University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015

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1 Uiversity of Wasigto Departmet of Cemistry Cemistry 453 Witer Quarter 15 Lecture 14. /11/15 Recommeded Text Readig: Atkis DePaula: 9.1, 9., 9.3 A. Te Equipartitio Priciple & Eergy Quatizatio Te Equipartio Priciple is a direct cosequece of represetig te partitio fuctio as a itegral. We we do tat we assume we may vary te eergy cotiuously by varyig te coordiates ad mometa cotiuously. Eergy levels are terefore cotiuous ad te gap betwee adjacet eergy levels is ifiitely small for traslatios, rotatios, ad vibratios. We te gap betwee eergy levels is small eat will always be absorbed by te motios of te molecules ad so eergy is equally partitioed betwee tese motios. So log as te eergy is a sum of squares of mometa ad coordiates, te partitio fuctio will deped o te same stadard itegral expressio ax 1 e dx ad a factor of T will be geerated for eac a occurrece of tis itegral i q. Tis is te matematical origi of te Equipartitio Priciple. Te oly trouble is...tis does ot work a lot of te time. As outlied i te last lecture, te Equipartitio Priciple (EP)fails to predict termodyamic properties of systems related to vibratioal ad rotatioal motios. at low temperatures. As a result eat capacities ad oter pysical properties wit teir origis i atomic vibratioal motios i crystals ad ideal diatomic gases are iaccurately predicted. A clue as to wat is goig wrog ere is afforded by te vibratioal ad rotatioal absorptio spectra of diatomic molecules i te gas pase. Figure 11.1 is a spectrum of te frequecies of radiatio absorbed by te vibratioal ad rotatioal motios of te HCl molecule: 8

2 Figure 14.1: Te spectrum of frequcies absorbed by te vibratioal ad rotatioal motios of HCl i te gas pase. Te fact tat discrete frequecies are absorbed implies tat eergy levels for tese motios are discotiuous. Te spectrum i Figure 14.1 is explaied if we assume te rotatioal ad vibratioal eergies are discreetly spaced as sow i Figure 11. Figure 14. Vibratioal eergies ad rotatioal eergies are ot a cotiuum, but rater are quatized. So eergy trasitios tat occur upo te absorptio of radiatio produce discrete trasitios, idicated by te arrows, ad wic correspod to te absorptio lies tat we observe i te vibratioal spectrum of HCl. B. Some Basic Quatum Mecaical Postulates To calculate macroscopic properties like U, S, A, P, etc. we require eergies for atomic traslatios, vibratios, rotatios, as well as electroic eergies. Tese eergies are summed ito partitio fuctios wic i tur geerate te macroscopic properties tat we wat. Altoug te eergies we ave for traslatioal motios of gases seem to produce correct results we subjected to tis process, vibratioal ad rotatioal eergies are ot distributed i a cotiuum ad we tat is assumed we get results for 9

3 eat capacities etc. wic are especially iaccurate at low temperatures. Moreover, data like Figure 14.1 idicate te eergy cotiuum idea is ot applicable to all motios ad especially ot to atomic vibratios ad molecular rotatios. Quatum mecaics is a teory wic produces discrete or quatized eergy values for motios of atoms, molecules, electros, etc. Te pysical basis for discrete eergy levels ca be mostly easily explaied for systems tat display periodic motios. Figure 14.3: A simple view of ow te wave legt of a particle wave must be matced to te legt of te orbit. I te case of a circular orbit of radius a, te legt of te orbit must accommodate a itegral umber of wavelegts: A particle wose properties are govered by quatum mecaics as associated wit it a wavelegt tat is related to its mometum by debroglie s equatio p mv (14.1) were =Plack s costat=6.6x1-34 Js If te particle is boud i space its orbit must be compatible wit its wavelegt, else te wave will iterfere wit itself as sow i Figure 14.3 Because te eergies of mecaical motios at te atomic level are quatized, eergy ca oly be absorbed or emitted i discrete quatum called potos.if a quatum of eergy or poto is quatified by is, were is te frequecy of te eergy, ad te differece betwee adjacet eergy levels is E, te for absorptio or emissio of eergy to occur E (14.) 3

4 Te wave ature of a particle is quatified by a wave fuctio. Te eergies of te particle s quatized motio are E, were is a positive iteger. Correspodig to eac eergy is a particle wave fuctio for oe dimesioal motio is (x,). How do we obtai tis wave fuctio? A form for te quatum mecaical wave equatio is foud as follows. I classical mecaics we solve Newto s equatio F=Ma to obtai a particles positio ad mometum as a fuctio of time. Tese quatities may be obtaied wit arbitrarily ig precisio. Te eergy is explicitly a fuctio of positio ad mometum by p E Vx (14.3) m It is assumed tat i quatum mecaics equatio 14.3 still olds. However we we combie equatio 14.1 wit equatio 14.3 we obtai V x E m (14.4) I quatum mecaics we assume te wave equatio must agree wit equatio 14.4 ad like oter wave equatios it must be liear. Tis meas it must cotai te wave fuctio oly to te first power. Tis meas te wave equatio must ave te form: x Vx x E x m (14.5) I oter wave problems, a plausible form for te wave fuctio is x x x Asi Bcos (14.6) A oe dimesioal wave equatio tat is cosistet wit 14.5 ad 14.6 is x V x x E x (14.7) m x 34 were 1.51 Js Equatio 14.7 is te oe dimesioal, statioary Scroediger equatio. Te values ad form for te potetial eergy V depeds upo te ature of te problem. For some types of motio like electroic motio or molecular rotatios, Scrodiger s equatio is trasformed from Cartesia coordiates to sperical coordiates. Te pysical meaig of te wave fuctio was ot at first agreed upo. Te wave fuctio is ow iterpreted as a probability, i.e. te probability of observig x x dx ad te probability of fidig a a particle betwee x ad x+dx is particle betwee x=a ad x=b is x b a xdx. Te superscript * deotes a complex cojugate. Wave fuctios ca be complex umbers. But te probability is a real umber. Multiplyig a complex umber a+ib by its complex cojugate a- ib assures te product is real. 31

5 Ulike classical mecaics were te mometa ad coordiates of a particle ca be determied to arbitrarily ig precisio for all time, i quatum mecaics te mometum ad positio of a particle caot be simultaeously determied wit arbitrary precisio. Te ucertaities of positio x ad mometum p are itertwied by te Heiseberg Ucertaity Priciple: xp x (14.7) I quatum mecaics kowledge of te wave fuctio allows us to calculate a particle s average positio ad average mometum. Tese quatities are called expectatio values i quatum mecaics. I te ext 3 lectures we will explore te properties of quatized traslatios, vibratios, ad rotatios. C. Particle i a Box; Simple Approac A particle wit potetial eergy V(x)= for <x<l ad V(x) ifiite oterwise, see Figure 14.4, is referred to as a particle i a box. Figure 14.4: Potetial eergy profile for a particle i a box. I suc a potetial defied as if x a V( x) (14.8) oterwise te particle must stay betwee x= ad x=l because it would eed a ifiite amout of eergy to escape from tis regio. As log as te particle remais witi te box it ca go aywere i te box ad will ave ay eergy 1 E K mv. Quatum mecaics produces a differet outcome, wic we ca obtai simply as follows. Te particle is associated wit a wave were te wave must ave odes at x= ad x=l. Tis meas te particle ca ave ay wavelegt 3

6 . L ; 1,,3,4 (14.9) Puttig equatios 14.6 ito 14.1: L p (14.1) Equatio 14.7 leads to a expressio for te lietic eergy of a particle tat ca ave oly certai values: 1 p 1 E mv (14.11) m m 8mL D. Particle-i-a Box: Scroediger s Equatio Te simple use of te debroglie equatio (i.e. equatio 14.1) ad te ituitive treatmet of te particle waves preseted i te prior sectio ca be made more rigorous by solvig Scroediger s equatio for te particle-i-te-box. Te simple approac gave us a expressio for te eergy of te particle but gave o expressio for te particle s wave form. Usig te Scroediger Equatio we ca obtai te eergies ad wave fuctios for a particle i a box, wic is a model for quatized traslatios. Te wave fuctio (x) is determied from te time-idepedet Scroediger equatio: d V( x) ( x) E( x) (14.1) 8 mdx Usig te potetial V(x)= for te particle witi te box equatio becomes d d x me E( x) or x (14.13) 8 mdx dx 34 were 1.51 Js Te geeral solutio of equatio is: me me x Asi x Bcos x (14.14) L we get B= ad te quatizatio coditio: me or E (14.15) L 8mL Wave fuctios must obey a ormalizatio coditios like probabilities: Wit te boudary coditios: wic gives us A L Te wave fuctio is: a x dx 1 (14.16) 33

7 x x si L L (14.17) Figure 14.5: Particle i Box Wave fuctios for =1,,3 Particle i a Box =1,,3 1.5 Wave Fuctio L Te probability tat te particle i eergy state is at positio x is * si x P x x x x L L C. How to Fid a Particle i a Box x Wave Fuctios: x si, =1,, 3are picture below. L L Note te x poits were te wave fuctio is zero. Tese poits are called odes. Te umber of odes icreases wit. 34

8 Te probability of fidig a particle i te t eergy at a poit x is x of particle probabilities for =1,,3 are displayed below. Particle i a Box: =1,,3 x si L L. Plots Probability x Figure 14.6: Particle i Box probabilities for =1,,3 Terefore for ay te probability of fidig te particle betwee ay two poits is xb xb si x Pa b x dx dx L L (14.18) xa xa 35

Atomic Physics 4. Name: Date: 1. The de Broglie wavelength associated with a car moving with a speed of 20 m s 1 is of the order of. A m.

Atomic Physics 4. Name: Date: 1. The de Broglie wavelength associated with a car moving with a speed of 20 m s 1 is of the order of. A m. Name: Date: Atomic Pysics 4 1. Te de Broglie wavelegt associated wit a car movig wit a speed of 0 m s 1 is of te order of A. 10 38 m. B. 10 4 m. C. 10 4 m. D. 10 38 m.. Te diagram below sows tree eergy