Heat Engines, Entropy, and the Second Law of Thermodynamics

Transcription

1 Heat Engnes, Entropy, and te Seond Law o ermodynams HER OULINE.1 Heat Engnes and te Seond Law o ermodynams. Heat umps and Rergerators. Reversble and Irreversble roesses.4 e arnot Engne. Gasolne and esel Engnes.6 Entropy.7 Entropy anges n Irreversble roesses.8 Entropy on a Mrosop Sale NSERS O UESIONS.1 Frst, te eeny o te automoble ne annot exeed te arnot eeny: t s lmted by te temperature o burnng uel and te temperature o te envronment nto w te exaust s dumped. Seond, te ne blok annot be allowed to go over a ertan temperature. rd, any pratal ne as rton, nomplete burnng o uel, and lmts set by tmng and energy transer by eat. *. For any yl proess te total nput energy must be equal to te total output energy. s s a onsequene o te rst law o termodynams. It s satsed by proesses, v, v, v, v but not by proesses,, v. e seond law says tat a yl proess tat takes n energy by eat must put out some o te energy by eat. s s not satsed or proesses v, v, and v. us te answers are () b () a () b (v) a (v) (v) a (v) (v) d.. ger steam temperature means tat more energy an be extrated rom te steam. For a onstant temperature eat snk at, and steam at, te eeny o te power plant goes as 1 and s maxmzed or a g..4 No. e rst law o termodynams s a statement about energy onservaton, wle te seond s a statement about stable termal equlbrum. ey are by no means mutually exlusve. For te partular ase o a ylng eat ne, te rst law mples +, and te seond law mples > 0.. ake an automoble as an example. ordng to te rst law or te dea o energy onservaton, t must take n all te energy t puts out. Its energy soure s emal energy n gasolne. urng te ombuston proess, some o tat energy goes nto movng te pstons and eventually nto te meanal moton o te ar. e emal energy turnng nto nternal energy an be modeled as energy nput by eat. e seond law says tat not all o te energy nput an beome output meanal energy. Mu o te nput energy must and does beome energy output by eat, w, troug te oolng system, s dsspated nto te atmospere. Moreover, tere are numerous plaes were rton, bot meanal and lud, turns meanal energy nto eat. In even te most eent nternal ombuston ne ars, less tan 0% o te energy rom te uel atually goes nto movng te ar. e rest ends up as useless eat n te atmospere p ndd 71 1/8/07 7::4 M

2 7 apter *.6 nswer (b). In te reversble adabat expanson O, te gas does work aganst a pston, takes n no energy by eat, and so drops n nternal energy and n temperature. In te ree adabat expanson OB, tere s no pston, no work output, onstant nternal energy, and onstant temperature or te deal gas. e ponts O and B are on a yperbol soterm. e ponts O and are on an adabat, steeper tan an soterm by te ator..7 sle o ot pzza ools o. Road rton brngs a skddng ar to a stop. up alls to te loor and satters. Your at des. ny proess s rreversble t looks unny or rgtenng wen sown n a vdeotape runnng bakwards. e ree lgt o a projetle s nearly reversble..8 (a) en te two sdes o te semondutor are at derent temperatures, an eletr potental (voltage) s generated aross te materal, w an drve eletr urrent troug an external rut. e two ups at 0 ontan te same amount o nternal energy as te par o ot and old ups. But no energy lows by eat troug te onverter brdgng between tem and no voltage s generated aross te semondutors. (b) eat ne must put out exaust energy by eat. e old up provdes a snk to absorb output or wasted energy by eat, w as nowere to go between two ups o equally warm water. *.9 () nswer (a). e ar ondtoner operatng n a losed room takes n energy by eletr transmsson and turns t all nto energy put out by eat. at s ts wole net eet. () nswer (b). e rozen stu absorbs energy by eat rom te ar. But you ll te e trays wt tap water and put tem bak nto te reezer, te rergerator wll pump more eat nto te ar tan t extrats rom te water to make t reeze. *.10 () nswer (d). () nswer (d). e seond law says tat you must put n some work to pump eat rom a lower-temperature to a ger-temperature loaton. But t an be very lttle work te two temperatures are very nearly equal..11 One: Energy lows by eat rom a ot bowl o l nto te ooler surroundng ar. Heat lost by te ot stu s equal to eat ganed by te old stu, but te entropy derease o te ot stu s less tan te entropy nrease o te old stu. wo: s you nlate a sot ar tre at a serve staton, ar rom a tank at g pressure expands to ll a larger volume. at ar nreases n entropy and te surroundng atmospere undergoes no sgnant entropy ange. ree: e brakes o your ar get warm as you ome to a stop. e soes and drums nrease n entropy and notng loses energy by eat, so notng dereases n entropy..1 (a) For an expandng deal gas at onstant temperature, te nternal energy stays onstant. e gas must absorb by eat te same amount o energy tat t puts out by work. en ts V entropy ange s S nr ln V 1 (b) For a reversble adabat expanson 0, and S 0. n deal gas undergong an rreversble adabat expanson an ave any postve value or S up to te value gven n part (a). *.1 nswer (). e wole Unverse must ave an entropy ange o zero or more. e envronment around te system omprses te rest o te Unverse, and must ave an entropy ange o +8.0 J K, or more p ndd 7 1/8/07 7:: M

3 Heat Engnes, Entropy, and te Seond Law o ermodynams 7 *.14 () () onsder te area tat ts under ea o te arrows, between ts lne segment and te orzontal axs. ount t as postve or arrows to te rgt, zero or vertal arrows, and negatve or arrows tendng let. en E > F > G > H > > B >. e tn blue yperbol lnes are soterms. Ea s a set o ponts representng states wt te same nternal energy or te deal gas smple. n arrow tendng arter rom te orgn tan te BE yperbola represents a proess or w nternal energy nreases. So we ave E > > B F > G > H. () e arrows and G are along an adabat. Vsualze or sket n a set o tese urves, unormly steeper tan te blue soterms. e energy nput by eat s determned by ow ar above te startng adabat te proess arrow ends. e ave E > > F > G > B > H >. *.1 roesses and G are adabat. ey an be arred out reversbly. long tese arrows entropy does not ange. Vsualze or sket n a set o tese adabat urves, unormly steeper tan te blue soterms. e entropy ange s determned by ow ar above te startng adabat te proess arrow ends. e ave E > > F > G > B > H >. *.16 (a) e redued low rate o oolng water redues te amount o eat exaust tat te plant an put out ea seond. Even wt onstant eeny, te rate at w te turbnes an take n eat s redued and so s te rate at w tey an put out work to te generators. I anytng, te eeny wll drop, beause te smaller amount o water arryng te eat exaust wll tend to run otter. e steam gong troug te turbnes wll undergo a smaller temperature ange. us tere are two reasons or te work output to drop. (b) e neer s verson o events, as seen rom nsde te plant, s omplete and orret. Hot steam puses ard on te ront o a turbne blade. Stll-warm steam puses less ard on te bak o te blade, w turns n response to te pressure derene. Hger temperature at te eat exaust port n te lake works ts way bak to a orrespondng ger temperature o te steam leavng a turbne blade, a smaller temperature drop aross te blade, and a lower work output. *.17 nswer (d). Heat nput wll not neessarly produe an entropy nrease, beause a eat nput ould go on smultaneously wt a larger work output, to arry te gas to a lower-temperature, lower-entropy nal state. ork nput wll not neessarly produe an entropy nrease, beause work nput ould go on smultaneously wt eat output to arry te gas to a lower-volume, lower-entropy nal state. Eter temperature nrease at onstant volume, or volume nrease at onstant temperature, or smultaneous nreases n bot temperature and volume, wll neessarly end n a more dsordered, ger-entropy nal state..18 n analogy used by arnot s nstrutve: waterall ontnuously onverts meanal energy nto nternal energy. It ontnuously reates entropy as te organzed moton o te allng water turns nto dsorganzed moleular moton. e umans put turbnes nto te waterall, dvertng some o te energy stream to our use. ater lows spontaneously rom g to low elevaton and energy spontaneously lows by eat rom g to low temperature. Into te great low o solar radaton rom Sun to Eart, lvng tngs put temselves. ey lve on energy low, more tan just on energy. baskng snake dverts energy rom a g-temperature soure (te Sun) troug tsel temporarly, beore te energy nevtably s radated rom te body o te snake to a lowtemperature snk (outer spae). tree bulds organzed ellulose moleules and we buld lbrares and babes wo look lke ter grandmoters, all out o a tn dverted stream n te unversal low o energy rasng down to dsorder. e do not volate te seond law, or we buld loal redutons n te entropy o one tng wtn te nexorable nrease n te total entropy o te Unverse. Your roommate s exerse puts energy nto te room by eat p ndd 7 1/8/07 7::6 M

4 74 apter.19 Eter statement an be onsdered an nstrutve analogy. e oose to take te rst vew. ll proesses requre energy, eter as energy ontent or as energy nput. e knet energy w t possessed at ts ormaton ontnues to make te Eart go around. Energy released by nulear reatons n te ore o te Sun drves weater on te Eart and essentally all proesses n te bospere. e energy ntensty o sunlgt ontrols ow lus a orest or jungle an be and ow warm a planet s. ontnuous energy nput s not requred or te moton o te planet. ontnuous energy nput s requred or le beause energy tends to be ontnuously degraded, as eat lows nto lower-temperature snks. e ontnuously nreasng entropy o te Unverse s te ndex to energy-transers ompleted. rnold Sommereld suggested te dea or ts queston..0 Sakng opens up spaes between jellybeans. e smaller ones more oten an all down nto spaes below tem. e aumulaton o larger andes on top and smaller ones on te bottom mples a small nrease n order, a small derease n one ontrbuton to te total entropy, but te seond law s not volated. e total entropy nreases as te system warms up, ts nrease n nternal energy omng rom te work put nto sakng te box and also rom a bt o gravtatonal energy loss as te beans settle ompatly togeter. SOLUIONS O ROBLEMS Seton.1 Heat Engnes and te Seond Law o ermodynams. 0 J.1 (a) e 60 J or 694. % (b) 60 J. 0 J J *. e ne s output work we denty wt te knet energy o te bullet: e energy exaust s 1 1 K mv kg( 0 m s) 1 J e J J e (a) e ave e J 1 J J m J kg 1. 7 m 1.80 kg 448 J wt J, we ave kj (b) 667 J and rom t, we ave 667 J 000 J s 0. s 1794 p ndd 74 1/8/07 7::7 M

5 Heat Engnes, Entropy, and te Seond Law o ermodynams 7.4 (a) e nput energy ea our s 60 mn ( J revoluton)( 00 rev mn) J mplyng uel nput ( L ) J L 7 (b) +. For a ontnuous-transer proess we may dvde by tme to ave + Useul power output () (d) J J 00rev1mn revoluton revoluton mn 60 s p. 18 p 746 τω τ ω Js 1 rev 00 rev 60 s π rad ( ) t J 00 rev revoluton 60 s N m. e eat to melt 1.0 g o Hg s ml 1 10 kg J kg 177 J e energy absorbed to reeze 1.00 g o alumnum s ( )( ) ml ( 10 kg) J/ kg 97 J ( ) and te work output s 0 J e teoretal (arnot) eeny s 0 J e 0. 4, or.% 4 97 J 9 K 4.1 K % 9 K Seton. Heat umps and Rergerators.6 O ( rergerator) (a) I 10 J and O 00,. ten 4. 0 J (b) Heat expelled Heat removed + ork done J+ 4 J 144 J 1794 p ndd 7 1/8/07 7::7 M

6 76 apter.7 O 00.. ereore,. 00. e eat removed ea mnute s ( kg)( J kg )(. 0 ) + ( kg ). 10 J kg t kg 090 J kg ( 0.0 ) J mn +( )( ) ( ) or, J s t us, te work done per seond s Js (a) Btu 10 J 1 1 1Btu 600 s 1 Js 9. (b) e energy extrated by eat rom te old sde dvded by requred work nput s by denton te oeent o perormane or a rergerator: ( O) rergerator Btu Btu Btu () t EER, : k Btu Energy purased s ( 00. k)( 100 ) k ( )( ) ost k $ k $ t EER 10, 10 Btu Btu Btu : k 10 Btu Energy purased s (. )( ). ost k $ k $ 10 ( )( ) 1 00 k k us, te ost or ar ondtonng s al as mu or an ar ondtoner wt EER 10 ompared wt an ar ondtoner wt EER. Seton. Seton.4 Reversble and Irreversble roesses e arnot Engne.9 70 K 14 K (a) e % (b) J, s 4 J 8. 8 k 1794 p ndd 76 1/8/07 7::8 M

7 Heat Engnes, Entropy, and te Seond Law o ermodynams en e e, 1 (a) 1 ( ) ( ) and J 870 MJ t 1 ( )( 600 s) (b) t 8 8 t ( )( 600) J 0 MJ *.11 e use amounts o energy to nd te atual eeny. + 0 kj + 1. kj 1. kj e 1. kj 1. kj e use temperatures to nd te arnot eeny o a reversble ne e K 4 K e atual eeny o s less tan our-tents o te arnot eeny o *.1 (a) e (b) In e 1 we derentate to nd de d 0 ( 1) s s te nrease o eeny per degree o nrease n te temperature o te ot reservor. () In e 1 we derentate to nd de d en de ( d ) s s te nrease o eeny per degree o derease n te temperature o te old reservor. Note tat t s a better deal to ool te exaust tan to superarge te rebox..1 Isotermal expanson at K Isotermal ompresson at K Gas absorbs 1 00 J durng expanson. (a) J ( ) (b) J 49 J.14 e arnot summer eeny s e nd n wnter, ( 7 + 0) K ( 7 + 0) K s,. e w, en te atual wnter eeny s or.% p ndd 77 1/8/07 7::9 M

8 78 apter V V.1 (a) In an adabat proess, V V. lso, vdng te seond equaton by te rst yelds Sne or rgon, and we ave ( 1 07 K) a a ( 1) 64 K (b) Ent nv 0, so n V, and te power output s nv or t t ( kg ) ( 1 mol/0.099 kg)( )( J mol K) ( ) K s k () e 64 K or 47.% 1 07 K.16 (a) emax % 9 (b) Js ( )( ) 6 11 ereore, Js 600 s J 11 From e we nd e J J 7. J () s ossl-uel pres rse, ts way to use solar energy wll beome a good buy. *.17 (a) e Now So e + e e e + e e e ( ) e e ee 1 1 ontnued on next page 1794 p ndd 78 1/8/07 7::1 M

9 Heat Engnes, Entropy, and te Seond Law o ermodynams 79 (b) e e + e ee e ombnaton o reversble nes s tsel a reversble ne so t as te arnot eeny. No mprovement n net eeny as resulted. () t, e ( ) (d) e1 e ( ) + e *.18 (a) e atual eeny s two trds te arnot eeny reads as an equaton + 1. ll te s represent absolute temperatures. en M 0. K 8 K e domnatng n te bottom o ts raton means tat te exaust power dereases as te rebox temperature nreases. (b) () (d) t M 0. K 0.(107 K) M K 8 K ( 107 8) K 187. M e requre t M. M 0. K K K 8 K K K 68 K/ K e mnmum possble eat exaust power s approaed as te rebox temperature goes to nnty, and t s 1.40 M( 0. 1) 0.7 M. e eat exaust power annot be as small as (1 4)(1.87 M) M. So no answer exsts. e energy exaust annot be tat small p ndd 79 1/8/07 7:: M

10 80 apter ( O) rerg (a) Frst, onsder te adabat proess : V V so V V ka 10.0 L 1.0 L lso nr V or nr V V V 1 71 ka V V 70 K K 1.0 Now, onsder te sotermal proess : 49 K V V V V V V V VV ka ( L) 44 ka 4. 0 L( 1. 0 L) Next, onsder te adabat proess B : V V B B V But, VV rom above. lso onsderng te sotermal proess, V 1 B VB Hene, V V V V B V VV L( 4.0 L) 1 VV w redues to VB L V 1. 0 L B B Fnally, V B V 10.0 L ka 87ka 16.0 L 1 State (ka) V (L) (K) B ontnued on next page 1794 p ndd 80 1/8/07 7:: M

11 Heat Engnes, Entropy, and te Seond Law o ermodynams 81 (b) For te sotermal proess B: Ent nv 0 VB so nr ln V. 4 mol ( 8.14 J mol K ) 16 0 ( 70 K) ln. 68kJ For te adabat proess B : 0 Ent nv ( B). 4 mol ( J mol K) ( 49 70) K kj and + E nt 0 + ( kj) kj For te sotermal proess : Ent nv 0 V and nr ln V. 4 mol ( 8.14 J mol K ) 1 0 ( 49 K) ln Fnally, or te adabat proess : 0 0. kj Ent nv ( ). 4 mol ( J mol K) ( 70 49) K kj and + E nt kj kj roess (kj) (kj) E nt (kj) B B B e work done by te ne s te negatve o te work nput. e output work s gven by te work olumn n te table wt all sgns reversed. () B e 16. kj 0. 7 or.% kj B e or.% 7.1 (a) For a omplete yle, E nt 0 and e text sows tat or a arnot yle (and only or a reversble yle) ( ) 1 ereore, (b) e ave te denton o te oeent o perormane or a rergerator, O Usng te result rom part (a), ts beomes O 1794 p ndd 81 1/8/07 7:: M

12 8 apter. ( O) eat pump ( O) arnot rerg J per 1 J energy removed by eat..4 O O arnot yle or arnot eeny arnot yle K 9 K 68 K 117. us, 117. joules o energy enter te room by eat or ea joule o work done. FIG..4 *. O ( rergerator) K 40.0 K K m 60 K. 1.6 e ( ) O rergerator Seton. Gasolne and esel Engnes.7 (a) V V V V V ( a) 0. 0 m 00 m (b) dv V V V Integratng, V V V 1 6 (.0)( a) m 1 19 J ( ) m 00 m 44 ka p ndd 8 1/8/07 7:: M

13 Heat Engnes, Entropy, and te Seond Law o ermodynams 8.8 (a), (b) e quantty o gas s 6 V ( a) m n R ( Jmol K)( 9 K) ( ) mol 6 Ent, nr V ( a) ( m ) 1 J In proess B, V 140. B VB ( a )( 8. 00) a 6 6 V B B ( a) ( m 8. 00) B nr mol J mol K 67 K E nt, nr ( )( ) 0 00 mol 8 14 J mol K 67 K 87 J (. )(. )( ) B B so E 87 J 1 J 16 J 0 B 16 J nt, B out out roess B takes us to: 0 00 mol ( 8 14 J mol K )( 1 0 K) nr (. ). 6 V m Ent, nr ( mol) ( J mol K)( 1 0 K) 46 J E nt, B 46 J 87 J 149 J 0 out 6 a B 149 J In proess : V V ( ) a a V ( a) ( m ) 44 K nr mol J mol K ( )( ) E nt, nr ( 0 00 mol) ( 8 14 J mol K) 44.. ( K ) 190 J E 190 J 46 J 46 J 0 nt, out out 46 J and Ent, Ent, Ent, 1 J 190 J 6. 0 J out J For te entre yle, E nt, net 16 J e net work s net 16 J J J J J 84. J ontnued on next page 1794 p ndd 8 1/8/07 7::6 M

14 84 apter e tables look lke: State (K) (ka) V (m ) E nt (J) B roess (J) output (J) E nt (J) B B B () e nput energy s 149 J, te waste s 6. 0 J, and 84. J. (d) 84. J e eeny s: e 149 J 0. 6 (e) Let represent te angular speed o te ranksat. en s te requeny at w we obtan work n te amount o 84. J yle: Js 84 Jyle (. ) 000 Js. 7 rev s 84. Jyle rev mn.9 ompresson rato 6.00, 1.40 (a) V Eeny o an Otto-ne e 1 1 V e % (b) I atual eeny e 1.% 0 losses n system are e e 6.% Seton.6 Entropy.0 For a reezng proess, S (. kg ). ( J kg ) K *.1 e proess o rasng te temperature o te sample n ts way s reversble, beause an nntesmal ange would make δ negatve, and energy would low out nstead o n. en we may nd te entropy ange o te sample as d md S ds m m 610 JK ln ln ln mln( ) 1794 p ndd 84 1/8/07 7::7 M

15 Heat Engnes, Entropy, and te Seond Law o ermodynams 8 *. (a) (b) () e proess s sobar beause t takes plae under onstant atmosper pressure. s desrbed by Newton s trd law, te stewng syrup must exert te same ore on te ar as te ar exerts on t. e eatng proess s not adabat (beause energy goes n by eat), sotermal ( goes up), sovolumetr (t lkely expands a bt), yl (t s derent at te end), or sentrop (entropy nreases). It ould be made as nearly reversble as you ws, by not usng a kten stove but a eater kept always just nrementally ger n temperature tan te syrup. e proess would ten also be eternal, and mpratal or ood produton. e nal temperature s 0 0 F 1 F+ 8 F F F 104 For te mxture, m m ( 900 g 1 al g + 90 g 0.99 al g ) ( ) al J onsder te reversble eatng proess desrbed n part (a): d ( m m d 1 1+ ) S ( m + m ) 1 1 ln [ 900( 1) + 90( 0 99) ]( ) J. al ln 1 al 1K JK J K ( ) d md. S m ln S 0 g( 1.00 al g ) ln al K 19 J K Seton.7 Entropy anges n Irreversble roesses.4 S JK 7. JK e ar ends up n te same termodynam state as t started, so t undergoes zero anges n entropy. e orgnal knet energy o te ar s transerred by eat to te surroundng ar, addng to te nternal energy o te ar. Its ange n entropy s 1 mv 70( 0. 0) S J K 10. kj K p ndd 8 1/8/07 7::8 M

16 86 apter *.6 ene 1 emp ream K. ene emp oee K e nal temperature o te mxture s: e entropy ange due to ts mxng s S S + ( ) JK ln 840 JK ln ( ) S J K 1 ( 0. 0 g) + ( 00 g) 0 g 1 d V 0. 0 g 00 g ( ) + ( ) ( 84 0 ) K d V 8 + ( ) 8. KJln 840 JKln 78.7 Sttng ere wrtng, I onvert emal energy, n ordered moleules n ood, nto nternal energy tat leaves my body by eat nto te room-temperature surroundngs. My rate o energy output s equal to my metabol rate, al J kal d s 1 al 10 My body s n steady state, angng lttle n entropy, as te envronment nreases n entropy at te rate S K~ 1 K 9 K en usng powerul applanes or an automoble, my personal ontrbuton to entropy produton s mu greater tan te above estmate, based only on metabolsm..8 ron 448 Jkg ; water Jkg old : kg J kg kg 448 Jkg 900 ot ( ) w yelds. 06. K 06. K ( ) ( ) 06. K ( ) ( ) watermwaterd ronmrond S + 8 K 117 K 06 S waterm. water ln ronmron ln 117 S Jkg K kg J kg K kg ( 14. ) S ( )( )( ) + ( )( ) 718 JK V.9 S nrln R V ln. 76 J K ere s no ange n temperature or an deal gas. FIG p ndd 86 1/8/07 7::8 M

17 Heat Engnes, Entropy, and te Seond Law o ermodynams 87 V.40 S nrln R V ( )( ) ln S ( 8. 14) ln J K FIG..40 Seton.8 Entropy on a Mrosop Sale.41 (a) 1 an only be obtaned one way, as (b) 7 an be obtaned sx ways: 6 + 1, +, 4 +, + 4, +, (a) e table s sown below. On te bass o te table, te most probable reorded result o a toss s eads and tals. (b) e most ordered state s te least lkely marostate. us, on te bass o te table ts s eter all eads or all tals. () e most dsordered s te most lkely marostate. us, ts s eads and tals. Result ossble ombnatons otal ll eads HHHH 1 H, 1 HHH, HHH, HHH, HHH 4 H, HH, HH, HH, HH, HH, HH 6 1H, H, H, H, H 4 ll tals 1.4 (a) Result ossble ombnatons otal ll red RRR 1 R, 1G RRG, RGR, GRR 1R, G RGG, GRG, GGR ll green GGG 1 (b) Result ossble ombnatons otal ll red RRRRR 1 4R, 1G RRRRG, RRRGR, RRGRR, RGRRR, GRRRR R, G RRRGG, RRGRG, RGRRG, GRRRG, RRGGR, RGRGR, GRRGR, RGGRR, GRGRR, GGRRR 10 R, G GGGRR, GGRGR, GRGGR, RGGGR, GGRRG, GRGRG, RGGRG, GRRGG, RGRGG, RRGGG 10 1R, 4G RGGGG, GRGGG, GGRGG, GGGRG, GGGGR ll green GGGGG p ndd 87 1/8/07 7::40 M

18 88 apter ddtonal roblems.44 e onverson o gravtatonal potental energy nto knet energy as te water alls s reversble. But te subsequent onverson nto nternal energy s not. e magne arrvng at te same nal state by addng energy by eat, n amount mgy, to te water rom a stove at a temperature nntesmally above 0.0. en, d mgy ( )( S 000 m kg m m s )( 0. 0 m) JK 9 K *.4 For te arnot ne, lso, so e e K K 10 J 0 J e and 0 J 10 J 100 J (a) 10 J e S 14 J FIG J 10 J 64. J (b), net 14 J 0 J. 7 J, net 64. J 100 J. 7 J e net low o energy by eat rom te old to te ot reservor wtout work nput, s mpossble. () For ne S: e so J e S S J and + J+ 100 J J FIG..4(b) (d), net J 0 J 8. J net J 10 J 8. J,net 0 e output o 8. J o energy rom te eat ne by work n a yl proess wtout any exaust by eat s mpossble. FIG..4(d) ontnued on next page 1794 p ndd 88 1/8/07 7::4 M

19 Heat Engnes, Entropy, and te Seond Law o ermodynams 89 (e) Bot nes operate n yles, so S S S arnot 0 For te reservors, S and S + us, 8. J 0 Stotal SS + Sarnot + S + S K 00 K derease n total entropy s mpossble. JK V *.46 (a) Let state represent te gas beore ts ompresson and state aterwards, V. For a 8 datom deal gas, v R, R p, and p Next, V V V so 0. V 140. V V nr V 18. 4V V. nr nr ( ) Ent nv n R nr V 10( Nm) m 94. J Sne te proess s adabat, 0 and E nt + gves 9. 4 J 1 1 (b) e moment o nerta o te weel s I MR. 1 kg( m) kg m e want te lyweel to do work 9.4 J, so te work on te lyweel sould be 9.4 J: () Now we want 00. Krot K + K rot rot 1 Iω 9. 4 J (. J) ω kg m 6. 4 rad s 9. 4J kg m ω 1 ( 789 J) ω kg m 9 rad s 1794 p ndd 89 1/8/07 7::4 M

20 90 apter.47 (a) eletr H E so all te eletr energy s onverted nto nternal energy, te steady-state ondton o te ouse s desrbed by H E. ereore, eletr K (b) For a eat pump, ( O) arnot 7 K tual O 0. 6( 10. 9) 6. ereore, to brng 000 o energy nto te ouse only requres nput power 000 eat pump 76 O J.48 S ot 600 K J S old K (a) S S + U S (b) ot old JK e e ( J) 417 J () net 417 J 0 J 167 J 1 S U 0 K ( J K) 167 J V.49 (a) For an sotermal proess, nr ln V 1 ereore, 1 nr( )ln and nr For te onstant volume proesses, E nt, nr and 4 E nt, 4 nr e net energy by eat transerred s ten ( ) ln 1 4 ( ) ( ) FIG..49 or nr ln (b) postve value or eat represents energy transerred nto te system. ereore, 1+ 4 nr( 1+ ln ) Sne te ange n temperature or te omplete yle s zero, ereore, te eeny s E nt 0 and e ln 1 ( + ln ) p ndd 90 1/8/07 7::44 M

21 Heat Engnes, Entropy, and te Seond Law o ermodynams 91.0 (a) K S e water S body d 4. 6 g ( al g K) d ln ( ) body S al K system ( ( 4. 6)( 1. 00) ) al K (b) ( 4. 6)( 1) ( F 74. 8) ( )( 1) ( ) F al K us, ( ) 10 F ( 70. 0)( 10. 1) + ( ) ( 74. 8) 10 and F K F S e water 4. 6ln 4. al K S body ( ) ln al K S sys 1. 9 al K s s sgnantly less tan te estmate n part (a)..1 e 1 + / : m : / t : t ( 1 ) t m m ( ) 9 m ( )( 00 K) K J kg ( 600. ) ( ) kg s 1794 p ndd 91 1/8/07 7::46 M

22 9 apter. e 1 / / t 1 ( / ) m, were s te spe eat o water. m ereore, and m ( ) e test or dmensonal orretness by dentyng te unts o te rgt-and sde: Js kg ( ) kg s, as on te let and sde. nk o yoursel as a power-ompany Jkg ( ) J neer arrangng to ave enoug oolng water to arry o your termal polluton. I te plant power nreases, te requred low rate nreases n dret proporton. I envronmental regulatons requre a smaller temperature ange, ten te requred low rate nreases agan, now n nverse proporton. Next note tat s n te bottom o te raton. s means tat you an run te reator ore or rebox otter, te requred oolant low rate dereases! I te turbnes take n steam at ger temperature, tey an be made more eent to redue waste eat output p ndd 9 1/8/07 7::47 M

23 Heat Engnes, Entropy, and te Seond Law o ermodynams 9. Lke a rergerator, an ar ondtoner as as ts purpose te removal o energy by eat rom te old reservor. Its deal O s O arnot (a) Its atual O s ( 14. 0) K K ( 100. k) 660. and t 848. k (b) + : k k 1. k () e ar ondtoner operates n a yle, so te entropy o te workng lud does not ange. e ot reservor nreases n entropy by ( Js)( 600 s) JK 00 K e old room dereases n entropy by ( Js)( 600 s) S JK 80 K e net entropy ange s postve, as t must be: JK JK JK (d) e new deal O s O arnot e suppose te atual O s ( 11. ) K 11. K s a raton o te orgnal.60, ts s , so te ratonal ange s to 60. drop by 0.0% p ndd 9 1/8/07 7::47 M

24 94 apter *.4 (a) For te sotermal proess B, te work on te gas s V B B B V B ln V ( a )( m ) ln J were we ave used atm a and L m FIG..4 V ( ) a m J B B ( ) 0 and B B J 410. kj (b) Sne B s an sotermal proess, E nt, B 0 and J For an deal monatom gas, B B R V and R lso, B V B B nr ( )( ) R R ( )( ) V nr R R n R V R 608. kj so te total energy absorbed by eat s B kj kj 14. kj () B n ( nr ) B VB B ( ) ( ) J kj J (d) e 0.88 or 8.% J (e) B arnot ne operatng between ot 060 R and old 1010 R as eeny %. e tree-proess ne onsdered n ts problem as mu lower eeny p ndd 94 1/8/07 7::48 M

25 Heat Engnes, Entropy, and te Seond Law o ermodynams 9 *. t pont, V nr and n 100. mol t pont B, V nrb so B ( )( ) and 6 t pont, V nr t pont, V nr ( ) so R e eat or ea step n te yle s ound usng V R and : n nr ( ) ( ) ( ) ( ) B V n nR B n 6 6nR V n. 0nR (a) ereore, + enterng B 10. B nr FIG.. (b) + leavng 80. nr () tual eeny, e (d) arnot eeny, e e arnot eeny s mu ger d nd 1.6 S n d n ln n( ln ln) n ln S n V nr ln n nr V ln 1794 p ndd 9 1/8/07 7::49 M

26 96 apter.7 (a) e deal gas at onstant temperature keeps onstant nternal energy. s t puts out energy by work n expandng t must take n an equal amount o energy by eat. us ts entropy nreases. Let, V, represent te state o te gas beore te sotermal expanson. Let, V, represent te state ater ts proess, so tat V V. Let, V, represent te state ater te adabat ompresson. en V ( V) Substtutng V V gves VV 1 V en 1 1 V ( 1) V V and V e work output n te sotermal expanson s 1 V dv nr V dv nr ln nr V ( 1) ln( ) nr ln 1 s s also te nput eat, so te entropy ange s S nr 1 ln Sne + R we ave and V V ( 1) R, V R 1 en te result s S n ln V R 1 (b) e par o proesses onsdered ere arres te gas rom te ntal state n roblem 6 to te nal state tere. Entropy s a unton o state. Entropy ange does not depend on pat. ereore te entropy ange n roblem 6 equals Ssotermal + Sadabat n ts problem. Sne S adabat 0, te answers to roblems 6 and 7(a) must be te same. V *.8 (a) dv V dv nr ( 100. ) R ln V V R ln V V V (b) le t lasts, ts proess does onvert all o te energy nput nto work output. But te gas sample s n a derent state at te end tan t was at te begnnng. e proess annot be done over unless te gas s reompressed by a work nput. o be pratal, a eat ne must operate n a yle. e seond law reers to a eat ne operatng n a yle, so ts proess s onsstent wt te seond law o termodynams p ndd 96 1/8/07 7::0 M

27 Heat Engnes, Entropy, and te Seond Law o ermodynams 97.9 e eat transer over te pats and B s zero sne tey are adabat. ( ) > 0 Over pat B: n B B B dabat roesses ( ) < 0 Over pat : n V ereore, and e eeny s ten ( ) V e 1 1 ( B) e 1 1 B.60 Smply evaluate te maxmum (arnot) eeny. B V V FIG..9 V e 400. K K e proposal does not mert serous onsderaton. Operatng between tese temperatures, ts deve ould not attan so g an eeny. *.61 (a) 0. 0 (b) S mln + mln 100. kg( 419. kj kg K) ln ( 419. kj K) ln 8 0 () S J K + ln 1 (d) Yes, te mxng s rreversble. Entropy as nreased p ndd 97 1/8/07 7::1 M

28 98 apter.6 (a) Use te equaton o state or an deal gas V nr 1. 00( 8. 14)( 600) V 0( ) ( 8. 14)( 400) V m m Sne B s sotermal, V V B B FIG..6 and sne B s adabat, V V ombnng tese expressons, V B B B V V 1 ( 1) V B m ( m ) ( ) m Smlarly, V V V 1 ( 1) ( ) m m 140. ( ) or V m Sne B s sotermal, V V B B and V B V m. 0 atm atm m lso, s an sotermal and V V.8 10 m 100. atm atm m Solvng part () beore part (b): () For ts arnot yle, e (b) B K K Energy s added by eat to te gas durng te proess B. For te sotermal proess, E nt 0. V B and te rst law gves B B nr ln V 11 9 or B 1 00 ( )( 600 ).. mol 8.14 J mol K K ln kj en, rom e te net work done per yle s e 0. ( kj). 99 kj 1794 p ndd 98 1/8/07 7:: M

29 Heat Engnes, Entropy, and te Seond Law o ermodynams 99 NSERS O EVEN ROBLEMS (a) 9.4 L (b) 18 p () 7 N. m (d) (a) 4.0 J (b) 144 J.8 (a).9 (b) oeent o perormane or a rergerator () e ost or ar ondtonng s al as mu or an ar ondtoner wt EER 10 ompared wt an ar ondtoner wt EER..10 (a) 870 MJ (b) 0 MJ.1 (a) 0.00 (b) () %.16 (a).1% (b).7 J () s ossl-uel pres rse, ts way to use solar energy wll beome a good buy..18 (a) 700 k( K) ( 8 K) e exaust power dereases as te rebox temperature nreases. (b) 1.87 M () K (d) No answer exsts. e energy exaust annot be tat small..0 (a) State (ka) V (L) (K) B J (b) roess (kj) (kj) E nt (kj) B B B ().7%; see te soluton.8 (a), (b) see te soluton () 149 J; 6. 0 J; 84. J (d) 6.% (e) rev mn J K 1794 p ndd 99 1/8/07 7:: M

30 600 apter. (a) e proess s sobar beause t takes plae under onstant atmosper pressure. e eatng proess s not adabat (beause energy goes n by eat), sotermal ( goes up), sovolumetr (t lkely expands a bt), yl (t s derent at te end), or sentrop (entropy nreases). It ould be made as nearly reversble as you ws, by not usng a kten stove but a eater kept always just nrementally ger n temperature tan te syrup. (b) 40 kj () 1.0 kj K.4.7 J K J K J K J K.4 (a) eads and tals (b) ll eads or all tals () eads and tals MJ K.46 (a) 9.4 J (b) 6.4 rad s 6 rev mn () 9 rad s 790 rev mn.48 (a) J K (b) 417 J () net 1 S U 167 J.0 (a).19 al K (b) 98.19ºF,.9 al K s s sgnantly less tan te estmate n part (a).. ( ).4 (a) 4.10 kj (b) 14. kj () 10.1 kj (d) 8.8% (e) e tree-proess ne onsdered n ts problem as mu lower eeny tan te arnot eeny..6 n p ln.8 (a) see te soluton (b) le t lasts, ts proess does onvert all o te energy nput nto work output. But te gas sample s n a derent state at te end tan t was at te begnnng. e proess annot be done over unless te gas s reompressed by a work nput. o be pratal, a eat ne must operate n a yle. e seond law reers to a eat ne operatng n a yle, so ts proess s onsstent wt te seond law o termodynams..60 e proposal does not mert serous onsderaton. Operatng between tese temperatures, ts deve ould not attan so g an eeny..6 (a), atm V, L B (b).99kj ().% 1794 p ndd 600 1/8/07 7:: M