CHEM 333 QUANTUM THEORY AND SPECTROSCOPY PROBLEM SET I SOLUTION KEY

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CHEM QUANTUM THEORY AND SPECTROSCOPY PROLEM SET I SOLUTION KEY. T for onstant of O is N.

vibrational frquny in - units? v OO O g ol 6 μ 6g / ol.89 π μ g g + 6. O O N.66 6 v s π.

Considr a wav ad of t first two aronis πut π πut π π u(, t) os sin + os + sin l l l l Sow

1 CHEM QUANTUM THEORY AND SPECTROSCOPY PROLEM SET I SOLUTION KEY. T for onstant of O is N. -. Assuing a aroni osillator odl for t vibrational otion of O, wat is t diatoi vibrational frquny in - units? v OO O g ol 6 μ 6g / ol.89 π μ g g + 6. O O N.66 6 v s π.89 g wavnubr: v s v.9989 s Considr a wav ad of t first two aronis πut π πut π π u(, t) os sin + os + sin l l l l Sow graially tat t surosition of ts standing wavs is a travling wav. π π u, t) A( t)sin + ( t)sin l l ( wit ( t) os( ω t) π and ( t) os ωt + A Construt A (t) + (t)

2 CONCORDIA UNIVERSITY CHEM Probl St I Solution Ky ω t) os( ω t) t π ( ( t) os ωt + A π / / - π / π / -/ π - T su of two standing wavs is a travling wav.

3 CONCORDIA UNIVERSITY CHEM Probl St I Solution Ky. In PLANCK s tory of blabody radiation, t dnsity of radiativ nrgy btwn v and v + dv is givn by 8πv dv ρ( v, T ) dv / T v a. Wat is t basi non-lassial assution tat lad PLANCK to tis rssion? E nv wit n nonzro ositiv intgr. T nrgy of t ltroni atoi osillators tat it radiation an only ta disrt valus roortional to t osillation frquny, i.. t nrgy is quantizd. b. Sow tat, for sall frqunis, PLANCK's rssion rdus to t lassial RAYLEIGH-JEANS rssion. 8π T ρ ( v, T ) dv v dv n Trunating t Taylor sris ansion to nd ordr wit!! n! v T 8πv 8πv 8πv T 8π T ρ ( v, T ) ~ v v / T ( + v / T + ) v Can you lain wy tis is t as in sil trs? T ltroni atoi osillators tat ar ativ, i.. it radiation, ust av a iniu nrgy v ( n ). For sall frqunis, tr is noug tral nrgy for all t osillators to av t sall rquird iniu nrgy v and trfor to b ativ. Classial ysis gnrally ovrounts t nubr of osillators, but for sall frqunis (or ig traturs) all osillators ar ativ and quantu rssions rdu to lassial ons.. Driv t STEFAN-OLTZMANN law R E v σt, wr R is t radiation nrgy r unit ara and ti and E v is t total radiation nrgy dnsity. Coar your rsult for t STEFAN-OLTZMANN onstant σ to t rintal valu of J - K - s -.

4 CONCORDIA UNIVERSITY CHEM Probl St I Solution Ky E R V 8π v dv π v dv ρ( v, T ) dv / v T v / T v Introduing t variabl, on obtains T R π T d π T d π T π 5 π 5 5 T σt 5 5 π π (.866 JK ) J K 8 wit σ s 5 5 (.9989 s ) (6.66 Js). T a of ission fro a ot tal in a stl furna ours around 6 n. Estiat t tratur of t stl. Win dislant law: T Js.9989 s JK a.9 K.9 K.9 K T 8K ~ 5 C 9 6 a 5. T rlativ ositions of t stral lins obsrvd in t Hydrogn ato ission stru an b rrodud by t RYDERG forula v R H n < n a. Aording to t or odl of t ydrogn ato, t RYDERG onstant an b drivd as μ R H 8ε Wat nurial valu do you obtain for t RYDERG onstant? How dos it oar to t rintal valu of 9, μ μ g 9.6 8ε + R H

5 CONCORDIA UNIVERSITY CHEM Probl St I Solution Ky R H g (.6 C) 9,677 8 (8.85 C s g ) (6.66 ) (.9989 s ) wi agrs vry wll wit t rintally dtrind valu of t Rydbrg onstant! b. Calulat t ionization nrgy of t Hydrogn ato (in V). T ionization nrgy is t nrgy ndd to rov t ltron fro t H ato, i.. t ga btwn t ground stat ( n ) and an infinitly itd stat ( ) orrsonding to t saratd ltron and roton. 9, 678 RH v RH 8.8 J IE v v 6.66 Js.9989 s V 9.6 J / V. For n, alulat t wavlngt (in n) for t first two lins givn by t RYDERG forula. Also alulat t wavlngt for t n sris liit. Wat art of t ltroagnti stru do ts lins blong to? 5 6 v RH n v v RH.8.8 8n 5 v sris liit: 6 v RH n v Ts lins blong to t nar-ir rgion of t ltroagnti stru (IR los to visibl) and tis sris is alld t Pasn sris. 6. a. Calulat t radius of t first ltroni orbit in t OHR odl of t Hydrogn ato. Calulat t vloity of an ltron in t first orbit. H ato: Z First orbit (ground stat): n 5

6 CONCORDIA UNIVERSITY CHEM Probl St I Solution Ky ε n ( 8.85 C s g ) (6.66 ) r Å 9 πμz π (9.6 g) (.6 C) v / πε μr or 6.66 Js 6.9 n v s ~% t sd of ligt! μr 9.6 g 5.9 b. Calulat t wavlngt assoiatd wit tat ltron. Wat an you onlud of t wav natur of t ltron in t Hydrogn ato? Drogli ostulat μv.66 Js 9.6 g s.å T ltron wavlngt is of siilar agnitud as t ysial dinsion of t ato (radius of first orbit ~.5Å), so t wav natur of an ltron tyially travling at % t sd of ligt in an ato is signifiant. As a attr of fat, t wav natur of ltrons was rovd by intrfrn and diffration rints. T wav-artil duality alis to irosoi objts su as ltrons, but not to arosoi objts. T wavlngt of ltrons (~Å) at tyial vloitis in atos lis in t X-ray rgion of t ltroagnti stru, trfor X-rays will it ltrons in atos; onvrsly, alrating ltrons in atos will it X-rays.. Wat is t unrtainty in t ontu of an ltron tat w wis to loat witin 5? Wat an you onlud? Δ 5. 5Å Using Hisnbrg unrtainty rinil Δ Tis translats into an unrtainty in t vloity Δ Js. g s Δ Δv g s 7.5 s g wi is of t sa ordr of agnitud as (it is atually largr tan) t vloity of an ltron in an ato 6 itslf (rall v.9 s ). T or odl of t H ato violats t Hisnbrg unrtainty rinil, as it infrs at nowldg of t osition and ontu of t ltron in t H ato. As tis alulation sows, if w wis to loat t ltron in an ato wit rasonabl auray ( Δ. 5Å, wi is onsistnt wit t ysial dinsion of t ato), tn w nd u wit ug unrtainty in t ltron vloity. T Hisnbrg unrtainty rinil alis to irosoi objts travling at ig vloitis su as ltrons in atos, but not to arosoi objts. 6

7 CONCORDIA UNIVERSITY CHEM Probl St I Solution Ky 7. a. Wat is t wor funtion of a tal? T wor funtion of a tal is dfind as t iniu nrgy rquird to trat an ltron fro t tal surfa. It is t bul analog of ionization nrgis for atos and oluls and rrsnts t binding nrgy of t ltron to a tal. b. T wor funtion of Nil is 5 V. Calulat t inti nrgy (in V) of t ltrons jtd fro t tal following radiation by ligt of wavlngt 5 and n? φ 5V.6 5V V 9 J 8. 9 J v + φ v 5n Js.9989 s 9 v.98 J. 8V 9 5 T inidnt oton dos not av noug nrgy to trat an ltron fro t Nil surfa. No otoltri fft will b obsrvd. n Js.9989 s 8 v.99 J. V 9 v v φ.v 5V 7. V T inti nrgy of t jtd ltrons is 7.V.. Wat is t na of tis ysial nonon? Ejtion of ltrons fro a tal lat uon irradiation is alld t otoltri fft, first disovrd by Hrtz in t lat 88s. d. Wo obtaind t Nobl Priz in Pysis in 9 for roviding an lanation to tis nonon (in 95)? Albrt Einstin 7

8 CONCORDIA UNIVERSITY CHEM Probl St I Solution Ky Constants π Equations J.s s C μ + ε C.s.g J.K - N A g g O 6. g/ol V v a T v.965 v π μ Units V.6-9 J Å - Elnts of Matatis v + φ v v r ε n πμz v / πε r μ l μvr n + +! +! n n! +... Δ Δ π d 5 8

9 CONCORDIA UNIVERSITY CHEM Probl St I Solution Ky Figur: t ltroagnti stru Adatd fro Quantis, Rudints of Quantu Pysis, Lévy-Lblond and alibar, Elsvir, 989. Visibl: tyially -9 n 9

Chapter 40 Introduction to Quantum

Chapter 40 Introduction to Quantum Catr 0 Introdution to Quantu Pysis 900-90 A nw tory alld quantu anis was igly sussful in xlaining t bavior of artils of irosoi siz. Baus sintists larn t wav and artil naturs of ligt in 9 t ntury, ty roos