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Chemistry 163B Refrigerators and Generalization of Ideal Gas Carnot Chemistry 163B Winter 2020 Refrigerators and Generalization of Carnot Cycle statements of the Second Law of Thermodynamics Lecture 10 1. Macroscopic properties of an isolated system eventually assume Chemistry 163B constant values (e.g. pressure in two bulbs of gas becomes constant; two block of metal reach same T) [Andrews. p37] Refrigerators and 2. It is impossible to construct a device that operates in cycles and that converts heat into work without producing some other change Generalization of in the surroundings. Kelvin’s Statement [Raff p 157]; Carnot Cycle 3. It is impossible to have a natural process which produces no other Ideal Gas Carnot effect than absorption of heat from a colder body and discharge of heat to a warmer body. Clausius’s Statement, refrigerator (four steps to exactitude) 4. In the neighborhood of any prescribed initial state there are states which cannot be reached by any adiabatic process E&R pp 125-129, 132-139 Raff pp. 159-164 4th ~ Caratheodory’s statement [Andrews p. 58] E&R3rd pp 86-91, 109-111 1 2 perpetual motion of the first kind (produce work with no heat input) perpetual motion and the 1st and 2nd Laws of thermodynamics Escher “Waterfall” cyclic machines: cyclic machine 1st Law U=0 q=-w =w (water ‘flows downhill’ sys surr to pillar on left; ‘round and ‘round) no energy input nd 2 Law qin > wsurr work on surroundings It is impossible to construct a device that operates in cycles and that converts heat into work without VIOLATES 1st LAW producing some other change in the surroundings. Kelvin’s Statement [Raff p 157]; Carnot Cycle 3 4 perpetual motion of the second kind (produce work extracting heat remember Carnot engine from cooler source to run machine at warmer temperature) Brownian Ratchet • does net work on surroundings BUT • extracts heat from surroundings at TU qU(I) < 0 • gives off heat to surroundings at TL qL(III) > 0 get work only if T1 > T2 https://en.wikipedia.org/wiki/Brownian_ratchet. 5 6 1 Chemistry 163B Winter 2020 Refrigerators and Generalization of Carnot Cycle roadmap for second law goals for lecture[s] 1. Phenomenological statements (what is ALWAYS observed) • Carnot in reverse: refrigerators and heat pumps 2. Ideal gas Carnot [reversible] cycle efficiency of heat → work (Carnot cycle transfers heat only at TU and TL ) • Show that εideal gas rev Carnot εany other machine otherwise one of 3. Any cyclic engine operating between TU and TL must have an the phenomenological statements violated equal or lower efficiency than Carnot OR VIOLATE one of the phenomenological statements (observations) • Not only for ideal gas but εideal gas rev Carnot = εany other rev Carnot 4. Generalize Carnot to any reversible cycle (E&R fig 5.4) 5. Show that for this REVERSIBLE cycle dq dq • dSreversible dS reversible 0 qqUL0 (dq inexact differential) TT but and Sis STATE FUNCTION q q dq UrevL 0( something special about ) TTUL T 6. S, entropy and spontaneous changes 7 8 remember Carnot engine Carnot engine arithmetic (for below table see handout) • does net work on surroundings and • net heat qU(I) + qL(III) = -wtotal 9 10 refrigerators and heat pumps: running the Carnot cycle in REVERSE heat pumps and refrigerators (fig. 5.20 E&R4th) 5.17 ER3rd Figure 5.20 Application • it only makes ‘sense’ to talk about of the principle of a reverse Carnot heat running a process in REVERSE for engine to refrigeration and household heating. (q ) <0 reversible processes (on a PV The reverse Carnot heat revcarnot U engine can be used to diagram there is no ‘reverse’ process induce heat flow from a wrevcarnot>0 cold reservoir to a hot (qrevcarnot)L>0 for irreversible expansion against reservoir with the input w <0 i.e. w of work. The hot surr in constant Pext) reservoir in both (a) and (b) is a room. The cold reservoir can be configured as the inside • however the Carnot cycle is a of a refrigerator or outside ambient air, combination of reversible processes earth, or water for a heat pump refrigerator heat pump 11 12 2 Chemistry 163B Winter 2020 Refrigerators and Generalization of Carnot Cycle Carnot refrigerator arithmetic (just negatives of Carnot engine) reverse Carnot cycle: a refrigerator for below table see handout wsys=-wsurr > 0 qU < 0 heat out at TU work done ON system qL >0 heat in at TL cyclic process IV adiabatic expansion III isothermal expansion q Coefficient of performance surr_ III qTIII L II adiabatic compression C of refrigerator: rR I isothermal compression wwTTtotal total U L 13 eqn 5.69 ER4th 5.45 E&R3rd 14 reverse Carnot cycle: a heat pump “goodness of performance” w =-w > 0 sys surr Since refrigerators and heat pumps are just Carnot machines qU < 0 heat out at TU work done ON system in reverse, the relationship for =-wtotal/qI = (TU-TL)/TU holds for these cycles. However only for “engines” does it represent the efficiency or “goodness of performance” (worktotal out/heat in). For a refrigerator performance is (heatin at TL/worktotal) q from freezer qTIII L C eqn 5.69 ER4th 5.45 E&R3rd rR wwTT total total U L † HW5 #28 (E&R4th P5.33) For a heat pump performance is (heat /work ) qL >0 heat in at TL cyclic process out at TU total IV adiabatic expansion q q T C given off to room I U ( > 1) III isothermal expansion hp HP wwTTtotal total U L II adiabatic compression eqn 5.68 ER4th 5.44 E&R3rd I isothermal compression 15 16 one more important FACTOID about Carnot machine one more important FACTOID about Carnot machine for complete reversible Carnot cycle: P nRT ln 1 dq U P constant TTrev 2 ∮ dU= U = 0? U T I T U ∮ dH= H = 0? dq adibatic rev 0 ∮ dq =q = ?0 T dqrev rev II 0 ∮ dw =w = ?0 ? rev P1 nRTL ln T dq P cycle constant TTrev 2 BUT ALSO LET’S LOOK AT: L T III T L dq adibatic rev 0 T dqrev IV have we uncovered a new state function entropy (S) ?? T dq cycle SdS rev does it just apply T to ideal gas Carnot cycle ??? 17 18 3 Chemistry 163B Winter 2020 Refrigerators and Generalization of Carnot Cycle Generalization of Ideal Gas Carnot Cycle Carnot machine Our work on the (reversible) Carnot Cycle using an ideal 1. gas as the ‘working substance” lead to the relationship Carnot machine: cyclic, reversible, machine exchanging heat with environment at only -wtotal T U - T L == two temperatures TU, TL qTUU However, the second law applies to machines with any ‘working substance’ and general cycles. Carnot engine: ‘forward’ Carnot cycle 2. The following presentation shows that a (reversible) Carnot machine Carnot heat pump: ‘reverse’ Carnot cycle with ‘any working substance’ cannot have rev any ws > carnot ideal gas . 3. REVERSING the directions of the ideal gas and ‘any rev Carnot’ (e.g. Raff, pp 160-162) shows that = T w rev any ws carnot ideal gas 1 L =total applies to both forward and TqUU 4. One can also show (E&R4th Fig 5.15 Fig 5.433rd , Raff 162-164 ) that any Carnot machine has ∮ dqrev /T =0 and that any reversible reverse Carnot cycles machine cycle can be expressed as a sum of Carnot cycles. 19 20 can x > c for any Carnot machine X vs ideal gas Carnot machine C ??? CAN εx> εCarnot : worksheet for numerical example (all wx goes into wCarnot ) (generalizing εrev=1-TL/TU that was derived for reversible ideal gas Carnot cycle) machine X: any ‘working substance” (not necessarily ideal gas) TRY: εx=0.6 > εcarnot =0.5 ; specify heat into x (qu)x=100 J and all (wout)x= (win)carnot exchanges heat at only two temperatures TU and TL x=0.6 +100J (wt)x=-60J -120J nd heat pump engine heat pump engine viol0.6ation of 2 Law 0.5 -40J -60J +60J +60J (qu)C=-120J C=0.5 the machine X operates as a Carnot engine doing work, while the machine C operates as a reverse Carnot ideal gas engine -w wtotal=0 acting as a heat pump. t -w t q=U Clausius In this example the work done by X, (wt)x is totally used by C: ε (q ) = -20J (into T ) ε= U=qUt +qL +w =0 U total u (wt)c = (wt)x q w=-tU q ε U (qL)total=+20J (out of TL) 21 22 for each of the cyclic processes CAN εx> εCarnot : worksheet for numerical example (all wx goes into wCarnot ) Clausius Statement of Second Law (numerical example) TRY: εx=0.6 > εcarnot =0.5 ; specify heat into x (qu)x=100 J and all (wout)x= (win)carnot we assumed an εx > εCig for ideal gas Carnot and calculated wtotal=0 +100J -120J (qL)total=+20J (into ‘combo engine’ out of cold reservoir at Tlower) 0.6 0.5 (qU)total=-20J (out of ‘combo engine’ into heat reservoir at Tupper) -40J -60J +60J +60J BUT: It is impossible to have a natural process which produces no other effect than absorption of heat from a colder body and discharge of heat to a warmer body. Clausius’s Statement -w t wtotal=0 -w t q=U ε= ε U=qUt +qL +w =0 (qU)total= -20J (into Tu) specific example showed: q w=-tU q ε if second law holds > cannot be true U (qL)total=+20J (out of TL) x cig nd 23 for x= any Carnot ideal gas or not 24 for each of the cyclic processes violation of 2 Law 4 Chemistry 163B Winter 2020 Refrigerators and Generalization of Carnot Cycle general proof: can x > c for Carnot machine X vs ideal gas Carnot machine C ??? general proof: can x > c for Carnot machine X vs ideal gas Carnot machine C ??? So (qU)total < 0 and (qL)total = - (qU)total > 0 (q ) < 0 (q ) > 0 Uc Ux with wtotal=(wt)x + (wt)c=0 -(w ) (w ) -(w ) tc tx tx ε c == ε x = so w hat’s the problem ??? (qUc ) (q Uc ) (qUx ) (wtx ) = (q Uc ) ε c (wtx ) = -(q Ux ) ε x q transferred TL → TU with no net work (qLc ) + (q Uc ) + (w tc ) = 0 (qLx ) + (q Ux ) + (w tx ) = 0 violates Clausius statement of second law so w hat’s the solution ??? -(qUx ) εεCx to give -(w)=(w),tx tC , and (q) UC (q) U x (qUC ) εε x C x > c is not a condition consistent with the Second Law ε (q ) (q ) (q ) (q )1 x w h ic h w ill be < 0 , fo r ε > ε Utotal Ux UC UxxC so any Carnot machine, ideal gas or not ε C (q ) (q ) (q ) -(q ) - (w ) + -(q ) + (w ) (q ) Ltotal L x LC U x tx UC tx U total 25 26 any carnot carnot ideal gasTTT U L U so far setup for (HW prob #23) ε efficiency greater than Carnot HW#4 Prob.
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