Articles
Ilidiall Journal ofChclllical Tcc hno logy
Vol. tJ, S.:plclllbcr 2002, pp, 402-406
,
, I
The the rmodynamics of systems at negative absolute temperatures
Jaime Wi sniak '"
I kparlmclli o r 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' th e 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'o nvcrt 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.
'
Th c possihilit y of the cx istence of ncgative absolut c
tc mpcraturcs has bccn discussed In a prc\ IOU'>
pub licati on I, It ha s bccn shown that for ordina ry
'>\ ~ t CIll thc ahsol utc tc mpcra 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 1III 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\\' th c d lrcctioll, and llll 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 leas t ~()() s, II \\ ill ta\... ' hct \\ CL'1l ri \ L' to
thin) m illu tcs ulltil thc Spill '.uh'.) "kill \\ ill rct ll rII to
the rm al cqu i libriu m with th e maill '.y'>tcm'l
It Gill be said that the Ilcgati vc tc mpc ralLlrc COllCL'p t
lllay bc applicd to thc cspeci al ca-;c,> wherc thc
additi on of cllerg) rrom Il 'illum l crcatl' S a pse ud ocCJuilih riulll subsystem or illvcrtcd Icvek Whether it
is approp riate to u<;e the term nega ti vc temperature or
pscudo-temperaturc is a question of terminology, ot
olll ) that, Ilowadays masers ane! lasers are best
approximated as thermodYIlamic syste m s that ex ist at
5
llegative absolute temperalLlres
A lso. at negati e
ahso lutc temperatures most res ista nces are negati ve .
thu s an elec tro mag netic wave wi ll be amplified
instead o f being absorbed,
", For corrc;.poncll:ncc: ( E-ma il : wi slli ak @hglllllail. bg u.ac.i l)
All yIH)\\ . it i <; o f int cres t to di,>cuss hO\\ th c L aw s 0"
th crlll odyna mic'. ap pl) to the'.c si tuation s an d ir the
pro pn ti cs or thc,c sy <.,[c ms bcha vc in thc salllL'
IllallIlL'1 as to thc (lIlC, Ihat h:l \'C pos iti vc tem peraturc,
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 ..... 100, , ,
+ i n fi 11 it) . - i n ri nil), .. '. - I 00 ... - \( I..... - I . -() I,
It must hc ulltkrqll()d thai the Zerotll and thl' Fil"-l
I ,a\\ or tl1L'rmlld) n:lmic'> apply ~quall) tt' hoth
tc mpc raturc dClmai n\ hcc,l lI '>e th ey are ind epelldcn t 01
th c sig n
thc te mperat urc, Th e Z croth L I\\
d e tcrmin e~
that t\\ () sv ... tc m,> are In thermal
equ il ibriu m w hen th cy ha ve th c sa me temperaturL'.
and the First Law represents the encrgy halan ee shcct
or the change ,
or
Now {'oliowing three statcments of thc Seco nd L I\\
sha ll be investigated:
(a) Heat fl ows spo ntan eously from a hot s()urce to a
cold one (Cla usius) or, it is impo sible to co nst ruct an
engin e th at ope rat es in a reve rsib le manner and the
so le effect or its operation is th e transfer of heat from
a co ld to a hot source,
(b) It i s imposs i ble to con . truct an engine tha t
wi thdraws heat from a th erm al so urce and converts it
co mpl etel y into work ,
Wi sniak:
TherlllUllynalllic~
(c) Any irreversible cha nge in an iso lated sys tem
res ults in an i ncrease in th e entropy of the system.
Ba sed on th e definition or heat gi ven above, clearl y
th e fi rst part or statement (a) is also rul fi lied in th e
nega ti ve temperature domai n if the hotter source is
defined as the one having the highest o"solilte va lue
or th e tem pera ture. Alternatively, as shown in th e
previous publication. if two bodies arc brough t into
thermal contact. the hott er is th e one that releases
heat. The fi rst definition will be appropriate for
phenomena occu rrin g in onl y one domain of th e
temperature; the second, fo r phenomena that tak e
place between the two doma in s.
Statement (c) will be first anal yzed and th en 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
... (2)
dS>O
for an ad iabatic sys tem. The sy mbol 8 is used to
indica te th at th e differential is not exact.
ow, two equili brium stat es of a sys tem are
co nsidered, very close one to the olher, and each at a
Il egative absolute temperature. Appro priate amount of
thermal energy, 8Q , is now added to ca use the sys tem
to evo l ve from one state to th e oth er, once by a
revers ible change and th en by an irreversible one.
Since th e internal energy is a state property, thu s In
the absence of kinetic and pot ential effects one has,
dU
=r8Q -
8W J"'I
=r8Q -
8W Jill
(3)
1'1 '
8Qill'('I' - 8Q"'I' = 8Wirrel. - 8Wn 'I' = 8~
8Wir"'I' >8W"'I"
~
8Q
TdS ~ dU
+8W
... (6)
A n immediate co nseq uence is that if a heat engine
is con nected that w ithdraws an amount o f heat Q I'rom
a so urce at temperature -T, th e source wi ll experiment
a positi ve change in entropy (Qln and hence there is
no impediment for tra nsform i ng co mpletel y heat into
work , a result that negates the Kelvin-Pl anck
statement of th e Second Law . But now the reverse
process of conve rtin g work co mpl etely into heat
beco mes i mposs i ble because it is accompanied by a
decreose in entropy. In oth er word s, in th e domain of
negati ve temperatures th e Kel vi n-Planck statement
reverses itsel f: (a) hea t w ithdrawn from a source can
be compl etely co nverted into wo rk and , (b) it is
impossible to co nstru ct an engine that rece ives work
and co nverts it co mpl etely into heat.
L et now two sys tem s A and B be co nsidered, very
close to each oth er, at temperatures T,1 and TIJ. We
assume th at system A is hotter th an system B and that
it tran sfers to B th e amount of heat 8QJ\ by means of a
qULlsistati c irrel'e rsible process . T he total change in
entropy w ill be,
.. . (7)
For th e amount of heat 8QJ\ tran sferred from A to B it
mu st be that,
.. , (8)
Since
for
a
reversi bl e process TdS " 'I' = c)Qrel' one mu st have,
TdS
has been added) it must be that dSi,rel' > 0, th e same as
in the domain of positive absolute tem peratures. It can
th en be generall y sa id that entropy will always
Increase during a process that takes place in an
iso lated sys tem, independen t of th e sign of th e
absolute temperature. Hence. in th e domain of
negati ve temperatures the co rrespondi ng expression
for Eq. ( I ) w ill be,
(4)
The quantity 8~ mu st necessaril y be positive since
it co rresponds to the wo rk over th e cycle co mposed of
the reversible and irrevers ibl e path s in seri es, at th e
expense of th e heat provided. Consequently,
oQirr"I' >8Qr"I' and
Articles
of systems at negati ve absolute tempera tures
.. . (5)
where th e eq ual sign co rresponds to the reversible
process. Since T < 0 and 8Qirr"I' > 0 (thermal energy
... (9)
Equation 9 can lead to some interestin g situati ons
c1 ependi ng on the signs of TA and T B . For ex ampl e:
(a) If TA > TIJ >0, then 8QJI < 0, that is, heat wi ll
fl ow from the hott es t sys tem (highes t temperature) to
the coldest one (lowest temperature). Thi s result
corresponds to the Clausi us statement of the Second
Law.
403
Articles
India n .I. Chcrn . T cchllDI. . Septcmber 2002
(b) l f 0 > T,\ > T/i th en again 8Q,\ < 0, and again hea t
w ill flow from the sys tem of absolllle higher K el vi n
tem perature to that o f absolllle lowe r temperature
(C lau siu s statement of the Second Law).
(c) II' T,\ >0> Tu then 8QII > 0 and heat w ill fl ow
from th e system with negative Kelvin temperature to
that of pos iti ve K el vin temperature. In re lation to th e
l
ob servation s reported earlier , thi s resu lt implies that
negati ve Kel vi n temperatures are lI o ll er than posi ti ve
Kel vin temperatures.
By mean s of the criteria of equili briu m it can be
show n that - 00 K and + 00 K are identica l leve ls o f
tempe rature, although - 0 K and +0 K are not because
th ey co rrespond to comp lete ly dU/crelll phy sical
states. A sys tem cannot become hotter than - 0 K
since it ca nnot absorb more energy; a sys tem at +0 K
can not become co ldcr sin ce energy can no longer be
abs trac ted from it. This res ult indi cates that th e Third
La w o f th erm ody nami cs does not negate th e
poss ibility of negative abso lute tcmperatures; it onl y
negates th e possibility of th ermal co mmun icat ion
between both domains through abso lute zero.
Stability of
temperatures
a
system
at
Ilegative
absolute
Let us app ly Eq. (6) to system in whic h the onl y
wo rk interactions are ex pan sion and compress ion. For
an irreversibl e process.
7dS <dV + Pdll
.. . ( I I )
In th e doma in of pos itive absolu te tempe ratures a
sys tem at constant tempe rature and vo lume wi II
achieve its stat e of equilibrium whe n th e va lue of the
Helmholt z fun cti on A ach ieves its minimum va lue. If
the process occurs at con stant temperature and
pressure, it will do so when th e Gibbs func ti on G
reaches its minimum value. Let us now in ves ti gate
w hat happens to these criteria in the domain o f
negative absolute temperatures. By definitio n A = V TS so that,
404
dA > - SdT - PdV
. .. ( 12)
. . . ( 13)
Usi ng simil ar arguments we can show that
dG> - SdT+ VdP
... ( 14)
Eqs ( 13) and ( 14) indicate, respec ti ely, th at in th e
neg ative temperature domain for an i.lOlliem/(/lisoc/lO ric process th e Helmho lt z func ti on must
increase and reach its maximum valu e when th e
sys tem comes to equi librium . For an isolll erlllolisobaric process th e G ibbs run cti on mus t increase and
achi eve its max imum va lue when the sys tem reaches
eq uilibrium . Mathematicall y ,
L\A < O,8A = 0,8 " A <0
( 15)
... ( 16)
These co nditi ons for stability can be written in an
alternative form. L:~ t two equil ibriu m states (V,\, SA,
V/Io P A , 7: \) and ( V fI, SII, V II, P II , Tfj) be con sidered and
the intermediate non-equilibrium state (Vii, Su, 1111 , PA ,
T,\). Us ing th e defin ition of the Gibbs function and
Eq. ( 16) one has,
( 17)
... ( 10)
It can be seen immediately that th e state of
eq uilibrium for an iso lated system for which V and V
are cons tan t co rresponds to max i mum entropy .
Mathemat icall y,
dA = dV - SdT - TdS
Combin ing Eqs ( 10) and ( 12) yie lds
( 18)
( 19)
[f the va lue ze ro is ass igned arbitrari ly f or the Gibbs
energy of an eq uilibrium state.
Sub tractin g Eq. ( 18) from Eq. ( 19) one gets
. . . (20)
w here t..T = Til - Tn and t..P = p,\ - Pu. Similarly.
subtracti ng Eq. ( 17) from Eq. ( 19) y ields,
.. . (2 1)
Finall y, addin g Eqs (20) and (2 1) yie lds,
t1 Tt1S - LlPt1 II < 0
... (22)
For a di fferential change from state A to state B th e
i nequality Eq. ( 17) wi ll be satisfied for two situati ons
Articles
Wi sniak : Thermody nami cs o f systcms at ncga ti ve absolute temperatures
(a)
(aT ] =~< o
as
I'
or
(ap] < 0
('/'
uV
... (23)
T
or
(b)
CI'
> 0 or
(ap)
<0
av T
. . . (24)
I f a simil ar procedure is used for 8 "C<O one obtains
that th e necessary conditi ons for stabi lity are,
apJ
(av
('1 , >0 or -_-
<0
... (2 5)
(a) Til 2: T/J >0. In thi s situation th e effi ciency is
bounded by 0 ~ r] " .\. < I . T hat is, th e efficiency of th e
engine w ill always be lower th an one. Thi s is the
stand ard si tuation for engi nes operati ng in the domai n
o f pos iti ve temperatures .
(b) 0> Til> T/J. Now th e bounds o f th e effici ency
arc - 00 < 17 < 0, that is, th e effi ciency of a reversible
engine operating in th e doma in o f nega ti ve absolute
temperatures will not onl y be lI ega live , it will be
capabl e llf havi ng ve rv la rge lI ega lil 'e va lues. Wh at is
th e meanin g of thi s res ult? The answer is i mmedi ate if
th e definiti on of r] is remembered,
r]
w
=-
.. . (28)
Q
T
Th e first co nditi on represe nts the co nditi on for
th erm al stabi lity and the second th e co nditi on for
6
mechani ca l stability o f th e sys tem .
Summari zing, the co nditi ons for equilibrium
stability in a system w ith negati ve abso lute
temperature are exac tly th e sam e as those for th e
sys tem w i th pos iti ve temperature: the speci fic heat at
CO li 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.
Efficiel/ cy of Heat EI/gil/ es
Let us assume now a heat engine operating betwee n
two reservo i rs at Til and Til. The engine recei ves an
amount Q'I of heat from th e reservo ir at T,\, transforms
part of it into work W, and deli vers th e difference, QII,
to th e second reservo ir. For th e general case th e total
change of entropy of the sys tem must be larger or
equal to zero:
. . . (26)
Th e equal sign corres ponds to the reversi bl e
process and the co rrespondi ng ex pression is th e basi s
for definin g th e Kel vin sc al e ' . The efficiency o f a
reversihl e Carnot engine ( 17) is,
. . . (27)
Aga in , one ca n di stin gui sh three cases of interest,
each of which sati sfies th e co nditi on that TA is hotter
th an TIJ:
For th e efficiency to take a negati ve valu e it is
necessary for W to be negati ve. th at is, for th e engine
to receive heat from th e hotter negati ve reservo ir and
deli ver it to th e negativ e temperature sink , it will ha ve
to be supplied w ith work (a statement oppos ite to that
o f C lau sius for th e Seco nd Law). For th e engine to
produce wo rk the heat transfer wi ll have to be in the
oppos ite direction .
(c) 7;1 > 0 > TA . Now th e bounds o f the efficiency
are + 00 > 17 > I . Th e effi ciency of a reversible Carno t
engine operating between a hOI reservo ir at negati ve
K el vin temperature and a co lder rese rvo ir at a pos iti ve
K el vin temperature is larger than unity. The examp le
o f th e magnetizat ion o f a sys tem o f spin s can be used
to show that it is imposs ible to build a Carnot
reversible cyc le that wi ll operate between th e two
temperature domains . In th e pos iti ve domain (or in th e
negative one) one can increase th e temperature by
ad iabati c magnetizati on as much as one wants but one
ca nnot make it to cross into th e negati ve domain .
Similarl y, demagneti zati on wi ll coo l the system if it is
in the positi ve domain but it wi ll heat it i f is In the
negative domai n.
What happens now if th e engine operates in an
irreversibl e manner? On e ca n use th e same arguments
app li ed for the domain o f positive absol ute
1
h :
temperatures7- ) to conc Iude tat
(a) The effici ency of an irreversibl e engine
operatin g In th e domain o f p osilive K el vin
temperatures is less than that of a reversible engine
operatin g between the same two reservo irs. In both
cases thi s effic iency w ill be less than unity .
(b) The efficiency of an irreversible eng ll1e
operatin g 111 th e domain of lI egative K el vin
405
Articles
tem peratures is IIl1lllerica lly g reeller than that of a
reversible heat engi ne opera ti ng between th e same
rese rvo irs. When this cyc le is operated in such a way
that the machine p er/orllls work whil e transferring
hcat from th e co ld to the hot reservoi r, then its
clli ciency will be pos itive and less than unity. The
irreve rsible engi ne wi II dissipate part o f th e incomi ng
energy to overcome fri cti on effects.
(c) The efficiency of an irre ve rsib le heat engine
ope ratin g between a hot reservo ir at negati ve absolute
temperature and a co lder reservoir at a positi ve
absulute tempe rature is less than that o f a reve rsible
eng ine operatin g between the same rese rvo irs. This
~fflc iell c\' I/WV be g r eole r l /w l/ IIl1il.".
Summari zing, how do the above res ults refl ect on
the Clausi us and Kelvin- Planck stat ements of th e
Second Law '?
(a) Claus ius stateme nt remain s unchanged, we
eit her say that heat flows spontaneousl y from th e
hotter to th e co lder temperature, or that it is
impossib le to cons truct an eng ine operating in a
closed cyc le that wi II prod uce no oth er effec t than th e
transfer o f' heat from a co lder to a hotter body .
(b) The Kelvin-Planck statement mu st be modified:
it is i mpossi ble to cons tru ct an eng ine that wi II
operate in a cyc le and produce no other effec t that (i)
ex traction of heat from a posilil 'e te mperature
rese rvoir and its comp lete conversion to work or, (ii)
the reject ion of heat into a lI egalil 'e temperature
rese rvo ir with th e correspo ndin g wo rk being done 0 11
the engine.
The res ults arri ved at for negati ve temperatures,
wh ich seem bi zarre, have no practical significan ce in
the ri eld of ene rgy production. Systems at nega tive
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 tempe ratures there is no benefit in
lowerin g th e temperature of the sink to increase the
406
Indi:lIl
J. C helll . T echno !. , Sept ember :2002
efficie ncy of a reversible Carnot e ngine rsee Eq.
(26)]. The amount of work required to perform thi s
task will be 01 l eosl eqllal to th e increase in work
res ulting from the hi gher efficienc y. [n the same
manner, th ere wi ll no benefit in co nsum in g work to
produce a reservoi r at a negative absolute temperature
and use it to operate a more e rfici ent eng ine .
Conclusion s
The dotnains or positive and negative absolute
te mperatures behave simi larl y with respect to th e
Zeroth and First Law o f th erm odyna mics and th e
pri nci pi e o f entropy increase durin g an irreversible
process . In th e world of negati ve absolute
temperatures it is poss ibl e to convert heat co mpl etely
into work but wo rk ca nn ot be totall y converted itHO
heat.
Although the poss ibilities of ac hie ving absol ut e
nega tive te mperatures arc very limited, neve rth eless,
th e application o f th ermodynami cs to the phenomena
offers an exce ll ent tea ching too l to facilitate the
unde rstandin g of a hot versus a colel body, and use of
th e principl e or entropy increase to determine th e
viability of a process.
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