Results of Final Exam

Transcription

1 Results of Fial Exa # of studets Grade Poits A > C C D F < poits

2 Proble 1 (partitio fuctio) The groud level of the eutral lithiu ato is doubly degeerate (that is, d ). The first excited level is 6-fold degeerate (d 1 6) ad is at a eergy 1. ev above the groud level. (a) (1) I the outer atosphere of the Su, which is at a teperature of about 6 K, what fractio of the eutral lithiu is i the excited level? Sice all the other levels of Li are at a uch higher eergy, it is safe to assue that they are ot sigificatly occupied. (b) (5) Fid the average eergy of Li ato at teperature T (agai, cosider oly the groud state ad the first excited level). (c) (15) Fid the cotributio of these levels to the specific heat per ole, C V, ad setch C V as a fuctio of T. If the groud level eergy is defied as zero ad E is the eergy of excited level: ( β ε ) + 6 ( β E) Z d The probability that the ato is i its excited level: P ( E) i 6 i i ( β E) 3( β E) 3 Z 1+ 3( β E) 3 + ( β E) E 1. ev, T 6K (~.5 ev), βe.3, (βe ) 1: P( E). 3

3 .75 1 ε The average eergy per ato: 1 Z Z ε 6 β + 6 The specific heat: ( βε ) 3ε ( βε ) ( βε ) + 3 ε E/ i ε βε x i 1 C V ε T V 3ε ε ( βε ) [ ( βε ) + 3] β T ( βε ) ( βε ) [ ( βε ) + 3] C / Vf i βε x i 1

4 P (a) Proble (blacbody radiatio) Plaet ercury revolves ad rotates at the sae rate, so oe side of the plaet is always facig the Su. ercury is a distace of 5.8 x 1 1 fro the Su, ad has a radius of.44 x 1 6. The radius of the Su is ad its total power output is 4 x 1 6 W. I this proble treat the plaet as if it were a blac body. a) (5) What is the eergy flux of the Su s radiatio at ercury's orbit? b) (5) What is the total power absorbed by ercury? [Hit: Cosider that it appears as a flat dis to the Su ad it absorbs all of the icidet radiatio.] c) (1) If ercury is i therodyaic equilibriu, it will eit the sae total power as it receives fro the Su. Assuig that the teperature of the "hot side of ercury is uifor, fid this teperature. d) (5) What is the pea frequecy of the radiatio absorbed by ercury? e) (5) What is the pea frequecy of the radiatio eitted by ercury? J P 4πR 6 Su W / orbit 4π 4 1 W 1 ( ) (b) P J πr W / π (.44 1 ) W ercury (c) ercury ercury 4 πrercuryσtercury 1/4 1/4 17 ercury W T ercury ercuryσ - hei-sphere P πr π 6 (.44 1 ) W / K K

5 Proble (blacbody radiatio), cot. (d) T 1/ 4 1/ 4 6 Su 4 1 W Su 5, Suσ P 4πR 4π 8 ( 7 1 ) W / K K ν 3 TSu J / K 5,795K h Js received 14 ax 1 Hz (e) ν 3 Tercury J / K 535K h Js eitted 13 ax 1 Hz

6 Proble 3 (oltza) (a) (15) Withi the odel of isotheral atosphere, fid the ratio of the ass of the atosphere to the ass of the plaet. Assue that the gravitatioal field is uifor. The acceleratio of the free fall, g, ad the pressure at the plaet s surface, P, are ow. Calculate this ratio for the Earth. (Hit: gg pl /R pl, where pl ad R pl are the ass of the plaet ad its radius, respectively, the gravitatioal costat G6.67*1-11 N /g ). gh T H # of olecules withi dh: dn( h h dh) ( area) dh the ass of the olecule, pl ad R pl - the ass of the plaet ad its radius, respectively, the desity of olecules at the plaet s surface. + h dh at at gh T 4π Rpl dh 4πR pl ( x) dx T g 4 4πR plt 4πG plp g g 11 5 πgp 4π N / g 1 Pa g 1 / s at pl ( ) The sae result you would get by cosiderig equilibriu coditio: F where F is the total force exerted o the atosphere by the plaet s surface F P 4π G Rpl atg pl P 4π atg g at 4 pl g P G π πr pl at g g T

7 Proble 3 (oltza) cot. (b) (1) Fid the average potetial eergy of the olecules i the atosphere. U gh gh area dh T ( ) gh area dh T ( ) gh T y ( T ) y ( y) ( y) T g T dy g dy T (c) (5) Fid the heat capacity per olecule i the atosphere (do t forget the ietic eergy). E 5 ( T ) K( T ) + U ( T ) T T + de 7 C C P dt

8 Proble 4 (Feri-ose) (1) For a syste of particles at roo teperature, how large ust ε-μ be for the Feri- Dirac, oltza, ad ose-eistei distributios agree withi 1%? () Estiate the desity of a syste of obile electros i a seicoductor that ca be treated at roo teperature equally well (with 1% accuracy) usig all three distributios. Assue that the effective electro ass is the sae as a free electro ass, ad that you ca use for this estiate the ressio for μ i a ideal classical gas. The oltza distributio lies betwee the Feri-Dirac ad ose-eistei distributios. It is sufficiet to request: ε μ ε μ T T ε μ T E FD ε μ + 1 T 1.1 ε μ 1 T ε μ > l 5.3 T This iequality ust be satisfied for all ε, icludig ε: Q Q μ > 5.3 T μoltza T l l > 5.3 < Q Q 3/ 3/ 31 3 π / 3 T g J K K h ( ) Js 3 < 6.3 1

9 Proble 5 (D Feri systes) Cosider a o-relativistic Feri gas i two diesios: N electros cofied to a square area AL. (a) (1) The desity of electroic states i two diesios per uit area: g D ε Show how to derive this forula, do t worry about a exact uerical pre-factor, focus o the depedece o eergy ad ass. (b) (5) Fid the Feri eergy (i ters of N ad A). (c) (5) Fid the average eergy per electro at T (i ters of E F ). (d) (5) Calculate E F for the electro desity N/A (a typical desity of electros i a field-effect trasistor), assue that the effective ass is.x(free electro ass). Is the electro gas i a FET degeerate at roo teperature? (a) For quatu particles cofied i a D box : N ( ) 1 4 π π π L L x y ( area) 4π 4π π ( ) G( ε ) G π x y x y x + Lx Ly 1 ε 4π h ( ε ) ( s + 1) ( ) π h - does ot deped o ε y g D π h π h E F π h N F F π h A (b) The Feri eergy: N g( ε )( area) dε A E E (c) The average F eergy per electro: U tot N ε g( ε )( area) E EF ( ) ( area) 1 area ε dε ε dε πh πh E F

10 Proble 5 (D Feri systes) (cot.) U (d) tot E F F A A 1 A 1 N ε ε g( ε )( area) dε ε dε E E F ε E F πh N πh πh E 34 ( Js) J K π h N π EF A g F At 3K, the syste is ot degeerate.