What characteristics distinguish irreversible (real) transformations from reversible (idealized) transformations?

Transcription

1 bbs Free Eergy: ptaety ad Equlbrum What haratersts dstgush rreversble real trasfrmats frm reversble dealzed trasfrmats? t every stage f a reversble trasfrmat the system departs frm ulbrum ly ftesmally. he system s trasfrmed yet remas effetvely at ulbrum thrughut the reversble hage f state. he dt fr reversblty s a dt fr ulbrum he dt fr reversblty s: d Dq rev the dt fr ulbrum s: d Dq rev he dt plaed a rreversble hage f state s the lausus Dqrrev ualty: r the frm: d > Dq rrev r lude bth d Dq Frst law says Dq d Dw d d Dw r - d Dw d 0 hs wrk ludes all kds Dw - et d Dw ther hus -d - et d Dw ther d 0 > sptaeus reversble dts fr ulbrum ad sptaety uder strats 1. rasfrmats slated system d 0 Dw 0 Dq 0 d 0 1

2 . rasfrmats at stat e d 0 the - d Dw d d 0 -d- - Dw Defe elmhltz free eergy ad -d -Dw 3. rasfrmats at stat ad stat at stat pressure et the ulbrum pressure f the system se d 0 ad d 0 the -d d Dw ther d 0 -d d -Dw ther -d -Dw ther Defe bbs free eergy -d -Dw ther d Dw ther rev - -w ther -d 0 s sptaeus hage 0 ulbrum s atural dret s ppste Fr stat ad sptaeus hages a tue utl 0

3 ummary strat dt fr sptaety dt fr ulbrum e -d d d Dw ther -d d d Dw ther 0 slated system d d 0 stat d Dw - d Dw 0 ad stat w ther 0 st w ther 0 st Mamum wrk that a system a d: d Dw ther - d Dw ther 0 d - d 0 d - d 0 Frst law says Dq d Dw Dqrev e d d Dq rev But f t reversble d > Dq rrev herefre d d - Dw r Dw d d ualty sg stads fr reversblty he mst egatve value f Dw ad therefre the mamum magtude f Dw s gve by d d. t stat w ma - hs says that sme ases depedg the sg f d t all the hage teral eergy f the system may be avalable fr dg wrk. If the hage urs wth a derease f the system the the rhs f the uat s t as egatve as tself ad the mamum wrk s less tha. w ma - If the hage urs wth a derease f the system fr the hage t be sptaeus sme f the eergy must esape as heat s as t geerate eugh etrpy the surrudgs t verme the redut etrpy the system. 3

4 I ths ase NRE demads a ta teral eergy as t s verted t wrk that passes arss the budary t the surrudgs. t stat - But we fud abve w ma -. hs fdg s uvalet t: s that part f the hage teral eergy that we are free t use t d wrk sder a hage state f the system suh that -10kJ ad -1 kj a we get as muh as 10 kj f wrk t appear the surrudgs? What are the lmtats? arry ut the hage reversbly: d Dqrev qrev 1kJ q rev w rev - 10 kj - 1 w rev w rev - 9 kj w ma - ere d < 0 s the hage state a ly happe f heat q s passed t the surrudgs t make surr > 0 s that tt 0. d ths ase tt 0 reversble B arry ut the hage rreversbly: d 1kJ stll true laussus ualty: d > Dq rrev the q rrev < -1 kj e.g. q rrev - 1. kj q rrev w rrev s stll true w rrev kj. Less wrk appears the surrudgs fr the rreversble press tt > 0. 4

5 sder a hage state f the system suh that -10kJ ad 1 kj arry ut the hage reversbly: d Dqrev qrev 1kJ q rev w rev - 10 kj 1 w rev w rev - 11 kj w ma - ere d > 0 ad heat q s take frm the surrudgs surr - 1 kj s tt 0 reversble B arry ut the hage rreversbly: d 1kJ stll true laussus ualty: d > Dq rrev the q rrev < 1 kj e.g. q rrev 0.8 kj q rrev w rrev s stll true w rrev kj. Less wrk appears the surrudgs fr the rreversble press tt > 0. he fudametal uats f hermdyams; Mawell relats rpertes f a system Mehaal: Fudametal prpertes defed by the laws f hermdyams: mpste prpertes: Frm the 1st ad d Laws: d d d I addt fr a lsed system ulbrum: 5

6 6 We a get frm the abve by dfferetat: d d d d d d d - d d d d d d d Nw we trdue d d d frm the 1st ad d Laws t eah d d d d d d d d d d d - d -d d d d d d d d d d d d therefre d d d d d d d -d d d d d hese are alled bbs Equats r the fur fudametal uats f hermdyams fr lsed system reversble press wrk ly e eah f these uats s a eat dfferetal the rss dervatves are ual frm whh we bta the fur Mawell relats: I terms f atural varables

7 7 d therefre d d d d d ad ad d d d d d ad ad d d d d d v ad v ad d d d d d ad ad Legedre rasfrmats I mathematal terms f 1 3. f s the depedet varable ad 1 3. are the depedet varables. he mplete dfferetal f f s: df f 1 d 1 f d f d where f f Nw sder the fut: g f f 1 1 O dfferetat: dg df df 1 1 df f 1 d 1 1 df 1 d the dg - 1 df 1 f d f d We have thus haged a depedet varable frm 1 t f 1 a the depedet varable frm f t g. We kw that ad d - d d I ard t g f f 1 1 d d d

8 8 mlarly se d d d d - d d Fally se d d d d d d Mathematally the trdut f the ew thermdyam futs s aheved by perfrmg the Legedre trasfrmat f the bas he prpertes f ; the bbs-elmhltz uat We prved abve that ad hese are tw f the mst mprtat pees f frmat hermdyams. he frst e says rease temperature dereases the bbs free eergy f pressure s kept stat. he sed e says rease pressure reases the bbs free eergy at stat temperature. If we tegrate we wll get the marsp hage at stat. d d ' 1 bar stadard state pressure hs uat takes dfferet frm fr lquds ad slds ad gases. " d

9 he vlume f gaseus system reases appreably wth pressure. I alulatg the varat f m wth at stat we hse the reversble path ay path etg the same tal ad fal states gves the same result s a state fut R " ' d ' d R l m - as 0 hs s ardae wth the vlume depedee f R l f at st. s 0 he vlume f the gas s mamzed as therefre as 0 d the - - vetal mlar bbs eerges m m m m m fr a pure elemet ts stadard state m 0 Obta a epress fr f m prdut ν m rea ta t m elemets ly m prdut f ν m hs m value dffers frm the rrespdg value fr the ethalpy m f 9

10 arat f free eergy ad ethalpy f a pure substae wth at stat few mre - he etrp trbut t s greater fr hgher temperatures hemal trasfrmat s always sptaeus f < 0 ad > 0 hemal trasfrmat s ever sptaeus f > 0 ad < 0 Fr all ther ases the relatve magtudes f ad determe the sptaety f the trasfrmat 10

11 11 emperature depedee f the free eergy sder : 1 ubsttute 1 Reall that hus bbs-elmhltz Equat ther dervat: 1 But ad hus bbs-elmhltz Equat d se t apples t bth f ad ls a be wrtte the frm 1 1

12 1 he bbs eergy f a mture Fr a pure substae r fr a mture f fed mpst the fudametal uat f bbs free eergy s d - d d If the mle umbers 1 3 f the substaes preset vary the 1 3 he ttal dfferetal d s: d d d d If the system des t suffer ay hage mpst d 1 0 d 0 et... d d d d ad suetly we get We defe j µ hemal ptetal d d d d d µ r d d d d µ fudametal uat f hemal thermdyams Beause the mpst s stat µ I s stat d d 0 ' 0 " µ ad µ s a etesve prperty m m ] [ µ

13 Fr a gve spees trasprt wll ur sptaeusly frm a reg f hgh hemal ptetal t e f lw hemal ptetal. I II I II d µ d µ d µ µ d < 0 - sptaeus he flw f materal wll tue utl the hemal ptetal has the same value all regs f the mture. t ulbrum the hemal ptetal f eah dvdual spees s the same thrughut a mture. bbs eergy f a gas a mture pure mture µ µ dt fr ulbrum pure mture µ l µ R µ he hemal ptetal f a gas a mture depeds lgarthmally ts partal pressure. Fr a gas : ad the abve uat bemes: mture pure µ µ R l R l µ R l Is the hemal ptetal f a gas a mture greater r smaller tha that f a pure gas? bbs eergy f mg fr deal gases mg f - f Xe e e Xe Ne e R l r R l Xe e Xe e Ne e Ne Ne Ne R l Ne r r r r Xe Xe R l r 13

14 mg f R l R l mg < 0 beause every term s egatve s mg at stat ad s sptaeus mg mg R l mg arses purely frm the depedee f at stat. What s the drvg fre fr the mg f gases? 0 beause the mleules a deal gas d t mg mg mg terat. mg s drve etrely by mg mg ls mg 0 Ideal mtures are frmed wthut ay vlume hage 14

15 hemal ulbrum a mture f deal gases sder a lsed system at stat ad. he system ssts f a mture f several hemal spees that a reat ardg t 0 ν where ν are the sthmetr effets ad the hemal frmulae. e ad are stat as the reat advaes d results frm the hemal reat advaemet d µ d Let the reat advaes by ξ mles: tal ν ξ ad d ν ξ d d ν µ dξ ν µ reat ξ at ulbrum : 0 ν µ 0 t ay mpst f the mture has the frm: µ We add ad subtrat µ the hemal ptetal f the pure spees I eah term f the sum: µ µ µ µ µ µ pure m By dfferetatg: pure ξ ξ t ulbrum: ξ m pure ξ ξ m 15

16 w reat s alulated? Dstgush betwee reat fr the stadard state at 1 bar ad reat - depedet ν reat f sder a hemal reat: α βb δd reat αµ αr l ν µ µ R l B βµ B βr l δµ D δr l reat D R l α D B d β δ D Q α β B ad reat reat R lq d at ulbrum: reat 0 ad therefre reat reat R l K r l K R K thermdyam ulbrum stat lke K s dmesless reat are futs f ly. 16

17 17 Epressg K terms f mle frat r mlarty β α δ β α δ β α δ β α δ β α δ B D B D B D B D K ν K K Beause the mlarty s defed as R we a wrte that R ad t tue wrkg wth dmesless quattes we trdue the rat ad R ν β α δ β α δ β α δ B D B D R K R K ν R K K arat f K wth temperature reat K R l Reall bbs-elmhltz uat:

18 We apply t t R l K ad we get reat d l K reat va t ff ulbrum uat d R d 1 d l K ad d l K d r R R d1 R We plt lk vs 1 ad the slpe s R If reat < 0 etherm reat lk ad therefre K tself dereases as the temperature rses If reat s depedet f temperature the reat 1 1 l K f l K 98.15K R f 98.15K Equlbra vlvg deal gases ad sld ad lqud phases ao 3 s ao s O g O reat µ ao s µ O g ao3 s R l µ Le hâteler s rple It predts that a system at ulbrum wll ted t shft twards the dret that absrbs heat f the temperature s rased. Fr a etherm reat a rease temperature wll favr the reatats. hus l K dereases wth reasg temperature fr a etherm reat. hs s eatly what we had derved usg ly the deft f ad 18

19 hemal ptetal f a real gas f µ µ R l f s alled fugaty he stadard state f a real gas s a hypthetal state whh the gas s at pressure ad behavg perfetly f φ φ s the fugaty effet dmesless ad t depeds the detty f the gas pressure ad temperature µ µ R l R lφ µ refers t the hypthetal gas the term l s the same as r a perfet gas ad the term lφ R epresses the etre effet f all termleular fres φ 1 as 0 he fugaty effet φ s gve by: Z 1 lφ d ad 0 Z R m 19