points Points <40. Results of. Final Exam. Grade C D,F C B

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Randall Harrington
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1 Results of inal Exa points Grade C D, Points A C < # of students

2 Proble (che. equilibriu) Consider the following reaction: CO(g) + H O(g) CO (g) + H (g) In equilibriu at 6K, n.8m n.8m n.7 M n CO H CO H (a) (5) Calculate the equilibriu constant and the standard Gibbs energy for this reaction at a given teperature. (b) () o the above equilibriu syste, we add an extra.m of CO. What are the new equilibriu concentrations of each coponent?.7 M K (b) (a) he ass action law: [ CO ][ H ] [ CO ][ H O ] start change finish x.8 G.8 G CO H O CO H x -x + x + x.7-x.7 - x.8+x.8+x.5 x +.8 x.65 ±.7 ln (.8 ).6 J (.8 + x ) (.7 x )(.7 x ).8 x.8. x he first root,.9, is greater than. thus, only the second root,.6, is eaningful..8 New equilibriu: n.866 M n CO.866 M n H. M n CO H. M

3 Proble (partition function, average energy) Consider a syste of distinguishable particles with five icrostates with energies,,,, and ( ev ) in equilibriu with a reservoir at teperature.5 ev.. () ind the partition function of the syste.. (5) ind the ean energy of the syste.. () What is the energy of N such particles? Z the ean energy of a single particle: E ev.. ev the energy of N such particles: U N E. ev.ev

4 dn (b) (a) Proble (oltzann distr.) Consider an ideal gas of atos with ass at teperature. (a) (7) Using the Maxwell-oltzann distribution for the speed v, find the corresponding distribution for the inetic energy (don t forget to transfor dv into d). (b) (8) ind the ost probable value of the inetic energy. (c) (5) Does this value of energy correspond to the ost probable value of speed? Explain. () v ND () v dv N π v dv dv d π / ( ) ND ( ) d v dv d N d D dn ( ) π / ( ) / D / π / + / v π / 8 π ax ax (c) the ost probable value of speed ( v ) dd v ax dv the in. energy that corresponds to the ost probable value of speed v v ax doesn t correspond

5 Proble (degenerate eri gas) When the copper atos for a crystal lattice with the density of atos of , each ato donates electron in the conduction band. (a) (5) Assuing that the effective ass of the conduction electrons is the sae as the free electron ass, calculate the eri energy. Express your answer in ev. (b) (7) he electrons participate in the current flow if their energies correspond to the occupancy n() that is not too close to (no epty states available for the accelerated electrons) and not too sall (no electrons to accelerate). At K, calculate the energy interval that is occupied by the electrons that participate in the current flow, assuing that for these electrons the occupancy varies between. and.9. (c) (8) Using the assuptions of (b), calculate the ratio N /N where N is the nuber of current-carrying electrons, N is the total nuber of electrons in the conduction band. Assue that within the range where the occupancy varies between. and.9, the occupancy varies linearly with energy (see the igure), and the density of states is alost energy-independent. he density of states for the threediensional eri gas: occupancy N g / ( ) E - E

6 (a) (b) n ( ) E Proble (degenerate eri gas) cont. h 8 N π V / ( 6.6 ) 8 9. π /. 8 J 6.7 ev E.9 9 E ln 9 E + E + + E. E ln 9 9 E. ev + ln 9 (c) N E + / N n / E E / E ( ) g ( ) d E.5 N N. hus, at K, the ratio of the current-carrying electrons to all electrons in the conduction is. or. %.

7 Proble 5 (blacbody radiation) (a) (5) he blac body radiation fills a cavity of volue V. he radiation energy is U σ, the radiation pressure is ( ), V V c Consider an isentropic (quasi-static and adiabatic) process of the cavity ansion (dsdu+pdv). he radiation pressure perfors wor during the ansion and the teperature of radiation will drop. ind how and V are related for this process. (b) (5) Assue that the cosic icrowave bacground (CM) radiation was decoupled fro the atter when both were at K. Currently, the teperature of CM radiation is.7 K. What was the radius of the universe at the oent of decoupling, copared to now? Consider the process of ansion as isentropic. P σ c (a) he equation that describes the isentropic (quasi-static adiabatic) process for the photon gas: σ ds du PdV P c σ σ σ σ 6 σ V d + dv dv 6 V d c c c c c d dv V U ( ) V const σ, V V c (b) V i R i f or R const hus, at the oent of decoupling, the radius V f R f i of the universe was ~ tie saller dv

8 Proble 6 (EC) Consider a non-interacting gas of hydrogen atos (bosons) with the density of -. a) (5) ind the teperature of ose-einstein condensation, C, for this syste. b) (5) Draw a qualitative graph of the nuber of atos as a function of energy of the atos for the cases: >> C and.5 C. If the total nuber of atos is, how any atos occupy the ground state at.5 C? c) (5) elow C, the pressure in a degenerate ose gas is proportional to 5/. Do you ect the teperature dependence of pressure to be stronger or weaer at > C? Explain and draw a qualitative graph of the teperature dependence of pressure over the teperature range < < C. (a) C / ( 6.6 ) / ( ). K.5 h N π V.8 π.7 (b) n() < C >> C n() / / N N N.5.65 C N [ ]

9 Proble 6 (EC) (cont.) (c) he atos in the ground state do not contribute to pressure. At < C, two factors contribute to the fast increase of P with teperature: (i) an increase of the nuber of atos in the excited states, and (b) an increase of the average speed of atos with teperature. Above C, only the latter factor contributes to P(), and the rate of the pressure increase with teperature becoes saller than that at < C. P() C

Solutions to the problems in Chapter 6 and 7

Solutions to the problems in Chapter 6 and 7 Solutions to the probles in Chapter 6 and 7 6.3 Pressure of a Feri gas at zero teperature The nuber of electrons N and the internal energy U, inthevoluev,are N = V D(ε)f(ε)dε, U = V εd(ε)f(ε)dε, () The