Temperature and the Kinetic Theory of Gases

Transcription

1 HAPTER OUTLNE 6. Tmpratur and th Zroth Law o Thrmodynamcs 6. Thrmomtrs and Tmpratur Scals 6. Thrmal Expanson o Solds and Lquds 6.4 Macroscopc Dscrpton o an dal Gas 6.5 Th Kntc Thory o Gass 6.6 Dstrbuton o Molcular Spds 6.7 ontxt onncton Th Atmosphrc Laps Rat Tmpratur and th Kntc Thory o Gass ANSWERS TO QUESTONS Q6. Th coppr s tmpratur drops and th watr tmpratur rss untl both tmpraturs ar th sam. Thn th mtal and th watr ar n thrmal qulbrum. Q6. Th astronaut s rrrn to th tmpratur o th lunar surac, spccally a 4 drnc. A thrmomtr would rstr th tmpratur o th thrmomtr lqud. Snc thr s no atmosphr n th moon, th thrmomtr wll not rad a ralstc tmpratur unlss t s placd nto th lunar sol. Q6. th amalam had a larr cocnt o xpanson than your tooth, t would xpand mor than th caty n your tooth whn you tak a sp o your r-blod co, rsultn n a brokn or crackd tooth! As you c down your now xcrucatnly panul brokn tooth, th amalam would contract mor than th caty n your tooth and all out, lan th nr roots xposd. sn t t nc that your dntst knows thrmodynamcs? Q6.4 Th masurmnts mad wth th hatd stl tap wll b too short but only by a actor o 5 5 o th masurd lnth. Q6.5 (a) On mol o H has a mass o.6. On mol o H has a mass o (c) On mol o O has a mass o 8.. Q6.6 Th dal as law, PV nrt prdcts zro olum at absolut zro. Ths s ncorrct bcaus th dal as law cannot work all th way down to or blow th tmpratur at whch as turns to lqud, or n th cas o O, a sold. Q6.7 all th procss sobarc cooln or sobarc contracton. Th rubbr wall s asy to strtch. Th ar nsd s narly at atmosphrc prssur ornally and stays at atmosphrc prssur as th wall mos n, ust mantann qualty o prssur outsd and nsd. Th ar s narly an dal as to start wth, but PV nrt soon als. Volum wll drop by a larr actor than tmpratur as th watr apor lqus and thn rzs, as th carbon doxd turns to snow, as th aron turns to slush, and as th oxyn lqus. rom th outsd, you s contracton to a small racton o th ornal olum. 47

2 48 Tmpratur and th Kntc Thory o Gass Q6.8 ylndr A must b at lowr prssur. th as s thn, t wll b at on-thrd th absolut prssur o B. Q6.9 At hh tmpratur and prssur, th stam nsd xrts lar orcs on th pot and cor. Stron latchs hold thm tothr, but thy would xplod apart you trd to opn th hot cookr. Q6. (a) Th watr ll n th ca rss by a smallr dstanc than th watr outsd, as th trappd ar s comprssd. Ar can scap rom th ca th rock s not compltly artht, and also by dssoln n th watr. Th dal ca stays compltly ull o watr at low td. Th watr n th ca s supportd by atmosphrc prssur on th r watr surac outsd. (a) G. Q6. Q6. Th sphr xpands whn hatd, so that t no lonr ts throuh th rn. Wth th sphr stll hot, you can sparat th sphr and rn by hatn th rn. Ths mor surprsn rsult occurs bcaus th thrmal xpanson o th rn s not lk th nlaton o a bloodprssur cu. Rathr, t s lk a photoraphc nlarmnt; ry lnar dmnson, ncludn th hol damtr, ncrass by th sam actor. Th rason or ths s that th atoms rywhr, ncludn thos around th nnr crcumrnc, push away rom ach othr. Th only way that th atoms can accommodat th ratr dstancs s or th crcumrnc and corrspondn damtr to row. Ths proprty was onc usd to t mtal rms to woodn waon and hors-buy whls. th rn s hatd and th sphr lt at room tmpratur, th sphr would pass throuh th rn wth mor spac to spar. G. Q6. Q6. Th cocnt o xpanson o mtal s larr than that o lass. Whn hot watr s run or th ar, both th lass and th ld xpand, but at drnt rats. Snc all dmnsons xpand, thr wll b a crtan tmpratur at whch th nnr damtr o th ld has xpandd mor than th top o th ar, and th ld wll b asr to rmo. Q6. Th alcohol aporats, absorbn nry rom th skn to lowr th skn tmpratur. Q6.4 Th molculs o all drnt knds colld wth th walls o th contanr, so molculs o all drnt knds xrt partal prssurs that contrbut to th total prssur. Th molculs can b so small that thy colld wth on anothr rlatly rarly and ach knd xrts partal prssur as th othr knds o molculs wr absnt. th molculs colld wth on anothr otn, th collsons xactly consr momntum and so do not act th nt orc on th walls.

3 haptr 6 49 Q6.5 Th olum o th balloon wll dcras. Th prssur nsd th balloon s narly qual to th constant xtror atmosphrc prssur. Thn rom PV nrt, olum must dcras n proporton to th absolut tmpratur. all th procss sobarc contracton. Q6.6 Th dry ar s dnsr. Snc th ar and th watr apor ar at th sam tmpratur, thy ha th sam kntc nry pr molcul. or a controlld xprmnt, th humd and dry ar ar at th sam prssur, so th numbr o molculs pr unt olum must b th sam or both. Th watr molcul has a smallr molcular mass (8. u) than any o th ass that mak up th ar, so th humd ar must ha th smallr mass pr unt olum. Q6.7 Suppos th balloon rss nto ar unorm n tmpratur. Th ar cannot b unorm n prssur bcaus th lowr layrs support th wht o all th ar abo thm. Th rubbr n a typcal balloon s asy to strtch and strtchs or contracts untl ntror and xtror prssurs ar narly qual. So as th balloon rss t xpands. Ths s an sothrmal xpanson, wth P dcrasn as V ncrass by th sam actor n PV nrt. th rubbr wall s ry stron t wll ntually contan th hlum at hhr prssur than th ar outsd but at th sam dnsty, so that th balloon wll stop rsn. Mor lkly, th rubbr wll strtch and brak, rlasn th hlum to kp rsn and bol out o th Earth s atmosphr. Q6.8 (a) Ara molcular kntc nry ncrass by a actor o. Th rms spd ncrass by a actor o. (c) Ara momntum chan ncrass by. (d) Rat o collsons ncrass by a actor o snc th man r path rmans unchand. () Prssur ncrass by a actor o. SOLUTONS TO PROBLEMS Scton 6. Tmpratur and th Zroth Law o Thrmodynamcs No problms n ths scton Scton 6. Thrmomtrs and Tmpratur Scals P6. Snc w ha a lnar raph, th prssur s rlatd to th tmpratur as P A + BT, whr A and B ar constants. To nd A and B, w us th data Soln () and () smultanously,. 9 atm A+ 8. B. 65 atm A+ 78. w nd A 7. atm a () a B () contnud on nxt pa

4 44 Tmpratur and th Kntc Thory o Gass and B atm Thror, P. 7 atm atm T (a) At absolut zro P. 7 atm atm T whch s T 7. At th rzn pont o watr P. 7 atm +. 7 atm. a. (c) And at th boln pont P. 7 atm atm. 74 atm b a 5 5 P6. (a) To conrt rom ahrnht to lsus, w us T T and th Kln tmpratur s ound as T T + 7 K n a ashon dntcal to that usd n (a), w nd T. 6 and T 5 K a 9 9 P6. (a) T T T T K P6.4 (a) T T K. K J 8. Scton 6. Thrmal Expanson o Solds and Lquds *P6.5 L α L T km km a b a. 6 Th xpanson can b compnsatd or by mountn th ppln on rollrs and placn Ω -shapd loops btwn straht sctons. Thy bnd as th stl chans lnth. P6.6 or th dmnsons to ncras, L α L T 4 a a. cm.. cm T. T 55. *P6.7 L α L T. 4 cm. 58 cm a a 6

5 haptr 6 44 *P6.8 W consdr th xpanson o th horzontal strp o nyl lyn btwn th snu nalhad and a nal nar th ar nd o th wall, whr th sdn panl can sld by.4 cm undr th nalhad: L α L T a a. 4 m α 5 m 8 5 a α 7. 5 P6.9 (a) L α L T 9.. cm mm 6 a a L α L T cm cm a a 6 4 H a a a K 6. π. 5 (c) V α V T 9. G cm J cm 4 P6. Th horzontal scton xpands accordn to L α L T. a a 6 x 7 8. cm cm Th rtcal scton xpands smlarly by a a 6 y 7 4 cm cm. G. P6. Th ctor dsplacmnt o th pp lbow has mantud a a r x + y 6. mm mm 66. mm and s drctd to th rht blow th horzontal at anl y θ H G K J H G tan tan K J. x mm.6 mm r 66. mm to th rht at 78. blow th horzontal P6. (a) A α A T : A m 5. 6 b a 5 A 9. m 9. cm Th lnth o ach sd o th hol has ncrasd. Thus, ths rprsnts an ncras n th ara o th hol. 4 6 b b a P6. V β α V T al al

6 44 Tmpratur and th Kntc Thory o Gass P6. (a) V V β T V β T β α V T b t t Al Al t Al 4 4 a cm 6. V cm orlows. Th whol nw olum o turpntn s a 4 cm + 9. cm 6. 8 cm so th racton lost s cm cm and ths racton o th cylndr s dpth wll b mpty upon cooln: a cm 94. cm 6 P6.4 (a) L La + α T: 5. 5 cm 5. cm + 4. T. T 47 a W must t L L or som T, or Al Brass Soln or T, T 8, so T b b L + α T L + α T, Al Al, Brass Brass cm + 4. T 55. cm + 9. Ths wll not work bcaus alumnum mlts at 66. T Scton 6.4 Macroscopc Dscrpton o an dal Gas b. k P6.5 Mass o old abradd: m k k 5 Each atom has mass m u u. k. u Now, m N m, and th numbr o atoms mssn s N m 45. m k 8. k atoms. contnud on nxt pa

7 haptr 6 44 Th rat o loss s N t N t 8. 5 H G K J H G K J H G K J yr H G K J atoms yr 87. atoms s d d 4 h h 6 mn mn 6 s P6.6 On mol o hlum contans Aoadro s numbr o molculs and has a mass o 4.. Lt us call m th mass o on atom, and w ha N A m 4. mol 4. mol or m molculs mol 4 molcul 7 m 664. k a a b. 8 a P6.7 (a) ntally, PV nrt. atm V nr K nally, P V n RT P V n R + K Ddn ths quatons, 8. P. 5 K. atm 8.5 K n P 95. atm or P 4. 5 Paaabs.. a b a Atr bn drn P.. 8V n R K P d d. P Pa 4 PV P6.8 PV NP V π r NP : N r P balloons π π.. a a a a w ha no spcal mans or squzn th last L o hlum out o th tank, th tank wll b ull o hlum at. atm whn th last balloon s nlatd. Th numbr o balloons s thn rducd. m to m πa P6.9 Th quaton o stat o an dal as s PV nrt so w nd to sol or th numbr o mols to nd N. 5 a a a PV. Nm. m. m. m 5 n 49. mol RT b8. 4 JmolK a9 K 5 9 N nn A. 49 mol 6. molculs mol 5. molculs

8 444 Tmpratur and th Kntc Thory o Gass *P6. onsdr th ar n th tank durn on dschar procss. W suppos that th procss s slow nouh that th tmpratur rmans constant. Thn as th prssur drops rom.4 atm to. atm, th olum o th ar doubls. Durn th rst dschar, th ar olum chans rom L to L. Just L o watr s xplld and L rmans. n th scond dschar, th ar olum chans rom L to 4 L and L o watr s sprayd out. n th thrd dschar, only th last L o watr coms out. Wr t not or mal pattrn dumbnss, ach prson could mor cntly us hs dc by startn wth th tank hal ull o watr. b bρout ρn4 P6. y : ρout V ρn V k m k Th dnsty o th ar outsd s 5. k m. r o ar r B rom PV nrt, n V P RT Th dnsty s nrsly proportonal to th tmpratur, and th dnsty o th hot ar s H G ρ n 5. k m 8 K T n 8 K Thn 5. k m T 8 K 4. T n n 4 m k r o k G. P6. 8 K 6. T n T n 47 K PV P6. (a) PV nrt n RT m nm PVM. Pa. m 8. 9 k mol RT 8. 4 JmolK K m 7. m 7. k 9.8 m s 5. mn k a 5 (c) PA. Nm. m. kn a b a 5 (d) Th molculs must b mon ry ast to ht th walls hard.

9 haptr *P6. (a) Th ar n th tub s ar rom lqucaton, so t bhas as an dal as. At th ocan surac t s dscrbd by PV t t nrt whr P t atm, V t Aa65. cm, and A s th cross-sctonal ara o th ntror o th tub. At th bottom o th d, PV b b nrt PA ba65. cm 7. cm. By dson, a Pb 8. cm a atm a65. cm Pb. Nm 7. Nm 8. Th salt watr ntrs th tub untl th ar prssur s qual to th watr prssur at dpth, whch s dscrbd by Pb Pt + ρ h Nm. Nm + km 9. 8 ms 4 7. k m m s h 7. m 4. s m k h Wth a ry thn tub, ar dos not bubbl out. At th bottom o th d, th tub s a ald radn n any orntaton. Th opn nd o th tub should b at th bottom atr th brd suracs, so that th watr wll dran away as th xpandn ar pushs t out. Studnts can mak th tubs and d wth thm n a swmmn pool, to obsr how dpndably thy work. P6.4 At dpth, P P + ρ h and PV nrt At th surac, PV nrt: Thror V V T T V V P h H G + ρ P H G K. cm K J 78 K 67. cm b 5 P PV ρ h V + T T 9. Pa + 5 k m 9. 8 m s 5. m 5. Pa a P6.5 PV nrt : m m so m m P P n PV RT P n RT PV P H G m m m m P P P. k 4. atm 6. atm K J 4. atm 49. k

10 446 Tmpratur and th Kntc Thory o Gass P6.6 My bdroom s 4 m lon, 4 m wd, and.4 m hh, nclosn ar at kpa and 9 K. Thnk o th ar as 8.% N and.% O. Aoadro s numbr o molculs has mass mol +.. mol. 8 8 k mol *P6.7 a b a b m H G M K J RT 5. Nm 8. 4 m b. 8 8 kmol b8. 4 JmolK a9 K Thn PV nrt s m PVM RT k ~ k Th O s ar rom lquacton, so atr t coms out o soluton t bhas as an dal as. ts molar mass s M. mol + b6. mol 44. mol. Th quantty o as n th cylndr s m n sampl M mol 48. mol Thn PV nrt s V nrt P P6.8 N PVN A RT m P6.9 PV nrt M RT m PV nrt M RT H G K J PVM m m R T T H G K J b a. 48 mol 8.4 J mol K 7 K + K 5. Nm 9 Pa. m 6. molculs mol 8. 4 JK mol K b a H G K J N m L J m 4. molculs 55. L Scton 6.5 Th Kntc Thory o Gass P6. onsdr th x axs to b prpndcular to th plan o th wndow. Thn, th ara orc xrtd on th wndow by th halstons s n mantud Nm t Nm x t x Thus, th prssur on th wndowpan s P Nm snθ A H G At k b m s P6. 4. N. s and P 4. N 7. 6 kpa. 4 A 8. m b H G K J Nm snθ snθ Nm snθ t t..

11 haptr P6. (a) PV nrt Nm Th total translatonal kntc nry s Nm E trans : 5 Etrans PV kj m kt B RT 84. N 6. A a a 6. P6. (a) PV Nk B T : N PV. Pa. 5 m kt B. 8 JK a9 K πa 5 4 J 54. atoms a J J K k B T mol (c) or hlum, th atomc mass s m molculs mol 4 molcul m 7 m 664. k molcul kt B k B T: rms 5. km s m 4 4 d d *P6.4 Lt d r rprsnt th damtr o th partcl. ts mass s m V r H G ρπ ρ ρ π ρ π K J 6 Thn m (a) rms rms kt s ρπ d rms kt 6 kt JK K H G 8 8 d K J. 8 a9 J 6 ρπ k m π m K 96. J 4 ms. 6 x x m t 4 t 9.6 m s 4. ms (c) 7 k k m π d 6 contnud on nxt pa d 5. m H G a K J H a K 8kT 8. 8 JK 9 K. d G ρπ k m π.5 m J rms 5. m t 88. s yr. m s Ths moton s too slow to obsr. ms

12 448 Tmpratur and th Kntc Thory o Gass (d) 8kT d 8kT d ρπ d s ρπ s d a k m π K 8. 8 JK 9 K s J Brownan moton s bst obsrd wth polln rans, smok partcls, or latx sphrs smallr than ths 9.7-µm sz. Thn thy can ttr about conncnly, shown rlatly lar acclratons sral tms pr scond. a P6.5 (a) K k B T 8. JK 4K 876. J K m 876. rms J 5 m so rms 75. J m () 4. mol or hlum, m molculs mol 4 molcul 7 m 664. k molcul 9. 9 mol Smlarly or aron, m molculs mol molcul 6 m 66. k molcul Substtutn n () abo, w nd or hlum, rms 6. km s and or aron, rms 54 m s Scton 6.6 Dstrbuton o Molcular Spds P6.6 (a) a a a a a a. n N 5 a ms n m s a N so rms.. a ms (c) mp 7. ms

13 haptr P6.7 n th Maxwll Boltzmann spd dstrbuton uncton tak dn d to nd 4π N m π kt xp m kt and sol or to nd th most probabl spd. Rct as solutons and m Rtan only kt B m k T B B B Thn kt B mp m P6.8 Th most probabl spd s mp a kt. 8 JK 4. K B 7 m 664. k ms. kt B P6.9 (a) rom a 8 π m 7 4 π 664. k. m s w nd th tmpratur as T 8. 8 JmolK 7 π k. 7 m s T 8. 8 JmolK 6. K 7. 4 K Scton 6.7 ontxt onncton Th Atmosphrc Laps Rat P6.4 or a unorm laps rat, th dntty T T T y y mpls P6.4 (a) dt dy T b a. T T + y y 6.5 km.66 km 6 γ M ms 89. mol γ R 4. b8. 4 JmollK 97. Km 97. km b k J k m s Ar contans watr apor. Ar dos not bha as an dal as. As a parcl o ar rss n th atmosphr and ts tmpratur drops, ts ablty to contan watr apor dcrass, so watr wll lkly condns out as lqud drops or as c crystals. (Th condnsat may or may not b sbl as clouds.) Th condnsat rlass ts hat o aporzaton, rasn th ar tmpratur abo th alu that would b xpctd accordn to part (a). contnud on nxt pa

14 45 Tmpratur and th Kntc Thory o Gass (c) (d) GMMarsm or an obct o mass m on Mars, wht orc o plant s raty: m or rmars GMMars r dt dy Mars γ M. γ R N m k 6. 4 k 77. ms 6 7. m b. 77 ms. 44 kmol 8. 4 JmolK T y dt dy : T 6 y 4 dt dy 46. km b a 44. km 46. Km 46. km () Th dust n th atmosphr absorbs and scattrs nry rom th lctromantc radaton comn throuh th atmosphr rom th sun. Th dust contrbuts nry to th as molculs hh n th atmosphr, rsultn n an ncras n th ntrnal nry o th atmosphr alot and a smallr dcras n tmpratur wth hht, than n th cas whr thr s no absorpton o sunlht. Th larr th amount o dust, th mor th laps rat wll dat rom th thortcal alu n part (c). Thus t was dustr durn th Marnr lhts n 969. Addtonal Problms rod tap b brass stl a a a a a a L m m P6.4 Th xcss xpanson o th brass s L L α α L T (a) Th rod contracts mor than tap to a lnth radn. 95 m. 66 m m. 95 m m. 95 m P6.4 At,. allons o asoln has mass, rom ρ m V b m ρ V 7. 8 m k m. al. al Th asoln wll xpand n olum by 7. 7 k V β V T 96.. al.. 9. al 4 b a contnud on nxt pa

15 haptr 6 45 At.,. 9 al 7. 7 k. al. al 7. 7 k 7. k.9 al H G Th xtra mass contand n. allons at. s 7. 7 k 7. k. 5 k. P6.44 (a) Th olum o th lqud ncrass as Vl V β T. Th olum o th lask ncrass as V α V T. Thror, th orlow n th capllary s Vc V Tbβ α; and n th capllary Vc A h. b. V Thror, h β α T A or a mrcury thrmomtr βbh 8. 4 and or lass, α. 6 Thus β α β wthn 5% or α << β. P6.45 (a) ρ m V and dρ m V dv or ry small chans n V and ρ, ths can b xprssd as m V ρ ρβ T. V V Th nat sn mans that any ncras n tmpratur causs th dnsty to dcras and c rsa. ρ or watr w ha β ρ T. cm cm. cma P6.46 Th astronauts xhal ths much O :. mol astronauts days a b mol. msampl 9 k n M astronaut day k Thn 5 mol o mthan s nratd. t s ar rom lquacton and bhas as an dal as. b a. P nrt 5 mol 8.4 J mol K 7 K 45 K V 5 m Pa

16 45 Tmpratur and th Kntc Thory o Gass P6.47 (a) W assum that ar at atmosphrc prssur s abo th pston. n qulbrum Thror, or m Pas A + P nrt ha whr w ha usd V m A + P nrt h m P A + ha as th olum o th as. rom th data n, b a. mol 8.4 J K mol 4 K h 5. k 9.8 m s +. N m. 8 m 66. m G. P6.47 P6.48 Th anl o bndn θ, btwn tannts to th two nds o th strp, s qual to th anl th strp subtnds at ts cntr o curatur. (Th anls ar qual bcaus thr sds ar prpndcular, rht sd to th rht sd and lt sd to lt sd.) (a) Th dnton o radan masur s L + L θ r and L + L θ r By subtracton, L L r r θb α L T α L T θ r b α α L T θ r G. P6.48 n th xprsson rom part (a), θ s drctly proportonal to T and also to bα α. Thror θ s zro whn thr o ths quantts bcoms zro. (c) (d) Th matral that xpands mor whn hatd contracts mor whn coold, so th bmtallc strp bnds th othr way. t s un to dmonstrat ths wth lqud ntron. 6 6 a a b α α 9 9 L θ T. mm r 5. mm rad 8 π K J. 8 rad

17 haptr 6 45 P6.49 rom th daram w s that th chan n ara s A l w+ w l+ w l. Snc l and w ar ach small quantts, th product w l wll b ry small. Thror, w assum w l. Snc w wα T and l lα T, w thn ha A lwα T + lwα T G. P6.49 P6.5 (a) T and snc A l w, A α A T. Th approxmaton assums w l, or α T. Anothr way o statn ths s α T <<. a L T. s 9. 8 m s π so L. 48 m 4π 4π L αl T m m T b a 6 5 L + L 48. m π π. 95 s 9.8 m s T 95. n on wk, th tm lost s tm lost wk s lost pr scond P6.5 z r dm or α T << w nd thus 5 s 86 4 s tm lost b H G K J 7. dwk.. d tm lost s lost a bb and snc rt rt + α T a a at b T b T T b T + α T + α T α T 95 5 s lost s K J (a) Wth α 7. 6 and T w nd or u: 6 a Wth α 4. 6 and T w nd or Al: 7.. 4% 6 a %

18 454 Tmpratur and th Kntc Thory o Gass *P6.5 (a) B ρ V P P + d ρp V B P b ρp V P + ρ d ρ PV PV (c) Snc d s n th dnomnator, B must dcras as th dpth ncrass. (Th olum o th balloon bcoms smallr wth ncrasn prssur.) a Bd ρpv P + ρd B ρp V P P + ρ d P P d ρ b P P + ρd 5. Nm. k m 98. m s. m P6.5 Atr xpanson, th lnth o on o th spans s L L + α T 5 m m. 6 b a L, y, and th ornal 5 m lnth o ths span orm a rht tranl wth y as th alttud. Usn th Pythaoran thorm s: a5. m y + a5 m yldn y 74. m. P6.54 Atr xpanson, th lnth o on o th spans s L L + α T. L, y, and th ornal lnth L o ths span orm a rht tranl wth y as th alttud. Usn th Pythaoran thorm s L L + y, or y L L L + α T L α T+ α T Snc α T <<, y L α T. b b b Th hht o th cntr o th buckln brd s drctly proportonal to th lnth. A small brd s omtrcally smlar to a lar on. Th hht s proportonal to th squar root o th tmpratur ncras. Doubln T maks y ncras by only 4%. A small alu o T can ha a surprsnly lar ct. n unts, th quaton rads m mb, so t s dmnsonally corrct. *P6.55 (a) No torqu acts on th dsk so ts anular momntum s constant. ts momnt o nrta dcrass as t contracts so ts anular spd must ncras. ω ω MR ω MRω M R + Rα T ω MR α T ω ω ω α T 5. rad s 5. rad s rad s

19 P6.56 (a) rom PV nrt, th olum s: V H G K J nr P T Thror, whn prssur s hld constant, V nr V T P T Thus, V V T β H G K J H G K J V V T, or β T haptr At T 7 K, ths prdcts β 66. K 7 K Exprmntal alus ar: β H. 665 K and β ar 67. K Thy ar wthn.6% and.%, rspctly. m P6.57 (a) Lt m rprsnt th sampl mass. Th numbr o mols s n and th dnsty s ρ m M V. So PV nrt bcoms PV m M RT or PM m V RT. m Thn, ρ V PM ρ RT PM RT. 5 b. Nm. kmol 8. 4 JmolK 9 K b a. k m P6.58 (a) Wth pston alon: T constant, so PV P V b b or PAh P Ah Wth A constant, P P h h H G mp But, P P + A whr m p s th mass o th pston. Thus, whch rducs to P h mp + P h A H G h h + 5. cm mp. k9.8 m s PA + 5. Pa.4 m πa G. P cm h + contnud on nxt pa Wth th man o mass M on th pston, a ry smlar calculaton (rplacn m p by mp + M ) s: h 5. cm mp + M 95. k9.8 m s PA +. Pa π.4 m a cm

20 456 Tmpratur and th Kntc Thory o Gass Thus, whn th man stps on th pston, t mos downward by h h h cm 49. cm. 76 cm 7. 6 mm. P const, so V T V or T n T T h h Ah Ah T T K J H G K J K K (or 4 ) P6.59 Som as wll pass throuh th porous plu rom th racton chambr to th rsror as th racton chambr s hatd, but th nt quantty o as stays constant accordn to PV n + n n + n. Assumn th as s dal, w apply n to ach trm: RT atm b a a a 5 4 K K J + 67 K K K J PV P V PV P V K R K R 67 K R K R a b P P. atm P6.6 Lt θ rprsnt th anl th curd ral subtnds. W ha L + L θr L + α T L and snθ R R L L Thus, θ + α T + α T θ R b b b sn G. P6.6 b b and w must sol th transcndntal quaton θ + α T sn θ. 5 5 sn θ Homn n on th non-zro soluton s, to our dts, θ. 8 6 rad. 4 5 Now, h R Rcosθ Ths ylds h 454. m, a rmarkably lar alu compard to L 55. cm. L a cosθ snθ *P6.6 (a) Lt xl rprsnt th dstanc o th statonary ln blow th top d o th plat. Th normal orc on th lowr part o th plat s ma xcosθ and th orc o kntc rcton on t s µ kma xcos θ up th roo. Aan, µ kmx cos θ acts down th roo on th uppr part o th plat. Th nar-qulbrum o th plat rqurs x kt xl moton kb tmpratur rsn G. P6.6(a) contnud on nxt pa

21 µ mx cosθ + µ m x cosθ m snθ k µ mx cosθ m snθ µ m cosθ k µ x µ tanθ k k tanθ x µ k k a k L and th statonary ln s ndd blow th top d by xl tanθ k K J µ. haptr (c) Wth th tmpratur alln, th plat contracts astr than th roo. Th uppr part slds down and ls an upward rctonal orc µ kma xcos θ. Th lowr part slds up and ls downward rctonal orc µ k mx cos θ. Th quaton x s thn th sam as n part (a) and th statonary ln s abo th L bottom d by xl. µ tanθ k K J Start thnkn about th plat at dawn, as th tmpratur starts to rs. As n part (a), a ln at dstanc xl blow th top d o th plat stays statonary rlat to th roo as lon as th tmpratur rss. Th pont P on th plat at dstanc xl abo th bottom d s dstnd to bcom th xd pont whn th tmpratur starts alln. As th tmpratur rss, ths pont mos down th roo bcaus o th xpanson o th cntral part o th plat. ts dsplacmnt or th day s b a L α α L xl xl T L L tanθ b O α α L Th Tc NM k QP b µ L tanθ b H G α α bth Tc. µ k kt moton xl tmpratur alln G. P6.6 xl xl P G. P6.6(c) kb At dawn th nxt day th pont P s arthr down th roo by th dstanc L. t rprsnts th dsplacmnt o ry othr pont on th plat. (d) α α H G b b L tan θ µ. mtan. Th Tc K J. mm k 4. () α α <, th daram n part (a) appls to tmpratur alln and th daram n part appls to tmpratur rsn. Th wht o th plat stll pulls t stp by stp down th roo. Th sam xprsson dscrbs how ar t mos ach day.

22 458 Tmpratur and th Kntc Thory o Gass P6.6 Th prssur o th as n th luns o th dr must b th sam as th absolut prssur o th watr at ths dpth o 5. mtrs. Ths s: P P + ρ h. atm +. k m 9. 8 m s 5. m a K J or P atm. atm. Pa atm 5. Pa th partal prssur du to th oxyn n th as mxtur s to b. atmosphr (or th racton o th total prssur) oxyn molculs should mak up only o th total numbr o molculs. Ths wll b tru. mol o oxyn s usd or ry 4.98 mol o hlum. Th rato by wht s thn a a b mol H 4. mol H. mol O mol O b b H G P6.6 N N m m 4π xp π kt k T Not that B mp H G kt B m K J B. 6. Thus, N N m a mp 4π H G π kt And or N N a mp H G mp K J mp 5 N N B mp a b mp H G 5 K J. 5 9 Th othr alus ar computd smlarly, wth th ollown rsults: To nd th last alu, not: a a b N mp N a mp lo 5 ln 499 ln lo ln lo ln 8.94

23 haptr P6.64 (a) Maxwll s spd dstrbuton uncton s N N m 4π π kt H G Wth N. 4, B m kt B m M. k 6. N A T 5 K 5. 6 k and k B 8. J molcul K ths bcoms N To th rht s a plot o ths uncton or th ran 5 ms. G. P6.64(a) Th most probabl spd occurs whr N s a maxmum. rom th raph, mp 5 ms (c) 8kT 8 B. 8 a5 a 575 ms π m 6 π 5. Also, kt B. 8 a5 rms 64 ms 6 m 5. (d) Th racton o partcls n th ran ms 6 s z 6 Nd whr N 4 N ms and th ntral o N s rad rom th raph as th ara undr th cur. Ths s approxmatly 4 4 and th racton s.44 or 44%. ANSWERS TO EVEN PROBLEMS P6. (a) 7., K ;. 6, 5 P6.4 (a) 8 ; 45 K K P a P

24 46 Tmpratur and th Kntc Thory o Gass P6..66 mm to th rht at 78. blow th horzontal P al P6.4 (a) 47 ;, no, alumnum mlts at 66. P P balloons k P6. s th soluton P6. (a). 7 k ;.5 mn; (c). kn; (d) molculs must b mon ry ast P cm P6.6 ~ k P molculs P6. Nm snθ t wndow, Nm K J prpndcular to th snθ At P6.8 m/s P P6.4 (a) m; 95. m V P6.44 (a) h β α T A b ; α << β P Pa bα αl T P6.48 (a) θ ; s th soluton; r (c) s th soluton; (d). 8 P6.5 (a) ncrass by s ; s lost P6.5 (a) P6.54 y L α T b ρp V P + up; dcras; (c). m ρd P6.56 (a) ; 66. K, t ars wthn T.% or bttr P6.58 (a) 7.6 mm; 97 K P6. (a).8 kj; 6. J P m P6.4 (a) (c). (d) m s ;.4 ms; ms, 88. s; P6.6 (a) 6.8 m/s; 7.4 m/s; (c) 7. m/s m P6.6.6 P6.64 (a) s th soluton; mp 5 ms; (c) 575 m/s, 64 m/s; (d) 44%