Review of Thermodynamics

Transcription

1 Revew of hermodynamcs from Statstcal Physcs usng Mathematca James J. Kelly, We revew the laws of thermodynamcs and some of the technques for dervaton of thermodynamc relatonshs. Introducton Equlbrum thermodynamcs s the branch of hyscs whch studes the equlbrum roertes of bulk matter usng macroscoc varables. he strength of the dsclne s ts ablty to derve general relatonshs based uon a few fundamental ostulates and a relatvely small amount of emrcal nformaton wthout the need to nvestgate mcroscoc structure on the atomc scale. However, ths dsregard of mcroscoc structure s also the fundamental lmtaton of the method. Whereas t s not ossble to redct the equaton of state wthout knowng the mcroscoc structure of a system, t s nevertheless ossble to redct many aarently unrelated macroscoc quanttes and the relatonshs between them gven the fundamental relaton between ts state varables. We are so confdent n the rncles of thermodynamcs that the subject s often resented as a system of axoms and relatonshs derved therefrom are attrbuted mathematcal certanty wthout need of exermental verfcaton. Statstcal mechancs s the branch of hyscs whch ales statstcal methods to redct the thermodynamc roertes of equlbrum states of a system from the mcroscoc structure of ts consttuents and the laws of mechancs or quantum mechancs governng ther behavor. he laws of equlbrum thermodynamcs can be derved usng qute general methods of statstcal mechancs. However, understandng the roertes of real hyscal systems usually requres alcaton of arorate aroxmaton methods. he methods of statstcal hyscs can be aled to systems as small as an atom n a radaton feld or as large as a neutron star, from mcrokelvn temeratures to the bg bang, from condensed matter to nuclear matter. Rarely has a subject offered so much understandng for the rce of so few assumtons. In ths chater we rovde a bref revew of equlbrum thermodynamcs wth artcular emhass uon the technques for manulatng state functons needed to exlot statstcal mechancs fully. Another mortant objectve s to establsh termnology and notaton. We assume that the reader has already comleted an undergraduate ntroducton to thermodynamcs, so omt roofs of many roostons that are often based uon analyses of dealed heat engnes.

2 2 Revewhermodynamcs.nb Macroscoc descrton of thermodynamc systems A thermodynamc system s any body of matter or radaton large enough to be descrbed by macroscoc arameters wthout reference to ndvdual (atomc or subatomc) consttuents. A comlete secfcaton of the system requres a descrton not only of ts contents but also of ts boundary and the nteractons wth ts envronment ermtted by the roertes of the boundary. Boundares need not be menetrable and may ermt assage of matter or energy n ether drecton or to any degree. An solated system exchanges nether energy nor mass wth ts envronment. A closed system can exchange energy wth ts envronment but not matter, whle oen systems also exchange matter. Flexble or movable walls ermt transfer of energy n the form of mechancal work, whle rgd walls do not. Dathermal walls ermt the transfer of heat wthout work, whle adathermal walls do not transmt heat. wo systems searated by dathermal walls are sad to be n thermal contact and as such can exchange energy n the form of ether heat or radaton. Systems for whch the rmary mode of work s mechancal comresson or exanson are consdered smle comressble systems. Permeable walls ermt the transfer of matter, erhas selectvely by chemcal seces, whle mermeable walls do not ermt matter to cross the boundary. wo systems searated by a ermeable wall are sad to be n dffusve contact. Also note that ermeable walls usually ermt energy transfer, but the tradtonal dstncton between work and heat can become blurred under these crcumstances. hermodynamc arameters are macroscoc varables whch descrbe the macrostate of the system. he macrostates of systems n thermodynamc equlbrum can be descrbed n terms of a relatvely small number of state varables. For examle, the macrostate of a smle comressble system can be secfed comletely by ts mass, ressure, and volume. Quanttes whch are ndeendent of the mass of the system are classfed as ntensve, whereas quanttes whch are roortonal to mass are classfed as extensve. For examle, temerature ( ) s ntensve whle nternal energy (U ) s extensve. Quanttes whch are exressed er unt mass are descrbed as secfc, whereas smlar quanttes exressed er mole are descrbed as molar. For examle, the secfc (molar) heat caactes measure the amount of heat requred to rase the temerature of a gram (mole) of materal by one degree under secfed condtons, such as constant volume or constant ressure. A system s n thermodynamc equlbrum when ts state varables are constant n the macroscoc sense. he condton of thermodynamc equlbrum does not requre that all thermodynamc arameters be rgorously ndeendent of tme n a mathematcal sense. Any thermodynamc system s comosed of a vast number of mcroscoc consttuents n constant moton. he thermodynamc arameters are macroscoc averages over mcroscoc moton and thus exhbt eretual fluctuatons. However, the relatve magntudes of these fluctuatons s neglgbly small for macroscoc systems (excet ossbly near hase transtons). he ntensve thermodynamc arameters of a homogeneous system are the same everywhere wthn the system, whereas an nhomogeneous system exhbts satal varatons n one or more of ts arameters. Under some crcumstances an nhomogeneous system may consst of several dstnct hases of the same substance searated by hase boundares such that each hase s homogeneous wthn ts regon. For examle, at the trle ont of water one fnds n a gravtatonal feld the lqud, sold, and vaor hases coexstng n equlbrum wth the hases searated by densty dfferences. Each hase can then be treated as an oen system n dffusve and thermal contact wth ts neghbors. Neglectng small varatons due to gravty, each subsystem s effectvely homogeneous. Alternatvely, consder a system n whch the temerature or densty has a slow satal varaton but s constant n tme. Although such systems are not n true thermodynamc equlbrum, one can often aly equlbrum thermodynamcs to analye the local roertes by subdvdng the system nto small arcels whch are nearly unform. Even though the boundares for each such subsystem are magnary, drawn somewhat arbtrarly for analytcal uroses and not reresentng any sgnfcant hyscal boundary, thermodynamc reasonng can

3 Revewhermodynamcs.nb 3 stll be aled to these oen systems. he defnton of a thermodynamc system s qute flexble and can be adjusted to meet the requrements of the roblem at hand. he central roblem of thermodynamcs s to ascertan the equlbrum condton reached when the external constrants uon a system are changed. he arorate external varables are determned by the nature of the system and ts boundary. o be secfc, suose that the system s contaned wthn rgd, mermeable, adathermal walls. hese boundary condtons secfy the volume V, artcle number N, and nternal energy U. Now suose that the volume of the system s changed by movng a ston that mght comrse one of the walls. he artcle number remans fxed, but the change n nternal energy deends uon how the volume changes and must be measured. he roblem s then to determne the temerature and ressure for the fnal equlbrum state of the system. he deendence of nternal varables uon the external (varable) constrants s reresented by one or more equatons of state. An equaton of state s a functonal relatonsh between the arameters of a system n equlbrum. Suose that the state of some artcular system s comletely descrbed by the arameters, V, and. he equaton of state then takes the form V, D = 0 and reduces the number of ndeendent varables by one. An equlbrum state may be reresented as a ont on the surface descrbed by the equaton of state, called the equlbrum surface. A ont not on ths surface reresents a nonequlbrum state of the system. A state dagram s a rojecton of some curve that les on the equlbrum surface. For examle, the ndcator dagram reresents the relatonsh between ressure and volume for equlbrum states of a smle comressble system. he fgure below llustrates the equlbrum surface U = ÅÅÅÅ 3 V for an deal gas wth fxed artcle number. he 2 ressure and energy can be controlled usng a ston and a heater, whle the volume of the gas resonds to changes n these varables n a manner determned by ts nternal dynamcs. For examle, f we heat the system whle mantanng constant ressure, the volume wll exand. Any equlbrum state s reresented by a ont on the equlbrum surface, whereas nonequlbrum states generally requre more varables (such as the satal deendence of densty) and cannot be reresented smly by a ont on ths surface. A sequence of equlbrum states obtaned by nfntesmal changes of the macroscoc varables descrbes a curve on the equlbrum surface. hus, much of the formal develoment of thermodynamcs entals studyng the relatonshs between the corresondng curves on surfaces constructed by changes of varables. A Quasstatc ransformaton U V

4 4 Revewhermodynamcs.nb Reversble and Irreversble ransformatons A thermodynamc transformaton s effected by changes n the constrants or external condtons whch result n a change of macrostate. hese transformatons may be classfed as reversble or rreversble accordng to the effect of reversng the changes made n these external condtons. Any transformaton whch cannot be undone by smly reversng the change n external condtons s classfed as rreversble. If the system returns to ts ntal state, the transformaton s consdered reversble. Although rreversble transformatons are the most common and general varety, reversble transformatons lay a central role n the develoment of thermodynamc theory. A necessary but not suffcent condton for reversblty s that the transformaton be quasstatc. A quasstatc (or adabatc) transformaton s one whch occurs so slowly that the system s always arbtrarly close to equlbrum. Hence, quasstatc transformatons are reresented by curves uon the equlbrum surface. More general transformatons deart from the equlbrum surface and may even deart from the state sace because nonequlbrum states generally requre more varables than equlbrum states. For examle, consder two equal volumes searated by a rgd menetrable wall. Intally one artton contans a gas of red molecules and the other a gas of blue molecules. he two subsystems are n thermal equlbrum wth each other. If the artton s removed, the two seces wll mx throughout the combned volume as a new equlbrum condton s reached. However, the orgnal state s not restored when the artton s relaced. Hence, ths transformaton s rreversble. An Irreversble ransformaton hermodynamc transformatons often are rreversble because the constrants are changed too radly. Suose that an solated volume of gas s confned to an nsulated vessel equed wth a movable ston. If the ston s suddenly moved outwards more radly than the gas can exand, the gas does no work on the ston. Because the nsulatng walls allow no heat to enter the vessel, the nternal energy of the system s unchanged by such a rad exanson of the volume, but the ressure and temerature wll change. If the ston s now returned to ts orgnal oston, t must erform work uon the gas. herefore, the nternal energy of the fnal state s dfferent from that of the ntal state even though the constrants have been returned to ther ntal condtons. Such a transformaton s agan rreversble. On the other hand, f the ston were to be moved slowly enough to allow the gas ressure to equale throughout the volume durng the entre rocess, the gas wll return to ts ntal state when returned to ts ntal volume. For ths system, reversblty can be acheved by varyng the volume suffcently slowly to ensure quasstatc condtons. he necessty of the requrement that a reversble transformaton be quasstatc follows from the requrement that the state of the system be unquely descrbed by the thermodynamc arameters that descrbe ts equlbrum state. Nonequlbrum states are not fully descrbed by ths restrcted set of varables. We must be able to reresent the hstory of the system by a trajectory uon ts equlbrum surface. However, the nsuffcency of quasstats can be llustrated by a famlar examle: the magnetaton M of a ferromagnetc materal subject to a magnetng feld H exhbts the henomenon of hysteress.

5 Revewhermodynamcs.nb 5 It s generally observed that a system not n equlbrum wll eventually reach equlbrum f the external condtons reman constant long enough. he tme requred to reach equlbrum s call the relaxaton tme. Relaxaton tmes are extremely varable and can be qute dffcult to estmate. For some systems, t mght be as short as 10-6 s, whle for other systems t mght be a century or longer. In the examles above, the relaxaton tme for the gas and ston system s robably mllseconds, whle the relaxaton tme for the ferromagnet mght be many years. In ths sense, hysteress occurs when the relaxaton tme s much longer than our atence, such that "slow" fals to concde wth "quasstatc". Laws of hermodynamcs à 0 th Law of hermodynamcs Consder two solated systems, A and B, whch have been allowed to reach equlbrum searately. Now brng these systems nto thermal contact wth each other. Intally, they need not be n equlbrum wth each other. Eventually, the combned system, A+B, wll reach a new equlbrum state. Some changes n both A and B wll generally have occurred, usually ncludng a transfer of energy. In the fnal equlbrum state of the combned system we say that the subsystems are n equlbrum wth each other. If a thrd system, C, can now be brought nto thermal contact wth A wthout any changes occurrng n ether A or C, then C s n equlbrum wth not only wth A but wth B also. hs ostulate may be exressed as: 0 th Law of hermodynamcs: If two systems are searately n equlbrum wth a thrd, then they must also be n equlbrum wth each other. he eroth law may be arahrased to say the equlbrum relatonsh s transtve. he transtvty of equlbrum condtons does not deend uon the nature of the systems nvolved and can obvously be extended to an arbtrary number of systems. he converse of the 0 th law Converse of 0 th Law of hermodynamcs: If three or more systems are n thermal contact wth each other and all are n equlbrum together, then any ar s searately n equlbrum. s easly demonstrated. If the state of the combned system s n equlbrum, ts roertes are constant; but f a ar of subsystems s not n equlbrum wth each other and are allowed to nteract, ther states wll change. hs result contradcts the ntal hyothess, thereby rovng the equvalence between the 0 th law and ts converse. he concet of temerature s based uon the 0 th law of thermodynamcs. For smlcty, consder three systems (A,B,C) each descrbed by the varables 8, V, œ 8A, B, C<<. he condton of equlbrum between systems A and C may be exressed as an equaton of the form F A, V A, C, V C D = 0

6 6 Revewhermodynamcs.nb whch may be solved for C as C = f A, V A, V C D = 0 Smlarly, the equlbrum between systems B and C yelds C = f B, V B, V C D = 0 so that f A, V A, V C D = f B, V B, V C D Fnally, the equlbrum between A and B can be exressed as F A, V A, B, V B D = 0 If these last two equatons are to exress the same equlbrum condton, we must be able to elmnate V C from the former equaton to obtan f A, V A D =f B, V B D where f A or f B deend only uon the state varables of systems A or B ndeendently. he democracy amongst the three systems can then be used to extend the argument to f C, such that f A, V A D =f B, V B D =f C, V C D for three systems n mutual equlbrum. herefore, there must exst a state functon that has the same value for all systems n thermal equlbrum wth each other. For each system, ths state functon deends only uon the thermodynamc arameters of that system and s ndeendent of the rocess by whch equlbrum was acheved and s also ndeendent of the envronment. hs functon wll, of course, be dfferent for dssmlar systems. An emrcal temerature scale can now be establshed by selectng a convenent thermometrc roerty of a standard system, S, and correlatng ts equlbrum states wth an emrcal temerature q n the form f@8x S <D =q, where 8x S < reresents a comlete set of thermodynamc arameters (other than temerature) for the standard system. he equaton of state of any test system, A, can now be determned. Mantanng the standard system n a constant state, we vary the arameters of the test system n such a way as to mantan equlbrum between S and A. hs set of varatons then determnes f@8x A <D =q as an equlbrum surface of A. he locus of all onts H A, V A L whch reman n equlbrum wth S at an emrcal temerature q descrbes a curve n the V dagram called an sotherm. Several such sotherms for an deal gas are sketched and labelled by ther emrcal temeratures n the ndcator dagram below. Ideal Gas Isotherms ressure H a r b. u nt sl q 5 q 4 q 3 q 2 q 1 volume Harb. untsl

7 Revewhermodynamcs.nb 7 he 0 th law of thermodynamcs requres that the form of the sotherms be ndeendent of the nature of the standard system S. If we had chosen a dfferent standard system, S, at the same emrcal temerature as determned by equlbrum between S and S, t would also have to be n equlbrum wth A for all states along the sotherm. herefore, the sotherms descrbe a roerty of the system of nterest and are ndeendent of subsdary systems. Several examles of sutable thermometrc roertes come readly to mnd. Frst, consder the heght of a mercury column n an evacuated tube, a common household thermometer. he mercury exands when heated and contracts when cooled. We may defne an emrcal temerature scale as a lnear functon of the heght of the column. Second, Boyle's law states that the sotherms of a dlute gas may be descrbed by V ê n = constant, where n s the number of moles. herefore, a sutable temerature scale can be defned by any functon assocated wth these sotherms by V = nf@qd. he smlest and most convenent choce, V = nrq, s known as the deal gas law. Wth a sutable choce of the gas constant, R, ths emrcal temerature scale s dentcal, n the lmt Ø 0, wth the absolute temerature scale to be dscussed shortly. It s mortant to reale that an emrcal temerature scale bears no necessary relatonsh to any scale of hotness. A rgorous macroscoc concet of heat deends uon the frst law of thermodynamcs whle the relatonsh between the drecton of heat flow and temerature deends uon the second law. à 1 st Law of hermodynamcs he frst law of thermodynamcs reresents an adataton of the law of conservaton of energy to thermodynamcs, where energy can be stored n nternal degrees of freedom. For thermodynamc uroses, t s convenent to defne the nternal energy, U, as the energy of the system n the rest frame of ts center-of-mass. he knetc energy of the center-ofmass s a roblem n mechancs that we wll generally gnore. Ordnarly, one states energy conservaton n the form of some varant of the statement: he energy of any solated system s constant. However, alcatons of ths law must nclude the concet of heat, whch has not yet been develoed. herefore, t s concetually clearer to roceed by a more crcutous route. akng the concet of mechancal work as the foundaton, we state the frst law as Frst Law of hermodynamcs: he work requred to change the state of an otherwse solated system deends solely uon the ntal and fnal states nvolved and s ndeendent of the method used to accomlsh ths change. he stulaton that the system nteracts wth the envronment only through a measurable source of work s crucal no other source of energy, such as heat or radaton, s ermtted by the thermodynamc transformaton descrbed. An mmedate consequence of the frst law s the exstence of a state functon we dentfy as the nternal energy, U. It s customary to denote the work erformed uon the system as W. he frst law states that the work erformed uon the system durng any adathermal transformaton deends only uon the change of states effected and s ndeendent of the ntermedate states through whch the system asses. Equlbrum condtons are not requred. herefore, the nternal energy s a state functon whose change durng an adathermal transformaton s gven by DU = W. We now consder a more general transformaton durng whch the system may exchange energy wth ts envronment wthout the erformance of work. Although DU s then no longer equal to W, the nternal energy s stll a state functon. he ablty to change the energy of a system wthout erformng mechancal work does not reresent a defect n the law of conservaton of energy. Rather, t demonstrates that energy may be transferred n more than one form. hs observaton leads to the defnton of heat as: Defnton of heat:

8 8 Revewhermodynamcs.nb he quantty of heat Q absorbed by a system s the change n ts nternal energy not due to work. Wth ths defnton, energy conservaton can be exressed n the form DU = Q + W hese consderatons have thus roduced a quanttatve concet of heat that s smlar to our commonlace notons. At frst glance, the argument resented above may aear crcular. We stated that energy s conserved, but then defned heat n a manner that guarantees that ths s true. However, ths crcularty s llusory. he crux of the matter s that nternal energy s ostulated to be a state functon. hs s a hyscal statement subject to exermental verfcaton, rather than merely a defnton. Suose that the state of some system s changed n an arbtrary manner from state A to state B. In rncle, we can now nsulate the system and return to state A by erformng a measurable quantty of work. he frst stage of ths cycle can now be reeated n some other arbtrary fashon. However, the quantty of work requred to reeat the nsulated return orton of the cycle must be the same f the frst law s vald. Often t s mortant to dstngush between roer and mroer dfferentals. Consder a fxed quantty of gas contaned wthn a ston. If the gas s exanded or comressed by movng the ston suffcently slowly so that the ressure equales throughout the volume, the dfferental work done on the gas by a nfntesmal quasstatc dslacement of the ston can be exressed as quasstatc ï dw = - V On the other hand, f the ston s wthdrawn so radly that there s no gas n contact wth the ston durng ts moton, then the gas erforms no work on the ston, such that dw Ø 0 even though the volume changes. herefore, we use to ndcate a roer dfferental that deends only uon the change of state or d to ndcate an mroer dfferental that also deends uon the rocess used to change the state. he change n nternal energy U = dq + dw s a roer dfferental because U s a state functon even though both the heat absorbed by and work done on the system are mroer or rocess-deendent dfferentals. à 2 nd Law of hermodynamcs Although the frst law greatly restrcts the thermodynamc transformatons that are ossble, t s a fact of common exerence that many rocesses consstent wth energy conservaton never occur n nature. For examle, suose that an ce cube s laced n a glass of warm water. Although the frst law ermts the ce cube to surrender some of ts nternal energy so that the water s warmed whle the ce cools, such an event n fact never occurs. Smlarly, f we mx equal arts of two gases we never fnd the two gases to have searated sontaneously at some later tme. Nor do we exect scrambled eggs to rentegrate themselves. he fact that these tyes of transformatons are never observed to occur sontaneously s the bass of the second law of thermodynamcs. he second law can be formulated n many equvalent ways. he two statements generally consdered to be the most clear are those due to Clausus and to Kelvn. Clausus statement: here exsts no thermodynamc transformaton whose sole effect s to transfer heat from a colder reservor to a warmer reservor. Kelvn statement:

9 Revewhermodynamcs.nb 9 here exsts no thermodynamc transformaton whose sole effect s to extract heat from a reservor and to convert that heat entrely nto work. Parahrasng, the Clausus statement exresses the common exerence that heat naturally flows downhll from hot to cold, whereas the Kelvn statement says that no heat engne can be erfectly effcent. Both statements merely exress facts of common exerence n thermodynamc language. It s relatvely easy to demonstrate that these alternatve statements are, n fact, equvalent to each other. Although these ostulates may aear somewhat mundane, ther consequences are qute rofound; most notably, the second law rovdes the bass for the thermodynamc concet of entroy. One mght be nclned to regard the Kelvn statement as more fundamental because t does not requre that an emrcal temerature scale be related drectly to drecton of heat flow or to the noton of hotness. However, such a relatonsh s easly establshed n ractce and s so basc to our ntutve notons of heat and temerature that such a crtcsm s regarded as merely edantc. here are also more abstract formulatons of the second law, notably that of Caratheodory, that can be used to establsh the exstence and roertes of entroy wth fewer assumtons, but less obvous connecton to hyscal henomena. Perhas the most common dervaton of entroy s based uon an analyss of heat engnes, whch are also used to demonstrate that the Kelvn and Clausus statements are equvalent. hs aroach can be found n any undergraduate thermodynamcs text. We refer to emloy a somewhat dfferent argument that s closer to our man toc of statstcal mechancs. hs aroach s based uon an alternatve statement of the second law. Maxmum entroy rncle: here exsts a state functon of the extensve arameters of any thermodynamc system, called entroy S, wth the followng roertes: 1. the values assumed by the extensve varables are those whch maxme S consstent wth the external constrants; and 2. the entroy of a comoste system s the sum of the entroes of ts consttuent subsystems. Usng smlar arguments based uon heat engnes, one can show that the maxmum entroy rncle s equvalent to the Kelvn and Clausus statements, but we shall not dgress to rovde that demonstraton. Instead, we take the maxmum entroy rncle as our rmary statement of the second law. Consder two systems searated by a rgd dathermal wall; n other words, the two systems are n thermal contact and can exchange heat but cannot erform work on each other. Further, suose that the combned system s solated, so that U 1 + U 2 = constant. Suose that the two systems exchange an nfntesmally small quantty of heat, such that U 2 =- U 1. Snce no work s done, ths energy exchange must be n the form of heat, U =dq, where we use x to reresent a roer dfferental and dx to reresent an mroer dfferental that s nfntesmally small but deends uon the nature of the rocess. he net change n entroy durng ths rocess s S = S 1 + S 2 = k j S 1 y ÅÅÅÅÅÅÅÅÅÅÅÅÅ U 1 + U 1 { V 1,N 1 k j S 2 y ÅÅÅÅÅÅÅÅÅÅÅÅÅ U 2 = J ÅÅÅÅÅÅÅ 1 - ÅÅÅÅÅÅÅ 1 N dq 1 U 2 { V 2,N 2 t 1 t 2 where we have defned 1 ÅÅÅÅÅ t = k ÅÅÅÅ U { V,N for each system n terms of the artal dervatve of entroy wth resect to energy holdng all other extensve varables (volume, artcle numbers, etc.) constant. Maxmaton of entroy at equlbrum then requres S = 0 ï t 1 =t 2

10 10 Revewhermodynamcs.nb for arbtrary (nonero) values of U 1. However, equlbrum also requres that the temeratures of the two systems be equal. herefore, we are nclned to dentfy t wth temerature ; after all, we have the exstence of emrcal temerature scales but have not yet made a defnton of absolute temerature. o demonstrate that ths dentfcaton s consstent wth more rmtve notons of temerature, suose that t 1 >t 2. We then exect that heat wll flow sontaneously from the hotter to the cooler system, such that dq 1 < 0. We then fnd that t 1 >t 2, dq 1 < 0 ï S > 0 demonstrates that the net entroy ncreases when heat flows n ts natural drecton from hot to cold. Heat contnues to flow untl the temeratures become equal, equlbrum s reached, and entroy s maxmed. herefore, ths relatonsh between temerature and entroy s consstent wth the second law of thermodynamcs. Now suose that system 1 s the workng substance of a heat engne whle system 2 conssts of a collecton of heat reservors that can be used to suly or accet heat at constant temerature. Imagne a cycle n whch system 1 starts n a well-defned ntal state n equlbrum wth a reservor at temerature and s then manulated such that at the end of the cycle t once agan returns to the same ntal state at fnal temerature f =. Hence, the nternal energy and entroy return to ther ntal values when the system s returned to ts ntal state. Intermedate states of system 1 need not be n equlbrum and may requre a larger set of varables for comlete characteraton. Although such states do not have a unque temerature, we magne that any exchange of heat wth the external envronment s made by thermal contact wth a reservor that does have a well-defned temerature at every stage of the cycle and that the reservors are large enough to make the necessary exchange wth neglgble change of temerature. If necessary, we could magne that a large collecton of reservors s used for ths urose. Accordng to the maxmum entroy rncle, the change n entroy s nonnegatve, such that DS =DS 1 + DS 2 0 hus, havng stulated a closed cycle for system 1, requrng ts ntal and fnal entroes to be dentcal, we conclude that the entroy of the envronment cannot decrease when system 1 executes a closed cycle, such that DS 1 = 0 ï DS 2 0 he entroy change for the envronment s related to the heat absorbed by the system accordng to ds 2 = - dq 1 ÅÅÅÅÅÅÅÅÅÅÅ Å ï dq 1 ÅÅÅÅÅÅÅÅÅÅÅ Å Focussng uon the system of nterest, whch undergoes a closed cycle, we can exress ths result n the form ÅÅÅÅÅÅÅÅ dq 0 where we nterret dq as the heat absorbed by the system and as the temerature of the reservor that sules that heat; the temerature of the system need not be the same as that of the reservor and s often not even unque. In the secal case that the reservor also undergoes a closed cycle, such that DS 2 = 0, we fnd dq ÅÅÅÅÅÅÅÅ = 0 f and only f the cycle s reversble where, accordng to the recedng analyss, reversble heat exchange requres the temeratures of both bodes must be the same. Clausus' theorem hese results are summared by Clausus' theorem

11 Revewhermodynamcs.nb 11 For any closed cycle, ò ÅÅÅÅÅÅÅ dq 0, where d Q s the heat absorbed at temerature. Equalty holds f and only f the cycle s reversble. If a closed cycle conssts only of reversble transformatons, we can use dq rev ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å = 0 to assocate changes S n the state functon S to reversble heat exchange accordng to S = dq rev ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å ï S =dq rev hs dentfcaton does not aly to rreversble heat exchange. Suose that a reversble transformaton between states A and B s followed by an rreversble return from B to A, such that dq ÅÅÅÅÅÅÅÅ 0 ï A B dq rev ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å A B B dq rev + S 0 ï S ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å B A A hus, a reversble transformaton entals the smallest ossble change n entroy. If the states A and are taken as nfntesmally close together, then the dfferentals satsfy the same relatonsh, such that S dq rev ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å where equalty ales only to reversble transformatons. hs relatonsh can now be used to rovde an exermental defnton for the change n entroy of a system. he entroy of state A can be related to that of a standard or reference state R by measurng reversble heat exchanges accordng to S A = S R + R A dq rev ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å he heat caacty subject to secfed constrants can now defned by C x = J dq rev ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å d where x descrbes the rocess and dq rev = S s the heat absorbed durng a reversble rocess n whch the temerature change s d. However, because the second law ostulates that S s a state functon that s ndeendent of rocess, be t reversble or rreversble, the form C x = k { x s more fundamental t does not deend uon the detals of rocess, just the change n state that s actually accomlshed. herefore changes n the entroy of a system can be deduced from heat caacty measurements usng DS = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ C@D where the heat caacty generally deends uon temerature. Changes n the nternal energy of a smle comressble system can be exressed n the form U =dq +dw where we dentfy dq rev = S as the heat absorbed and dw =- V as the work erformed uon the system durng an nfntesmal reversble rocess. herefore, we can exress the energy dfferental as U = S - V N x

12 12 Revewhermodynamcs.nb where the use of state functons lberates us from the restrcton to reversble rocess; n other words, by exressng the energy dfferental n terms of state functon, t becomes a erfect dfferental that ales to any rocess, reversble or not. We can now defne the sochorc heat caacty as the heat absorbed for an nfntesmal temerature change under condtons of constant volume as V = 0 ï S = U ï C V = k = { V k ÅÅÅÅ U y { V where the fnal exresson n terms of state functons s consdered the fundamental defnton. Smlarly, the sobarc heat caacty under condtons of constant ressure s easly obtaned by defnng enthaly H = U + V ï H = S + V such that = 0 ï S = H ï C = k { = k ÅÅÅÅ H y { à 3 rd Law of hermodynamcs For most alcatons of thermodynamcs t s suffcent to analye changes n entroy rather than entroy tself, but determnaton of absolute entroy requres an ntegraton constant or, equvalently, knowledge of the absolute entroy for a standard or reference state of the system. Hstorcally, attemts to formulate a general rncle for determnaton of absolute entroy were motvated to a large extent by the need of chemsts to calculate equlbrum constants for chemcal reactons from thermal data alone. Several smlar, but not qute equvalent, statements of the thrd law have been roosed. he Nernst heat theorem states: a chemcal reacton between ure crystallne hases that occurs at absolute ero roduces no entroy change. hs formulaton was later generaled n the form of the Nernst-Smon statement of the thrd law. Nernst-Smon statement of thrd law: he change n entroy that results from any sothermal reversble transformaton of a condensed system aroaches ero as the temerature aroaches ero. he Nernst-Smon statement can be reresented mathematcally by the lmt lm HDSL = 0 Ø0 An mmedate consequence of the Nernst-Smon statement s that the entroy of any system n thermal equlbrum must aroach a unque constant at absolute ero because sothermal transformatons are assumed not to affect entroy at absolute ero. hs observaton forms the bass of the Planck statement of the thrd law. Planck statement of thrd law: As Ø0, the entroy of any system n equlbrum aroaches a constant that s ndeendent of all other thermodynamc varables. Planck actually generaled ths statement even further by hyothesng that lm Ø0 S = 0. However, ths strong form of the Planck statement requres that the quantum mechancal degeneracy of the ground state of any system must be unty. hus, the strong form mlctly assumes that there always exst nteractons that slt the degeneracy of the ground state,

13 Revewhermodynamcs.nb 13 but the energy slttng mght be so small as to be rrelevant at any attanable temerature. herefore, we wll not requre the strong form and wll content to aly the weaker form stated above. Fnally, the thrd law s often resented n the form of the unattanablty theorem. Unattanablty theorem: here exsts no rocess, no matter how dealed, caable of reducng the temerature of any system to absolute ero n a fnte number of stes. For the uroses of ths theorem, a ste s nterreted as ether an sentroc or an sothermal transformaton. he followng fgure llustrates the meanng of the unattanablty theorem (the temerature and entroy unts are arbtrary). Suose that entroy S@, xd deends uon temerature and an external varable x, wth the uer curve corresondng to x = 0 and the lower curve to a large value of x. For examle, ncreasng the external magnetc feld aled to a aramagnetc materal ncreases the algnment of atomc magnetc moments and reduces the entroy of the materal. Suose that the samle s ntally n thermal equlbrum at x = 0 wth a reservor at temerature 0. We slowly ncrease x whle mantanng thermal equlbrum at constant temerature. Next we adabatcally reduce x back to ero, thereby coolng the samle. Consder the rght sde of the fgure, n whch the entroy at Ø 0 deends uon x. In ths case an dealed sequence of alternatng sothermal and sentroc stes aears to be caable of reachng ero temerature, but the deendence S@0,xD ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0 volates the thrd law. herefore, the thrd law asserts that the entroy for hyscal systems must aroach a x constant as Ø 0 that s ndeendent of any external varable. In the fgure on left, an nfnte number of stes would be needed to reach absolute ero because the entroy at ero temerature s ndeendent of external varables, such that S@0,xD ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = 0, and s consstent wth the thrd law. x Unattanablty heorem Allowed Not Allowed S S Adabatc demagnetaton s an mortant technque for reachng low temeratures that reles on a rocess smlar to that sketched above. A aramagnetc salt s a crystallne sold n whch some of the ons ossess ermanent magnetc dole moments. In a tycal samle the magnetc ons consttute a small fracton of the materal and are well searated so that the sn-sn nteractons are very weak and the Cure temerature very low. (he Cure temerature marks the hase transton at whch sontaneous magnetaton s obtaned as a aramagnetc materal s cooled.) For the uroses of thermodynamc analyss, we can consder the magnetc doles to consttute one subsystem whle the nonmagnetc degrees of freedom consttute a second subsystem, even though both subsystems are comosed of the same atoms. hese nterenetratng systems are n thermal contact wth each other, but the nteracton between magnetc and nonmagnetc degrees of freedom s generally qute weak so that equlbraton can take a relatvely long tme. he frst ste s to cool the samle wth B = 0 by conventonal thermal contact. In a tycal demagnetaton cell the samle s surrounded by helum gas. Around that s a dewar contanng lqud helum bolng under reduced ressure at a temerature of about 1 kelvn. he gas serves to exchange heat between the samle and the lqud, mantanng the system at constant temerature. Next, the external magnetc feld s ncreased, algnng the magnetc doles and reducng the entroy of the samle under sothermal condtons. he helum gas s then removed and the system s thermally solated. he feld s then reduced under essentally adabatc condtons, reducng the temerature of the magnetc subsystem whle the coulng between magnetc and nonmagnetc subsystems cools the lattce. hs ste must be slow enough for heat to be transferred reversbly between

14 14 Revewhermodynamcs.nb nonmagnetc and magnetc subsystems. Nevertheless, t s ossble to reach temeratures n the mllkelvn range usng adabatc demagnetaton that exlots onc magnetc moments, lmted by the atomc Cure temerature c. Even lower temerature, n the mcrokelvn range, can be reached by addng a nuclear demagnetaton ste; the Cure temerature s lower by the rato of nuclear and atomc magnetc moments, whch s aroxmately the rato m e ê m between electron and roton masses. hermodynamc Potentals Suose that the macrostates of a thermodynamc system deend uon r extensve varables 8X, = 1, r<. here are then r + 1 ndeendent extensve varables consstng of the set 8X < sulemented by ether S or U, such that the fundamental relaton for the system can be exressed n ether entroc form, S = S@U, X 1,, X r D or energetc form U = U@S, X 1,, X r D hese relatonshs descrbe all ossble equlbrum states of the system. Changes n these state functons can then be exressed n terms of total dfferentals U = S + P X S = ÅÅÅÅÅ 1 U + Q X where for reversble transformatons varatons of the external constrants va X erform work P X uon the system (whch may be negatve, of course) whle S reresents energy suled to hdden nternal degrees of freedom n the form of heat. However, by exressng the total dfferental solely n terms of state varables, the fundamental relaton ales to any change of state, reversble or not, ndeendent of the ath by whch that change was made. herefore, we can dentfy the coeffcents wth artal dervatves, whereby = k ÅÅÅÅ U y = S { X k ÅÅÅÅ S -1 y U { P = k ÅÅÅÅ U y X { S,X j X X Q = k ÅÅÅÅ X { are ntensve arameters whch satsfy equatons of state of the form X 1,, X r D P = X 1,, X r D Q = X 1,, X r D U,X j X he equaton for s called the thermal equaton of state whle the equatons for P or Q are mechancal and/or chemcal equatons of state deendng uon the hyscal meanng of the conjugate varable X. Note that the sets of P and Q are not ndeendent because there are only r + 1 ndeendent equatons of state whch comletely determne the fundamental relaton. If we choose to emloy the energy reresentaton, U = U@S, 8X <D, the arorate choce of ntensve arameters would be H, 8P <L and any Q can be exressed n terms of H, 8P <L.

15 Revewhermodynamcs.nb 15 For examle, a smle comressble system s charactered comletely by ts energy (or entroy), volume, and artcle number for a sngle seces. he fundamental relaton n the energy reresentaton then takes the form U = U@S, V, ND ï U = S - V + m N where the ntensve arameter conjugate to -V s clearly the ressure and the ntensve arameter conjugate to N s dentfed as the chemcal otental m, such that = k ÅÅÅÅ U y =- S { V,N k ÅÅÅÅ U y m= V { S,N k ÅÅÅÅ U y N { S,V Alternatvely, n the entroy reresentaton we wrte S = S@U, V, ND ï S = U + V - m N such that 1 ÅÅÅÅÅ = k ÅÅÅÅ U { V,N ÅÅÅÅÅ = k ÅÅÅ V { U,N m ÅÅÅÅÅ =- k ÅÅÅÅ N { U,V where we have emloyed the customary defntons for the energetc and entroc ntensve arameters. More comlex systems requre a greater number of extensve varables and assocated ntensve arameters. A ar of varables, x and X, are sad to be conjugate when X s extensve, x ntensve, and the roduct x X has dmensons of energy and aears n the fundamental relaton for U or S. It s straghtforward to generale these develoments for systems wth dfferent or addtonal modes of mechancal work. For examle, the conjugate ar H-, VL for a comressble system mght be relaced by Hg, AL for a membrane wth area A subject to surface tenson g. A change X n one of the extensve varables requres work P X where P = ÅÅÅÅÅÅÅÅ U X can be nterreted as a thermodynamc force. [Note that the absence of a negatve sgn s related to the conventon that ostve work s erformed uon the system, ncreasng ts nternal energy.] hus, the fundamental relaton for an elastc membrane s wrtten n the form U = S +g A, where the surface tenson g = ÅÅÅÅÅÅÅ U lays the role of a thermodynamc force actng through generaled dslacement A to erform work g A uon the A membrane. he fundamental relaton must be exressed n terms of ts roer varables to be comlete. hus, the energy features entroy, rather than temerature, as one of ts roer varables. However, entroy s not a convenent varable exermentally there exsts no meter to measure nor knob to vary entroy drectly. herefore, t s often more convenent to construct other related quanttes n whch entroy s a deendent nstead of an ndeendent varable. hs goal can be accomlshed by means of the Legendre transformaton n whch a quantty of the form x X s added to the fundamental relatonsh n order to nterchange the roles of a ar of conjugate varables. For examle, we defne the Helmholt free energy as F = U - S so that for a smle comressble system we obtan a comlete dfferental of the form F = U - S ï F =-S - V +m N n whch s now the ndeendent varable and S the deendent varable. he fundamental relaton now assumes the form F = F@, V, ND where S =- k ÅÅÅ F y =- { V,N k ÅÅÅ F y m= V {,N k ÅÅÅÅ F y N {,V hs state functon s clearly much more amenable to exermental manulaton than the nternal energy even f U occues a more central role n our thnkng. On the other hand, a smle ntutve nterretaton of F s avalable also. Consder a smle comressble system enclosed by mermeable walls equed wth a movable ston and suose that the system s n thermal contact wth a heat reservor that mantans constant temerature. he change n Helmholt free energy when the volume s adjusted sothermally s then

16 16 Revewhermodynamcs.nb = 0, N = 0 ï F =- V If the change of state s erformed reversbly and quasstatcally, the change n free energy s equal to the work erformed uon the system, such that F = dw. herefore, the Helmholt free energy can be nterreted as the caacty of the system to erform work under sothermal condtons. he free energy s less than the nternal energy because some of the nternal energy must be used to mantan constant temerature and s not avalable for external work. Hence, the free energy s often more relevant exermentally than the nternal energy. State functons obtaned by means of Legendre transformaton of a fundamental relaton are called thermodynamc otentals because the roles they lay n thermodynamcs are analogous to the role of the otental energy n mechancs. Each of these otentals rovdes a comlete and equvalent descrton of the equlbrum states of the system because they are all derved from a fundamental relaton that contans all there s to know about those states. However, the functon that rovdes the most convenent ackagng of ths nformaton deends uon the selecton of ndeendent varables that s most arorate to a artcular alcaton. For examle, t s often easer to control the ressure uon a system that ts volume, esecally f t s nearly ncomressble. Under those condtons t mght be more convenent to emloy enthaly H = U + V ï H = S + V +m N or the Gbbs free enthaly G = F + V ï G =-S + V +m N than the corresondng nternal or free energes. he deendent varables are then dentfed n terms of the coeffcents that aear n the dfferental forms of the fundamental relaton chosen. hus, for the Gbbs reresentaton we dentfy entroy, volume, and chemcal otental as S =- k ÅÅÅ G y V = {,N k ÅÅÅ G y m= { k ÅÅÅÅ G y N {, Note that the Helmholt free energy or Gbbs free enthaly are often referred to as Helmholt or Gbbs otentals, somewhat more alatable aellatons. More comlcated systems wth addtonal work modes requre addtonal terms n the fundamental relaton and gve rse to a corresondngly greater number of thermodynamc otentals. Smlarly, f t s more natural n some alcaton to control the chemcal otental than the artcle number, a related set of thermodynamc otentals can be constructed by subtractng m N from any of the otentals whch emloy N as one of ts ndeendent varables. If any of the thermodynamc otentals s known as a functon of ts roer varables, then comlete knowledge of the equlbrum roertes of the system s avalable because any thermodynamc arameter can be comuted from the fundamental relaton. For examle, suose we have G@, D. he nternal energy U = G + S - V can then be exressed n terms of temerature and ressure as U = G@, D - k ÅÅÅ G y - { k ÅÅÅ G y { where S and V are obtaned as arorate artal dervatves of G. An almost lmtless varety of smlar relatonshs can be develoed easly usng these technques. In fact, much of the formal develoment of thermodynamcs s devoted to fndng relatonshs between the quanttes desred and those that are most accessble exermentally. hus, the sklled thermodynamcst must be able to manulate related rates of change subject to varous constrants. Several useful technques are develoed n the next secton.,n

17 Revewhermodynamcs.nb 17 hermodynamc Relatonshs à Maxwell relatons An mortant class of thermodynamc relatonshs, known as Maxwell relatons, can be develoed usng the comleteness of fundamental relatons to examne the dervatves of one member of a conjugate ar wth resect to a member of another ar. Consder a smle comressble system wth fundamental relaton U = k ÅÅÅÅ U y S + S { V k ÅÅÅÅ U y V = S - V V { S he comleteness of the total dfferental U requres the two mxed second artal dervatves to be equal, such that 2 U ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅ V S = 2 U ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅ S V ï k ÅÅÅ y V { S k ÅÅÅÅ U y = S { V k y S { V k ÅÅÅÅ U y V { S herefore, uon dentfcaton of the temerature and ressure wth arorate frst dervatves, we obtan the Maxwell relaton j y ÅÅÅÅÅÅÅÅ ÅÅÅ =- k V { S k y S { V he ower of ths relatonsh stems from the fact that t deends only uon the statement that the nternal energy s a functon solely of S and V; hence, ts valdty s ndeendent of any other secfc roertes of the system. Smlar Maxwell relatons can be develoed from each of the thermodynamc otentals exressed as functons of a dfferent ar of ndeendent varables. he four Maxwell relatons below are easly derved (verfy!) for smle comressble systems. U = U@S, VD ï k ÅÅÅ y =- V { S k y S { V H = H@S, D ï k y { F = F@, VD ï k ÅÅÅ V { G = G@, D ï k { S = k ÅÅÅ V y S { = k y { V = - k V y ÅÅÅ { Each of these relatonshs s an essental consequence of the comleteness of the fundamental relaton and several of these relatonshs mose constrants uon the behavor of quanttes of ractcal, exermental sgnfcance. In each relatonsh two ndeendent varables aear n the denomnators, one as for dfferentaton and the other as a constrant, whle n each dervatve the deendent varable from the ooste conjugate ar aears n the numerator. he sgn s determned by the relatve sgn that aears n the fundamental relaton for the thermodynamc otental arorate to the chosen ar of ndeendent varables. Any remanng ndeendent varables (such as artcle number, N) should aear n the constrants for both sdes, but are often left mlct.

18 18 Revewhermodynamcs.nb o llustrate the usefulness of Maxwell relatons, consder the dervatve of the sobarc heat caacty, C, wth resect to ressure for constant temerature. hs quantty s drectly accessble to measurement. Exressng C n terms of S and usng the commutatve roerty of artal dervatves, we fnd k ÅÅÅÅÅ C y { = j y ÅÅÅÅÅÅÅÅ k { k S y { = k y { k { Use of Maxwell relaton now yelds k { =- k ÅÅÅ V y ï { k ÅÅÅÅÅ C y { =- j 2 V y ÅÅÅÅÅÅÅÅÅÅÅÅÅ k 2 { Smlarly, wth the ad of another Maxwell relaton, the volume deendence of the sochorc heat caacty becomes k ÅÅÅ = V { k y ï { V k j C V y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = j 2 y ÅÅÅÅÅÅÅÅÅÅÅÅÅ V { k 2 { V Note that both of these dervatves vansh for an deal gas but n general wll be fnte for real gases. Hence, the rncal heat caactes for an deal gas deend only uon temerature and are ndeendent of ressure and volume, whereas for real gases one can use the emrcal equaton of state to evaluate the ressure and volume deendences of these quanttes. he beauty of thermodynamcs s that many useful relatonshs between aarently unrelated roertes of a system can be develoed from general rncles wthout detaled knowledge of the dynamcs of the system on a mcroscoc level. Smlar, but more comlcated, relatonshs can also be develoed by treatng one of the thermodynamc otentals as an ndeendent varable. For examle, we can use the comleteness of S = ÅÅÅÅÅ 1 U + ÅÅÅÅÅ V to dentfy 1 ÅÅÅÅÅ = k ÅÅÅÅ U { V ÅÅÅÅÅ = k ÅÅÅ V { U and thereby obtan the Maxwell relaton S = S@U, VD ï k j ÅÅÅÅÅÅÅÅÅÅÅ V 1 y ÅÅÅÅÅ { U = k j ÅÅÅÅÅÅÅÅÅÅÅÅ U y ÅÅÅÅÅ { V à Chan and cyclc rules for artal dervatves Consder the set of state varables 8x, y, <, of whch any two may be consdered ndeendent. he dfferentals x and y then become x = k x y y { y + k j ÅÅÅÅÅÅÅ x y { y y = k y y x { x + k y y { x so that x = k x y y { k y Ä y x + x { ÇÅ k j ÅÅÅÅÅÅÅ x y + { y k x y y { k y É y { xöñ hs relatonsh ales to any 8 x, y<. In artcular, f we choose to make = 0, then the coeffcents of x must be equal. Hence, we obtan the famlar chan rule