The Kinetic Theory of Gases

Transcription

1 he Knetc heory o Gases CHER OULNE Molecular Model o an deal Gas Molar Specc Heat o an deal Gas dabatc rocesses or an deal Gas 4 he Equpartton o Enery he oltzann Dstrbuton Law 6 Dstrbuton o Molecular Speeds 7 Mean ree ath NSWERS O QUESONS Q he olecules o all derent knds collde wth the walls o the contaner, so olecules o all derent knds exert partal pressures that contrbute to the total pressure he olecules can be so sall that they collde wth one another relately rarely and each knd exerts partal pressure as the other knds o olecules were absent the olecules collde wth one another oten, the collsons exactly consere oentu and so do not aect the net orce on the walls Q he helu ust hae the hher rs speed ccordn to Equaton 4, the as wth the saller ass per ato ust hae the hher aerae speed-squared and thus the hher rs speed Q Yes s soon as the ases are xed, they coe to theral equlbru Equaton 4 predcts that the lhter helu atos wll on aerae hae a reater speed than the heaer ntroen olecules Collsons between the derent knds o olecules es each knd the sae aerae knetc enery o translaton Q4 the aerae elocty were non-zero, then the bulk saple o as would be on n the drecton o the aerae elocty n a closed tank, ths oton would result n a pressure derence wthn the tank that could not be sustaned Q he alcohol eaporates, absorbn enery ro the skn to lower the skn teperature Q6 artally eacuatn the contaner s equalent to lettn the reann as expand hs eans that the as does work, akn ts nternal enery and hence ts teperature decrease he lqud n the contaner wll eentually reach theral equlbru wth the low pressure as hs eect o an expandn as decreasn n teperature s a key process n your rererator or ar condtoner Q7 Snce the olue s xed, the densty o the cooled as cannot chane, so the ean ree path does not chane he collson requency decreases snce each olecule o the as has a lower aerae speed Q8 he ean ree path decreases as the densty o the as ncreases Q9 he olue o the balloon wll decrease he pressure nsde the balloon s nearly equal to the constant exteror atospherc pressure hen ro nr, olue ust decrease n proporton to the absolute teperature Call the process sobarc contracton 6

2 6 he Knetc heory o Gases Q he dry ar s ore dense Snce the ar and the water apor are at the sae teperature, they hae the sae knetc enery per olecule or a controlled experent, the hud and dry ar are at the sae pressure, so the nuber o olecules per unt olue ust be the sae or both he water olecule has a saller olecular ass (8 u) than any o the ases that ake up the ar, so the hud ar ust hae the saller ass per unt olue Q Suppose the balloon rses nto ar unor n teperature he ar cannot be unor n pressure because the lower layers support the weht o all the ar aboe the 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 sotheral expanson, wth decreasn as ncreases by the sae actor n nr the rubber wall s ery stron t wll eentually contan the helu at hher pressure than the ar outsde but at the sae densty, so that the balloon wll stop rsn More lkely, the rubber wll stretch and break, releasn the helu to keep rsn and bol out o the Earth s atosphere Q datoc as has ore derees o reedo those o braton and rotaton than a onatoc as he enery content per ole s proportonal to the nuber o derees o reedo Q (a) erae olecular knetc enery ncreases by a actor o he rs speed ncreases by a actor o (c) erae oentu chane ncreases by (d) Rate o collsons ncreases by a actor o snce the ean ree path reans unchaned (e) ressure ncreases by a actor o Q4 hey can, as ths possblty s not contradcted by any o our descrptons o the oton o ases the essel contans ore than a ew olecules, t s hhly probable that all wll hae the sae speed Collsons wll ake ther speeds scatter accordn to the oltzann dstrbuton law Q Collsons between olecules are edated by electrcal nteractons aon ther electrons On an atoc leel, collsons o bllard balls work the sae way Collsons between as olecules are perectly elastc Collsons between acroscopc spheres can be ery nearly elastc So the hardsphere odel s ery ood On the other hand, an ato s not sold, but has sall-ass electrons on throuh epty space as they orbt the nucleus Q6 s a parcel o ar s pushed upward, t oes nto a reon o lower pressure, so t expands and does work on ts surroundns ts und o nternal enery drops, and so does ts teperature s entoned n the queston, the low theral conductty o ar eans that ery lttle heat wll be conducted nto the now-cool parcel ro the denser but warer ar below t Q7 ore asse datoc or polyatoc olecule wll enerally hae a lower requency o braton t roo teperature, braton has a hher probablty o ben excted than n a less asse olecule he absorpton o enery nto braton shows up n hher specc heats SOLUONS O ROLEMS Secton Molecular Model o an deal Gas N t e k 7 N 7 a j a 8 sn 4 8 sn 4 s s 94 N

3 6 e j e468 k jb s s and 4 N 7 6 ka N Chapter 6 We rst nd the pressure exerted by the as on the wall o the contaner Nk N k R 84 9 b N ol K a K 9 8 hus, the orce on one o the walls o the cubcal contaner s 4 Use Equaton, K K K 4 e je j 9 a 4 6 N N H G, so that where N nn N N a a8 at e a at je j a N ol 6 olecules ol a b a e j Jolecule a N N KE KE d Equaton e je4 j d e6 j N olecules n N 6 olecules ol 4 ol 4 olecules 6 One ole o helu contans oadro s nuber o olecules and has a ass o 4 Let us call the ass o one ato, and we hae N 4 ol 4 ol 4 or 664 olecule 6 olecules ol k 7 (a) Nk : N a k e8 JK ja9 K 4 K k πa e ja J J 4 atos 4 ol (c) or helu, the atoc ass s olecules ol k olecule k k : rs k s 4 olecule

4 64 he Knetc heory o Gases k 8 O MHe 4 He MO 8 s O 477 s 8 9 (a) K k e8 JK ja4 K 876 J K rs 876 J so rs 7 J 4 ol or helu, olecules ol k olecule 9 9 ol Slarly or aron, 66 6 olecules ol 66 6 k olecule Substtutn n () aboe, we nd or helu, rs 6 k s 4 olecule olecule () and or aron, rs 4 s (a) nr N he total translatonal knetc enery s N E trans : E e je j 8 (a) a a trans kj k R 84 N 6 a N a a a e j J N K J 6 J or a onatoc deal as, Ent nr or any deal as, the enery o olecular translaton s the sae, Etrans nr hus, the enery per olue s Etrans J

5 Chapter 6 Secton Molar Specc Heat o an deal Gas Ent nr Ent nr a ol 8 4 J ol K K 74 8 J b a We us the tabulated alues or C and C (a) Q nc ol 88 J ol K 4 K 46 kj b a b a Ent nc ol 4 J ol K K 4 kj (c) W Q+ E nt 46 kj + 4 kj kj 4 he pston oes to keep pressure constant Snce nr, then and nr nr or a constant pressure process Q Q nc nbc + R so nc + R Q 7nR Q 7 H G K J 7 Q 7 nr e ja 44 J L ol 84 J ol K K a b a hus, + L + L 7 L n ol, K Q n R + R b b L Q 7nR Snce constant, W (a) Ent Q+ W 9 J + 9 J (c) E nc n R nt H G K J a a b E 9 so nr ol 84 J ol K nt J + K K 7 K 6 8 K

6 66 he Knetc heory o Gases R 6 (a) Consder heatn t at constant pressure Oxyen and ntroen are datoc, so C 7 U y 7 7 Q nc nr Q H G K J e N je j a 7 K K 8kJ U y 8 J 6 98 s e j k *7 (a) We assue that the bulb does not expand hen ths s a constant-olue heatn process he quantty o the as s n he enery nput s Q t nc so R t tr nc C tr he nal teperature s + + C he nal pressure s H tr + C 6 J 4 s 84 J ol K atg+ J s ol K N 4 J 8 (a) C R b JolK H G Mn M H G R K J b πa K 8 at 84 ol 79 J k K 79 kj k 8 9 k K e j b a K 89 a k ol 84 JolK 8 k (c) (d) We consder a constant olue process where no work s done Q C 8 k 79 kj k K 7 K K kj b a We now consder a constant pressure process where the nternal enery o the as s ncreased and work s done b + H G K J H G K J L O NM b a Q 7 Q C C R R 7 C Q 8 k 7 79 kj k K 4 K 7 kj

7 Chapter 67 9 Consder 8 c o (laored) water at 9 C xn wth c o datoc deal as at C: Qcold Qhot or c c d a ar c, ar d, ar bρ c a ar, ar 9 C C wcw bρ wwcw ar, ar, ar w w w a w where we hae antcpated that the nal teperature o the xture wll be close to 9 C he olar specc heat o ar s C R, ar 7 So the specc heat per ra s 7 R ol c, ar JolK J C H G 7 M K J 84 b H G 89 a e c jec jb J Ca7 C w e c je8 c jb4 86 J k C a C or w he chane o teperature or the water s between C and C b b sobarc sooluetrc Q nc + nc n the sobarc process, doubles so ust double, to n the sooluetrc process, trples so chanes ro to 6 b b Q n R n R nr H G 7 K J + H G K J 6 n the sooluetrc process, W and Q nc J J J or H G R n K Jb a + nr a J K + 4 K ol 84 J ol K n the sobarc process C, a b b J nr Q nc C hus, a J J (a) C 4 K nr ol 84 Jol K a b 6 K he work done on the as durn the sobarc process s W nr ol 8 4 J ol K 6 K 4 J or b a b a C C W C + J he work done on the as n the sooluetrc process s zero, so n total W on as + J

8 68 he Knetc heory o Gases * (a) t any pont n the heatn process, k and k nr t the end, nr and 4 nr nr z z nr he work nput s W d d nr nr 4 nr he chane n nternal enery, s Ent nc n R + 4 nr 8 s Q E nt W nr 9 a ol R (a) he heat requred to produce a teperature chane s Q n C + n C he nuber o olecules s N Q n + n C b, so C nc n + nc + n K J Q nc n C C nc n e j b he heat nput + N, so the nuber o oles o the xture s n + n and Secton dabatc rocesses or an deal Gas 4 (a) so H G K J H G K J 7 8 H G K J H G K J a a 8 (c) Snce the process s adabatc, Q C R+ C Snce 4, C R C C and Ent nc b ol H G 6 K J b 8 4 J ol K K J a and W Q+ E nt + J + J

9 Chapter 69 (a) H G K J H G K J 4 at 9 at e je j a nr ol 84 J ol K b e a je j olb84 J ol K 9 nr (c) he process s adabatc: Q 6 K K C R+ C 4, C R C C Ent nc ol J ol K K K kj b8 4 a K J W E Q466 kj 466 kj nt 6 π 4 4 he quantty o ar we nd ro nr 4 e je j n R n 997 a 4 84 JolK K b a ol dabatc copresson: ka + 8 ka 9 ka (a) nr 4 H G 4 K J 9 H G K J b 7 H G K J 7 K 6 K 9 (c) he work put nto the as n copressn t s E nc contnued on next pae b nt e j b a W 997 ol 8 4 J ol K 6 K W 9 J

10 6 he Knetc heory o Gases Now ane ths enery ben shared wth the nner wall as the as s held at constant olue he pup wall has outer daeter + + 9, and olue L NM e j e j 6 π 4 π Q 6 e je j and ass ρ 7 86 k 6 79 he oerall warn process s descrbed by 9 J nc + c 9 J 9 97 ol 84 J ol K K + k 448 J k K K 9 J 7 J K + 9 J K K K 4 K e jb d b d O e j b d 7 H G K J H G K J 4 K, then 7 K *8 (a) n we hae hen H G K J 4 H G 7 K J he actor o ncrease n teperature s the sae as the actor o ncrease n nternal enery, accordn to E nt nc hen E E nt, nt, a n H G K J H G K J we hae 7 H G K J

11 Chapter 6 9 (a) See the dara at the rht C C 7 C 9 C C e j e j a 9 4 L 877 L (c) nr nr K 9 K a 4 L dabatc C C (L) (d) ter one whole cycle, K G 9 b a (e) n, Q nc n R nr H G K J Q C as ths process s adabatc b a nr 9 9 nr C C C so 9 C b a QC nc n R nr H G 7 K J 9 47 or the whole cycle, Q Q + Q + Q 47 nr 89 nr b E Q + W W Q 89 nr 89 W C C C nt C C C a a a a C C C a e je j 89 a 4 6 J (a) See the dara at the rht C C dabatc C 7 C 9 (c) nr nr (d) ter one whole cycle, C C al G contnued on next pae

12 6 he Knetc heory o Gases b a (e) n, Q nc n R nr H G K J or the whole cycle, Q C as ths process s ababatc b b nr 9 9 nrso 9 C C C C QC nc n R nr H G 7 K J 9 47 Q Q + Q + Q 47 nr 8 nr b C C C E Q + W nt (a) he work done on the as s b W z d C C C W Q 8 nr 8 C C ab or the sotheral process, a a W W ab ab b nraz H G K J d a a H G b b a a b a nra ln H G nr ln hus, W ab ol 84 J ol K 9K ln W ab 8 kj G or the adabatc process, we ust rst nd the nal teperature, b Snce ar conssts prarly o datoc olecules, we shall use ar 4 and C hen, or the adabatc preocess H G a R 84 8, ar JolK a a 4 b a 9 K 76 K b hus, the work done on the as durn the adabatc process s b b b olb8 J ol K a76 9 K 46 kj W Q+ E + nc nc ab nt ab ab b a or W ab contnued on next pae

13 Chapter 6 (c) or the sotheral process, we hae H G b b a a a hus, a b a b at at or the adabatc process, we hae H G b b a a a hus, a 4 b a at at b We suppose the ar plus burnt asolne behaes lke a datoc deal as We nd ts nal absolute pressure: e j e j 7 7 at c 4 c 7 H G K J at 4 at 8 Now Q and W E nc nt d W nr nr d G N 6 e j e j c at e j W 4 at 4 c at c W J he output work s W + J he te or ths stroke s 4 H G K J n 6 s n 6 s W J t 6 s kw

14 64 he Knetc heory o Gases Secton 4 he Equpartton o Enery he heat capacty at constant olue s nc n deal as o datoc olecules has three derees o reedo or translaton n the x, y, and z drectons we take the y axs alon the axs o a olecule, then outsde orces cannot excte rotaton about ths axs, snce they hae no leer ars Collsons wll set the olecule spnnn only about the x and z axes (a) the olecules do not brate, they hae e derees o reedo Rando collsons put equal aounts o enery k nto all e knds o oton he aerae enery o one olecule s k he nternal enery o the two-ole saple s K J H G K J H G K J N k nn k n R nc he olar heat capacty s C R and the saple s heat capacty s H G K J b nc n R ol 84 JolK nc 4 6 JK or the heat capacty at constant pressure we hae 7 nc n C + R n R + R nr nc 8 JK K J 7 b ol b84 JolK K J n braton wth the center o ass xed, both atos are always on n opposte drectons wth equal speeds braton adds two ore derees o reedo or two ore ters n the olecular enery, or knetc and or elastc potental enery We hae 7 nc n R H G K J 9 and nc n R H G K J k nr 4 () Ent N H G K J H G K J () C dent R n d H G K J () C C + R + R (4) C + C b 8 J K 74 8 J K K J

15 Chapter 6 Rotatonal Knetc Enery ω r, 67 7 k, r 7 4 k ω s K ω rot J Cl G Cl Secton Secton 6 he oltzann Dstrbuton Law Dstrbuton o Molecular Speeds 6 (a) he rato o the nuber at hher enery to the nuber at lower enery s e E s the enery derence Here, and at C, E a e 6 e 9 J 6 8 e ja k 8 JK 7 K 77 J J Ek where Snce ths s uch less than the exctaton enery, nearly all the atos wll be n the round state and the nuber excted s 8 6 J 4 7 exp 7 e j e je 77 J hs nuber s uch less than one, so alost all o the te no ato s excted t C, he nuber excted s 9 e j k 8 JK 7K 4 J 8 6 J 7 exp e j e je 4 J

16 66 he Knetc heory o Gases 7 (a) a a a a a a n N a s n e j 4 9 s a N so rs e j a s (c) p 7 s 8 (a) rs, rs, 7 R M R M7 7 ol H G K J ol he lhter ato, Cl, oes aster 9 n the Maxwell oltzann speed dstrbuton uncton take dn d to nd 4π N π k exp k k and sole or to nd the ost probable speed Reject as solutons and Retan only k k hen p 4 he ost probable speed s p e ja k 8 JK 4 K k s 4 (a) ro a 8 k π 7 4 e je j π 6 64 k s we nd the teperature as 8 8 JolK 7 e je j π 6 64 k 7 s 8 8 JolK e j e 6 K j 7 4 K 4 t C, rs k t the hher teperature, b rs k a K 9 K 89 C

17 Chapter 67 *4 (a) ro the oltzann dstrbuton law, the nuber densty o olecules wth ratatonal enery y s ne y k hese are the olecules wth heht y, so ths s the nuber per olue at heht y as a uncton o y b ny n y k My N k My R e e e e e je je j b a 8 9 k ol 9 8 s 8 4 J ol K 9 K 79 e 78 *44 (a) We calculate z z y k dy y k e dy e k y k k e y k k k a Usn able 6 n the appendx y k k z! ye dy k H G K J z z y k ye dy k k hen y y k k e dy b b k R 84 J 8 K s y 8 MN M ol K 89 k 98 b Secton 7 Mean ree ath 4 (a) N N N H G K J and N N R so that R e ja a e j 6 84 a a olecules n πd Nπd olecules π 779 k e j e j a (c) 64 4 s

18 68 he Knetc heory o Gases 46 he aerae olecular speed s 8k 8kN π π N 8R π M b e JolK K π 6 k ol 78 s j (a) he ean ree path s π d n π 6 8 e 9 j he ean ree te s s yr 78 s Now n s 6 tes larer, to ake saller by 6 tes: 6 hus, 7 s yr 47 ro Equaton, or an deal as, n π d n N k k hereore,, as requred π d 48 π d n d 6 n n k e8 ja9 69 8, or about 9 olecular daeters

19 49 Usn n k, Equaton becoes (a) e8 JK ja9 K πe je j a k π d 96 8 Chapter 69 () Equaton () shows that akn ro (a) and wth, we nd (c) or, we hae 8 a at e9 6 j 8 96 at a 8 at e 9 6 j at ddtonal robles ( a )( 4 ) (a) n ol R ( 84 JolK )( 9K) N nn ol 6 olecules ol N 789 e je j 6 olecules e jb nm ol 8 9 k ol 7 9 k (c) e ja k 8 J k 9 K 67 J olecule (d) or one olecule, M 89 k ol 6 48 k olecule N 6 olecules ol rs e j Jolecule k olecule s (e),() Ent nc n R H G K J e je j E nt a 7 98 MJ

20 6 he Knetc heory o Gases (a) ka 4 K nr ol 84 J ol K 4K a b a L a a b a a b a Ent nr ol 84 J ol K K 8 kj W nr ol 84 J ol K K 66 kj Q E nt W 8 kj + 66 kj 748 kj 4 K nr ol 84 J ol K K a H G K J H G ka K J 4 K K b a ka W d z L snce constant E nt 8 kj as n part (a) Q E nt W 8 kj 8 kj (c) ka K ka H G L K J ka L Ent nr W d nr d nr nr H G W z z ln ln ka a H G ol b84 J ol K ak ln K J + Q E W 9 J 99 J nt ka H G a snce constant 99 J C C + R R+ R 4 9 (d) ka C C R 7 : so 79 ka H G L K J ka L H G K J ka H G K J H G 4 L K K J K ka 499 L E nr ol 8 4 J ol K 4 K 7 J nt Q a a b a badabatc process W Q+ E + 7 J + 7 J nt

21 Chapter 6 (a) he aerae speed a s just the wehted aerae o all the speeds a a a a a a a a a (c) rst nd the aerae o the square o the speeds, a a a a a a a a he root-ean square speed s then 99 he ost probable speed s the one that ost o the partcles hae; e, e partcles hae speed rs a 9 (d) Na hereore, a 6 H G 9 (e) he aerae knetc enery or each partcle s e j K a z z (a) k So, W d k d de dq dw nt + and dq or an adabatc process hereore, W + E nt nc d o show consstency between these equatons, consder that C C hereore, C R Usn ths, the result ound n part (a) becoes and C C R d W C R lso, or an deal as R n so that W ncd

22 6 he Knetc heory o Gases *4 (a) W nc d J ol 8 4 JolK K K d nr b b nr b b H G K J 6 at at H G K J H G K J b * Let the subscrpts and reer to the hot and cold copartents, respectely he pressure s hher n the hot copartent, thereore the hot copartent expands and the cold copartent contracts he work done by the adabatcally expandn as s equal and opposte to the work done by the adabatcally copressed as d d nr nr K () Consder the adabatc chanes o the ases and H G K J, snce and nr H G nr K J nr H G K J nr, usn the deal as law, snce and H G K J H G K J 4 K K 76 () Soln equatons () and () sultaneously es K, 9 K

23 Chapter 6 *6 he work done by the as on the bullet becoes ts knetc enery: b k s 79 J he work on the as s d 79 J lso L N M So 79 J 4 H G K J O Q nd c + c c c hen 6 79 Ja4 c L O NM Q 74 6 a 66 at 4 c c c h 7 he pressure o the as n the luns o the der ust be the sae as the absolute pressure o the water at ths depth o eters hs s: + ρ h at + k 9 8 s e je ja K J at or at + a 98 at a the partal pressure due to the oxyen n the as xture s to be atosphere (or the racton o the total pressure) oxyen olecules should ake up only o the total nuber o olecules hs wll be true ole o oxyen s used or eery 498 ole o helu he rato by weht s then a b 498 ol He 4 ol He ol O 999 ol O b b 6 8 (a) Maxwell s speed dstrbuton uncton s N 4π N H G π k k e Wth N 4, M k N 6 K and k 8 J olecule K ths becoes N e 6 e j e j 6 o the rht s a plot o ths uncton or the rane s contnued on next pae k G 8(a)

24 64 he Knetc heory o Gases he ost probable speed occurs where N s a axu ro the raph, p s (c) e ja a 7 s lso, 8k 8 8 π 6 π k 8 6 e e ja rs 64 s j (d) he racton o partcles n the rane s 6 s s z 6 Nd where N 4 and the nteral o N s read ro the raph as the area under the cure hs s approxately 4 4 and the racton s 44 or 44% N 9 (a) Snce pressure ncreases as olue decreases (and ce ersa), d d < and or an deal as, nr d nr and κ d the copresson s sotheral, s constant and H G K J κ L N M d d O Q > nr H G K J (c) or an adabatc copresson, C (where C s a constant) and (d) κ at at C C κ a κ d C C + + d b H G K J H G and or a onatoc deal as,, so that at a at

25 6 (a) he speed o sound s where d ρ d ccordn to roble 9, n an adabatc process, ths s nm nr M lso, ρ R n the deal as s s a a ρ M R R M H G K J b a 4 84 JolK 9K 89 k ol κ Chapter 6 where s s the saple ass hen, the speed o sound R M 44 s hs nearly arees wth the 4 /s lsted n able 7 (c) We use k R N and M N R kn k : M N k he ost probable olecular speed s, 8k the aerae speed s, and the rs speed s π k ll are soewhat larer than the speed o sound k 6 n 4 ol M 8 9 k ol (a) nr (c) a b a 4 ol 8 4 J ol K 98 K a so nr H G 4 e H G K J 4 j e je j 4 a 6 4 ol 8 4 J ol K a b z z (d) W d C d K e j H G K J H G K J W a 4 L O NM e 6 j a 4 Q 4 8 L a O NM b e j J (e) Ent nc ol J ol K Q K 6 Ent 8 J Q E 6 6 W 8 J J 8 J 8 MJ nt

26 66 he Knetc heory o Gases 6 he ball loses enery b a a b a 4 ae8 4 j 47 b84 JolK a9k 4 k 47 4 s 9 9 J he ar olue s π and ts quantty s n R he ar absorbs enery accordn to Q nc Q 9 9 J So nc 47 ol 7 84 J ol K 6 N 4π N a Note that H G exp π k k p H G k K J chb hus, Na e pj 4π N H G e π k nd or N N a e j p H G p K J p N N e e p a b e e pj H G K J he other alues are coputed slarly, wth the ollown results: o nd the last alue, note: a 499 e e a b j 9 ol 96 C N p N a e pj lo ln 499 ln lo 499 ln lo 499 ln 894 e

27 Chapter (a) he eect o hh anular speed s lke the eect o a ery hh ratatonal eld on an atosphere he result s: he larer-ass olecules settle to the outsde whle the reon at saller r has a hher concentraton o low-ass olecules Consder a snle knd o olecules, all o ass o cause the centrpetal acceleraton o the olecules between r and r+ dr, the pressure ust ncrease outward accordn to r ar hus, a b e ω j + d n dr r where n s the nuber o olecules per unt olue and s the area o any cylndrcal surace hs reduces to d nω rdr ut also nk, so d kdn hereore, the equaton becoes dn n ln n ln ω k rdr n n an 6 rst nd a hen, a n n as ω k H G ω k r a N r r n z n dn n ω k rdr or z r and soln or n: n n e r ω k z z N d Let a k 4Nπ a 4 a d π k e 4a π N 8a a he root-ean square speed s then rs a k o nd the aerae speed, we hae N N d Na a a a z 4 π e d N z 4 π a e j 8k π *66 We want to ealuate d or the uncton pled by d nr constant, and also or the derent uncton pled by constant We can use plct derentaton: ro constant d + d d d ro constant + d d d d hereore, d K J H G d K J he theore s proed adabat sother d d K J sother d d K J adabat

28 68 he Knetc heory o Gases 67 (a) n R H G C H G e je j a 84 JolK K C C b a H G K K J 9 K 9 K H G L 9 K J L ol a b a (c) Ent, nr ol 8 4 J ol K K 76 J Ent, Ent, C nr a ol 8 4 J ol K 9 K 8 kj b a (d) (at) (L) (K) E nt (kj) C 9 8 (e) G 67 or the process, lock the pston n place and put the cylnder nto an oen at 9 K or C, keep the saple n the oen whle radually lettn the as expand to lt a load on the pston as ar as t can or C, carry the cylnder back nto the roo at K and let the as cool wthout touchn the pston () or : W Ent Ent, Ent, 8 76 kj kj Q E nt W kj or C: E nt, W nr C H G ln a b a a W ol 8 4 J ol K 9 K ln 67 kj Q E W nt 67 kj a a b a or C: Ent Ent, Ent, C 76 8 kj kj W nr ol 8 4 J ol K 6 K kj Q E W kj kj kj nt a () We add the aounts o enery or each process to nd the or the whole cycle Q W b C C E nt + kj + 67 kj kj 66 kj 67 kj + kj 66 kj C + kj + kj

29 68 (a) b ol 6 olecules 4 8 ol 6 olecules Chapter 69 ter one day, o the ornal olecules would rean ter two days, the racton would be, and so on ter 6 days, only o the ornal olecules would lkely rean, and ater 7 days, lkely none (c) k he soup s ths racton o the hydrosphere: k hereore, today s soup lkely contans ths racton o the ornal olecules he nuber o ornal olecules lkely n the pot aan today s: e j k 4 6 olecules 6 olecules k 69 (a) or escape, GM GM Snce the ree-all acceleraton at the surace s, ths can RE RE also be wrtten as: GM RE R E or O, the ass o one olecule s k ol 6 olecules ol 6 k olecule k hen, R E H G K J, the teperature s R E k 6 6 e k je98 s je67 j 6 8 JolK 7 (a) or sodu atos (wth a olar ass M ol) e j 4 K k M N k rs 4 b e j R 8 4 JolK 4 K M k s d t s rs s

30 6 he Knetc heory o Gases NSWERS O EEN ROLEMS 7 6 ka 4 89 C 4 Jolecule 44 (a) see the soluton; 8 k s k (a) 8 kj; 6 J 74 8 J 4 7 L 6 (a) 8 kj; 6 k 8 (a) 79 J k K ; 8 k ; (c) kj; (d) 7 kj (a) 4 ; 9 ol 46 (a) 6 8 ; 9 yr ; 6 ; yr 48 9 olecular daeters (a) olecules; 7 9 k; (c) 67 Jolecule; (d) s; (e) 798 MJ; () 798 MJ (a) 6; 99; (c) ; (d) 6 ; (e) (a) K ; at a a R 8 (a) see the soluton; s; 4 (a) 8; ; (c) ; J; J 6 (a) ; 6 K ; (c) 4K 8 (a) ; 7 (c) a 7 s; rs 64 s; (d) 44% 6 (a) see the soluton; 44 s nearly areen wth the tabulated alue; (c) see the soluton; soewhat saller than each (a) see the soluton; 9 ; (c) ; (d) 6 96 C ; (e) 8 64 see the soluton kw 4 see the soluton 6 (a) No ato, alost all the te; 7 8 (a) ; Cl 66 see the soluton 68 (a) 4 6 olecules ; durn the 7th day; (c) 6 olecules 7 (a) s ; s 4 s