Announcements: The Rydberg formula describes. A Hydrogen-like ion is an ion that
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1 Q: A Hydogelike io is a io that The Boh odel A) is cheically vey siila to Hydoge ios B) has the sae optical spectu as Hydoge C) has the sae ube of potos as Hydoge ) has the sae ube of electos as a Hydoge io ) is a ato that has lost all but oe of its electos Aouceets: Readig fo Fiday:.9 &.0 xa is ext week (Tue. 7:309p) Topics: TZ, Chaptes 35,.9,.0 Pactice exa is posted o CUlea. egy Suay of ipotat Ideas ) lectos i atos ae foud at specific eegy levels ) iffeet set of eegy levels fo diffeet atos 3) Oe photo eitted pe electo jup dow betwee eegy levels. Photo colo deteied by eegy diffeece betwee the two levels. 4) If electo ot boud to a ato: Ca have ay eegy. (Fo istace kietic eegy of fee electos i the P effect.) Hydoge Lithiu lecto eegy levels i diffeet atos: Levels have diffeet spacig (explais uique colos fo each type of ato. (ot to scale) Atos with oe tha oe electo: lowest levels filled. Q: The Rydbeg foula descibes a. the colos of light eitted fo all types of atos. b. the agula distibutio of paticles scatteig off of atos that Ruthefod obseved i his faous expeiet. c. the colos of light eitted by hydoge atos. d. the eegy distibutio of cathode ays e. the fequecies of light eitted by heliu atos. Now we kow about the eegy levels i atos. But how ca we calculate/pedict the? Need a odel Step : Make pecise, quatitative obsevatios! Step : Be ceative & coe up with a odel. Step 3: Put you odel to the test.
2 Bale seies: A close look at the spectu of hydoge Bale (885) oticed wavelegths followed a pogessio whee 3,4,5, 6,. As gets lage, what happes to wavelegths of eitted light? λ gets salle ad salle, but it appoaches a liit. So this gets salle Bale coectly pedicted yet udiscoveed spectal lies. λ λ gets salle ad salle, but it appoaches a liit whee 3,4,5,6,. gets salle as icease gets lage as icease, but o lage tha /4 4* liit Bale seies: A close look at the spectu of hydoge Bale (885) oticed wavelegths followed a pogessio Hydoge ato Rydbeg foula oes geealizig Bale s foula wok? Yes! It coectly pedicts additioal lies i HYROGN. Rydbeg s geeal foula Pedicts λ of tasitio: Hydoge eegy levels Hydoge ato Lya Seies Rydbeg s foula Pedicts λ of tasitio: (>) Hydoge eegy levels Ca Rydbeg s foula tell us what goud state eegy is? (>) (,,3..) (,,3..) BaleRydbeg foula Hydoge eegy levels Hydoge eegy levels Look at eegy fo a tasitio betwee ifiity ad A oe geeal case: What is the eegy of each level ( ) i hydoge? hc hc λ iitial fial hc fial eV hc hc λ iitial fial hc fial 3.6eV 0
3 The Bale/Rydbeg foula coectly descibes the hydoge spectu! Is it a good odel? The Bale/Rydbeg foula is a atheatical epesetatio of a epiical obsevatio. It does t eally explai aythig. How ca we explai (ot oly calculate) the eegy levels i the hydoge ato? Next step: A seiclassical explaatio of the atoic specta (Boh odel) Ruthefod shot alpha paticles at atos ad he figued out that a tiy, positive, had coe is suouded by egative chage vey fa away fo the coe. Oe possible odel: Ato is like a sola syste: electos ciclig the ucleus like plaets ciclig the su The poble is that acceleatig electos should adiate light ad spial ito the ucleus: *lapsed tie: ~0 secods Q3: Why do't plaets spial ito the su? A. Well, they do, but vey, vey slowly. B. Because plaets obey quatu echaics, ot classical echaics. C. Because plaets obey classical echaics, ot quatu echaics.. Because plaets ae uch bigge tha electos they do t eit aythig. Aswe is A. Gavitatioal adiatio (i.e. gavitatioal waves) is uch, uch weake tha electoagetic adiatio. ( Tidal foces likely doiate the eegy loss of the plaets). Nucleus lectostatic potetial eegy lecto Highe egy Whe a electo oves to locatio futhe away fo the ucleus its eegy iceases because eegy is equied to sepaate positive ad egative chages, ad thee is a icease i the electostatic potetial eegy of the electo. egy levels Foce o electo is less, but Potetial egy is highe! lectos at highe eegy levels ae futhe fo the ucleus! Potetial eegy of the electo i hydoge F + We defie electo s P as 0 whe fa, fa away fo the poto! lecto's P wok doe by electic field fo Coulob s costat kqelectqpot F d d P kq elect q pot d kq elect ke e e (fo hydoge) (k / ( 4πε0 ) : Coulob foce cost.) q pot
4 Potetial eegy of a sigle electo i a ato P of a electo at distace fo the poto is ke P, ke P ke(ze) + + (Fo Z potos) potetial eegy 0 distace fo poto How ca we calculate the eegy levels i hydoge? Step : Make pecise, quatitative obsevatios! Step : Be ceative & coe up with a odel. How to avoid the KaBoo? *lapsed tie: ~0 secods Boh Model Whe Boh saw Bale s foula, he cae up with a ew odel that would pedict it ad 'solve' the poble of electos spialig ito the ucleus. The Boh odel has soe pobles, but it's still useful. Why does t the electo fall ito the ucleus? Accodig to classical physics, It should! Accodig to Boh, It just does t. Mode QM will give a oe satisfyig aswe, but you ll have to wait till ext week. Oigial pape: Niels Boh: O the Costitutio of Atos ad Molecules, Philosophical Magazie, Seies 6, Volue 6, p. 5, July 93.) Boh's appoach: #: Teat the echaics classical (electo spiig aoud a poto): Newto's laws assued to be valid Coulob foces povide cetipetal acceleatio. #: Boh's hypothesis (Boh had o poof fo this; he just assued it leads to coect esults!): The agula oetu of the electos is quatized i ultiples of ћ. The lowest agula oetu is ћ. ћ h / π Boh Model. # : Classical echaics The cetipetal acceleatio a v / is povided by the coulob foce F k e /. (k /( 4πε 0 ): Coulob foce cost.) Newto's secod law v / k e / o: v k e / The electo's kietic eegy is K ½ v The electo's potetial eegy is P ke / K + P ½ ke / ½ P Theefoe: If we kow, we kow ad v, etc Fk e / v + cicula obit of electo Boh Model. #: Quatized agula oetu Boh assued that the agula oetu of the electo could oly have the quatized values of: L ћ Ad theefoe: v ћ, (,,3 ) o: v ћ/() Substitutig this ito v k e / leads to: B R, with B h 5.9 p ke ( ke ) /, with R h 3.6eV Fk e / v, B : Boh adius cicula obit of electo, R : Rydbeg egy
5 Boh Model. Results B R, with B h 5.9 p ke ( ke ) /, with R h 3.6eV, B : Boh adius, R : Rydbeg egy The Boh odel ot oly pedicts a easoable atoic adius B, but it also pedicts the eegy levels i hydoge to 4 digits accuacy! Possible photo eegies: γ R ( > ) Successes of Boh Model 'xplais' souce of Bale foula ad pedicts epiical costat R (Rydbeg costat) fo fudaetal costats: R / 9. k e 4 /(4πch 3 ) xplais why R is diffeet fo diffeet sigle electo atos (called hydogelike ios). Pedicts appoxiate size of hydoge ato xplais (sot of) why atos eit discete spectal lies xplais (sot of) why electo does t spial ito ucleus The Boh odel 'explais' the Rydbeg foula!! Shotcoigs of the Boh odel: Why is agula oetu quatized yet Newto s laws still wok? Why do t electos adiate whe they ae i fixed obitals yet Coulob s law still woks? No way to kow a pioi which ules to keep ad which to thow out Ca't explai shapes of olecula obits ad how bods wok Ca t explai doublet spectal lies Questios? Q4: Which of the followig piciples of classical physics is violated i the Boh odel? A. Coulob s law B. Newto s F a C. Acceleatig chages adiate eegy.. Paticles always have a welldefied positio ad oetu (Heisebeg s ucetaity piciple). All of the above. Note that both A & B ae used i deivatio of Boh odel. Ideas fo how to esolve these pobles? Aswes to clicke questios Q: Q: C Q3: A Gavitatioal adiatio (i.e. gavitatioal waves) is uch, uch weake tha electoagetic adiatio. ( Tidal foces likely doiate the eegy loss of the plaets). Q4: C Matte waves: TZ Chapte 6
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