he Knetc heory o Gases CHE OULIE. Molecular Model o an Ideal Gas. Molar Specc Heat o an Ideal Gas. dabatc rocesses or an Ideal Gas.4 he Equpartton o Energy.5 Dstrbuton o Molecular Speeds SWES O QUESIOS Q. he molecules o all derent knds collde wth the walls o the contaner, so molecules o all derent knds exert partal pressures that contrbute to the total pressure. he molecules can be so small that they collde wth one another relatvely rarely and each knd exerts partal pressure as the other knds o molecules were absent. I the molecules collde wth one another oten, the collsons exactly conserve momentum and so do not aect the net orce on the walls. Q. he helum must have the hgher rms speed. ccordng to Equaton (.4), the gas wth the smaller mass per atom must have the hgher average speed-squared and thus the hgher rms speed. Q. he alcohol evaporates, absorbng energy rom the skn to lower the skn temperature. *Q.4 () () Statements a, d, and e are correct statements that descrbe the temperature ncrease o a gas. Statement b s true the molecules have any sze at all, but molecular collsons wth other molecules have nothng to do wth temperature. () Statement c s ncorrect. he molecular collsons are perectly elastc. emperature s determned by how ast molecules are movng through space, not by anythng gong on nsde a molecule. *Q.5 () b. he volume o the balloon wll decrease. () c. he pressure nsde the balloon s nearly equal to the constant exteror atmospherc pressure. Snap the mouth o the balloon over an absolute pressure gauge to demonstrate ths act. hen rom n, volume must decrease n proporton to the absolute temperature. Call the process sobarc contracton. *Q.6 t K, mv rms k. t the hgher temperature, m ( v rms ) k hen 4 4( K) 8 K. nswer (d). *Q.7 nswer c > a > b > e > d. he average vector velocty s zero n a sample macroscopcally at rest. s adjacent equatons n the text note, the asymmetrc dstrbuton o molecular speeds makes the average speed greater than the most probable speed, and the rms speed greater stll. he most probable speed s ( M) and the speed o sound s ( M), necessarly smaller. Sound represents an organzed dsturbance superposed on the dsorganzed thermal moton o molecules, and movng at a lower speed. 54 794 ch_p54-57.ndd 54 /7/6 :4:44 M
544 Chapter *Q.8 nswer (b). he two samples have the same temperature and molecular mass, and so the same rms molecular speed. hese are all ntrnsc quanttes. he volume, number o moles, and sample mass are extrnsc quanttes that vary ndependently, dependng on the sample sze. Q.9 he dry ar s more dense. Snce the ar and the water vapor are at the same temperature, they have the same knetc energy per molecule. For a controlled experment, the humd and dry ar are at the same pressure, so the number o molecules per unt volume must be the same or both. he water molecule has a smaller molecular mass (8. u) than any o the gases that make up the ar, so the humd ar must have the smaller mass per unt volume. Q. Suppose the balloon rses nto ar unorm n temperature. he ar cannot be unorm n pressure because the lower layers support the weght o all the ar above them. he rubber n a typcal balloon s easy to stretch and stretches or contracts untl nteror and exteror pressures are nearly equal. So as the balloon rses t expands. hs s an sothermal expanson, wth decreasng as ncreases by the same actor n n. I the rubber wall s very strong t wll eventually contan the helum at hgher pressure than the ar outsde but at the same densty, so that the balloon wll stop rsng. More lkely, the rubber wll stretch and break, releasng the helum to keep rsng and bol out o the Earth s atmosphere. Q. datomc gas has more degrees o reedom those o molecular vbraton and rotaton than a monatomc gas. he energy content per mole s proportonal to the number o degrees o reedom. *Q. () nswer (b). verage molecular knetc energy ncreases by a actor o. () nswer (c). he rms speed ncreases by a actor o. () nswer (c). verage momentum change ncreases by. (v) nswer (c). ate o collsons ncreases by a actor o snce the mean ree path remans unchanged. (v) nswer (b). ressure ncreases by a actor o. hs s the product o the answers to and v. Q. s a parcel o ar s pushed upward, t moves nto a regon o lower pressure, so t expands and does work on ts surroundngs. Its und o nternal energy drops, and so does ts temperature. s mentoned n the queston, the low thermal conductvty o ar means that very lttle energy wll be conducted by heat nto the now-cool parcel rom the denser but warmer ar below t. *Q.4 nswer (a), temperature 9 K. he area under the curve represents the number o molecules n the sample, whch must be as labeled. Wth a molecular mass larger than that o ntrogen by a actor o, and the same speed dstrbuton, krypton wll have ( )m o v ( )k average molecular knetc energy larger by a actor o. hen ts temperature must be hgher by a actor o than that o the sample o ntrogen at K. 794 ch_p54-57.ndd 544 //7 :45:8 M
he Knetc heory o Gases 545 SOLUIOS O OLEMS Secton. Molecular Model o an Ideal Gas. Use u 66. 4 g 4 4 (a) For He, m 4..66 g u 6 64 u. g 4.66 g (b) For Fe, m 55 9 9 9. u. g u 4 66. (c) For b, m 7 u u g 44. g *. ecause each mole o a chemcal compound contans vogadro s number o molecules, the number o molecules n a sample s tmes the number o moles, as descrbed by n, and the molar mass s tmes the molecular mass, as descrbed by M m. he denton o the molar mass mples that the sample mass s the number o moles tmes the molar mass, as descrbed by m nm. hen the sample mass must also be the number o molecules tmes the molecular mass, accordng to m nm n m m. he equatons are true or chemcal compounds n sold, lqud, and gaseous phases ths ncludes elements. We apply the equatons also to ar by nterpretng M as the mass o vogadro s number o the varous molecules n the mxture. v. F m 5 5. t F 57. m 57. a [ 8. sn 45. ( 8. sn 45. )] ( kg). s ms. 94 ( ) 6 v 5. 4 68.4 F m. kg m s 4. and t. s F 4. 4 8. m 7. 6 ka *.5 and so that (. )( )(. ) 6. ( 8. 4). molecules mv.6 Use the equaton descrbng the knetc-theory account or pressure:. hen K K mv where n av av ( ) ( 8 ) 5. atm. a atm 5. m ( mol) 6. molecules mol K 55. J molecule av 794 ch_p54-57.ndd 545 /7/6 :4:46 M
546 Chapter.7 ( KE ) rom the knetc-theory account or pressure. 5 ( ( KE). )( 4. ) ( 6. ) 4. molecules n 6. molecules mol.. mol 4 molecules mv.8 (a) n he total translatonal knetc energy s m v E trans : E trans ( 5 )(.. 5. ).8 kj (b) mv k ( 8 4) 6.. ) 6. J (.9 (a) k : 5 4. a π. 5 m 54. k ( 8. JK)( 9 K) atoms (b) K k ( 8. )( 9) J 67. J 4. gmol (c) For helum, the atomc mass s m 6 664. 4 g molecule. molecules mol 7 m 664 k m v k : v rms m m J. (a) a ( a) Jm a m. kg molecule. 5 km s (b) For a monatomc deal gas, Ent n For any deal gas, the energy o molecular translaton s the same, E trans n hus, the energy per volume s E trans 794 ch_p54-57.ndd 546 /7/6 :4:47 M
he Knetc heory o Gases 547. (a) K k ( 8. JK)( 4K) 876. J (b) K mv rms 876. J 75. J so v rms m () 4. gmol For helum, m 6 664. 4 g molecule. molecules mol 7 m 664. kg molecule 9. 9 gmol Smlarly or argon, m 6 66. g molecule. molecules mol 6 m 66 Substtutng n () above, we nd or helum, v rms 6. km s. kg molecule and or argon, v rms 54 m s Secton. Molar Specc Heat o an Ideal Gas. n. mol, K (b) Snce constant, W (a) Ent Q+ W 9 J+ 9 J (c) Ent nc n Ent ( 9 J) so n (. mol) 8. 4 J mol K 6. 8 + K+ 6. 8 K 7 K K. We use the tabulated values or C and C (a) Q nc. mol 8.8 J mol K ( 4 ) K. 46 kj (b) Ent nc. mol(.4 J mol K)( K).45 kj (c) W Q+ E nt 46. kj + 45. kj. kj 794 ch_p54-57.ndd 547 /7/6 :4:48 M
548 Chapter.4 (a) Consder warmng t at constant pressure. Oxygen and ntrogen are datomc, so C 7 7 7 Q nc n 7. 5 m ( m ) Q (. K ) 8 kj K (b) U g mgy m U g gy 5 8. J 6. kg 9.8 m s. m.5 Consder 8 cm o (lavored) water at 9. C mxng wth cm o datomc deal gas at. C: Q Q cold hot m c ( ) ( ) or m c m c ar, ar, ar w w w ( ) w mc ar, ar ar ar w w, ρ c,ar( 9. C. C) c ρ w w w where we have antcpated that the nal temperature o the mxture wll be close to 9. C. he molar specc heat o ar s C 7, ar 7 mol So the specc heat per gram s c,ar J mol K M 7 ( ) 8 4.. Jg C 8.9 g. ( ) ( ). gcm cm (. Jg C)( 7. C) w (. gcm )( 8 cm ) ( 4. 86 Jkg C) or ( ) w 55. C he change o temperature or the water s between C and C..6 (a) C 5 5 ( 8 4 ). mol. J mol K.8 9 kg 79 J kg K. 79 kj kg K (b) m Mn M m (. 8 9 kg mol) ( 8. 4 J mol K)( K) a (.5 m ). 8 kg (c) (d) We consder a constant volume process where no work s done. Q mc. 8 kg.79 kj kg K ( 7 K K ) kj We now consder a constant pressure process where the nternal energy o the gas s ncreased and work s done. Q mc m C m m C ( + ) 7 7 5 Q. 8 kg 7 (. 79 kj kg K ) 4 K 7 k 5 J 794 ch_p54-57.ndd 548 /7/6 :4:49 M
he Knetc heory o Gases 549.7 Q nc nc + sobarc sovolumetrc In the sobarc process, doubles so must double, to. In the sovolumetrc process, trples so changes rom to 6. Q n 7 ( ) + n 5 ( 6 ). 5n. 5 Secton. dabatc rocesses or an Ideal Gas.8 (a) so.. (b) (. ) (.8 ) (c) Snce the process s adabatc, Q C + C Snce 4. C C, C 5 57. 8 5. and 5. 5. Ent nc ( mol) 5. 6 ( 8. 4 J mol K) [. 5( K) ] 5 J and W Q+ E nt + 5 J + 5 J..9 (a) (b) 4.. 5. atm 9. atm. ( ) 5 5.. a. m n. mol 8.4 J mol K ( ) 5. 9. a. m n. mol 8.4 J mol K (c) he process s adabatc: Q 66 K 5 K C + C 4., C C C 5 5 Ent nc. mol ( 8. 4 J mol K ) ( 5 K 66 K) 4. 66 kj W E Q 466. kj 4.66 kj nt 794 ch_p54-57.ndd 549 /7/6 :4:5 M
55 Chapter. π 5. m 4. 5 m. 45 m he quantty o ar we nd rom n n 5 4 (. a). 45 m ( 8. 4 Jmol K)( K) n 997. mol dabatc compresson:. ka + 8 ka 9. ka (a) 45. 5 55. m 4 m. 9. 57 (b) n ( 57). K 56 K 9. (c) he work put nto the gas n compressng t s E nc ( ) nt 5 W ( 997. mol) ( 8. 4 J mol K) ( 56 ) K W 5. 9 J ow magne ths energy beng shared wth the nner wall as the gas s held at constant volume. he pump wall has outer dameter 5. mm +. mm +. mm 9. mm, and volume ( ) π 4. 5 π. 5 4. m m 6 ρ 7. 86 kg m 6. 79 m 5. g ( ) m 679. m and mass 6 he overall warmng process s descrbed by 5. 9 J nc + mc 5 5. 9 J ( 9. 97 mol) ( 8. 4 Jmol K) K + 5. kg 448 JkgK K 5. 9 J. 7 J K+. 9 J K K K. 4 K ( ) ( ).. 4 I K, then 7 K. 794 ch_p54-57.ndd 55 /7/6 :4:5 M
he Knetc heory o Gases 55. We suppose the ar plus burnt gasolne behaves lke a datomc deal gas. We nd ts nal absolute pressure: ow Q. atm 5. cm 4 cm. atm 8 and W E nc nt 75 75 75 4. atm 5 5 5 W n n 5 W 5J W 4. atm ( 4 cm ). atm ( 5. cm ) he output work s W + 5 J he tme or ths stroke s mn 6 s 6 s 4 5 mn. FIG.. 5. m 6 m atm ( cm ) W 5 J t 6. s 5. kw. (a) See the dagram at the rght. (b) C C C 5 7 ( ) ( ). 9 C dabatc C. 9( 4. L) 8. 77 L (c) n n ( ) K 9 K C (d) ter one whole cycle, K. 4L (e) In, Q nc n n 5 ( ) ( 5. ) Q C as ths process s adabatc FIG.. C C nc 9. 9. n so C 9. QC nc n 7 ( 9. ) ( 47. ) n For the whole cycle, Q Q + Q + Q ( 5. 4. 7) n (. 89) n W C C C + W E Q W nt C C C C Q (. 89) n (. 89) C ( 89. ). C 5 (. ) a 4 m 6 J C (L) 794 ch_p54-57.ndd 55 /7/6 :4:5 M
55 Chapter.4 (a) See the dagram at the rght. (b) C C C dabatc 5 7. 9 C (c) n n (d) ter one whole cycle, (e) In, Q nc n 5 ( ) ( 5. ) n Q C as ths process s adabatc n ( 9. ) 9. n so 9. C C C C Q 7 nc n C ( 9. )47. n For the whole cycle, Q Q + Q + Q ( 5. 4. 7) n. 8n C C C + E Q W W nt C C C C Q 8. n. 8 C FIG..4 C C (L).5 (a) he work done on the gas s W ab For the sothermal process, W W ab ab n n b a a a b d a d b ln a n ln a b hus, W ab 5. mol( 8.4 J mol K)( 9 K) ln (. ) FIG..5 W ab 8. kj contnued on next page 794 ch_p54-57.ndd 55 /7/6 :4:5 M
he Knetc heory o Gases 55 (b) For the adabatc process, we must rst nd the nal temperature, b. Snce ar conssts prmarly o datomc molecules, we shall use 5 5 ar 4. and C,ar ( 8. 4 ). 8 J mol K hen, or the adabatc process a. 4 b a 9 K(. ) 76 K b hus, the work done on the gas durng the adabatc process s W Q+ E nc nc ( + ) ( ) ab nt ab ab b a or W ab 5. mol.8 J mol K ( 76 9) K 46. kj (c) For the sothermal process, we have b b a a hus a b a. atm (. ). atm b For the adabatc process, we have b b a a hus a 4. b a. atm (.) 5. atm b Secton.4 he Equpartton o Energy k n.6 () Ent () C dent n d () C C + ( + ) C (4) C + contnued on next page 794 ch_p54-57.ndd 55 /7/6 :4:54 M
554 Chapter.7 he sample s total heat capacty at constant volume s nc. n deal gas o datomc molecules has three degrees o reedom or translaton n the x, y, and z drectons. I we take the y axs along the axs o a molecule, then outsde orces cannot excte rotaton about ths axs, snce they have no lever arms. Collsons wll set the molecule spnnng only about the x and z axes. (a) I the molecules do not vbrate, they have ve degrees o reedom. andom collsons put equal amounts o energy k nto all ve knds o moton. he average energy o one molecule s 5 k. he nternal energy o the two-mole sample s 5 k 5 5 n k n nc he molar heat capacty s C 5 and the sample s heat capacty s nc n 5 5 mol ( 8. 4 J mol K ) nc 46. J K For the heat capacty at constant pressure we have nc n( C + ) n 5 + n 7 mol 7 ( 8. 4 Jmol K ) nc 58. JK (b) In vbraton wth the center o mass xed, both atoms are always movng n opposte drectons wth equal speeds. braton adds two more degrees o reedom or two more terms n the molecular energy, or knetc and or elastc potental energy. We have nc n 7 58. J K and nc n 9 74. 8 J K.8 otatonal Knetc Energy Iω Cl I m r 7, m 5.. 67 kg, r m I 7. 45 kg m ω. s Krot Iω. J Cl FIG..8 *.9 Sulur doxde s the gas wth the greatest molecular mass o those lsted. I the eectve sprng constants or varous chemcal bonds are comparable, SO can then be expected to have low requences o atomc vbraton. braton can be excted at lower temperature than or the other gases. Some vbraton may be gong on at K. Wth more degrees o reedom or molecular moton, the materal has hgher specc heat. 794 ch_p54-57.ndd 554 /7/6 :4:55 M
he Knetc heory o Gases 555 *. (a) (b) (c) he energy o one molecule can be represented as ( )m v x + ( )m v y + ( )m v z + ( )Iω x + ( )Iω z Its average value s ( )k + ( )k + ( )k + ( )k + ( )k (5 )k he energy o one mole s obtaned by multplyng by vogadro s number, E nt n (5 ) nd the molar heat capacty at constant volume s E nt n (5/) he energy o one molecule can be represented as ( )m v x + ( )m v y + ( )m v z + ( )Iω x + ( )Iω z + ( )Iω y Its average value s ( )k + ( )k + ( )k + ( )k + ( )k + ( )k k he energy o one mole s obtaned by multplyng by vogadro s number, E nt n nd the molar heat capacty at constant volume s E nt n Let the modes o vbraton be denoted by and. he energy o one molecule can be represented as.5m [v x + v y + v z ] +.5Iω x +.5Iω z + [.5µ v rel +.5kx ] + [.5µv rel +.5kx ] Its average value s ( )k + ( )k + ( )k + ( )k + ( )k + ( )k + ( )k (9 )k he energy o one mole s obtaned by multplyng by vogadro s number, E nt n (9 ) nd the molar heat capacty at constant volume s E nt n (9/) (d) he energy o one molecule can be represented as.5m [v x + v y + v z ] +.5Iω x +.5Iω z +.5Iω y + [.5µv rel +.5kx ] + [.5µv rel +.5kx ] Its average value s ( )k + ( )k + ( )k + ( )k + ( )k + ( )k (5)k he energy o one mole s obtaned by multplyng by vogadro s number, E nt n 5 nd the molar heat capacty at constant volume s E nt n 5 (e) Measure the constant-volume specc heat o the gas as a uncton o temperature and look or plateaus on the graph, as shown n Fgure.7. I the rst jump goes rom to 5, the molecules can be dagnosed as lnear. I the rst jump goes rom to, the molecules must be nonlnear. he tabulated data at one temperature are nsucent or the determnaton. t room temperature some o the heaver molecules appear to be vbratng. Secton.5 Dstrbuton o Molecular Speeds. (a) v v n + + + + + ( av [ 5 47 9 ) ] 68. m s 5 (b) n v ( v ) 54. 9 av so v v rms (c) v mp 7. ms av m s 54. 9 7. 4 ms 794 ch_p54-57.ndd 555 /7/6 :4:56 M
556 Chapter. (a) he rato o the number at hgher energy to the number at lower energy s e Ek where E s the energy derence. Here, 9 6. J E. e 6. J e and at C, k ( 8. JK)( 7K) 77. J Snce ths s much less than the exctaton energy, nearly all the atoms wll be n the ground state and the number excted s 8 5 6. J ( 7. ) exp.77 J 5 7 ) 4. e hs number s much less than one, so almost all o the tme no atom s excted. (b) t C, he number excted s 9 k 8. JK7K 4. J 8 5 6. J 7. exp 9.4 J ( ).. 7 e. 7 5 5. In the Maxwell oltzmann speed dstrbuton uncton take d v to nd dv m 4π m v m v π k exp k v k and solve or v to nd the most probable speed. eject as solutons v and v hey descrbe mnmally probable speeds. m etan only k v hen v mp k m.4 (a) rms, 5 rms, 7 / M 5 7. / M 5. 7 gmol gmol. (b) he lghter atom, 5 Cl, moves aster..5 (a) From v av 8 k π m ( m s) π 6. 64 kg. we nd the temperature as 8. 8 J mol K ( m s) π 6. 64 kg. 7 (b) 8. 8 J mol K 7 7 4 6. K 7. 4 K 794 ch_p54-57.ndd 556 /7/6 :4:58 M
he Knetc heory o Gases 557 *.6 For a molecule o datomc ntrogen the mass s m M (8. kg mol) (6. molecules mol) 4.65 6 kg molecule k (. 8 J/molecule K) ( 9 K) (a) v mp m 4.65 kg/molecule 6 7 8k 8(. 8 J/molecule K) ( 9 K) (b) v avg πm π 4.65 kg/molecule 6 85 k (. 8 J/molecule K) ( 9 K) (c) v rms m 4.65 kg/molecule 6 895 m/s m/s m/s (d) he graph appears to be drawn correctly wthn about m/s..7 (a) From the oltzmann dstrbuton law, the number densty o molecules wth gravtatonal energy m gy s ne mgyk. hese are the molecules wth heght y, so ths s the number per volume at heght y as a uncton o y. (b) n( y) mgyk Mgy k Mgy e e e n ( 8. 9 )( 9 8 )( ) 8 4 e kg mol. m s m (. JmolK )( 9 K) 79 e. 78..8 (a) We calculate mgyk mgyk m gdy k e dy e k mg y Usng able.6 n the appendx, k k e mgyk k ( ) mg mg mg mgyk! k ye dy ( mgk ) mg hen y mgyk ye dy ( k mg ) mgyk k m g e dy k mg (b) y k M g Mg 8. 4 J 8 K s 8. mol K 8.9 kg 9.8 m m 794 ch_p54-57.ndd 557 /7/6 :4:59 M
558 Chapter ddtonal roblems.9 (a) ka 4 K n. mol( 8.4 J mol K)( 4 K). 66 5 m 66. 5 L a Ent (. 5) n. 5(. mol) ( 8. 4 J mol K)( K) 5. 8 kj W n (. mol) 8. 4 J mol K ( K) 66. kj Q E nt W 58. kj + 66. kj 748. kj (b) 4 K n. mol( 8.4 J mol K)( K) 499. m 499. L a 4 K ka K ka W d snce constant E nt 58. kj as n part (a) Q E nt W 58. kj 58. kj (c) ka K ka 49. 9 L 4. 6 L Ent ( 5. ) n snce constant ka W d n d n n ln ln ka W (. mol) ( 84. J mol K) ( K) ln 99 J ka + Q E W 99 J 99 J nt C C + 5. + 45. 9 (d) ka C C 5. 5. 7 : so ka 49. 9 L ka 7 9 4. L ka K 4. L K ka 49.9 L E ( 5. ) n 5. (. mol) ( 8. 4 J mol K)(. 4 K) 7 J Q nt ( adabatc process) W Q+ E + 7 J + 7 J nt 794 ch_p54-57.ndd 558 /7/6 :4:59 M
he Knetc heory o Gases 559 *.4 (a) (. 5 )(. 4.. 5 ) n a m m m. mol (. 8 4 Jmol K)( 9 K) n. mol 6. molecules mol) ( 789. 6 molecules (b) m nm. mol. 8 9 kg mol 7. 9 kg (c) (d) mv k ( 8. Jk)( 9K) 67. J molecule For one molecule, M. 8 9 kg mol 6 m 4. 8 kg molecule 6. molecules mol ( 6. 7 J molecule) vrms 5 ms 6 48. kg molecule (e), () Ent nc n 5 5 5 5 E nt (. a)(. 5 m ) 7. 98 MJ he smaller mass o warmer ar at 5 C contans the same nternal energy as the cooler ar. When the urnace operates, ar expands and leaves the room. *.4 For a pure metallc element, one atom s one molecule. Its energy can be represented as ( )m v x + ( )m v y + ( )m v z + ( )k x x + ( )k y y + ( )k z z Its average value s ( )k + ( )k + ( )k + ( )k + ( )k + ( )k k he energy o one mole s obtaned by multplyng by vogadro s number, E nt n nd the molar heat capacty at constant volume s E nt n (b) (8.4 J mole K) 8.4 J [55.845 kg] K 447 J kg K 447 J/kg C. hs agrees wth the tabulated value o 448 J/kg C wthn.%. (c) (8.4 J mole K) 8.4 J [97 kg] K 7 J kg K 7 J/kg C. hs agrees wth the tabulated value o 9 J/kg C wthn %..4 (a) he average speed v avg s just the weghted average o all the speeds. v v v v v v v v [ + ( ) + 5( ) + 4( 4 ) + ( 5 ) + ( 6 ) + ( 7 )] avg ( + + 5+ 4+ + + ) (b) Frst nd the average o the square o the speeds, v v 5 v 4 4v 5v ( v ) + + + + + ( 6v) + ( 7v) avg + + 5+ 4+ + + he root-mean square speed s then v ( rms v ). avg 99v 65. v 5. 95v (c) he most probable speed s the one that most o the partcles have;.e., ve partcles have speed. v. contnued on next page 794 ch_p54-57.ndd 559 /7/6 :5: M
56 Chapter (d) m v av hereore, m ( 5. 95) v m v 6 (e) he average knetc energy or each partcle s K mvav m( 5. 95v ) 7. 98mv *.4 (a) k. So W d k d k For k we can substtute and also to have W (b) de dq dw nt + and dq or an adabatc process. hereore, W + E nc nt o show consstency between these two equatons, consder that C C hereore, C Usng ths, the result ound n part (a) becomes W C and C C. lso, or an deal gas n so that W nc ( ), as ound n part (b)..44 (a) W nc 5 J mol 8. 4 JmolK 5 K K (b) n n ( ) ( ) 6. atm 5 ( 5) 5 ( ). atm 794 ch_p54-57.ndd 56 /7/6 6:5:5 M
he Knetc heory o Gases 56.45 Let the subscrpts and reer to the hot and cold compartments, respectvely. he pressure s hgher n the hot compartment, thereore the hot compartment expands and the cold compartment contracts. he work done by the adabatcally expandng gas s equal and opposte to the work done by the adabatcally compressed gas. n n ( ) ( ) hereore Consder the adabatc changes o the gases. and + + K () 8, snce and n n n n, usng the deal gas law, snce and 4. 55 K.756 () 5 K Solvng equatons () and () smultaneously gves 5 K, 9 K.46 he net work done by the gas on the bullet becomes the bullet s knetc energy: mv. kg( m s) 79. J he ar n ront o the bullet does work (. 5 m )(.5 m)(. 4 m ).5 J he hot gas behnd the bullet then must do output work +W n +W.5 J 7.9 J W 8.7 J. he nput work on the hot gas s 8.7 J ( ) 87. J lso So 87. J 4. nd cm + 5 cm. cm. 5 cm hen 6 87. J(.4) cm m 6 585. a. 4. 5 cm ( /. 5) cm 577. atm 794 ch_p54-57.ndd 56 /7/6 :5: M
56 Chapter.47 he pressure o the gas n the lungs o the dver must be the same as the absolute pressure o the water at ths depth o 5. meters. hs s: + ρ gh. atm +. kg m 9. 8 m s or +. atm. atm 5. 5 5 a 5. a 598. atm ( 5. m) I the partal pressure due to the oxygen n the gas mxture s to be. atmosphere (or the racton o the total pressure) oxygen molecules should make up only o the total number o 598. 598. molecules. hs wll be true. mole o oxygen s used or every 4.98 mole o helum. he rato by weght s then ( 4. 98 mol He)( 4. g mol He) g 6. (. mol O )( 5. 999 gmolo ) g.48 (a) Maxwell s speed dstrbuton uncton s m v 4π v π k Wth. 4, M. kg m 6. 6 5. kg 5 K e m v k and k 8. J molecule K v ths becomes ( 7 4 v 85 e 6.. ) v o the rght s a plot o ths uncton or the range v 5 ms. FIG..48(a) (b) he most probable speed occurs where v s a maxmum. From the graph, v mp 5 ms 8k 8(. 8 )( 5) (c) v av 575 m s 6 π m π 5. lso, k (. 8 )( 5) v rms 6 m 5. 64ms (d) he racton o partcles n the range ms v 6 ms s 6 dv where 4 and the ntegral o v s read rom the graph as the area under the curve. hs s approxmately ( + 6.5 + 6.5 + 5)( 4)() 4 4 and the racton s.44 or 44%. v 794 ch_p54-57.ndd 56 /7/6 :5: M
he Knetc heory o Gases 56.49 (a) Snce pressure ncreases as volume decreases (and vce versa), d d < and d > d (b) For an deal gas, n and κ d n d I the compresson s sothermal, s constant and n κ (c) For an adabatc compresson, C (where C s a constant) and d κ C C ( )+ d (d) κ. 5 atm (. atm ) C C κ and or a monatomc deal gas, 5, so that (. atm ). atm 5 *.5 (a) he speed o sound s v ρ d where d + ccordng to roblem 49, n an adabatc process, ths s lso, κ m ρ s nm n M ( ) M (b) v where m s s the sample mass. hen, the speed o sound n the deal gas s v ρ. 4( 8. 4 Jmol K)( 9 K). 8 9 kg mol M 44 ms M hs agrees wthn.% wth the 4 m/s lsted n able 7.. (c) We use k k k and M m : v M m m he most probable molecular speed s and the rms speed s k m. k, the average speed s m 8k, π m he speed o sound s somewhat less than each measure o molecular speed. Sound propagaton s orderly moton overlad on the dsorder o molecular moton. 794 ch_p54-57.ndd 56 /7/6 6:7: M
564 Chapter *.5 (a) he latent heat o evaporaton per molecule s J J 8 g mol 4 4. J/molecule g g mol 6. molecule 77. I the molecule has just broken ree, we assume that t possesses the energy as translatonal knetc energy. (b) Consder one gram o these molecules: K ( )mv 4 J ( )( kg) v v (4 86 m s ). m s (c) he total translatonal knetc energy o an deal gas s ( )n, so we have (4 J g)(8. g mol) ( )( mol)(8.4 J mol K).5 K he evaporatng molecules are exceptonal, at the hgh-speed tal o the dstrbuton o molecular speeds. he average speed o molecules n the lqud and n the vapor s approprate just to room temperature. *.5 (a) Let d r represent the dameter o the partcle. Its mass s d d m ρ ρ 4 π r ρ 4 π ρπ. hen mv rms k gves ρπ d v k 6 rms so 6 J/K 9 v rms 8k 8. 8 K / ρπ d kg/m π d 48. / (b) v d t [4.8 m 5 s]d d t d t 8. s m [4.8 m 5/ /s] / d 5 / d 5 / JK (c) v rms 8k 8. 8 ( 9 K) 96 6. ρπ d ( kg m ) π ( m) 4 ms v x 6 x m t 4. ms t 4 v 9.6 m s π d (d) 7 kg kg m d. 5 m 6 JK v rms 8k 8(. 8 )( 9 K) ρπ d kg m.5 m. π m 5/ s d / ms. 5 m t. 88 s yr hs moton s too slow to observe.. m s (e) 5 8k d 8k d ρπ d s ρπ s 8. 8 J K 9 K s d kg m π 5 5 97. m rownan moton s best observed wth pollen grans, smoke partcles, or latex spheres smaller than ths 9.7-µm sze. hen they can jtter about convncngly, showng relatvely large acceleratons several tmes per second. smple rule s to use the smallest partcles that you can clearly see wth some partcular mcroscopc technque. 794 ch_p54-57.ndd 564 /7/6 :5:5 M
he Knetc heory o Gases 565 m. kg.5 n 4. 5 mol M.8 9 kg mol n (a) ( 4. 5 mol )( 8. 4 J mol K )( 98 K ) a. 54 m (b) (c) so ( ) 4. 54 m.6 4 a. 6 m n ( 4. 5 mol) 8.4 J mol K (d) W d C d 8. K m ( ) a W ( 6m 5. ) ( 5. 4 m) 4 8 J.54 m. (e) Ent nc ( 4. mol) 5 5 ( J mol K ) 8. 4 (. 8 98) K E nt 8. 6 Q E W. 8 J + 4. 8 5 J. 8 6 J. 8 MJ.54 he ball loses energy nt 6 J mv mv. 4 kg 47 4 5.. m s 9. 9 J he ar volume s π. 7 m 9. 4 m. 8 4 m 5. a (.8 4 m ) and ts quantty s n ( 8. 4 J mol K)( 9 ) 47. mol K he ar absorbs energy as t were warmed over a stove accordng to Q nc So Q 9. 9 J 7 nc. 47 mol 8. 4 ( J mol K). 96 C 794 ch_p54-57.ndd 565 /7/6 :5:6 M
566 Chapter m.55 v m v k v v 4π exp π k ote that v mp k m hus, m v ( v) 4π π k nd For v e where exp(x) represents e x ( v vmp ) v ( v) v v v e v ( v ) ( mp ) v mp mp v v mp 5 v ( v) ( 5) e 9. v 5 v mp he other values are computed smlarly, wth the ollowng results: o nd the last value, we note: 5 499 ( 5) e 5e 5 ( ) 49 e v v mp ( v) v v v ( mp ) 9. 5 69.. 59.. 99. 5. 5 log ln ( 9 ln ) log 5 499 ln log 5 499 ln 8. 94. 96 8 4 8.56 (a) he eect o hgh angular speed s lke the eect o a very hgh gravtatonal eld on an atmosphere. he result s: he larger-mass molecules settle to the outsde whle the regon at smaller r has a hgher concentraton o low-mass molecules. (b) Consder a sngle knd o molecules, all o mass m. o cause the centrpetal acceleraton o the molecules between r and r + dr, the pressure must ncrease outward accordng to Fr m ar. hus, + d nm dr rω ( ) where n s the number o molecules per unt volume and s the area o any cylndrcal surace. hs reduces to d nmω rdr. ut also nk, so d kdn. hereore, the equaton becomes dn n ln n n m ω k rdr m ω k r gvng n n dn n mω k r rdr or and solvng or n gves n n e mr k ω n mω r ln( n) n k r 794 ch_p54-57.ndd 566 /7/6 :5:6 M
he Knetc heory o Gases 567.57 Frst nd v av as v m av v d v v. Let a k hen, v v v av 4π a 4 a d π k v e 4a π 8a a m he root-mean square speed s then v o nd the average speed, we have rms v av k m ( ) v v av v v v 4a a 4a π d v e dv a π 8k π m.58 We want to evaluate d or the uncton mpled by d n constant, and also or the derent uncton mpled by constant. We can use mplct derentaton: From constant d + d d d From constant d + d hereore, he theorem s proved. d d adabat d d 5 (. a) 5. m *.59 (a) n ( 8. 4 Jmol K)( K). mol sotherm d d d d sotherm adabat (b) C. K. 9 K 9 K C C 5. L 9 5. L (c) Ent, n (. mol) ( 8. 4 J mol K) ( K) 76 J Ent, Ent, C n (. mol) ( 8. 4 J mol K)( 9 K). 8 kj (d) (atm) (L) (K) E nt (kj). 5..76. 5. 9.8 C. 5. 9.8 FIG..59 contnued on next page 794 ch_p54-57.ndd 567 /7/6 :5:7 M
568 Chapter (e) For the process, lock the pston n place and put the cylnder nto an oven at 9 K. For C, keep the sample n the oven whle gradually lettng the gas expand to lt a load on the pston as ar as t can. For C, carry the cylnder back nto the room at K and let the gas cool wthout touchng the pston. () For : W E E E ( 8. 76. ) 5. kj Q E nt W nt nt, nt, kj 5. kj For C: E nt, W n C ln W (. mol) ( 8. 4 J mol K)( 9 K ) ln (. ) 67. kj Q E W nt 67. kj For C: E E E (. 76. 8). 5 kj nt nt, nt, C kj W n (. mol) 8. 4 J mol K ( 6 K). kj Q E W. 5 kj. kj. 5 kj nt (g) We add the amounts o energy or each process to nd them or the whole cycle..6 (a) g Q W C C +. 5 kj +. 67 kj. 5 kj. 656 kj. 67 kj +. kj. 656 kj ( E ) + 5. kj + 5. kj nt C. mol 6. molecules 8. g. mol. 4 6 molecules (b) ter one day, o the orgnal molecules would reman. ter two days, the racton would be, and so on. ter 6 days, only o the orgnal molecules would lkely reman, and ater 7 days, lkely none.. kg (c) he soup s ths racton o the hydrosphere:. kg hereore, today s soup lkely contans ths racton o the orgnal molecules. he number o orgnal molecules lkely n the pot agan today s:. kg 4. 6 molecule. kg ( s) 5. 6 molecules.6 (a) For escape, GmM GM m v. Snce the ree-all acceleraton at the surace s g, ths can also be wrtten as: E E GmM m v E mg E contnued on next page 794 ch_p54-57.ndd 568 /7/6 :5:8 M
he Knetc heory o Gases 569 (b) For O, the mass o one molecule s. kg mol m 6 5. 6 kg molecule. molecules mol k hen, mg E, the temperature s 6 6 mg ( 5. kg)( 98. m s ) 67 E (. m) 6. 5 5. 8 J mol K k.6 (a) For sodum atoms (wth a molar mass M. g mol) mv k M v k (b) 4 ( K) 8. 4 Jmol K. 4 vrms M. kg d. m t ms.5 m s v rms 5. m s 4 K SWES O EE OLEMS. ecause each mole o a chemcal compound contans vogadro s number o molecules, the number o molecules n a sample s tmes the number o moles, as descrbed by n, and the molar mass s tmes the molecular mass, as descrbed by M m. he denton o the molar mass mples that the sample mass s the number o moles tmes the molar mass, as descrbed by m nm. hen the sample mass must also be the number o molecules tmes the molecular mass, accordng to m nm n m m. he equatons are true or chemcal compounds n sold, lqud, and gaseous phases ths ncludes elements. We apply the equatons also to ar by nterpretng M as the mass o vogadro s number o the varous molecules n the mxture..4 7.6 ka.6 55. J molecule.8 (a).8 kj (b) 6.. see the soluton. (a) 9 J (b) (c) 7 K.4 (a) 8 kj (b) 6. kg.6 (a) 79 J kg K (b).8 kg (c) kj (d) 7 kj.8 (a).8 (b).5 (c) ; +5 J; +5 J J. (a) 5. 5 5 m (b) 56 K (c).4 K. 5. kw.4 (a) see the soluton (b).9 (c) (d) (e).8 794 ch_p54-57.ndd 569 /7/6 :5:9 M
57 Chapter.6 see the soluton.8. J. (a) 5 (b) (c) 9 (d) 5 (e) Measure the constant-volume specc heat o the gas as a uncton o temperature and look or plateaus on the graph, as shown n Fgure.7. I the rst jump goes rom to 5, the molecules can be dagnosed as lnear. I the rst jump goes rom to, the molecules must be nonlnear. he tabulated data at one temperature are nsucent or the determnaton. t room temperature some o the heaver molecules appear to be vbratng.. (a) o atom, almost all the tme (b). 7.4 (a). (b) 5 Cl.6 (a) 7 m s (b) 85 m s (c) 895 m s (d) he graph appears to be drawn correctly wthn about m s..8 (a) see the soluton (b) 8. km.4 (a) 7. 89 6 molecules (b) 7.9 kg (c) 6. 7 J molecule (d) 5 m s (e) 7.98 MJ () 7.98 MJ he smaller mass o warmer ar contans the same nternal energy as the cooler ar. When the urnace operates, ar expands and leaves the room. mv.4 (a).65v (b).99v (c).v (d) 6 (e) 798. mv.44 (a) K (b). atm.46 5.85 Ma.48 (a) see the soluton (b) 5. ms (c) v av 575 ms; v rms 64 ms (d) 44%.5 (a) see the soluton (b) 44 m s, n good agreement wth able 7. (c) he speed o sound s somewhat less than each measure o molecular speed. Sound propagaton s orderly moton overlad on the dsorder o molecular moton..5 (a) [8 k πρ d ] [4.8 m 5 s]d (b) [.8 s m 5 ]d 5 (c).96 mm s and.4 ms (d). m s and.88 s (e) 9.7µm It s good to use the smallest partcles that you can clearly see wth some partcular mcroscopc technque..54.96 C.56 (a) he eect o hgh angular speed s lke the eect o a very hgh gravtatonal eld on an atmosphere. he result s that the larger-mass molecules settle to the outsde whle the regon at smaller r has a hgher concentraton o low-mass molecules. (b) see the soluton.58 see the soluton.6 (a). 4 6 molecules (b) durng the 7th day (c). 5 6 molecules.6 (a). 5 m s (b) ms 794 ch_p54-57.ndd 57 /7/6 :5:9 M