Articles Ilidiall Journal ofChclllical Tcc hn ology Vol. tJ, S.:plclllbcr 2002, pp, 402-406 , , I

The thermodynamics of systems at negative absolute

Jaime Wisniak'"

I kparlmclli or Chcmic" En gi nceri ng, Ikn-C/urion Uni \ c r ~ il y or Ihe Negc\', Beer-She\',:. Israel X4 10)

Ueceil'cd 27 Jill.'" 2001 : IIIH'fI/('d II MOl' 2002

The application of the laws of thermodynamics is a nalyzed fOl' the case that a system exists in the domain of ncgatiyc ahsolute tcmpcrature, It is shown thaI irreversible proccsses arc accompanied by an increase in cntropy, but that it is possihlc to l'onvcrt totally into work. In addition, it is impossihlc to convert work totaHy int o heat and work llIust hc added to the S)'stCIll to transfer thermal energy frolll a source to a hot one. '

Thc possihility of the cx istence of ncgative absolutc AllyIH)\\ . it i <; o f intcres t to di,>cuss hO\\ th c L aws 0" tcmpcraturcs has bccn discussed In a prc\ IOU'> th crlllodynamic'. appl) to the'.c si tuations an d ir the publicati on I, It ha s bccn shown that for ordinary propn ti cs or thc,c sy <.,[c ms bcha vc in thc salllL' '>\ ~ t CIll thc ahsol utc tcmpcraturc lllU <., t hc pos itive IllallIlL'1 as to thc (lIlC, Ihat h:l \'C pos iti vc tem peraturc, bccausc It ha'> an UPP C! hllUI1l! tll cllcrgy, II()\\L'\ n. HerOIC dlllllg '.0 It i'> agrccd that \\orl. ami hca t ha\ L' \lrdillar) S) slcl11~, tlndcr \cry spL'clfil ~'lrCUllhlallCL'S, tilL '>: 1111 L' dc'i' illl'll III hOlh klllpL'rallllL' domain la) 1ll;1) cOlltaill subs~ "tC1l1\ til,lt illlcrdL't \\ L'a\...l)' wilh thc hC:lt is thL' cllcrg) thai rim\· ... hct\\'ccll 'hL' '>y'-.lcm alld 1l1dill \) sIL'!)) ,lIld kl\'c a li!ll/li'd 1111111t),,'1 III L'll-:!';;) it ... ~Ilrruulldi :,', 11C-::nl"L' llf .' tCIllPl,,',IlLll'C Lill'i'L'rl'lll' . IL'\'L'I, TIlL' !L'ljUif'Clllcllt of \h',d illlL'!actilll, I" ;IIHi (h) \\(11'1 : ... :111 i'llc'i',,,:ti(lIl l'l,ll t,,\...c· ... pldl'c' IIL'C'l'S~;II~ '>ll Ih:1I tllL'llll:ti L'l\uilih!iul~) I)c d!l:ll'11C,; \ L'I') i)L't\\I'l'1l lill' S)' c'1l1 .Ilhl it ... urrnulldi 1;\ i/O: CllhL'd

.,I()\\ I), I( thL IIIL':lI1al Llluilihl'lulll II! tIl'.' ,ul,,-) .1L'1ll I)) " Il'I11P':I,lturC (\;1'1' 'I'll,','. I" .!cittil ,In. di :i'll'ti 11 'L': III laPldl). thL"; 11, tl'lnpLTatll c' \\111 k dillL'IL'n! \\ "I Iw lll:"k hL'I\\ 'L'll ;, i1()j ,111.1 , L'oicl h,l(:\ h' I 1'1 1III 1I1:lt ll!" thl' main \)\tL'm t\ \\lll knll\\llt'\,lIllpk IflIILiIl~ at till' dllL'l"I(11l (II hL'd' tr;lll .. kr: heat \\ ii' or thi, ,ltuati()1l i'. '{ 'L't or Ilul'll·.!! '>pin (\1' litillulll :11\\;1)', n,,\\ II' lill ,hL' h'llll'r tIl th,,' clll"L'1 hpti) \\ il~'11 1111, iii a L'l\ ,t,lI of lithiulll riulll'id', II' a rL'\ n\l' Cdll,>idL'l':Il~' 110th I .'~atl\ l' aIlII Jlfl ... iti\ L' :lh,,(llutl' magllL'tiL' ficld i appliL'd thc ckctroll'> \\ ill Illlt he ahk Il'lllpnaturL's. thl'sc \\ ill progrc\'. 'r(l1l1 "colder" l<' to Julio\\' th c dlrcctioll, and llll l'. I \\ ill rCIll:lill oricllted "hollL'r" III the \l'LfUl'I1CC: (). I ... , I 0 ..... 100, , , :llllipa!allci. III thi" cry'.tal Spill-Spill, rcla\~ltioll tilllL'\ + i n fi 11 it) . - i n ri nil), .. '. - I 00 ... - \( I..... - I . -() I, arl' ahout 10 ' S, \\ hill' CI') stal "pill-Ja1licc rcia.\atidll It must hc ulltkrqll()d thai the Zerotll and thl' Fil"-l tllllC arc at least ~()() s, II \\ ill ta\... ' hct \\ CL'1l ri \ L' to I ,a\\ or tl1L'rmlld) n:lmic'> apply ~quall) tt' hoth thin) millutcs ulltil thc Spill '.uh'.) "kill \\ ill rctllrII to tcmpc raturc dClmai n\ hcc,l lI '>e th ey are indepelldcnt 01 the rmal cqu ilibriu m with th e maill '.y'>tcm'l th c sign or thc te mperat urc, The Zcroth L I\\ It Gill be said that the Ilcgati vc tcmpcralLlrc CO llCL'p t d e tcrmin e~ that t\\ () sv ... tcm ,> are In thermal lllay bc applicd to thc cspeci al ca-; c ,> wherc thc equ ilibrium w hen th cy ha ve thc sa me temperaturL'. addition of cllerg) rrom Il 'illuml crcatl'S a pse udo­ and the First Law represents the encrgy halanee shcct cCJuilih riulll subsystem or illvcrtcd Icvek Whether it or the change, is appropriate to u <; e the term negati vc temperature or Now {'oliowing three statcments of thc Second L I\\ pscudo-temperaturc is a question of terminology, ot sha ll be investigated: olll) that, Ilowadays masers ane! are best (a) Heat flows spontaneously from a hot s()urce to a approximated as thermodYIlamic systems that ex ist at cold one (Clausius) or, it is impo sible to const ruct an llegative absolute temperalLlres 5 A lso. at negati e engine th at operat es in a reversible manner and the ahso lutc temperatures most res ista nces are negati ve . so le effect or its operation is th e transfer of hea t from thu s an elec tro magnetic wave wi ll be amplified a cold to a hot source, instead o f being absorbed, (b) It is imposs ible to con. truct an engine that withdraws heat from a th ermal so urce and converts it ", For corrc;. poncl l:ncc: (E-mail : wi slliak @hglllllail. bg u.ac.il) completel y into work, Wi sniak: TherlllUllynalllic~ of systems at negati ve absolute temperatures Articles

(c) Any irreversible cha nge in an iso lated sys tem has been added) it must be that dSi,rel' > 0, th e same as res ults in an increase in th e of the system. in the domain of positive absolute tem perat ures. It can Ba se d on th e definition or heat given above, clea rl y th en be generally sa id that entropy will always th e fi rst part or statement (a) is also rul fi lied in th e Increase during a process that takes place in an nega ti ve domai n if the hotter source is iso lated sys tem, independen t of th e sign of th e defined as the one having the highest o"solilte va lue absolute temperature. Hence. in th e domain of or th e tem pera ture. Alternatively, as shown in th e negati ve temperatures the corres pondi ng expression previous publication. if two bodies arc brought into for Eq. ( I ) w ill be, thermal contact. the hotter is th e one that releases heat. The fi rst definition will be appropriate for TdS ~ dU +8W ... (6) phenomena occu rring in onl y one domain of th e temperature; the second, fo r phenomena that take A n immediate co nseq uence is that if a heat engine place between the two domain s. is con nected that w ithdraws an amount of heat Q I'rom Statement (c) will be first anal yzed and th en used to a so urce at temperature -T, th e so urce wi ll experiment investigate the other two. a positive change in entropy (Qln and hence there is In ordinary systems th e joint expression I'or the no impediment for transforming completely heat into First and Second laws is, work, a res ult that negates the -Planck statement of th e Second Law. But now the reverse TdS ? dU +8w .. . ( I ) process of converting work co mplete ly into heat becomes i mposs i ble because it is accompanied by a and decreose in entropy. In oth er word s, in th e domain of negati ve temperatures th e Kel vi n-Planck statement dS>O ... (2) reverses itse l f: (a) hea t w ithdrawn from a source can be complete ly converted into work and , (b) it is for an ad iabatic sys tem. The sy mbol 8 is used to impossible to co nstruct an engine that rece ives work indica te th at th e differential is not exact. and co nverts it completely into heat. ow, two equilibrium stat es of a sys tem are Let now two sys tem s A and B be considered, very considered, very close one to the olher, and each at a close to each oth er, at temperatures T,1 and TIJ. We Il egative absolute temperature. Appropriate amount of assume th at system A is hotter th an system B and that thermal energy, 8Q , is now added to ca use the sys tem it tran sfers to B th e amount of hea t 8QJ\ by mea ns of a to evolve from one state to th e oth er, once by a qULl sistati c irrel'ersible process . T he total change in reversible change and th en by an irreversible one. entropy w ill be, Since th e is a state property, thus In the absence of kinetic and potential effects one has, .. . (7)

For th e amount of heat 8QJ\ transferred from A to B it d U = r8Q - 8W J" 'I = r8Q - 8W Jill 1'1 ' (3) mu st be that,

8Qill'('I' - 8Q"'I' = 8Wirrel . - 8Wn 'I' = 8~ (4) .. , (8)

The quantity 8~ must necessa rily be positive since it co rres pond s to the work over th e cycle co mposed of the reversible and irreversible path s in series, at th e ... (9) expense of th e hea t provided. Consequently, oQirr"I' >8Qr"I' and 8W "'I' >8W"'I" Since for a ir Equation 9 can lead to some interesting situations reversi ble process TdS " 'I' = c)Qrel' one must have, c1 ependi ng on the signs of TA and TB. For ex ample: (a) If TA > TIJ >0, then 8QJI < 0, that is, hea t wi ll TdS ~ 8Q .. . (5) fl ow from the hott es t sys tem (highes t temperature) to the coldest one (lowest temperature). This res ult where th e eq ual sign co rresponds to the reversible corresponds to the Clausi us statement of the Second process. Since T < 0 and 8Qirr"I' > 0 (thermal energy Law.

403 Articles Indian .I. Chcrn . Tcc hllDI. . Se ptcmber 2002

(b) l f 0 > T,\ > T/i th en again 8Q,\ < 0, and again hea t Combining Eqs ( 10) and ( 12) yields w ill flow from the sys tem of absolllle higher Kelvi n tem perature to that of absolllle lower temperature dA > - SdT - PdV .. . ( 13) (C lau sius statement of the Second Law). (c) II' T,\ >0> Tu then 8QII > 0 and heat will flow Using similar arguments we can show that from th e system with negative Kelvin temperature to that of pos itive Kelvin temperature. In re lation to th e dG> - SdT+ VdP ... ( 14) l ob serva tions reported earlier , thi s resu lt implies that negati ve Kel vi n temperatures are lI oller than posi ti ve Eqs ( 13) and ( 14) indicate, respec ti ely, th at in th e Kelvin temperatures. neg ative temperature domain for an i.lOlliem/(/l­ By mean s of the criteria of equilibriu m it can be isoc/lOric process th e Helmholtz functi on must shown that - 00 K and + 00 K are identical leve ls of increa se and reac h its maximum value when th e tempe rature, although - 0 K and +0 K are not beca use sys tem comes to equi librium. For an isolll erlllol­ th ey co rres pond to complete ly dU/crelll physical isobaric process th e G ibbs run cti on must increase and states. A sys tem cann ot become hotter than - 0 K achieve its maximum va lue when the sys tem rea ches since it ca nn ot abso rb more energy; a sys tem at +0 K eq uilibrium. Mathematically, can not become co ldcr sin ce energy can no longer be abs trac ted from it. This res ult indicates that th e Third L\A < O,8A = 0,8 " A <0 ( 15) Law of th erm odynami cs does not negate th e poss ibility of negative abso lute tcmperatures; it onl y ... ( 16) negates th e poss ibility of th ermal commun icat ion between both domains through abso lute zero. These conditions for stability can be written in an Stability of a system at Il egative absolute alternative form. L:~ t two equilibrium states (V,\, SA, temperatures V/Io PA , 7:\) and ( V fI, SII, VII , PII, Tfj) be con sidered and Let us app ly Eq. (6) to system in which the onl y the intermediate non-equilibrium state (Vii, Su, 1111, PA, work interactions are ex pan sion and compression. For T,\). Using th e definition of the Gibbs function and an irreversible process. Eq. ( 16) one has,

( 17) 7dS

( 18) It can be seen immediately that th e state of eq uilibrium for an iso lated system for which V and V ( 19 ) are cons tan t co rresponds to max i mum entropy. Mathematically, [f the va lue ze ro is ass igned arbitrarily for the Gibbs energy of an eq uilibrium state. .. . ( I I ) Subtracting Eq. ( 18) from Eq. ( 19) one gets

In th e domain of pos itive absolute temperatures a ... (20) sys tem at constant temperature and volume wi II achieve its stat e of equilibrium when th e va lue of the where t..T = Til - Tn and t..P = p,\ - Pu. Similarly. Helmholtz fun cti on A ach ieves its minimum va lue. If subtracti ng Eq. ( 17) from Eq. ( 19) y ields, the process occurs at constant temperature and pressure, it will do so when th e Gibbs functi on G .. . (2 1) reaches its minimum value. Let us now inves ti gate Finally, adding Eqs (20) and (2 1) yields, what happens to these criteria in the domain of negative abso lute temperatures. By definition A = V - t1 T t1S - LlP t1 II < 0 ... (22) TS so that,

For a di fferential change from state A to state B th e dA = dV- SdT- TdS ... ( 12) inequality Eq. ( 17) wi ll be satisfied for two situations

404 Wisniak: Thermodynamics of systcms at ncga ti ve absolute temperatures Articles

(a) Til 2: T/J >0. In this situation th e efficiency is (a) or (ap] < 0 ... (23) (aT ] =~< o bounded by 0 ~ r] " .\. < I . That is, th e efficiency of th e as I' ('/' uV T engine will always be lower th an one. This is the standard si tuation for engi nes operati ng in the domai n or of pos iti ve temperatures . (b) 0> Til> T/J. Now th e bounds of th e efficiency

arc - 00 < 17 < 0, that is, th e efficiency of a reversible (b) CI' > 0 or (ap) < 0 . .. (24) engine operating in th e domain of nega ti ve absolute av T temperatures will not only be lI egalive, it will be capabl e llf havi ng verv la rge lI egalil'e va lu es. What is If a similar procedure is use d for 8 "C0 or -_a -pJ < 0 ... (25) r] = - .. . (28) ( av T Q

The first condition rep rese nts the condition for For th e efficiency to take a negati ve value it is th ermal stabi lity and the seco nd th e condition for necessa ry for W to be negative. th at is, for th e engine 6 mec hani ca l stability of th e sys tem . to rece ive heat from th e hotter negati ve reservoir and Summarizing, the conditions for equilibrium deli ver it to th e negative temperature sink, it will ha ve stability in a system with negati ve abso lute to be supplied with work (a statement oppos ite to that temperature are exactly th e sam e as those for th e of Clau sius for th e Seco nd Law). For th e engine to sys tem wi th pos iti ve temperature: the spec i fic hea t at produce work the wi ll have to be in the oppos ite direction. COli S ((I II I volullle must be pos iti ve and an isoth ermal co mpress ion lead s to a decrease in vo lume. (c) 7;1 > 0 > TA . Now th e bo un ds of the efficiency

are + 00 > 17 > I . The efficiency of a reversible Carnot Efficiel/ cy of Heat EI/gil/ es engine operating between a hOI reservo ir at negati ve Let us ass ume now a heat engine operating between Kelvin temperature and a co lder rese rvoir at a pos iti ve two rese rvoirs at Til and Til. The engine rece ives an Kelvin temperature is larger than unity. The example amount Q'I of hea t from th e re servoir at T,\, transforms of th e magnetization of a sys tem of spins can be used part of it into work W, and delivers th e difference, QII, to show that it is imposs ible to build a Carnot to th e seco nd reservo ir. For th e general case th e total reversible cycle that wi ll operate between th e two change of entropy of the sys tem must be larger or temperature domains. In th e pos itive domain (or in th e equal to zero: negative one) one can increase th e temperature by ad iabatic magnetization as much as one wants but one ca nn ot make it to cross into th e negati ve domain . ... (26) Similarly, demagneti zati on will cool the system if it is in the positive domain but it wi ll heat it if is In the negative domai n. The equal sign corres pond s to the reversi bl e process and the corres pondi ng ex press ion is th e basi s What happens now if th e engine operates in an irreversible manner? One ca n use th e same arguments for defining th e Kel vin sc al e'. The efficiency of a app lied for the domain of positive absol ute reversihle Carnot engine ( 17 ) is, 1 temperatures7 - ) to conc I u d e tat h : (a) The efficiency of an irreversible engine .. . (27) operating In th e domain of posilive Kelvin temperatures is less than that of a reversible engine operating between the same two reservoirs. In both Again, one ca n distingui sh three cases of interes t, cases this efficiency will be less than unity. each of which sati sfies th e condition that TA is hotter (b) The efficiency of an irreversible eng ll1 e th an TIJ: operating 111 th e domain of lI egative Kelvin

405 Articles Indi:lIl J. C helll. T echno !. , Se ptember :2002 temperatures is IIl1lllerically g reeller than that of a efficiency of a reversible Carnot engin e rsee Eq. reversi ble heat engi ne operati ng between th e same (26)]. The amount of work required to perform thi s rese rvoirs. When this cycle is operated in such a way task will be 01 leosl eqllal to th e increase in work that the machine p er/orllls work while transferring resulting from the hi gher efficiency. [n the same hcat from th e co ld to the hot reservoi r, then its manner, th ere wi ll no benefit in consum in g work to clli ciency will be positive and less than unity. The produce a reservoi r at a negative absolute temperature irreversi bl e engi ne wi II dissi pate part of th e i ncomi ng and use it to operate a more erfici ent eng ine. energy to overcome fri cti on effects. (c) The efficiency of an irreve rsible heat engin e Conclusions operating between a hot reservo ir at negati ve absolute The dotnains or positive and negative absolute temperature and a cold er reservoir at a positi ve temperatures behave simi larl y with respect to th e absulute temperature is less than that of a reversibl e Zeroth and First Law of th erm odynamics and th e engin e operatin g between the same reservo irs. This pri nci pi e of entropy in crease during an i rreversi ble ~fflc i e ll c\' I/WV be g r eoler l /w l/ IIl1il.". process. In th e world of negati ve absolute Summarizin g, how do the above res ults refl ect on temperatures it is poss ibl e to convert heat compl etely the Clausi us and Kelvin- Planck stat ements of th e into work but work cann ot be totall y converted itHO Second Law '? heat. (a) Clausius statement remains unchanged, we Although the poss ibilities of ac hie ving absol ut e either say that heat flows spontaneously from th e negative temperatures arc very limited, neverth eless, hotter to th e co lder temperature, or that it is th e application of th ermodynami cs to the phenomena impossible to construct an engine operating in a offers an excell ent teaching tool to facilitate the closed cycle that wi II prod uce no oth er effect than th e understandin g of a hot versus a colel body, and use of transfer o f' heat from a co ld er to a hotter body. th e principle or entropy increase to determine th e (b) The Kelvin-Planck statement mu st be modified: viability of a process. it is i mpossi ble to constru ct an engi ne that wi II operate in a cycle and produce no other effect that (i) References ex traction of heat from a posilil'e temperature I Wisniak J, J Chl'lIl Edll('. 77 (2000) 5 1g. reservoir and its complet e conversion to work or, (ii) 2 Purc..: 11 E M & Poulld R V . PIn's Rel,.g l ( 1t)5 I ) 27t). the rejection of heat into a lI egalil'e temperature :l Ram s..:y N F. /Jhr.l· ReI'. I ()] ( I t)56) 20. -I I\bragalll A & Proctor W C . Phrs Rei'. IOl) ( 195 X) 1-1-11. reservo ir with th e correspondin g work being done 0 11 the engin e. 5 Ralllsey F. In Modem DI'I'l'/oplIll'IIf.l' ill 7IleI'IIIOdrIlWllic,l. ed ited by B Cal-O r (J Wiley. ew York ). 197-1. 107. The res ults arri ved at for negati ve temperatures, 6 Pri goginc I & Dera y R. Chl'lIlim/ Thl'l'Il/I)(/rllwlli('.I' wh ich seem bi zarre, have no practical significance in ( L ong lllans. L ondo n). 195-1. the ri eld of energy production. Systems at negative 7 Barrow G M . PhY.I'iw/ Chellli.l'IiT. 5th cd. (M cGraw-Hili. absolute temperatures satisfy the First law and foll ow ew York), tt)XH . 2 10. X Levine I N . Ph.1'.I'iC!l/ Chl'lllislr\'. 3rcl cd. (McCraw-I Iii!. NelV the Second Law and its corollaries. In th e domain of York), 19XX . n . pos iti ve absolute temperatures there is no benefit in l) Sonntag R E. Borgnakke C & van W y len Ci J. FIlIlt/(fIlIt'II W/.I' lowerin g th e temperature of the sink to increase the o/,T/I('/'II/{.'(/I'I/wllic.l'. -"t h cd. (J Wi ley. New York). 199X. :20:2 .

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