On determining absolute entropy without quantum theory or the third law of thermodynamics

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1 PAPER OPEN ACCESS On dtrmnng absolut ntropy wthout quantum thory or th thrd law of thrmodynamcs To ct ths artcl: Andrw M Stan 2016 Nw J. Phys Rlatd contnt - Quantum Statstcal Mchancs: Exampls and applcatons: qulbrum P Attard - Quantum Statstcal Mchancs: Probablty oprator for non-qulbrum systms P Attard - Sackur Ttrod quaton n th lab Francsco José Paños and Enrc Pérz Vw th artcl onln for updats and nhancmnts. Rcnt ctatons - Effct of ntropy on anomalous transport n ITG-mods of magnto-plasma M. Yaqub Khan t al Ths contnt was downloadd from IP addrss on 06/06/2018 at 22:18

2 do: / /18/4/ OPEN ACCESS RECEIVED 22 July 2015 REVISED 25 January 2016 ACCEPTED FOR PULICATION 28 January 2016 PULISHED 15 Aprl 2016 PAPER On dtrmnng absolut ntropy wthout quantum thory or th thrd law of thrmodynamcs Andrw M Stan Dpartmnt of Atomc and Lasr Physcs, Clarndon Laboratory, Parks Road, Oxford OX1 3PU, England E-mal: a.stan@physcs.ox.ac.uk Kywords: ntropy, thrd law of thrmodynamcs, Saha quaton, Gbbs Duhm quaton, Sackur Ttrod quaton Orgnal contnt from ths work may b usd undr th trms of th Cratv Commons Attrbuton 3.0 lcnc. Any furthr dstrbuton of ths work must mantan attrbuton to th author(s) and th ttl of th work, journal ctaton and DOI. Abstract W mploy classcal thrmodynamcs to gan nformaton about absolut ntropy, wthout rcours to statstcal mthods, quantum mchancs or th thrd law of thrmodynamcs. Th Gbbs Duhm quaton ylds varous smpl mthods to dtrmn th absolut ntropy of a flud. W also study th ntropy of an dal gas and th onzaton of a plasma n thrmal qulbrum. A sngl masurmnt of th dgr of onzaton can b usd to dtrmn an unknown constant n th ntropy quaton, and thus dtrmn th absolut ntropy of a gas. It follows from all ths xampls that th valu of ntropy at absolut zro tmpratur dos not nd to b assgnd by postulat, but can b dducd mprcally. Th am of ths papr s to clarfy som ssus n classcal thrmodynamcs, concrnng absolut ntropy and th thrd law of Thrmodynamcs. Absolut ntropy s smply th total ntropy of a gvn systm, usually ndcatd by th symbol S. In many stuatons, on dos not nd to know th total ntropy, only th ntropy chang DS n som gvn procss. A smlar statmnt appls to othr xtnsv proprts such as mass and volum. Howvr, whras t s rlatvly straghtforward to masur total mass or total volum, and to undrstand th cas of zro mass or volum, t s lss straghtforward to masur total ntropy and to undrstand th cas of zro ntropy. For ths rason, th trm absolut ntropy s ntroducd to draw attnton to thos cass whr w ar ntrstd n th ntropy tslf, not just th chang. Th fact that condtons of zro ntropy ar not slf-vdnt s also th rason why th thrd law of thrmodynamcs s ntroducd. Howvr, ths dos not man that w cannot dtrmn absolut ntropy wthout th thrd law. It s smply that th thrd law s a usful summary: lk any law of physcs, t s at onc a gnralzaton from mprcal obsrvaton, and also part of a slf-consstnt and lgant thortcal framwork. Th thrd law has bn usful to physcsts n undrstandng hat capacts and othr rspons functons at low tmpratur, and t provd to b vry usful and nfluntal n th foundatons of chmstry. It undrpns svral tchnqus to valuat racton constants and affnts. Th thrd law, n ts modrn formulaton, assrts that th absolut ntropy tnds to zro, n th lmt as tmpratur tnds to absolut zro, for ach aspct of a systm that rmans n thrmal qulbrum. Th condtons ar mpossbl to ralz xactly, n a fnt amount of tm, but th statmnt s vry usful nonthlss bcaus n practc many aspcts of many systms approach suffcntly closly to thrmal qulbrum at low tmpratur that thr ntropy s nglgbl, at th lowst achvabl tmpraturs, compard to th ntropy at som othr tmpratur of ntrst. Thr ar systms, such as a glass, whch do not strctly attan a thrmal qulbrum stat, so th thrd law as statd abov dos not apply to thm. On may stll assocat a tmpratur wth a glass, and mak statmnts about th bhavour of ntropy n th low tmpratur lmt, but th prsnt papr s concrnd purly wth thrmal qulbrum so w do not nd to consdr such systms. Smlarly, w may for prsnt purposs gnor th cas of a dgnrat ground stat, by th argumnt that such a dgnracy would always b lftd n practc by som tny ntracton or loss of symmtry, so that th tru ground stat s not dgnrat. It s an opn 2016 IOP Publshng Ltd and Dutsch Physkalsch Gsllschaft

3 quston whthr that argumnt always appls, but n any cas, w shall consdr systms for whch t dos apply, and thn commnt brfly on th mpact of ground stat dgnracy at th nd. Th ntroducton of th thrd law nto th foundatons of classcal thrmodynamcs may lav on wth th mprsson that, wthout t, th absolut ntropy of a physcal systm could not b dtrmnd by classcal thrmodynamc mthods, wthout an appal to quantum thory. Th man purpos of ths papr s to show that ths s wrong. ut frst lt us xamn th usual account. Th standard way to dtrmn th absolut ntropy at a gvn tmpratur T f s to mploy an ntgral such as Tf đqrv ST ( f) - S( 0 ) =ò, () 1 0 T whr đq rv s th hat ntrng th systm by a rvrsbl procss at ach tmpratur T. It s oftn convnnt to us th quvalnt xprsson Tf C ST - S = ò + å T T L ( f) () 0 d, () 2 0 T whr C( T)s th hat capacty undr th condtons of th chang, and L ar latnt hats. If on assrts that S () 0 = 0on th bass of th thrd law, thn ST ( f )can b dtrmnd by valuatng th ntgral and sum on th rght-hand sd. In practc ths s don by a combnaton of mprcal obsrvaton and mathmatcal modllng. Th lattr s rqurd, for xampl, n ordr to carry out an xtrapolaton to T = 0. It s wdly blvd and assrtd that, wthout th thrd law, classcal thrmodynamcs only maks statmnts about ntropy changs, and cannot gv valus of absolut ntropy, bcaus t s basd on quatons such as du = TdS - pd V. () 3 W wll show that ths s not tru, n th followng sns. Only volum chang d V, not absolut volum, V, s nvolvd n th xprsson for work don on a smpl comprssbl flud, but t dos not follow from ths that thr s no rady dfnton of absolut volum, or mprcal mthod to dtrmn absolut volum, wthout appalng to a postulat concrnng, for xampl, th valu of V n th lmt of nfnt prssur. Smlarly, only ntropy chang d S, not absolut ntropy, S, s nvolvd n th xprsson for hat xchang, but t dos not follow from ths that thr s no rady dfnton of absolut ntropy, or mprcal mthod to masur t, wthout appalng to th thrd law. W wll prsnt svral smpl mprcal mthods that allow th absolut ntropy of a flud to b dducd wthout any us of quantum thory, and wthout calormtry or coolng to nar absolut zro. Ths wll clarfy what classcal thrmodynamcs can and cannot tll us about ntropy. In partcular, t s not tru to say that classcal thrmodynamcs can prdct ntropy changs but not th absolut ntropy of an dal gas, as f ntropy wr a spcal cas, unlk ntrnal nrgy, prssur, tmpratur and so on. In fact n ach of ths cass a combnaton of a modst amount of mprcal obsrvaton s combnd wth thrmodynamc rasonng n ordr to yld gnral nformaton about th systm proprts, and ths s just as tru of absolut ntropy as t s of othr proprts. Of cours quantum thory corrctly dscrbs th structur of any systm, wthn th lmts of our currnt undrstandng of physcs, and classcal thory dos not. An attmpt to modl a gas n trms of classcal partcls obyng Nwtonʼs laws of moton, for xampl, compltly fals to dscrb th low-tmpratur bhavour corrctly. So n ths sns quantum thory s ndd and cannot b avodd. Howvr, th argumnt of th prsnt papr, lk many thrmodynamc argumnts, s playng a dffrnt rol. It s showng how thrmodynamc rasonng can b nvokd to connct on physcal proprty to anothr, so that an mprcal masurmnt of on proprty allows us to nfr th othr, wthout th nd for a thortcal modl of th structur of th systm. That s th sns n whch w shall prsnt mthods to dtrmn absolut ntropy wthout th us of quantum thory. Th mthods to b dscussd do not nvok th thrd law. Thrfor, by also dtrmnng th valus of th ntgrals and sum on th rght-hand sd of (1) or (2), on can nfr th valu of S(0) wthout appalng to th thrd law. Th thrd law rmans a usful obsrvaton about th rsults of such studs. It follows from ths that th valu of th ntropy at absolut zro cannot b assgnd arbtrarly. It s not th cas that a unvrsal offst n th ntropy would hav no obsrvabl consquncs n classcal thrmodynamcs. Th stuaton s dffrnt from that of potntal nrgy, for xampl, and gaug frdom mor gnrally. Som txtbooks do not addrss ths ssu, but most, such as Calln (1985), Carrngton (1994) or lundll and lundll (2006) mply that th absolut ntropy s undtrmnd wthout a postulat such as th thrd law, and som, such as Wlks (1961) and Adkns (1983), nclud an xplct statmnt that thr s such a frdom, akn to gaug frdom. Carrngton, p 190 stats: Kt (th thrd law) mans that w may consstntly fx th constant of ntgraton n th dfnton of th ntropy of a smpl systm. KIn partcular, w can stablsh an absolut valu 2

4 for th ntropy of ach of th chmcal spcs K. Ths s not wrong, but t mpls that such fxng and stablshng could not b don wthout th thrd law. lundll and lundll p 193 stats: thus t sms that w ar only abl to larn about changs n ntropy, Kand w ar not abl to obtan an absolut masurmnt of th ntropy tslf. Th ln of argumnt s th sam as th on followd by Wlks, p 54, whr t s mor xplct: to obtan absolut valus of ntropy t s ncssary to mak us of th thrd law. Adkns, p 242 stats: th ssntal pont of th thrd law s that th constant s th sam for all systms, and t s strctly a mattr of convnnc to st t qual to zro. Calln p 279 stats: n th thrmodynamc contxt thr s no a pror manng to th absolut valu of th ntropy. Th argumnts of th prsnt papr mply that such clams ar fals. Scton 1 consdrs th ntropy of thrmal radaton; scton 2 th ntropy of an dal gas, and scton 3 that of a mor gnral smpl comprssbl systm. All th argumnts ar rlatd to th Gbbs Duhm quaton, but for th mddl scton of th papr that conncton s ndrct. Scton 2.1 uss thrmodynamc rasonng to obtan an xprsson for th total ntropy of an dal gas whos hat capacty s ndpndnt of tmpratur and has bn gvn; th xprsson contans a sngl unknown addtv constant. Scton 2.2 prsnts a drvaton of an quaton quvalnt to th Saha quaton (Saha 1920, Kngdon and Langmur 1923, Frdman 2008), whch nabls th unknown constant to b dtrmnd by th masurmnt of th dgr of onzaton of a hydrogn plasma. Scton 2.3 dscusss th valdty of th mthod and th conncton to statstcal mthods. Th man concluson of all th xampls s that absolut ntropy can b dtrmnd wthout appalng to thr a mcroscopc modl or th thrd law. Th papr concluds n scton 4 by prsntng th mplcatons for, and uss of, th thrd law of thrmodynamcs, n vw of ths. 1. Thrmal radaton Frst, consdr thrmal radaton. Not, w hav no nd of a partcl modl (or a wav modl for that mattr) of th radaton nsd a cavty n thrmal qulbrum. W rqur only th knowldg that th nrgy flux and th prssur ar both drctly rlatd to th nrgy dnsty. Thn, th frst shows, followng th thrmodynamc rasonng of Krchhoff (Adkns 1983), that th nrgy dnsty u s a functon of tmpratur alon, u = u( T) and th scond thn mpls that th prssur also s a functon of tmpratur alon. In fact on fnds p = u 3; ths can b rgardd thr as an mprcal obsrvaton or as followng from classcal lctromagntsm for sotropc radaton. Th ntropy S of a volum V of thrmal radaton n a cavty at a tmpratur T can b consdrd as a functon of V and T, so that w hav whr w usd a Maxwll rlaton and th fact that p = d = S + = S + T d S T V d S p V T T d d dt d V, V T V S V T p( T). Thrfor dp = d T. () 4 ut, snc S and V ar xtnsv and T s ntnsv, w hav ( S V) = S V. Usng ths n (4) gvs T d S = V p d T. () 5 Hnc, n ordr to obtan S for a gvn V, t suffcs to masur p as a functon of T. On can also, of cours, adopt a lss drct approach, for xampl by obtanng p = at 4 by a standard thrmodynamc argumnt, whr a s a constant rlatd to th Stfan oltzmann constant. y masurng th lattr, on obtans a and thus pt ( )and hnc S. Ths mthod maks us of th fact that th rlatonshp a = ( 4 3) s cbtwn a and th Stfan oltzmann constant σ dpnds only on nrgy, not ntropy, so dos not nvolv Planckʼs constant. W conclud that by masurng thr th prssur or th nrgy flux at som known tmpratur, w can fnd th absolut ntropy of a quantty of thrmal radaton wthout rcours to th thrd law or quantum thory or any othr modl of th mcroscopc structur of th radaton. 2. Th ntropy of an dal gas W now turn to th cas of an dal gas. W wll dscrb an xprmntal mthod to dtrmn S whch nvolvs gnral thrmodynamc rasonng and a sngl masurmnt of a nutral hydrogn plasma. Th mthod s qut dffrnt to th on nvokd for thrmal radaton n th prvous scton. It rls on a conncton btwn ntropy and chmcal potntal, and rqurs a gratr amount of thortcal dvlopmnt. Frst, n scton 2.1, 3

5 w dvlop an quaton for th ntropy of an dal gas, whch gvs th dpndnc on ntrnal nrgy, volum and partcl numbr, up to an unknown addtv constant. Thn, n scton 2.2, w show how masurmnt of a plasma, combnd wth th Saha quaton, ylds th constant Entropy as a functon of ts natural varabls Much of th argumnt of th prsnt scton s n standard thrmodynamcs txts, but som txts stop short of th full rsult. W bgn by consdrng a closd smpl comprssbl systm, and thn gnralz to opn systms. For a closd smpl comprssbl systm w hav that th ntropy may b consdrd a functon of two varabls, and w pck tmpratur and volum for convnnc, so that w may wrt = S + S CV = + S T V T V T T p d d d d dv () 6 T V T V usng th dfnton of constant-volum hat capacty C V, and a Maxwll rlaton. For a systm such as an dal gas, whr th quaton of stat s pv = Nk T, ths gvs CV S = ò + T dt Nk ln V. () 7 If w now furthr assum that th hat capacty s ndpndnt of tmpratur thn w hav S = S + C ln T + Nk ln V, 8 0 V () whr S 0 s an unknown constant. Ths s a standard txtbook rsult 1. Whn th hat capacty s ndpndnt of tmpratur for an dal gas, on can show that t s gvn by C = Nk ( g - 1 ) - 1, ( 9) V whr γ s th adabatc ndx. Th lss wdly known part of th argumnt gnralzs (8) to opn systms by obtanng th dpndnc of S 0 on N. Th mthod s to us th xtnsv natur of ntropy. If on bgns th argumnt afrsh, but tratng S = S( T, V, N), thn on obtans (8) agan but now S 0 s a functon of N, not a constant. To fnd th dpndnc, on may us th fact that S( lu, lv, ln) = ls( U, V, N)for any λ, and thrfor SU (, V, N) = ( N N0) SNU ( 0 N, NV 0 N, N0)by choosng l = N0 N. To mak us of ths, on nds also th rlatonshp btwn ntrnal nrgy and tmpratur, whch s U = U0 + CV T, ( 10) whr U 0 s a constant of ntgraton. Emprcally, ths U 0 s th ntrcpt wth th U axs of a ln tangntal to U ( T), but not, w do not rqur that (10) s vald all th way to absolut zro, and n fact t s not, bcaus th gas wll lqufy and/or soldfy and ntr th quantum dgnrat rgm at low tmpraturs. W hav ncludd U 0 hr (oftn t s st qual to zro) n ordr to b clar about what assumptons w do and do not nd to mak n th argumnt. Snc U and C V ar xtnsv, and T s not, t s clar that ths U 0 must also b xtnsv, whch w us n th followng. Aftr xprssng th tmpratur n trms of nrgy, and usng th xtnsv natur of ntropy, th rsult s 1 - g- = V U U0 1 SU (, V, N) Nk z ln, ( 11) N N whr ζ s a constant that dos not dpnd on U, V or N. Ths s th quaton gvng th ntropy of an dal gas of constant hat capacty, n trms of th natural varabls of S. Th complt thrmodynamc bhavour can b drvd from t. Whn th valu of th constant ζ s also gvn n trms of fundamntal quantts, ths s th Sackur Ttrod quaton. Our ntrst hr s n obtanng ζ from mprcal masurmnts Saha quaton W now turn to th consdraton of th thrmal qulbrum of a hydrogn gas. Such a gas s ordnarly consdrd to consst of a st of molculs, or, whn dssocatd, nutral atoms, n thrmal moton, and ths s a vry good approxmaton for a wd rang of dnsts and tmpraturs. Howvr, n fact thr s always som non-zro dgr of onzaton, and ths bcoms spcally mportant n th contxt of astrophyscs and plasma physcs, whr t s a major sourc of nformaton about th photosphrs of stars and othr plasmas. In th 1 A gnral dal gas (.., a systm that obys oylʼs law and Joulʼs law) nd not ncssarly hav a hat capacty that s ndpndnt of tmpratur, and n fact many gass that ar dal to good approxmaton ovr som rang of tmpratur hav a hat capacty that changs sgnfcantly wthn that rang (.g. ntrogn n th rang K at on atmosphr). Howvr th rstrcton to constant hat capacty appls to any gas ovr a suffcntly small rang of tmpratur, and dscrbs monatomc gass vry wll at tmpraturs not nar th bolng pont. 4

6 followng w consdr an atomc hydrogn gas, allowng for ts non-zro stat of onzaton n thrmal qulbrum, but n condtons whr th dgr of onzaton s small. W rcall th standard drvaton of th Saha quaton (Saha 1920, Kngdon and Langmur 1923, Frdman 2008) whch gvs th dgr of onzaton n trms of tmpratur and atomc proprts. Th onzaton/rcombnaton procss s and at qulbrum on has H p mh = mp + m, ( 12) whr th subscrpts rfr to hydrogn atoms, protons and lctrons, rspctvly. Ths condton can b xprssd n trms of othr quantts by tratng th protons, lctrons and hydrogn atoms as a mxtur of dal gass, but w must account corrctly for th bndng nrgy E R. Th ffct of th lattr can b undrstood as follows. Each gas n th mxtur obys quaton (10), so w hav U = U0, + CV, T. ( 13) Whr th subscrpt runs ovr th proton, lctron and nutral hydrogn gas. If on wr studyng any on gas n solaton, thn th valu of U 0, for that gas could b st arbtrarly, but whn th thr ntract, th valus of U 0, ar not ndpndnt. Snc ach U s xtnsv, w hav U0, = Nu, ( 14) whr u s that part of th nrgy pr partcl that dos not dpnd on tmpratur. y consrvaton of nrgy, w hav uh + ER = up + u. ( 15) Whr E R s th nrgy rqurd to onz a hydrogn atom. In th followng w wll tak t that ths bndng nrgy s gvn, to good approxmaton, by th ground stat onzaton nrgy, also calld Rydbrg nrgy. Ths approxmaton gnors th ntrnal thrmal xctaton of th hydrogn atoms; t assums that n ordr to onz an atom th full Rydbrg nrgy must b provdd. Ths s a good approxmaton for kt E; R w assum ths lmt for th rst of th argumnt. Snc ER k K, th approxmaton wll hold vry wll n typcal laboratory condtons (as wll as n th photosphrs of stars). Not, also, that th ntranc of E R nto th argumnt dos not mply that w hav assumd any partcular modl of hydrogn atoms, such as th on provdd by quantum physcs. W rqur only th mprcal obsrvaton that a fxd amount of nrgy s rqurd to onz ach atom. In a standard approach, on mght now ntroduc a formula such as m = kt ln ( nl 3 T) whr l T s th thrmal d rogl wavlngth. Ths formula can b obtand, for xampl, by a statstcal mchancal argumnt. Hr w wll not nd that argumnt, bcaus w can nstad obtan an xprsson for chmcal potntal from th ntropy. Consdr th Gbbs functon of a sngl-componnt dal gas: G = mn = U - TS + pv. ( 16) For a gas havng constant hat capacty w may us (10) and thrfor CV m = k N T T S U0 N N. ( 17 ) W now apply ths quaton sparatly to th lctrons, protons and hydrogn atoms n th plasma. Ths s justfd by th usual argumnt that, at low dnsty, dffrnt gass n a mx of dal gass contrbut thr ntrops ndpndntly, and hr t also nvolvs an assumpton of ovrall nutralty of th plasma, n ordr to justfy gnorng th contrbuton of th lctromagntc fld. Undr th approxmaton that th lattr can b nglctd, w may substtut (17) thr tms nto (12) and thus obtan 5 T( s + sp - sh) = kt + a, ( 18) 2 whr w usd C N = ( 3 2) k for ach spcs, s s th ntropy pr partcl n ach cas, and V a = up + u - uh = ER ( 19) Usng now th ntropy quaton, (11), to xprss s H and s p n trms of th constants z H, z p and othr quantts, w fnd whr n s numbr dnsty. z = 5 ER + + Hnp s k ln 2 kt z n, ( 20) p H 5

7 An altrnatv drvaton of quaton (20) s gvn n th appndx. Nxt, w argu that zh 2zp, to an accuracy of ordr on part pr thousand, bcaus whn th ffct of lctrcal charg s nutralzd by th prsnc of th lctrons, th gas of hydrogn atoms and th gas of protons ar alk, xcpt on has four nrgtcally avalabl ntrnal stats pr partcl, th othr two. Ths nglcts th slght dffrnc btwn th mass of a nutral hydrogn atom and th mass of a proton. Th factor 2 can b argud as a thrmodynamc statmnt, not an appal to oltzmannʼs formula S = k ln W, bcaus whn all th spn stats ar qually occupd on can always modl ach gas as an qual mxtur of dstngushabl gass, on n ach spn stat, and apply th ntropy formula to ach part n th mxtur. Ths pont s prsntd mor fully n th appndx. Hnc w fnd that, undr th statd approxmatons, th ntropy of a gas of lctrons formng a componnt of a nutral hydrogn plasma n thrmal qulbrum s gvn by S = N k 5 E + + R n ln 2 p 2 kt n, ( 21) H whr N s th numbr of lctrons, E R s th Rydbrg nrgy and n p, n H ar th numbr dnsts of protons and hydrogn atoms, rspctvly, n th plasma. Equaton (21) may b rgardd as a way of wrtng th Saha quaton that s convnnt to our purposs. Th sgnfcanc of our analyss s that t shows how th proprts of th plasma may b rlatd drctly to ntropy usng only classcal thrmodynamc rasonng, wthout th us of statstcal mchancs or quantum thory. Th xprmntal dtrmnaton of S now procds by masurng th dnsty and th dgr of onzaton at a known tmpratur, and dducng th ntropy of th lctron gas from (21). Ths s th total ntropy, wthout unknown addtv constants, somtms calld th absolut ntropy. y usng ths rsult to supply th valu of th othrws unknown constant ζ n quaton (11), on obtans th ntropy undr all condtons for an dal gas of lctrons n a nutral plasma. Equatons (21) and (11) can b combnd to gv an xplct formula for z n trms of th masurd quantts: z = 2 n n 3 kt p n H (In a nutral plasma, n = npand nh + np = constant.) -3 2 ER k T. ( 22) 2.3. Dscusson Equatons (22) and (11) gv our scond xampl of th man pont assrtd n ths papr. Thy show that, n ordr to fnd out how much ntropy a monatomc gas has got, undr ordnary condtons whr t bhavs to good approxmaton lk an dal gas, on dos not nd quantum thory and on dos not nd to carfully track th hat suppld as th systm s warmd rvrsbly from absolut zro. Ths s bcaus th law of mass acton (of whch th Saha quaton s an xampl) can b convrtd nto a statmnt about ntropy, nstad of th usual form n trms of chmcal potntal. Ths obsrvaton may not b compltly nw, snc smlar rasonng appls to a larg numbr of chmcal ractons and chmsts ar famlar wth ths typ of stratgy for avodng calormtry whn studyng chmcal potntal. Howvr, w hav brought out som mplcatons for th thrd law that appar to hav bn ovrlookd. Th thrmodynamc pont hr s that th calculaton of th absolut ntropy dos rqur mprcal nput, of cours t dos bcaus on cannot say anythng about a systm whch has not bn masurd or spcfd n som way but th amount of mprcal nput s small compard to th amount of physcal prdcton that can thn b dducd. Ths s smlar to th obsrvaton of othr proprts such as ntrnal nrgy and tmpratur. W prsntd th argumnt drctly n trms of ntropy, n ordr to arrv at quaton (21), but som radrs may fnd th followng approach mor ntutv. Frst w wrt down a fundamntal quaton rlatng to chmcal potntal of a sngl componnt systm: m = U S - T. ( 23) N N Nxt w us (10) and (11), to obtan TV, TV, U g m = + z g kt kt ln ln 1 N 1 n g ( g-1). ( 24) Ths draws attnton to th fact that ζ appars as an unknown offst n th quaton for μ, just as t dos n th quaton for S. Th ssnc of th argumnt s to s that w can obtan ths unknown offst by usng th fact that n chmcal qulbrum th chmcal potntals balanc (quaton (12)), n a stuaton whr th contrbuton of potntal nrgy to th nrgy pr partcl s known, quaton (15), and ths contrbuton has to b mad up by th translatonal dgrs of frdom n th chmcal potntal balanc. y substtung (24) thr tms nto (12) and mployng (15), on obtans (22) as bfor. 6

8 Fgur 1. Th ntropy of on mol of sotopcally pur non at on atmosphr, as a functon of tmpratur. Th full ln s th xprmntal rsult, th dashd ln s th prdcton of quaton (26). Th lattr gvs unphyscal ngatv valus as T 0, but t dscrbs th gasous bhavour vry wll. (Th mprcal curv was obtand by combnng nformaton about hat capacts and latnt hat.) Th fact that no gas s dal at low tmpraturs dos not nvaldat th mthod for masurng ntropy that w hav dscrbd. Indd, quaton (11) s compltly wrong at low tmpraturs; ths dos not nvaldat th argumnt but rathr srvs to mphasz that w hav no nd of th thrd law or hatng from absolut zro n ordr to arrv at our rsult. Equaton (11) s corrct n ts rgm of valdty (namly, low dnsty, and ovr modst changs n tmpratur), and th obsrvaton of th onzaton fracton wll dtrmn th constant ζ corrctly. To llustrat ths pont, fgur 1 shows th ntropy of on mol of non at on atmosphr, as a functon of tmpratur. Whn w modl non as an dal gas, w do not larn about ts ntropy at or blow th bolng pont, but ths dos not man that our quaton for th absolut ntropy (not just ntropy changs) lacks accuracy abov th bolng pont. Ths can b clarfd by wrtng quaton (20) anothr way: n n p H 1 2 R. ( 25) = s k -E k T 5 2 Wth th bnft of statstcal rasonng, w can ntrprt th rght-hand sd of ths formula as th product of a numbr of stats and a oltzmann factor. Th obsrvaton of th onzaton fracton nabls on to larn what th numbr of stats s, whn th nrgy E R and th tmpratur ar known. W wll now prsnt th conncton to statstcal mchancs mor fully. Accordng to classcal statstcal mchancs, th ntropy s gvn by oltzmannʼs formula S = k ln W, and W, th numbr of stats avalabl to th systm, s countd by dvdng th accssbl phas spac volum pr partcl by som constant h 3 whos valu has to b dtrmnd mprcally. Of cours quantum thory coms n and tlls us that ths constant h s non othr than Planckʼs constant that rlats nrgy to frquncy and momntum to wavlngth of d rogl wavs, but on dos not nd to hav that furthr nformaton n ordr to mak us of classcal statstcal mchancs combnd wth th mprcally dtrmnd unt of volum n phas spac. Whthr a classcal or a quantum tratmnt s adoptd, on fnds that n th cas of a monatomc gas at low prssur (whr partcl ntractons can b nglctd), th ntropy s whr l T s th thrmal d rogl wavlngth gvn by 5 S = Nk - l ln n 3 T, ( 26) 2 l T 2p 2 = mk T. ( 27) Ths mpls that for th monatomc gas, th constant ζ n (11) s z = 5 2 ( 3p 2 m) - 3 2, whr m s th mass of on atom. Hnc th masurmnt of ζ can b rgardd thr as a way to obtan th ntropy, or as a way to dtrmn. Onc ths has bn don n on cas (such as th lctron gas), on can thn us th formula to fnd th ntropy of any othr dal gas. In ordr to compar ths rsult wth quaton (22) on should not that (22) ncluds th contrbuton from th spn dgr of frdom for a gas of lctrons, whras (26) appls to a gas of spnlss partcls. As an ovrall consstncy chck, w quot hr th standard statmnt of th Saha quaton n th cas of hydrogn: 7

9 Fgur 2. Exprmntal mthod to fnd th ntropy of a narly pur vapour. Hr m s th concntraton of a solut n a solvnt and th barrr btwn th two chambrs s a smprmabl channl that allows solvnt but not solut to pass. Smprmabl mmbrans on ach sd of th channl support a prssur dffrnc, and th channl s thrmally nsulatng to good approxmaton (s man txt). Thr s phas qulbrum btwn th soluton and ts vapour n th nnr chambr. Th fxd p0, T0 nsur that μ s constant n th outr chambr. y allowng th concntraton m n th nnr chambr to chang, on adjusts th poston of th phas qulbrum ln as a functon of prssur and tmpratur. Ths allows on to xplor a rang of p, T valus all at th sam μ. Whn th vapour prssur of th solut s low, th vapour s almost a pur substanc, so on larns th ntropy of that pur substanc n ts vapour phas. Th vapour may hav any quaton of stat. n n p n H mkt 2p = -E R k T. ( 28) If on substtuts ths nto th rght-hand sd of (22) on fnds z Ttrod rsult for a gas of spn-half partcls. 3. A mor gnral systm = 25 2( 3p 2 m )- 3 2whch s th Sackur W turn now to th cas of a sampl of ordnary mattr (not radaton, and not ncssarly an dal gas). W wll rstrct our attnton to a sampl that can b tratd as an dal flud,.., a systm that xhbts prssur but not shr strss, but th sampl nd not b a gas and may hav any quaton of stat. Our proposd mthod s basd on th Gbbs Duhm rlaton. For an dal flud th Gbbs Duhm rlaton rads å SdT - Vdp + N dm = 0, ( 29) whr th symbols hav thr usual manngs. N s th numbr of partcls of th th componnt, m s th chmcal potntal pr partcl. In th cas of a sngl componnt systm, w hav p S = V T and m ( 30) m =- S N. ( 31) T oth of ths quatons offr ways to fnd S. Th tratmnt of thrmal radaton n scton 1 can b rgardd as an xampl of (30) f on modls thrmal radaton as a flud at zro chmcal potntal. Th argumnt from th Saha quaton prsntd n scton 2.2 could b rgardd as ndrctly or loosly connctd to quaton (31). W now prsnt a furthr applcaton of quaton (30). W can apply (30) straghtforwardly to any xprmnt n whch th prssur and tmpratur of a flud ar causd to vary quasstatcally, undr condtons of constant chmcal potntal. Such condtons ar not achvd undr ordnary crcumstancs. An dalzd partcl rsrvor can provd partcls wthout also provdng ntropy, and such a rsrvor s not ruld out n prncpl (Waldram 1985), but what on wants s a practcal mthod. Fgur 2 llustrats th man componnts of such a mthod. A chambr of adjustabl prssur and tmpratur p, T s surroundd by anothr whos prssur and tmpratur p0, T0 ar mantand constant. Th wall btwn th two chambrs conssts, n whol or n part, of a scton that s prmabl to a solvnt such as watr, but not to a solut such as common salt (sodum chlord). Ths scton could consst, for xampl, of a short channl covrd by sm-prmabl mmbrans. Th nnr chambr contans a pool of solvnt wth solut dssolvd, n phas qulbrum wth ts vapour. In ths crcumstanc, solvnt wll mov btwn th chambrs untl th chmcal potntal μ of th solvnt s th sam 8 p

10 n two chambrs, and at qulbrum thr wll b a unform tmpratur T = T 0, and a prssur dffrnc owng to osmotc prssur, such that p > p 0. W now chang th concntraton m and th tmpratur T n th nnr chambr, by small amounts dm, dt, whl kpng p 0 and T 0 fxd n th outr chambr. Owng to th fxd p0, T0, th solvntʼs chmcal potntal μ dos not chang n th outr chambr. Snc th sm-prmabl mmbrans allow solvnt to pass, th condton of chmcal qulbrum wll b attand whn μ s also unchangd n th nnr chambr. Onc ths has happnd, thr wll b a nw valu of th osmotc prssur, so p wll hav changd by som small amount dp. On thn has a chang n both p and T wthout a chang n μ, so by masurng both for a fw valus and takng th lmt, on obtans, subjct to a provso prsntd nxt, an mprcal valu for ( p T) m for th solvnt n ths xprmnt. for w can mak such a clam, th followng consdraton has to b takn nto account. At th nw valus p + dp and T + dt, th flud n th nnr chambr wll not b n a strct thrmal qulbrum, bcaus of hat transport to th outr chambr. On would lk th smprmabl channl to b also thrmally nsulatng, but ths s not strctly possbl, owng to hat transport by convcton as th solvnt passs btwn th chambrs. Howvr, on can hav a cas whr thr s a larg rato btwn th rlvant rlaxaton tms. Thr ar thr rlaxaton tms to b consdrd. Lt t m b th tm rqurd for th qualty of th chmcal potntal to b stablshd, t T b th tm rqurd for th flud n th nnr chambr to attan ts own thrmal qulbrum f t wr n complt thrmal solaton, and t c b th tm for tmpratur qulbrum to b stablshd btwn th chambrs n th actual systm. If th channl has a low coffcnt of thrmal transport (ncludng th contrbutons from both convcton and ordnary thrmal conducton) thn on can hav { t, t } t. ( 32) m T In ths stuaton, th condtons n th nnr chambr ar clos to thrmal qulbrum condtons at th nw tmpratur T + dt for tms t satsfyng { tm, tt} t tc. W hav now stablshd that, as long as th condton (32) s satsfd, thn th masurmnts of dt and dp can b ntrprtd n th ordnary way, and on can thus obtan an accurat masurmnt of ( p T) m. It only rmans to commnt on th rol of th vapour n th nnr chambr. Th chmcal soluton n th nnr chambr s a systm of two chmcal componnts, and so quaton (30) dos not apply to t. Instad anothr quaton appls, nvolvng th chmcal potntals of both th solvnt and th solut. Howvr, t s qut common for th vapour prssur of a solut to b vry much lowr than that of a solvnt, so that th vapour conssts almost ntrly of th solvnt. In ths cas th vapour s a on-componnt systm to whch (30) appls, and thus th xprmnt ylds th absolut ntropy of a gvn volum of th vapour (whr w hav usd th fact that, n phas qulbrum, μ s th sam for th two phass). Not that although w assumd th vapour dd not xhbt shr strss, w dd not nd to assum anythng about ts quaton of stat. In short, t nd not b an dal gas. Also, although w hav calld t a vapour n th abov, th scond phas of th mattr n th nnr chambr nd not b a gasous phas; w only rqur that t b a phas to whch th sngl-componnt Gbbs Duhm quaton can b appld to good approxmaton. Ths ncluds som solds as wll as many lquds and gass. Th xprmntal mthod w hav dscrbd s vry closly rlatd to th clbratd fountan ffct n lqud hlum. Hr, two chambrs contanng lqud hlum blow th lambda pont ar sparatd by a narrow tub or plug whch s porous to th suprflud but not th ordnary componnt of th lqud. In th prsnc of a tmpratur dffrnc btwn th chambrs, suprflud movs from th coldr to th hottr chambr untl a prssur dffrnc bulds up, gvn by quaton (30). On can us th quaton n two ways. If th ntropy s obtand by usng th thrd law and ntgratng th hat capacty, on prdcts th prssur dffrnc for a gvn tmpratur dffrnc. Altrnatvly, on may masur th prssur dffrnc, and us t to obtan th ntropy pr unt volum of th lqud hlum. 4. Th thrd law Th thrd law of thrmodynamcs maks a statmnt about th absolut valu of th ntropy of any physcal systm, as xpland n th ntroducton. Th usual applcaton of th thrd law s to say that snc S () 0 = 0, th rght-hand sd of xprssons (1) and (2) can b usd to dtrmn ST ( f ). Exprssons (11), (22) and (30) offr an altrnatv applcaton. If, by usng thos quatons, w larn ST ( f )for a gvn systm such as a mol of non at som tmpratur, and thn w us th thrd law to clam that S () 0 = 0, thn w obtan a prdcton for th valu of th rght-hand sd of quatons (1) and (2). Not, ths prdcton s not about th gas alon, but about an ntgral nvolvng th hat capacty of a sold and a lqud and a gas, and a sum nvolvng th valus of varous latnt hats and transton tmpraturs. An altrnatv applcaton of our rsults s to gnor th thrd law, and us (1) and (2) to dtrmn th valu of S(0). For many systms, on wll fnd th valu S () 0 = 0, and on wll nvr fnd a ngatv valu. It s not 9 c

11 ncssary to nvok thr classcal or quantum statstcal mchancs hr; on smply maks th masurmnt. Of cours, n ths approach on wll vntually formulat th thrd law as a usful summary of what s found n such xprmnts. It can b statd thr as w hav don, n trms of thrmal qulbrum and th valu S () 0 = 0, or on may nvok quantum thory and dscuss th ground stat and ts possbl dgnracy. In thr cas on fnds that, for any gvn systm, th ntropy n th lmt T 0 taks a sngl valu that dos not chang n sothrmal procsss, and on can us th rsults of ths papr to dscovr what that valu s. Th pont s, th valu s fxd absolutly by a masurmnt at hgh tmpratur and th us of calormtry to track th ntropy rducton as th tmpratur falls. It s somtms assrtd that th valu of th constant towards whch th ntropy tnds as T 0 s not fxd by thrmodynamc argumnts, and could n prncpl b som othr valu, as long as t s th sam for all aspcts of all systms. Such an assrton s n Adkns book, for xampl (Adkns 1983). Calln (1985) maks a smlar statmnt. On of th mportant faturs of th argumnt of ths papr s that t shows that such s not th cas. Th valu of ST ( )as T 0 s not arbtrary but can b ascrtand by masurmnt, wthout rqurng statstcal or quantum thory. Ths dos not man th thrd law s not usful, but t dos man that th valu of th ntropy at absolut zro s a masurabl, not an arbtrarly assgnd, quantty, wthn classcal thrmodynamcs. All such argumnts ar not ncssary to physcs f on assums that som fundamntal thory, such as quantum thory and th Standard Modl, s corrct, and f on also assums that w know how to apply such a modl n ordr to gan undrstandng. Howvr, thrmodynamc argumnts rman mportant bcaus thr rol s, n part, to show us what must b tru of th physcal world, undr a small numbr of assumptons, rrspctv of our mcroscopc modls. Thr rol s also, n part, to show us what concpts and ways of rasonng ar frutful. Acknowldgmnts I thank Stphn lundll and two anonymous rfrs for hlpful commnts on th papr. Appndx Altrnatv drvaton of quaton (20) Txtbook drvatons of th Saha quaton oftn adopt a mthod slghtly dffrnt to th on prsntd n scton 2.2, as follows. Consdr frst th dfnton of chmcal potntal m º U N SV,,{ Nj}. ( 33) For a gas, th ntrnal nrgy can b convnntly dvdd nto th kntc nrgy assocatd wth translatonal moton of th partcls, and th rst (potntal nrgy and th nrgy n ntrnal dgrs of frdom of th partcls thmslvs). Snc th ntrnal nrgy can thus b rgardd as a sum of two parts, so can th chmcal potntal. Lt us attach a star to th part assocatd purly wth translatonal kntc nrgy, gvng * m * º U N SV,,{ Nj}. ( 34) Thn for a cas whr th ntrnal dgrs of frdom contrbut nglgbly to th hat capacty, w hav U* = å U * whr U* = CV, T. ( 35) In th nutral hydrogn plasma, th nrgy rqurd to rmov on hydrogn atom and provd on proton and on lctron, wthout changng th systm ntropy, s - m* + m* + m* H p + E R. ( 36) Th symbol m* H hr can b ntrprtd as th nrgy lbratd whn a hydrogn atom n ts ground stat s rmovd from th mx wthout frst xctng t wth th nrgy E R that would b rqurd for onzaton. Th bndng nrgy E R s normally assocatd wth th hydrogn atom n quaton (36), so on dfns mh = m* H - ER but ths s not th only way to undrstand th stuaton: th nrgy E R s an ntracton nrgy so t rally blongs to th lctromagntc fld. For th purposs of book-kpng, t could b assgnd to any on of th partcls, or shard among thm. Th mportant pont s to wrt down corrctly what s th total nrgy consrvaton condton, allowng for all th nrgs nvolvd. Ths s don by assrtng that, n qulbrum, th chmcal potntals ar so arrangd that th sum n (36) s zro. That s 10

12 m* + m* + - m* p ER H = 0. ( 37) Ths s anothr way of wrtng quaton (12). Not that w hav carfully avodd hr any statmnts about partton functons. Indd, th fact that th partton functon for th lctronc xctatons of a sngl hydrogn atom n an nfnt volum s tslf nfnt s commonly gnord or glossd-ovr n txtbook tratmnts 2. Equaton (16) gvs, for th contrbuton to chmcal potntal from translatonal kntc nrgy CV, m * = k + - N T T S N. ( 38 ) y substtutng ths thr tms nto (37) on obtans quaton (20) as bfor. Justfcaton of zh 2zp Th fact that zh 2zp, that w usd n ordr to obtan quaton (21), s a standard part of th drvaton of th Saha quaton, but ordnarly t s justfd by th us of oltzmanʼs statstcal formula for th ntropy, whch w wsh to avod. To obtan t from thrmodynamcs alon, on may procd as follows. Consdr th gnral problm of a gas of N partcls wth Z qually lkly dstnct ntrnal stats pr partcl, whr th moton of ach partcl s ndpndnt of ts ntrnal stat. Th man concpt w nd s that such a gas has all th sam thrmodynamc proprts as an qual mxtur of Z gass of N/Z partcls ach, whr ach of th gass n th mxtur s of th sam typ as th unpolarzd gas, but fully polarzd, that s, wth all partcls n th sam ntrnal stat, and ach of th Z cass appars qually n th mxtur. Of cours th countng of th ntrnal stats s xpland by quantum thory, but all w nd s th mprcal fact that f on somhow prpars th mxtur just dscrbd, thn th systm that rsults wll b thrmodynamcally th sam as th unpolarzd gas. Lt S b th total ntropy of th gas, and S th ntropy of on of th componnts n th abov modl. Thn S = å S. Applyng quaton (11) to ach trm n ths quaton, w fnd Z V V Nk ln z k = ånk ln z k, ( 39) N N = 1 whr N = N Z and k = [ kt g g-1 ( )] ( ). Hr, ζ s th constant apparng n th formula for th ntropy of th whol gas, and z ar th constants apparng n th formula for th ntrops of th componnt gass. W now clam that all th z ar th sam, bcaus th motons of th partcls ar ndpndnt of thr ntrnal stat. Substtutng ths nto (39) gvs z = Zz. It follows that, for two gass that ar dntcal xcpt that on has 4 ntrnal stats pr partcl, th othr 2, on wll fnd th ζ of th frst wll b twc that of th scond. QED. Rfrncs Adkns C J 1983 Equlbrum Thrmodynamcs 3rd dn (Cambrdg: Cambrdg Unvrsty Prss) lundll S J and lundll K M 2006 Concpts n Thrmal Physcs (Oxford: Oxford Unvrsty Prss) Calln H 1985 Thrmodynamcs and an Introducton to Thrmostatstcs 2nd dn (Nw York: Wly) Carrngton G 1994 asc Thrmodynamcs (Oxford: Oxford Unvrsty Prss) Frdman A A 2008 Plasma Chmstry (Cambrdg: Cambrdg Unvrsty Prss) Kngdon K H and Langmur I 1923 Phys. Rv Saha M N 1920 Phl. Mag. Sr Waldram J R 1985 Th Thory of Thrmodynamcs (Cambrdg: Cambrdg Unvrsty Prss) Wlks J 1961 Th Thrd Law of Thrmodynamcs (Oxford: Oxford Unvrsty Prss) 2 For a sngl hydrogn atom n an nfnt volum, th partton functon assocatd purly wth th lctronc xctatons, vn wthout onzaton, s Z = ån= 1 2 n2 xp [(- E + E n2 0 R ) kt], whch s nfnt. Sttng ths qual to 2 wthout a thorough argumnt s hardly justfd! 11