Temperature. Chapter Heat Engine

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1 Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the smple case whch 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 0 we dscussed a more 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 method of Lagrange multplers was used, and provded the summatons can be done, a general method of solutons was presented. In Chapter 2 we looked at the mplcatons of the Prncple of Maxmum ntropy for physcal systems that adhere to the mult state model motvated by quantum mechancs, as outlned n Chapter. We found that the Lagrange multpler 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. 3. eat ngne 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 Lagrange multplers 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 /β, and ndeed of any constant tmes /β. (Actually ths statement s not true f one of the two values of s postve and the other s negatve; n ths case the resultng value of β s ntermedate but Author: Paul Penfeld, Jr. Verson.0.2, May 2, Copyrght c 2003 Massachusetts Insttute of Technology 2

2 3. eat ngne 3 Fgure 3.: Dpole moment example. (ach dpole can be ether up or down.) the resultng value of /β s not.) Note that /β can, by usng the formulas n Chapter 2, 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 T = (3.) k B β where k B = Joules per Kelvn s Boltzmann s constant. The probablty dstrbuton that comes from the use of the Prncple of Maxmum ntropy s, when wrtten n terms of T, p e β = e α (3.2) e /k B T = e α (3.3) 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 Lagrange multplers, from now on we wll express our results n terms of temperature. The absolute temperature T s expressed n unts of Kelvn. 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 Amerca, dffers by both an addtve constant and a multplcatve factor. For general nterest, Table 3. shows varous temperatures of nterest on the three scales, along wth β. K C F k B T = β (J) β (J ) Absolute Zero Outer Space (approx) Lqud elum bp Lqud Ntrogen bp Water mp Room Temperature (approx) Water bp Table 3.: Varous Temperatures of nterest bp = bolng pont, mp = meltng pont The magnetc dpole system we are consderng s shown n Fgure 3., where the two envronments are 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). Let us rewrte the formulas from Chapter2 wth the use of β replaced by temperature. Thus

3 3.2 nergy Converson Cycle 4 = p (3.4) = p (3.5) S = k B p ln p (3.6) e /k B T p = e α (3.7) α = ln e /k B T S = (3.8) kb kb T The dfferental formulas from Chapter 2 for the case of the dpole model where each state has an energy proportonal to become 0 = dp (3.9) d = () dp + d (3.0) T ds = d d (3.) [ ] dα = dt d (3.2) [ ] [ ] () dp = p dt d (3.3) [ ] [ ] d = p ( () ) 2 dt d + d (3.4) [ ] [ ] T ds = p ( () ) 2 dt d (3.5) and the change n energy can be attrbuted to the effects of work dw and heat dq 3.2 nergy Converson Cycle dw = d (3.6) dq = () dp (3.7) = T ds (3.8) Consder the cycle shown on the dagram below. Wthout loss of generalty we can treat the case where s postve. Assume that the left envronment has a temperature T 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

4 3.2 nergy Converson Cycle 5 the devce to work). A smple way to envson the cycle s to consder the plane formed by axes correspondng to S and T of the system. The cycle we are nterested n forms a rectangle, wth corners marked a, b, c, and d, and sdes correspondng to the values S, S 2, T, and T 2 : Fgure 3.2: Temperature Cycle 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 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 used to stretch a sprng or rase a weght aganst gravty, thereby storng ths energy. nergy 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 to T 2 wthout change n entropy. Ths s an adabatc step. By one of the equatons above, ncreasng T s accomplshed by ncreasng, whle not permttng the system to nteract wth ether of ts two envronments. 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, so 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 to S 2 at constant temperature T 2. Ths step, at constant temperature, s called sothermal. Accordng to one of the formulas above, ths s accomplshed by decreasng, whle the system s n contact wth the rght envronment, whch s assumed to be at temperature T 2. Durng ths leg the change n energy 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 ) and the work s postve snce the decreasng 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 sothermalleg 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

5 3.2 nergy Converson Cycle 6 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 3.2 summarzes the heat engne cycle. Leg Start nd Type ds dt eat 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 postve negatve Table 3.2: nergy cycle For each cycle the energy lost by the hgh temperature envronment s T 2 (S 2 S ) and the energy ganed by the low temperature envronment s T (S 2 S ) and so the net energy converted s the dfference (T 2 T )(S 2 S ). 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 = (3.9) hgh temperature heat n T 2 Ths rato s known as the Carnot effcency, named after the French physcst Sad Ncolas Léonard Carnot ( ). e 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. eat engnes run n ths reverse fashon act as refrgerators or heat pumps. For a bography check out groups.dcs.st andrews.ac.uk/ hstory/mathematcans/carnot Sad.html

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