. Th lins of th atoic hydrogn blonging to Balr sris (, β, γ,, and ) whos α wavlngth w ar going to dtrin, also appar in Figur, in thir rlativ positions with rspct to th lins of th rcury spctru. W study th spctrogra with a icroscop. Th icroscop bd can b ovd horizontally, along two prpndicular dirctions, using two scrws. Th ovnt along th spctru allows us to asur th δ ε 96
6 position of a spctru lin on a rulr, graduatd in and with a vrnir, th prcision bing 0.. To fix th position of dsird lin th ocular of th icroscop has a rticular wir. To accoplish th work, thr ar ncssary th spctrogra with th visibl atoic hydrogn spctru (Balr sris) and th rcury spctru, and a icroscop. 4. Working Procdur Looking through th ocular, adjust th icroscop irror to hav th bst illuination. If th icroscop has a lap, plug in its transforr. In ordr not to brak th spctrogra, th initial position of th icroscop ust hav th objct lns stuck to th spctrogra and thn gradually lift, until th spctral lins ar clarly visibl. Vrify th paralllis btwn th spctral lins and th rticular wir, th paralll stting of th rticular wir bing ad by rotating th ocular. Idntify th rcury and hydrogn spctra, looking first at th spctrogra with th nakd y and thn at th icroscop. Rad on th rulr (aftr suprposing th rticular wir on ach lin) th positions x i of th rcury lins (th wavlngths bing givn for ach of th). Rad also on th rulr th positions x j of th lins fro th hydrogn sris ( α, β, γ, δ, and ε ). Draw on a illitric papr th calibration curv for th rcury λ = λ( x). aving th positions x j of th hydrogn lins, th wavlngths of th lins α, β, γ, δ, and ε, ncssary to coput th Rydbrg constant, rsult fro th calibration curv. 5. Exprintal data procssing Th data asurd for rcury ar writtn in Tabl. Tabl. λ (Å) x ()... 97
7 Th asurd data for th lins α, β, γ, δ, and writtn in Tabl. Tabl. Lin x () λ (Å) n R (  ) 3 α ε ar Coput th Rydbrg constant according to th rlation (); th obtaind valu ar writtn in Tabl. Coput th an valu 5 R = R j (3) 5 and th standard rror σ R = 0 j= 5 ( R j R ) j= Th final rsult is prsntd in th for (  R ± σ ) R R. (4) =. (5) 6. Qustions. What is a spctral sris?. What is a spctral tr? 3. What postulats lay at th basis of Bohr's forula for th nrgy lvls of th hydrognoid atos / ions? 98
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