Chapter 12 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum Entropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 8 we dscussed the smple case that can be done analytcally, n whch there are three probabltes, one constrant n the form of an average value, and the fact that the probabltes add up to one. There are, then, two equatons and three unknowns, and t s straghtforward to express the entropy n terms of one of the unknowns, elmnatng the others, and fnd the maxmum. Ths approach also works f there are four probabltes and two average-value constrants, n whch case there s agan one fewer equaton than unknown. In Chapter 9 we dscussed a general case n whch there are many probabltes but only one average constrant, so that the entropy cannot be expressed n terms of a sngle probablty. The result prevously derved usng the method of Lagrange multplers was gven. In Chapter 11 we looked at the mplcatons of the Prncple of Maxmum Entropy for physcal systems that adhere to the mult-state model motvated by quantum mechancs, as outlned n Chapter 10. We found that the dual varable β plays a central role. Its value ndcates whether states wth hgh or low energy are occuped (or have a hgher probablty of beng occuped). From t all the other quanttes, ncludng the expected value of energy and the entropy, can be calculated. In ths chapter, we wll nterpret β further, and wll defne ts recprocal as (to wthn a scale factor) the temperature of the materal. Then we wll see that there are constrants on the effcency of energy converson that can be expressed naturally n terms of temperature. 12.1 Temperature Scales A heat engne s a machne that extracts heat from the envronment and produces work, typcally n mechancal or electrcal form. As we wll see, for a heat engne to functon there need to be two dfferent envronments avalable. The formulas below place restrctons on the effcency of energy converson, n terms of the dfferent values of β of the two envronments. We wll derve these restrctons. Frst, however, t s useful to start to deal wth the recprocal of β rather than β tself. Recall that β s an ntensve property: f two systems wth dfferent values of β are brought nto contact, they wll end up wth a common value of β, somewhere between the orgnal two values, and the overall entropy wll rse. The same s true of 1/β, and ndeed of any constant tmes 1/β. (Actually ths statement s not true f one of the two values of β s postve and the other s negatve; n ths case the resultng value of β s ntermedate but 131
12.2 Heat Engne 132 the resultng value of 1/β s not.) Note that 1/β can, by usng the formulas n Chapter 11, be nterpreted as a small change n energy dvded by the change n entropy that causes t, to wthn the scale factor k B. Let us defne the absolute temperature as 1 T = (12.1) k B β where k B = 1.381 10 23 Joules per Kelvn s Boltzmann s constant. The probablty dstrbuton that comes from the use of the Prncple of Maxmum Entropy s, when wrtten n terms of T, p = e α e βe (12.2) = e α e E/k B T (12.3) The nterpretaton of β n terms of temperature s consstent wth the everyday propertes of temperature, namely that two bodes at the same temperature do not exchange heat, and f two bodes at dfferent temperatures come nto contact one heats up and the other cools down so that ther temperatures approach each other. In ordnary experence absolute temperature s postve, and the correspondng value of β s also. Because temperature s a more famlar concept than dual varables or Lagrange multplers, from now on we wll express our results n terms of temperature. Absolute temperature T s measured n Kelvns (sometmes ncorrectly called degrees Kelvn), n honor of Wllam Thomson (1824 1907), who proposed an absolute temperature scale n 1848. 1 The Celsus scale, whch s commonly used by the general publc n most countres of the world, dffers from the Kelvn scale by an addtve constant, and the Fahrenhet scale, whch s n common use n the Unted States, dffers by both an addtve constant and a multplcatve factor. Fnally, to complete the roster of scales, Wllam Rankne (1820 1872) proposed a scale whch had 0 the same as the Kelvn scale, but the sze of the degrees was the same as n the Fahrenhet scale. More than one temperature scale s needed because temperature s used for both scentfc purposes (for whch the Kelvn scale s well suted) and everyday experence. Naturally, the early scales were desgned for use by the general publc. Gabrel Fahrenhet (1686 1736) wanted a scale where the hottest and coldest weather n Europe would le between 0 and 100. He realzed that most people can deal most easly wth numbers n that range. In 1742 Anders Celsus (1701 1744) decded that temperatures between 0 and 100 should cover the range where water s a lqud. In hs ntal Centgrade Scale, he represented the bolng pont of water as 0 degrees and the freezng pont as 100 degrees. Two years later t was suggested that these ponts be reversed. 2 The result, named after Celsus n 1948, s now used throughout the world. For general nterest, Table 12.1 shows a few temperatures of nterest on the four scales, along wth β. 12.2 Heat Engne The magnetc-dpole system we are consderng s shown n Fgure 12.1, where there are two envronments at dfferent temperatures, and the nteracton of each wth the system can be controlled by havng the barrers ether present or not (shown n the Fgure as present). Although Fgure 12.1 shows two dpoles n the system, the analyss here works wth only one dpole, or wth more than two, so long as there are many fewer dpoles n the system than n ether envronment. Now let us rewrte the formulas from Chapter 11 wth the use of β replaced by temperature. Thus Equatons 11.8 to 11.12 become 1 Thomson was a prolfc scentst/engneer at Glasgow Unversty n Scotland, wth major contrbutons to electromagnetsm, thermodynamcs, and ther ndustral applcatons. He nvented the name Maxwell s Demon. In 1892 he was created Baron Kelvn of Largs for hs work on the transatlantc cable. Kelvn s the name of the rver that flows through the Unversty. 2 Accordng to some accounts the suggeston was made by Carolus Lnnaeus (1707 1778), a colleague on the faculty of Uppsala Unversty and a protege of Celsus uncle. Lnnaeus s best known as the nventor of the scentfc notaton for plants and anmals that s used to ths day by botansts and zoologsts.
12.2 Heat Engne 133 Absolute Zero Outer Space (approx) Lqud Helum bp Lqud Ntrogen bp Water mp Room Temperature (approx) Water bp K C F 1 R k B T = β (J) β (J 1 ) 0-273.15-459.67 0 0 2.7-270 -455 4.9 3.73 10 23 2.68 10 22 4.22-268.93-452.07 7.6 5.83 10 23 1.72 10 22 77.34-195.81-320.46 139.2 1.07 10 21 9.36 10 20 273.15 0.00 32.00 491.67 3.73 10 21 2.65 10 20 290 17 62 520 4.00 10 21 2.50 10 20 373.15 100.00 212.00 671.67 5.15 10 21 1.94 10 20 Table 12.1: Varous temperatures of nterest (bp = bolng pont, mp = meltng pont) H Fgure 12.1: Dpole moment example. (Each dpole can be ether up or down.) 1 = p (12.4) E = p E (12.5) 1 S = k B p ln p (12.6) p = e α e E/k B T (12.7) α = ln e E/k B T S E = (12.8) kb kb T The dfferental formulas from Chapter 11 for the case of the dpole model where each state has an energy proportonal to H, Equatons 11.30 to 11.36 become
12.3 Energy-Converson Cycle 134 0 = dp (12.9) E de = E (H) dp + dh (12.10) H E T ds = de dh (12.11) H [ ] E 1 1 dα = dt dh (12.12) k B T T H [ ] [ ] E ( H) E 1 1 dp = p dt dh (12.13) kb T T H [ ] [ ] 1 1 1 E de = p (E (H) E) 2 dt dh + dh (12.14) k B T T H H [ ] [ ] 1 1 1 T ds = p (E (H) E) 2 dt dh (12.15) k B T T H and the change n energy can be attrbuted to the effects of work dw and heat dq E dw = dh (12.16) H 12.3 Energy-Converson Cycle dq = E (H) dp = T ds (12.17) Ths system can act as a heat engne f the nteracton of the system wth ts envronments, and the externally appled magnetc feld, are both controlled approprately. The dea s to make the system change n a way to be descrbed, so that s goes through a successon of states and returns to the startng state. Ths represents one cycle, whch can then be repeated many tmes. Durng one cycle heat s exchanged wth the two envronments, and work s exchanged between the system and the agent controllng the magnetc feld. If the system, over a sngle cycle, gets more energy n the form of heat from the envronments than t gves back to them, then energy must have been delvered to the agent controllng the magnetc feld n the form of work. The cycle of the heat engne s shown below n Fgure 12.2. Wthout loss of generalty we can treat the case where H s postve. Assume that the left envronment has a temperature T 1 whch s postve but less (.e., a hgher value of β) than the temperature T 2 for the rght envronment (the two temperatures must be dfferent for the devce to work). Ths cycle s shown on the plane formed by axes correspondng to S and T of the system, and forms a rectangle, wth corners marked a, b, c, and d, and sdes correspondng to the values S 1, S 2, T 1, and T 2. Snce the temperatures are assumed to be postve, the lower energy levels have a hgher probablty of beng occuped. Therefore, the way we have defned the energes here, the energy E s negatve. Thus as the feld gets stronger, the energy gets more negatve, whch means that energy actually gets delvered from the system to the magnetc apparatus. Thnk of the magnetc feld as ncreasng because a large permanent magnet s physcally moved toward the system. The magnetc dpoles n the system exert a force of attracton on that magnet so as to draw t toward the system, and ths force on the magnet as t s moved could be
12.3 Energy-Converson Cycle 135 Fgure 12.2: Temperature Cycle used to stretch a sprng or rase a weght aganst gravty, thereby storng ths energy. Energy that moves nto the system (or out of the system) of a form lke ths, that can come from (or be added to) an external source of energy s work (or negatve work). Frst consder the bottom leg of ths cycle, durng whch the temperature of the system s ncreased from T 1 to T 2 wthout change n entropy. An operaton wthout change n entropy s called adabatc. By Equaton 12.15 above, ncreasng T s accomplshed by ncreasng H, whle not permttng the system to nteract wth ether of ts two envronments. (In other words, the barrers preventng the dpoles n the system from nteractng wth those n ether of the two envronments are n place.) The energy of the system goes down (to a more negatve value) durng ths leg, so energy s beng gven to the external apparatus that produces the magnetc feld, and the work done on the system s negatve. Next, consder the rght-hand leg of ths cycle, durng whch the entropy s ncreased from S 1 to S 2 at constant temperature T 2. Ths step, at constant temperature, s called sothermal. Accordng to Equaton 12.15, ths s accomplshed by decreasng H, whle the system s n contact wth the rght envronment, whch s assumed to be at temperature T 2. (In other words, the barrer on the left n Fgure 12.1 s left n place but that on the rght s wthdrawn.) Durng ths leg the change n energy E arses from heat, flowng n from the hgh-temperature envronment, and work from the external magnetc apparatus. The heat s T 2 (S 2 S 1 ) and the work s postve snce the decreasng H durng ths leg drves the energy toward 0. The next two legs are smlar to the frst two except the work and heat are opposte n drecton,.e., the heat s negatve because energy flows from the system to the low-temperature envronment. Durng the top leg the system s solated from both envronments, so the acton s adabatc. Durng the left-hand sothermal leg the system nteracts wth the low-temperature envronment. After gong around ths cycle, the system s back where t started n terms of ts energy, magnetc feld, and entropy. The two envronments are slghtly changed but we assume that they are each so much larger than the system n terms of the number of dpoles present that they have not changed much. The net change s a slght loss of entropy for the hgh-temperature envronment and a gan of an equal amount of entropy for the low-temperature envronment. Because these are at dfferent temperatures, the energy that s transferred when the heat flow happens s dfferent t s proportonal to the temperature and therefore more energy leaves the hgh-temperature envronment than goes nto the low-temperature envronment. The dfference s a net negatve work whch shows up as energy at the magnetc apparatus. Thus heat from two envronments s converted to work. The amount converted s nonzero only f the two envronments are at dfferent temperatures. Table 12.2 summarzes the heat engne cycle.
12.3 Energy-Converson Cycle 136 Leg Start End Type ds dt H E Heat n Work n bottom a b adabatc 0 postve ncreases decreases 0 negatve rght b c sothermal postve 0 decreases ncreases postve postve top c d adabatc 0 negatve decreases ncreases 0 postve left d a sothermal negatve 0 ncreases decreases negatve negatve Total a a complete cycle 0 0 no change no change postve negatve Table 12.2: Energy cycle For each cycle the energy lost by the hgh-temperature envronment s T 2 (S 2 S 1 ) and the energy ganed by the low-temperature envronment s T 1 (S 2 S 1 ) and so the net energy converted s the dfference (T 2 T 1 )(S 2 S 1 ). It would be desrable for a heat engne to convert as much of the heat lost by the hgh-temperature envronment as possble to work. The machne here has effcency work out = T 2 T 1 (12.18) hgh-temperature heat n T 2 Ths rato s known as the Carnot effcency, named after the French physcst Sad Ncolas Léonard Carnot (1796-1832). 3 He was the frst to recognze that heat engnes could not have perfect effcency, and that the effcency lmt (whch was subsequently named after hm) apples to all types of reversble heat engnes. The operatons descrbed above are reversble,.e., the entre cycle can be run backwards, wth the result that heat s pumped from the low-temperature envronment to the one at hgh temperature. Ths acton does not occur naturally, and ndeed a smlar analyss shows that work must be delvered by the magnetc apparatus to the magnetc dpoles for ths to happen, so that more heat gets put nto the hgh-temperature envronment than s lost by the low-temperature envronment. Heat engnes run n ths reverse fashon act as refrgerators or heat pumps. 3 For a bography check out http://www-groups.dcs.st-andrews.ac.uk/ hstory/mathematcans/carnot Sad.html
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