Topic 6&7 Entropy and the 2 nd & 3 rd Laws

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Meredith Griffin
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1 pc 6&7 Entrpy and the 2 nd & rd Laws ntent & Objectves: hapters 6 and 7 cncern entrpy and ts calculatn n deal gas systems. hapter 6 dscusses the need fr a thermdynamc state functn capable f predctng the drectn f natural prcesses. he 2 nd Law ntrduces such a functn, the entrpy, whch prvdes a crtern fr spntanety and equlbrum n slated systems. We wll dscuss the mcrscpc bass fr entrpy (.e. the relatnshp t dsrder but the emphass n hapter 6 wll be n the thermdynamc cnsequences f the 2 nd Law, manly usng deal gases fr llustratn. he rd Law, whch establshes an abslute scale fr entrpy, s dscussed n hapter 7. ere we wll lk at the statstcal mechancal calculatn f abslute entrpes f deal gases n rder t better understand the nature f entrpy, as well as dscuss hw standard entrpes are btaned frm calrmetrc data. Readng: M&S hapters 5-6, Mathhapter E Other Surces: 2 nd Law: Rck h. 4; arrngtn h. 6 rd Law and Statstcal alculatns: Rck h. 6 & 14; Ptzer hapters 5-6 he Meanng f Entrpy: P. W. Atkns, he Secnd Law (Scentfc Amercan Lbrary, New Yrk, 1984 h. -4. Entrpy and he 2 nd Law Numerus expermental bservatns suggest a fundamental drectnalty t natural prcesses. strcally, these bservatns were summarzed n terms f restrctns n the peratn f heat engnes. w famus statements f the 2 nd Law can be paraphrased: lausus Statement (1850: N engne wrkng n a cycle can transfer energy frm a cld bdy t a ht bdy and have n ther effect. Planck-Kelvn Statement (1851: It s mpssble t cnstruct a devce whch perates n a cycle and has n ther effect than t prduce wrk by extractng heat frm a reservr at a sngle temperature. he mdern statement f the 2 nd Law f hermdynamcs we wll emply s here exsts an extensve state functn f a system, termed the entrpy, S, such that fr any change n thermdynamc state δq ds, where the equalty apples nly f the change s carred ut reversbly. Nte the mprtance f the cncept f reversblty n ths statement. Recall that a reversble (r quas-statc prcess s ne n whch the system evlves thrugh a successn f quasequlbrum states. be truly reversble a prcess must be carred ut nfntely slwly. All real prcesses are therefre rreversble t sme extent. 1

2 Fr any prcess ne can rewrte the lausus nequalty f the 2 nd Law as an equalty by ntrducng the dea f entrpy prductn δq ds = δ S prd + δs prd s the entrpy generated durng a prcess va the creatn and dsspatn f gradents n temperature r pressure, by frctnal effects, etc. In an (dealzed reversble prcess, wheren such effects are cmpletely elmnated, δqrev ds = Many real prcesses can be treated as beng reversble t a gd apprxmatn. In addtn, even f the prcess f nterest s nt reversble, t s ften useful t calculate the entrpy change between the endpnts f the prcess usng ths expressn fr sme reversble path. Interpretatns f Entrpy and Irreversblty he 1 st Law establshes the equalty f wrk and heat as frms f energy whereas the 2 nd Law recgnzes the dfferental utlty ( qualty f these energy frms. Mechancal wrk can be vewed as the cherent (rdered mtns f mlecules, whereas heat nvlves chatc (dsrdered mtn. he fundamental rreversblty f all natural prcesses ( tme s arrw stems frm the fact that change n the drectn rder dsrder s verwhelmngly mre prbable than s the reverse prcess. he physcal prncple behnd the 2 nd Law s that energy tends t dsperse bth spatally and n terms f ts cherence (Atkns. lausus (1865 succnctly summarzed the 1 st and 2 nd Laws n the statement: he energy f the unverse s cnstant; the entrpy tends tward a maxmum. he statstcal mechancal nterpretatn f entrpy n an slated system s: S( N,, U = k ln Ω( N,, U where Ω ( N,, U s the number f quantum states (each equally prbably cnsstent wth the macrscpc cnstrants f fxed N,, and U. Mre generally the entrpy can be expressed n any ensemble by: S = k p ln p General nsequences f the 2 nd Law In an slated system, δq=0 and the system entrpy prvdes a crtern fr spntanety and equlbrum: ds > 0 fr any spntaneus prcess n an slated system ds = 0 fr a slated system at equlbrum 2

3 .e. slated systems evlve twards a state f maxmal entrpy. Engnes, refrgeratrs, and heat pumps are devces that perate n a cyclc manner t effect transfrmatns between wrk and heat. Such devces are maxmally effcent when run reversbly. Fr reversble peratn, effcency s lmted slely by the tw temperatures between whch the devce perates. Smple analyss based n cmbned applcatn f the 1 st and 2 nd Laws t ne cycle f peratn f the devce rev rev U = w + q + ql = 0 (1 st rev rev q ql Law S = + = 0 (2 nd Law prvdes the theretcal maxmum effcences shwn n the fllwng schematc. L Effcences f hermdynamc evces = eat Engne surce q q L w w Effcency = q L L exhaust = Refrgeratr eat Pump Ar nd. ktchen huse utdrs q q L w q L w L L q w L q L w L L L ce bx utdrs huse Real devces have smaller effcences than these lmtng values and can nly apprach them when surces f entrpy prductn are mnmzed. Nte that these lmtng thermdynamc effcences are cmpletely ndependent f the partculars f the devces; they depend nly n the rat f the tw temperatures nvlved. Such effcences can therefre used t establsh the true thermdynamc scale f temperature.

4 he mbned 1 st and 2 nd Laws and Entrpy hanges Fr a reversble change n whch nly P wrk s perfrmed, δ w rev = Pd and the 1 st Law can be wrtten du = δ qrev Pd. Furthermre, fr a reversble prcess the 2 nd Law states that δ q rev = ds, s that a cmbnatn f the tw laws leads t: du = ds Pd hs equatn shfts attentn away frm the prcess rentatn f the 1 st and 2 nd Laws and places t n the state functns f a system n equlbrum. (he rev subscrpt s therefre rrelevant and mtted here. Usng ths relatn ne can easly determne hw entrpy changes wth,, and P: 1 U ds = d + P + d Of mst nterest nw are the relatns: S = ds = S P P 1 d + P = P dp whch allw ne t determne schrc r sbarc changes n S wth based n a knwledge f ( r P (. Entrpy hanges n Ideal Gases Fr an deal gas ( U / = ( / P = 0 s that fr any reversble prcess the abve equatns fr ds can be smplfed t: IG nr IG P nr ds = d + d ds = d dp P Fr a reversble sthermal expansn: S IG = nr ln( 2 / 1 = nr ln( P2 / P1 Fr a reversble adabatc expansn S = 0. (Any reversble adabatc expansn, whether f an deal gas r nt, s a cnstant entrpy r sentrpc prcess. If s ndependent f temperature (as fr a mnatmc deal gas ne has: S IG IG ( 2, 2 S ( 1, 1 = ln( 2 / 1 + nr ln( 2 / 1 IG IG S ( 2, P2 S ( 1, P1 = P ln( 2 / 1 nr ln( P2 / P1 Mxtures f deal gases prvde an mprtant reference fr thnkng abut mxtures f real substances. A result knwn as Gbbs therem states that the entrpy f a mxture f deal gases at a gven and sme ttal s the sum f the entrpes that each gas wuld have f t alne were t ccupy the ttal vlume. Usng ths result, the sthermal mxng f deal gases results n an entrpy f mxng mx S IG S ( mxture y S (pure = R y ln y 4

5 were y s the mle fractn f speces. he rd Law f hermdynamcs & Practcal Abslute Entrpes he rd Law f hermdynamcs serves t defne the zer f the entrpy scale. Several statements f ths law have been prpsed: Nernst (1906: Fr any sthermal prcess S 0 as 0. Nernst (1912: he abslute zer f temperature s unattanable. Planck (1912: he entrpy f a chemcally hmgeneus sld r lqud substance has the value zer at the abslute zer f temperature. We wll adpt the defntn ffered n the textbk: Every substance has a fnte pstve entrpy, but at zer Kelvn the entrpy may becme zer, and des s n the case f perfectly crystallne substances. hs last statement s n keepng wth the statstcal mechancal defntn f entrpy n terms f rder at 0 K systems n equlbrum shuld exst n ther lwest energy quantum states, whch shuld have a degeneracy (Ω f 1 (r at least f rder 1. Adptng the cnventn S=0 at =0 K defnes the scale f rd Law r practcal abslute entrpes. Gven ths cnventn the entrpy at any temperature can be determned frm an ntegratn f heat capacty data f the frm: P ( tr1 S ( = d tr1 trn P ( d hs expressn ncrprates ntegratns ver P ( n regns where a sngle phase exsts, as well as the dscntnuus changes that ccur at phase transtns tr S tr = tr he supercrpts n these expressns dente standard state cndtns (P = 1 bar and the standard state f the cmpund f nterest. he caveat practcal appled t the entrpes determned n ths manner s a remnder that tw surces f dsrder / entrpy are gnred n these rd -Law entrpes the entrpy asscated wth nuclear spns and the entrpy f stpc mxng. Snce nucle are cnserved n chemcal prcesses, t s nt necessary t accunt fr ether f these ptental surces f dsrder n thermchemcal calculatns (except perhaps n the case f 2. 5

6 Sme representatve P ( data and the resultng S ( data are pltted belw: 80 eat apacty f r 2 00 Entrpy f r 2 eat apacty P / (J ml -1 K fusn vaprzatn Entrpy S / (J ml -1 K fus S vap S emperature /K emperature /K ata taken frm M. W. hase, Jr., NIS-JANAF hermchemcal ables, 4 th Edtn (AS, AIP & NSRS, he dtted lnes at lw temperatures are smply guesses. Lw emperature eat apactes It s mprtant t understand the behavr f heat capactes as 0 because t s always necessary t extraplate avalable data t 0 n rder t calculate rd Law entrpes. he rd Law requrement that S 0 as 0 (r at least that S reman fnte mples that P 0 as 0 Statstcal mechancal mdels fr the behavr f slds at lw temperature prvde useful relatns fr the lmtng lw-temperature dependence f P (. Fr example, the ebye thery (1912 s an extensn f the Ensten mdel dscussed prevusly. It prvdes the crrect lmtng behavr f the heat capacty due t crystal lattce vbratns fr mst nncrystallne nnmetallc crystals. he predctns f the ebye mdel can be summarzed: ( = 9 R Θ Θ / 0 u 4 e u du u 2 ( e 1 where Θ s the ebye temperature, a characterstc f a gven sld, whch s usually treated as an emprcal cnstant. Examples f fts f expermental data t ths functnal frm are llustrated belw. 6

7 (frm K. S. Ptzer, hermdynamcs, rd Edtn (McGraw-ll, he lmtng behavr predcted by the ebye thery s: 4 12π / R 5Θ fr ( / Θ << 1 (and / R fr ( / Θ >> 1 Nte the scalng f by a sngle characterstc parameter Θ. hs scalng mples that ( / Θ s a unversal functn fr all slds that ft the ebye mdel. hs crrespndng states behavr s ndeed bserved fr mnatmc slds, but nt fr slds f mre cmplcated mlecules r ns. wever, the lmtng dependence as 0 s general. At lw enugh temperatures ther mechansms f energy strage may becme mprtant. One such mechansm s the electrnc heat capacty due t valence electrns n a metal. he cntrbutn f ths mechansm can be calculated usng an electrn gas mdel. he result s: 2 π = = ( el / R A n 2 F where n s the number f valence electrns and F s the Ferm temperature (ε F /k where ε F s the energy f the hghest ccuped state at 0 K; F ~10 4 K s typcal. he relatve magntudes f electrnc and lattce vbratnal heat capactes fr tw smple metals are tabulated belw: Electrnc eat apacty n ypcal Metals 2 K 0 K Sld F Abs (el ( el (el ( el /(10 4 K Acalc /(10-4 R ( tt /(10-4 R ( tt u Al ata frm Ptzer and N. W. Aschrft and N.. Mermn, Sld State Physcs (Suanders llege,

8 In lght f these tw mechansms, the lmtng lw-temperature behavr f crystals s usually: P a + b a S ( + b as 0 as 0 Fr nn-metals the 1 term s absent, and the entrpy s smply related t the bserved 1 heat capacty: S ( ( P as 0. rd -Law ntegratns therefre ften start 1 wth the value S = ( fr the lwest temperature measured. ( mn P mn Statstcal Mechancal alculatn f the Entrpes f Ideal Gases he machnery develped n hapters & 4 prvdes the means fr calculatng the entrpes f deal gases based n mlecular cnstants btaned frm spectrscpc measurements. Recall that the relatn between S, the ttal system parttn functn Q(N,,, and the N mlecular parttn functn q(, (where Q( N,, = q(, / N! can be wrtten: ln Q ln q S = k ln Q + k = Nk ln q k ln N! + Nk N, Usng Sterlng s apprxmatn, ln N! N ln N N, and chsng N=N A ths equatn becmes: q ln q S / R = 1+ ln + N A he mlecular parttn functn can be wrtten q(, = qtrans (, qrt ( qvb ( qel ( where the ndvdual peces were descrbed n pcs &4 : q trans 2πMk a (, = 2 h 2 / 2 = Λ q q rt rt 1 ( σ Θ rt 1/ 2 π ( σ Θ A rt Θ rt Θ rt 1/ 2 (lnear mlecules (nnlnear mlecules q vb ( vb e ( = (1 f fvb Θvb / 2 ( qh = ( Θvb / = 1 = 1 e As a result f ths factrzatn, the entrpy f an deal gas, just lke the energy, s a sum f translatnal, rtatnal, vbratnal, and electrnc cntrbutns: 8

9 Fr lnear mlecules: 5/ 2 N 5 S e e ln ln R = Θ N + + Λ A σθrt = 1 e and fr nnlnear mlecules: S R / vb Θ / ln(1 e vb + Θ vb / 1 1/ 2 5/ 2 1/ 2 / 2 N = e e ln ln + rt rt rt N + π Λ A A σ Θ Θ Θ 6 = 1 Θ e ln g / e1 vb Θ / ln(1 e vb + Θ vb / 1 ln g e1 Gas-phase entrpes calculated n ths way can be cmpared wth practcal abslute entrpes, derved frm ntegratn f calrmetrc data assumng S ( 0 K = 0. In mst cases the tw methds agree t wthn expermental uncertantes. Sme cmparsns are prvded belw. able 14-4 frm P. A. Rck, hemcal hermdynamcs (Unversty Press ks,

10 When actvatn barrers prevent cmplete relaxatn, frzen-n cnfguratnal entrpy can persst n the sld state dwn t 0 K. In such cases S ( 0 K 0 and as a result the entrpy deduced frm calrmetrc measurements at hgher temperatures s less than that btaned frm calculatns (r ther means. Mst nstances f frzen-n entrpy are fund n glassfrmng substances and n crystals f quas-symmetrc mlecules whch can adpt nearly equvalent rentatns n the sld state. Examples n the latter categry are als llustrated n part (b f the prevus tabulatn. Entrpes f Reactn r S he standard entrpy f reactn, r S (, s the entrpy asscated wth cmplete cnversn f reactants t prducts f a mlar quantty f the reactn (as wrtten. All speces are n pure frm at the standard pressure f 1 bar and at sme specfed temperature. entng a chemcal reactn by ν = 0, the standard entrpy f reactn can be A calculated frm practcal abslute entrpes va: r S ( = ν S ( Interpretng and Estmatng the Magntudes f S and r S Entrpy s a drect measure f the number f states avalable t a system va Ω = exp( S / R. A few generalzatns cncernng the magntude f S / R are: - he entrpes f slds cnfrmng t the ebye mdel can be estmated at temperatures greater than Θ frm the equatn: S / R ln( / Θ he entrpes f fusn f mnatmc slds are usually n the range f 1.0R t 1.7R. (Rare gases have values near 1.7R; mst metals have values nearer t 1.R. Lattces cmpsed f sngle ns behave as mnatmc slds wth entrpes f fusn n the range 1.2R 1.7R per n. Mlecular substances f smlar structure have apprxmately equal values f the sum f the entrpes f fusn and any sld-sld transtns that may exst. - he entrpes f vaprzatn at the blng temperature f a lqud and 1 bar pressure are clse t 10.5R fr mst nn-asscated substances. hs bservatn s knwn as rutn s rule (frmulated n he entrpes f smerc sets f mlecules can ften be ranked accrdng t the nature f the bndng present. Entrpy ncreases wth flexblty. yclc and branched structures tend t reduce the number f cnfguratns accessble t a mlecule and these bndng patterns therefre reduce the value f S / R. Snce the entrpes f gases are typcally much larger than the entrpes f cndensed phases under standard cndtns, the sgn and apprxmate magntude f reactn entrpes are ften determned by the change n the numbers f gas-phase speces n reactn. 10

_J _J J J J J J J J _. 7 particles in the blue state; 3 particles in the red state: 720 configurations _J J J _J J J J J J J J _

_J _J J J J J J J J _. 7 particles in the blue state; 3 particles in the red state: 720 configurations _J J J _J J J J J J J J _ Dsrder and Suppse I have 10 partcles that can be n ne f tw states ether the blue state r the red state. Hw many dfferent ways can we arrange thse partcles amng the states? All partcles n the blue state: