Physics 231 Lecture 35

Transcription

1 ysis 1 Leture 5 Main points of last leture: Heat engines and effiieny: eng e 1 Carnot yle and Carnot engine. eng e 1 is in Kelvin. Refrigerators CO eng Ideal refrigerator CO rev reversible Entropy ΔS

2 Computation of work: at is te work done by te gas going from ( i,v i ) to ( f,v f )? a) i (V i -V f ) b) f (V i -V f ) ) 0 d) ( i + f )(V f -V f )/ e) ( i + f )(V i -V f )/ A termal pat wi returns to its initial ondition is alled a yle. e work done by te gas on a lokwise yle is te area ontained in te pat. e work done by te gas on a ounterlokwise yle is te negative of te area in te pat. e work done on te gas is te negative of te work done by te gas.

3 Example e work done on te gas to ompress one mole of a monatomi ideal gas is 600 J. e temperature of te gas anges from 50 K to 550 K. How mu eat flows between te gas and its surroundings? Determine weter te eat flows into or out of te gas. ork done by te gas is : gas 600J Internal energy ange in te gas is: ΔU nr( Δ ) ΔU + gas nr( Δ ) 600J ( 8.1J / K )( K ) 600J 700J Heat flows out of te gas

4 Example e drawing refers to one mole of monatomi ideal gas and sows a proess tat as four steps, two isobari (A to B and C to D) and two isovolumetri (B to C and D to A). Complete te following table by alulating ΔU, Δ and Δ (inluding te algebrai signs) for ea of te four steps. Note tat te gas as returned to its initial state at te end of te proess, so tat te value for te total ΔU an be predited in advane witout any alulation. at ΔU Δ Δ A-B 5.0kJ.kJ 8.kJ B-C - 5.kJ 0-5.kJ C-D -.5kJ -1.7kJ -4.kJ D-A.5kJ 0.5kJ U U A A D A A B A B A B RA 4986J B C A U U 0 Δ ΔU + Δ ( V V ) V V R( ) B B B ( U B U 4J RB U 997J U A A ) A C D B B B C C C D ( U 166J RC U D 4986J U C A A D D U n 1 C B ) RD 49J D A

5 Example A person takes in a breat of 0 C air and olds it until it warms to 7.0 C. e air as an initial volume of L and a mass of 7.70 x 10 4 kg. Determine (a) te work done by te air on te lungs if te pressure remains onstant at 1 atm, (b) te ange in internal energy of te air, and () te energy added to te air by eat. Model te air as if it were a monatomi gas. f ΔU gas gas i atm atm ( V V ) 5 ( 1.01x10 a)(.0006m ) nr Δ ΔU + f 1.01x10 ( ) f gas i i 5 atm Assume monotoni gas : 5 V nr a i nri V V ( ) 0.5J f atm atm i f i f i atm 8.J f Vi i gas 1 1.J

6 Engines In a eat engine, termal energy is used to do work, eng. Some of te original termal energy esapes and ends up eating someting else A eat engine involves some working substane in a ylial proess ermal effiieny is defined as te ratio of te work done by te engine to te energy absorbed at te iger temperature. For simpliity bot an be omputed over one yle: eng e 1 e 1 (100% effiieny) only if 0 No energy expelled to old reservoir, wi is teoretially possible for 0, but pratially impossible.

7 Example e energy absorbed by an engine is tree times as large as te work it performs. (a) at is its termal effiieny? (b) at fration of te energy absorbed is expelled to te old reservoir? b) 1 a) + e

8 Reversible and Irreversible roesses reversible proess is one in wi every state along some pat is an equilibrium state. And one for wi te system an be returned to its initial state along te same pat. Volume and pressure anges are slow. en objets are brougt into termal ontat, tey are at te same temperatures. Carnot yle is an example of a reversible proess. An irreversible proess does not meet tese requirements Most natural proesses are irreversible Burning fueling in an automobile engine Dropping ie into warm water Heating water on a range Reversible proess are an idealization, but some real proesses are good approximations. If a proess is reversible, te anges in entropy ΔS i Δ/ i of te system add up to zero and te proess an be reversed witout a gain in entropy. Irreversible proesses always inrease te entropy and tus annot be reversed beause it is improbably tat you an do so.

9 Carnot Engine A Carnot engine is te most effiient possible engine tat takes eat from a ot reservoir at temperature and expels eat into a old reservoir at temperature. It uses gas as te working substane. It absorbs eat during an isotermal expansion wile in ontat wit te ot reservoir. It expands adiabatially a little furter. and te entropy ange are zero ere. It ompresses isotermally wile in ontat wit old reservoir. Here te entropy ange anels tat during te isoteral expansion. It ompresses adiabatially a little furter.

10 Carnot Cyle, A to B A to B is an isotermal expansion e gas is plaed in ontat wit te ig temperature reservoir e gas absorbs eat e gas does work AB in raising te piston AB nr ln V B at appens to te internal energy of te gas? V A a) It inreases b) It dereases ) It stays te same at appens to te pressure? a) It inreases b) It dereases ) It stays te same at appens to te entropy of te gas? a) It inreases b) It dereases ) It stays te same at appens te entropy of te ot reservoir? a) It is te same as for te gas b) It is minus tat of te gas. ) annot be determined. Fig. 1.1, p. 74 Slide

11 Carnot Cyle, B to C B to C is an adiabati expansion e base of te ylinder is replaed by a termally nononduting wall No eat enters or leaves te system e temperature falls from to e gas does work BC at appens to te pressure? a) inreases b) dereases ) stays te same at appens to te internal energy? a) inreases b) dereases ) stays te same. at appens to te entropy of te gas? a) inreases b) dereases ) stays te same. Fig. 1.1, p. 74 Slide

12 Carnot Cyle, C to D e gas is plaed in ontat wit te old temperature reservoir C to D is an isotermal ompression e gas expels energy C ork CD is done on te gas CD nr C ln V D V C at appens to te internal energy of te gas? a) It inreases b) It dereases ) It stays te same at appens to te pressure? a) It inreases b) It dereases ) It stays te same at appens to te entropy of te gas? a) It inreases b) It dereases ) It stays te same Fig. 1.1, p. 74 Slide

13 Carnot Cyle, D to A D to A is an adiabati ompression. e gas is again plaed against a termally nononduting wall So no eat is exanged wit te surroundings e temperature of te gas inreases from to e work done on te gas is CD at appens to te pressure? a) inreases b) dereases ) stays te same at appens to te internal energy? a) inreases b) dereases ) stays te same at appens to te entropy of te gas? a) inreases b) dereases ) stays te same Fig. 1.1, p. 74 Slide

14 Carnot yle effiieny e effiieny of te Carnot yle depends only on te temperatures of te ot and old reservoirs: eng e 1 No engine operating between tese two temperatures is more effiient tan te Carnot engine A eat engine operates between two reservoirs at temperatures of 0 C and 00 C. at is te maximum effiieny possible for tis engine? C 9 a) emax earnot H 0.49

15 Example An engine does 0900 J of work and rejets 70 J of eat into a old reservoir at 98K. at is te smallest possible temperature of te ot reservoir? a) e K J 80J J J e 1.74 arnot.6 98K.6 80J 1

16 Heat pumps and refrigerators Heat engines an run in reverse Send in energy Energy is extrated from te old reservoir Energy is transferred to te ot reservoir is proess means te eat engine is running as a eat pump A refrigerator is a ommon type of eat pump An air onditioner is anoter example of a eat pump In te sout or in Asia, people often use eat pumps to eat omes One usually rates refrigerators in terms of teir oefiient of performane CO CO 1 / 1 A reversible energy run as refrigerator as te igest possible CO CO rev 1 / 1

17 Example A Carnot refrigerator maintains te food inside it at 76 K wile te temperature of te kiten is 98 K. e refrigerator removes.00x 10 4 J of eat from te food. How mu eat is delivered to te kiten? As a e e Carnot engine run in reverse takes meanial energy to move and deposit arnot Carnot eat engine, 1 1 in te kiten. from te inside of we know tat x10 4 J te refrigerator.x10 4 J

18 ΔS reversible Entropy an only be alulated from a reversible pat, and must be done tat tat way even if te system atually follows an irreversible pat o alulate te entropy for an irreversible proess, model it as a reversible proess en energy is absorbed, is positive and entropy inreases en energy is expelled, is negative and entropy dereases S ln(probability). A disordered state wit energy and matter spread out everywere is more probable tan aving all of te energy stored in an organized way tat an be used to do work.

19 Example Δ S ΔS ΔS e surfae of te Sun is approximately at 5700 K, and te temperature of Eart s surfae is approximately 90 K. at entropy ange ours wen 1000 J of energy is transferred by eat from te Sun to Eart? e sun loses of sun eart ΔS sun e eart gains of sun eart e total entropy ange is: + ΔS eart eat and terefore dereases eat and terefore inreases its entropy by te amount 1 eart 1 sun 1000 J its entropy by te amount 1 90 K K.7J / K

20 Example A power plant as been proposed tat would make use of te temperature gradient in te oean. e system is to operate between 0.0 C (surfae-water temperature) and 5.00 C (water temperature at a dept of about 1 km). (a) at is te maximum effiieny of su a system? (b) If te useful power output of te plant is 75.0 M, ow mu energy is absorbed per our? () In view of your answer to (a), do you tink su a system is wortwile (onsidering tat tere is no arge for fuel)? a) b) e max e e arnot 1 1 ) at is te energy required e 78 9 ( 75M )( 600s) x10 to pump te water? 1 J

21 Story of Hawaiian deep water projet Keaole sits at a point were underwater land slopes sarply down into te sea, it was a plae were warm water an be piped from te surfae of te sea and old water an be piped from depts of about a alf-mile. A proess alled oean termal energy onversion, or OEC, used te temperature differene between ot and old sea water to produe 50 K of eletriity at Keaole in 199. e proess worked but it was uneonomial. KAILUA, HAAI'I Koyo USA Corp., a ompany selling deep-sea water from Keaole Hawai'i, is expanding its plant and as applied to sell te water in te United States e ompany is produing more tan 00,000 bottles a day and says it an't keep up wit demand in Japan, were it sells 1.5 liter bottles of its MaHaLo brand for $4 to $6 ea.