The thermodynamics of systems at negative absolute temperaturesnopr.niscair.res.in/bitstream/123456789/16819/1/IJCT 9(5... · 2016. 7. 20. · Wi sniak: TherlllUllynalllic~ of systems

Ilidiall Journal ofChclllical Tcchnology Vol. tJ, S.:plclllbcr 2002, pp, 402-406 ,

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Articles

The the rmodynamics of systems at negative absolute temperatures

Jaime Wi sniak '"

I kparlmclli or Chcmic" Engi 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 dom ain 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 heat totally into work. In addition, it is impossihlc to convert work tota Hy int o heat and work llIust hc added to the S)'stCIll to transfer thermal energy frolll a cold source to a hot one. '

Thc possihility of the cx istence of ncgative absolut c tcmpcraturcs has bccn discussed In a prc\ IOU'> pub licati on I, It has bccn shown that for ordinary '>\ ~ t CIll thc ahsol utc tcmpcra turc lllU <., t hc pos i tive bccausc It ha'> an UPPC! hllUI1l! tll cllcrgy, II ()\\L'\ n. \lrdillar) S) slcl11~, tlndcr \cry spL'clfil ~'lrCUllhlallCL'S,

1ll;1) cOlltaill subs~ "tC1l1\ til,lt illlcrdL't \\ L'a\...l)' wilh thc 1l1dill \) sIL'!)) ,lIld kl\'c a li!ll/li'd 1111111t),,'1 III L'll-:!';;) IL'\'L'I, TIlL' !L'ljUif'Clllcllt of \h',d illlL'!actilll, I" IIL'C'l'S~;II~ '>ll Ih:1I tllL'llll:ti L'l\uilih!iul~) I)c d!l:ll'11C,; \ L'I') .,I()\\ I), I( thL IIIL':lI1al Llluilihl'lulll II! tIl'.' ,ul,,-) .1L'1ll

'L': III laPldl). thL"; 11, tl'lnpLTatll c' \\111 k dillL'IL'n! I 1'1 1 III 1I1:lt ll!" thl' main \)\tL'm t\ \\lll knll\\llt'\,lIllpk or thi, ,ltuati()1l i'. '{ 'L't or Ilul'll·.!! '>pin (\1' litillulll 1111, iii a L'l\ ,t,lI of lithiulll riulll'id', II' a rL'\ n\l' magllL'tiL' ficld i appliL'd thc ckctroll'> \\ ill Illlt he ahk to Julio\\' thc dlrcctioll, and ll ll l'. I \\ ill rCIll:lill oricllted :llllipa!a ll ci. III thi" cry'.ta l Spill-Spill, rcla\~ltioll tilllL'\

arl' ahout 10 ' S, \\ hill' CI') stal "pill-Ja1licc rcia.\atidll tllllC arc at least ~()() s, II \\ ill ta\... ' hct \\ CL'1l ri \ L' to thin) mi llu tcs ulltil thc Spill '.uh'.) "kill \\ ill rct ll rII to

thermal cqu i l ibriu m with th e maill '.y'>tcm'l

It Gill be said that the Ilcgati vc tcmpcralLlrc COllCL'p t lllay bc applicd to thc cspeci al ca-;c ,> wherc thc additi on of cllerg) rrom Il 'illuml crcatl'S a pseudo­cCJuilih riulll subsystem or illvcrtcd Icvek Whether it is appropriate to u<;e the term negati vc temperature or pscudo-temperaturc is a question of terminology, ot olll ) that, Ilowadays masers ane! lasers are best approximated as thermodYIlamic systems that ex ist at llegative absolute temperalLlres 5 A lso. at negati e ahso lutc temperatures most res istances are negati ve. thu s an elec tro magnetic wave wi ll be amplified instead o f being absorbed,

", For corrc;.poncl l:ncc: (E-mail : wi slli ak @hglllllail. bgu.ac.i l)

All yIH)\\ . it i <; o f intcres t to di,>cuss hO\\ thc L aws 0"

thcrlll odynamic'. appl) to the'.c si tuation s and ir the propn ti cs or thc,c sy <.,[c ms bchavc in thc salllL' IllallIlL'1 as to thc (lIlC, Ihat h:l \'C pos iti vc temperaturc, HerOIC dlllllg '.0 It i'> agrccd that \\orl. ami hca t ha\ L' tilL '>: 1111 L' dc'i' illl'll III hOlh klllpL'rallllL' domain la) hC:lt is thL' cllcrg) thai rim\· ... hct\\'ccll 'hL' '>y'-.lcm alld it ... ~Ilrruulldi :,', 11C-::nl"L' llf .' tCIllPl,,',IlLll'C Lill'i'L'rl'lll' . ;IIHi (h) \\(11'1 : ... :111 i'llc'i',,,:ti(lIl l'l,ll t,,\...c· ... pldl'c' i)L't\\I'l'1l lill' S)' c'1l1 .Ilhl it ... urrnulldi 1;\ i/O: CllhL'd I)) " Il'I11P':I,lturC (\;1'1' 'I'll,','. I" .!cittil ,In. di :i'll'ti 11

\\ "I Iw lll:"k hL'I\\ 'L'll ;, i1()j ,111.1 , L'oicl h,l(:\ h' IflIILiIl~ at till' dllL'l"I(11l (II hL'd' tr;lll .. kr: heat \\ ii' :11\\;1)', n,,\\ II' lill ,hL' h'llll'r tIl th,,' clll"L'1 hpti) \\ il~'11 Cdll,>idL'l':Il~' 110th I .'~atl\ l' aIlI I Jlfl ... iti\ L' :lh,,(llutl' Il'lllpnaturL's. thl'sc \\ ill progrc\'. 'r(l1l1 "colder" l<'

"hollL'r" III the \l'LfUl'I1CC: (). I ... , I 0 ..... 1 00, , , + i n fi 11 it) . - i n ri nil), .. '. - I 00 ... - \( I ..... - I . -() I,

I t must hc ulltkrqll()d thai the Zerotll and thl' Fil"-l I ,a\\ or tl1L'rmlld) n:lmic'> apply ~quall) tt' hoth tcmpcraturc dClmai n\ hcc,l lI '>e they are indepelldcn t 01 thc sign or thc te mperat urc, The Zcroth L I\\ detcrmine~ that t\\ () sv ... tcm,> are In thermal equ il ibriu m when thcy have thc same temperaturL'. and the First Law represents the encrgy halanee shcct or the change ,

Now {'oliowing three statcments of thc Second L I\\ sha ll be investigated:

(a) Heat fl ows spontaneously from a hot s()urce to a cold one (Clausius) or, it is impo sible to const ruct an engine that operates in a reversib le manner and the sole effect or its operation is the transfer of heat from a co ld to a hot source,

(b) It is imposs ible to con. truct an engine tha t wi thdraws heat from a thermal source and converts it completel y into work ,

Wi sniak: TherlllUllynalllic~ of systems at negati ve absolute temperatures Articles

(c) Any irreversible change in an iso lated system results in an increase in the entropy of the system.

Based on th e definition or heat given above, clearl y the fi rst part or statement (a) is also r ul fi lied in the negati ve temperature domai n if the hotter source is defined as the one having the highest o"solilte va lue or the temperature. Alternatively, as shown in the previous publication. if two bodies arc brough t into thermal contact. the hotter is th e one that releases heat. The fi rst definition will be appropriate for phenomena occu rring in only one domain of the temperature; the second, fo r phenomena that take place between the two domain s.

Statement (c) will be first anal yzed and then used to investigate the other two.

In ord inary systems th e joint expression I'or the First and Second laws is,

TdS ? dU +8w .. . ( I )

and

dS>O ... (2)

for an ad iabatic system. The sy mbol 8 is used to indica te th at the differential is not exact.

ow, two equili brium states of a system are considered, very close one to the olher, and each at a Il egative absolute temperature. Appropriate amount of

thermal energy, 8Q, is now added to cause the system to evo lve from one state to the other, once by a revers ible change and then by an irreversible one. Since the internal energy is a state property, thus In

the absence of kinetic and potential effects one has,

d U = r 8Q - 8W J "'I = r 8Q - 8W J ill 1'1 ' (3)

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

The quantity 8~ must necessaril y be positive since it corresponds to the work over the cycle composed of the reversible and irrevers ible paths in series, at the expense of the heat provided. Consequently,

oQirr"I' >8Qr"I' and 8Wir"'I' >8W"'I" Since for a

reversi ble process TdS " 'I' = c)Qrel' one must have,

TdS ~ 8Q .. . (5)

where the equal sign corresponds to the reversible

process. Since T < 0 and 8Qirr"I' > 0 (thermal energy

has been added) it must be that dSi,rel' > 0, the same as in the domain of positive absolute temperatures. It can th en be generall y sa id that entropy will always Increase during a process that takes place in an iso lated system, independen t of the sign of the absolute temperature. Hence. in the domain of negati ve temperatures the correspondi ng expression for Eq. ( I ) w ill be,

TdS ~ dU +8W ... (6)

A n immediate consequence is that if a heat engine is connected that w ithdraws an amount o f heat Q I'rom a source at temperature -T, the source wi ll experiment a positi ve change in entropy (Qln and hence there is no impediment for transforming completel y heat into work , a result that negates the Kelvin-Planck statement of the Second Law. But now the reverse process of convertin g work completely into heat becomes i mposs i ble because it is accompanied by a decreose in entropy. In other words, in the domain of negati ve temperatures the Kel vi n-Planck statement reverses itse l f: (a) heat w ithdrawn from a source can be completely converted into work and, (b) it is impossible to construct an engine that rece ives work and converts it completely into heat.

Let now two systems A and B be considered, very close to each other, at temperatures T,1 and TIJ. We assume th at system A is hotter than system B and that

it tran sfers to B the amount of heat 8QJ\ by means of a qULlsistati c irrel'ersible process . T he total change in entropy w ill be,

.. . (7)

For the amount of heat 8QJ\ transferred from A to B it must be that,

.. , (8)

... (9)

Equation 9 can lead to some interesting situations c1 ependi ng on the signs of TA and TB. For example:

(a) If TA > TIJ >0, then 8QJI < 0, that is, heat wi ll fl ow from the hottes t system (highes t temperature) to the coldest one (lowest temperature). Thi s result corresponds to the Clausi us statement of the Second Law.

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(b) l f 0 > T,\ > T/i then again 8Q,\ < 0, and again heat w ill flow from the system of absolllle higher Kelvi n temperature to that o f absolllle lower temperature (C lausius statement of the Second Law).

(c) II' T,\ >0> Tu then 8QII > 0 and heat will fl ow from th e system with negative Kelvin temperature to that of pos iti ve Kelvin temperature. In re lation to the observations reported earlier l

, thi s resu lt implies that negati ve Kel vi n temperatures are lI o ller than posi ti ve Kelvin temperatures.

By mean s of the criteria of equili briu m it can be shown that - 00 K and + 00 K are identica l leve ls o f

temperature, although - 0 K and +0 K are not because they co rrespond to comp lete ly dU/crelll physical states. A system cannot become hotter than - 0 K since it cannot absorb more energy; a system at +0 K cannot become co ldcr since energy can no longer be abstrac ted from it. This result indicates that the Third Law o f th ermodynami cs does not negate the poss ibility of negative abso lute tcmperatures; it onl y negates the possibility of thermal commun icat ion between both domains through abso lute zero.

Stability of a system at Ilegative absolute temperatures

Let us app ly Eq. (6) to system in which the only work interactions are ex pansion and compress ion. For an irreversible process.

7dS <dV + Pdll ... ( 10)

It can be seen immediately that the state of eq uilibrium for an iso lated system for which V and V are cons tan t corresponds to max i mum entropy . Mathemat icall y,

.. . ( I I )

In the domain of pos itive absolu te temperatures a system at constant temperature and vo lume wi II achieve its stat e of equilibrium when the va lue of the Helmholtz fun cti on A ach ieves its minimum va lue. If the process occurs at constant temperature and pressure, it will do so when the Gibbs func ti on G reaches its minimum value. Let us now inves ti gate what happens to these criteria in the domain o f negative absolute temperatures. By definition A = V -TS so that,

dA = dV - SdT - TdS . .. ( 12)

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Indian .I. Chcrn . TcchllDI. . Septcmber 2002

Combin ing Eqs ( 10) and ( 12) yie lds

dA > - SdT - PdV . . . ( 13)

Using simil ar arguments we can show that

dG> - SdT + VdP ... ( 14)

Eqs ( 13) and ( 14) indicate, respecti ely, th at in the negative temperature domain for an i.lOlliem/(/l­isoc/lOric process the Helmholtz func ti on must increase and reach its maximum value when the system comes to equi librium. For an isolll erlllol­isobaric process the G ibbs runcti on must increase and achieve its maximum va lue when the system reaches eq uilibrium. Mathematicall y ,

L\A < O,8A = 0,8 " A <0 ( 15)

... ( 16)

These conditions for stability can be written in an alternative form. L:~ t two equil ibrium states (V,\, SA, V/Io PA , 7: \) and (V fI, SII, V II , PII, Tfj) be con sidered and the intermediate non-equilibrium state (Vii, Su, 1111, PA,

T,\). Using the defin ition of the Gibbs function and Eq. ( 16) one has,

( 17)

( 18)

( 19)

[f the va lue zero is ass igned arbitrari ly for the Gibbs energy of an equilibrium state.

Subtracting Eq. ( 18) from Eq. ( 19) one gets

. . . (20)

where t..T = Til - Tn and t..P = p,\ - Pu. Similarly. subtracti ng Eq. ( 17) from Eq. ( 19) y ields,

.. . (2 1)

Finally, adding Eqs (20) and (2 1) yie lds,

t1 T t1S - LlP t1 II < 0 ... (22)

For a di fferential change from state A to state B the inequality Eq. ( 17) wi ll be satisfied for two situations

Wi sniak : Thermodynamics of systcms at ncgati ve absolute temperatures Articles

(a) (aT ] =~<o as I' ('/'

or (ap] < 0 uV T

or

(b) CI' > 0 or (ap) < 0 av T

... (23)

. . . (24)

I f a similar procedure is used for 8 "C<O one obtains that the necessary conditions for stabi lity are,

('1, >0 or -_ - < 0 ( ap J av T

... (25)

The first condition represents the condition for thermal stabi lity and the second the condition for mechanica l stability o f the system6

.

Summari zing, the conditions for equilibrium stability in a system w ith negati ve abso lute temperature are exactly the same as those for the system w i th pos iti ve temperature: the speci fic heat at COli S ((I II I volullle must be pos iti ve and an isothermal compress ion leads to a decrease in vo lume.

Efficiel/ cy of Heat EI/gil/ es Let us assume now a heat engine operating between

two reservoi rs at Til and Til. The engine receives an amount Q'I of heat from the reservoir at T,\, transforms part of it into work W, and deli vers the difference, QII, to the second reservo ir. For the general case the total change of entropy of the system must be larger or equal to zero:

. . . (26)

The equal sign corresponds to the reversi ble process and the correspondi ng expression is the basi s for defining the Kel vin scale' . The efficiency o f a reversihle Carnot engine ( 17) is,

. . . (27)

Again, one can distingui sh three cases of interest, each of which sati sfies the condition that TA is hotter than TIJ:

(a) Til 2: T/J >0. In thi s situation the effi ciency is

bounded by 0 ~ r] " .\. < I . That is, the efficiency of the engine w ill always be lower th an one. Thi s is the standard si tuation for engi nes operati ng in the domai n o f pos iti ve temperatures .

(b) 0> Til> T/J. Now the bounds o f the effici ency arc - 00 < 17 < 0, that is, the effi ciency of a reversible engine operating in the domain o f negati ve absolute temperatures will not only be lI ega live , it will be capable llf havi ng verv la rge lI egalil 'e va lues. What is the meaning of thi s result? The answer is immediate if

the definiti on of r] is remembered,

w r] = -

Q .. . (28)

For the efficiency to take a negati ve value it is necessary for W to be negative. that is, for the engine to receive heat from the hotter negati ve reservoir and deli ver it to the negative temperature sink, it will have to be supplied with work (a statement oppos ite to that o f C lausius for the Second Law). For the engine to produce work the heat transfer wi ll have to be in the oppos ite direction .

(c) 7;1 > 0 > TA . Now the bounds o f the efficiency

are + 00 > 17 > I . The effi ciency of a reversible Carnot engine operating between a hOI reservo ir at negati ve Kelvin temperature and a co lder reservoir at a pos iti ve Kelvin temperature is larger than unity. The example o f the magnetizat ion o f a system of spins can be used to show that it is imposs ible to build a Carnot reversible cyc le that wi ll operate between the two temperature domains. In the pos iti ve domain (or in the negative one) one can increase the temperature by ad iabatic magnetization as much as one wants but one cannot make it to cross into the negati ve domain . Similarl y, demagneti zati on wi ll cool the system if i t is in the positi ve domain but it wi ll heat it i f is In the negative domai n.

What happens now if the engine operates in an irreversible manner? One can use the same arguments app lied for the domain o f positive absol ute

7-1) I d h temperatures to conc u e tat :

(a) The efficiency of an irreversible engine operating In the domain o f posilive Kelvin temperatures is less than that of a reversible engine operating between the same two reservoirs. In both cases thi s effic iency w ill be less than unity .

(b) The efficiency of an irreversible eng ll1e operating 111 the domain of lI egative Kelvin

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temperatures is IIl1lllerica lly g reeller than that of a reversi ble heat engi ne opera ti ng between the same rese rvoirs. When this cyc le is operated in such a way that the machine p er/orllls work while transferring hcat from the co ld to the hot reservoi r, then its clli ciency will be pos itive and less than unity. The irreversi ble engi ne wi II dissi pate part of the i ncomi ng energy to overcome fri cti on effects.

(c) The efficiency of an irreversib le heat engine operating between a hot reservoir at negati ve absolute temperature and a co lder reservoir at a positi ve absulute temperature is less than that of a reversible eng ine operating between the same rese rvo irs. This

~fflc iell c\' I/WV be g reole r l /w l/ IIl1il.".

Summarizing, how do the above res ults re fl ect on the Clausi us and Kelvin- Planck statements of the Second Law '?

(a) Claus ius statement remains unchanged, we either say that heat flows spontaneously from the hotter to the co lder temperature, or that it is impossib le to construct an engine operating in a closed cyc le that wi II prod uce no other effect than the transfer o f' heat from a co lder to a hotter body .

(b) The Kelvin-Planck statement must be modified: it is i mpossi ble to constru ct an eng i ne that wi II operate in a cyc le and produce no other effect that (i) ex traction of heat from a posilil 'e temperature rese rvoir and its complete conversion to work or, (ii) the reject ion of heat into a lI egalil 'e temperature reservoir with the correspondin g work being done 0 11

the engine. The res ults arri ved at for negati ve temperatures,

wh ich seem bi zarre, have no practical significance in the ri e ld of energy production. Systems at negative abso lute temperatures satisfy the First law and foll ow the Second Law and its corollaries. In th e domain of pos iti ve absolute temperatures there is no benefit in lowering the temperature of the sink to increase the

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Indi:lIl J. C helll . T echno !. , September :2002

efficie ncy of a reversible Carnot engine rsee Eq. (26)]. The amount of work required to perform thi s task will be 01 leosl eqllal to the increase in work resulting from the higher efficiency. [n the same manner, there wi ll no benefit in consum ing work to produce a reservoi r at a negative absolute temperature and use it to operate a more e rfici ent eng ine .

Conclusions The dotnains or positive and negative absolute

temperatures behave simi larl y with respect to the Zeroth and First Law of thermodyna mics and the pri nci pi e of entropy increase during an i rreversi ble process . In the world of negati ve absolute temperatures it is poss ibl e to convert heat compl etely into work but work cannot be totall y converted itHO heat.

Although the poss ibilities of ac hie ving absol ute negative temperatures arc very limited, nevertheless, the application of thermodynamics to the phenomena offers an exce ll ent teaching too l to facilitate the understandin g of a hot versus a co lel body, and use of the principle or entropy increase to determine the viability of a process.

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2 Purc..: 11 E M & Poulld R V . PIn's Rel ,.g l ( 1t)5 I ) 27t). :l Ram s..:y N F. /Jhr.l· ReI'. I ()] ( I t)56) 20. -I I\bragalll A & Proctor W C . Phrs Rei'. IOl) ( 195X) 1-1-11.

5 Ralllsey F. In Modem DI'I 'l'/oplIll'IIf.l' ill 7IleI'IIIOdrIlWllic,l.

ed ited by B Ca l-O r (J Wiley. ew York ). 197-1. 107. 6 Pri gog inc I & Deray R. Chl'lIlim/ Thl'l'Il/I)(/rllwlli(' .I'

(L ong lllans. L ondon). 195-1.

7 Barrow G M . PhY.I'iw/ Chellli.l'IiT. 5th cd. (M cGraw-Hili . ew York ), tt) XH . 2 10.

X Le v ine I N . Ph.1'.I'iC!l / Chl'lllislr\'. 3rcl cd. (McCraw-I Iii!. NelV York ), 19XX. n .

l) Sonntag R E. Borgnakke C & van W y len Ci J. FIlIl t/(f IlI t'IIW/.I'

o/,T/I('/'II/{.'(/I'I/wll ic.l' . -"t h cd. (J Wi ley. New York). 199X. :20:2 .