Chemistry 163B Refrigerators and Generalization of Ideal Gas Carnot (four steps to exactitude) E&R pp 86-91, Raff pp.

Size: px

Start display at page:

Download "Chemistry 163B Refrigerators and Generalization of Ideal Gas Carnot (four steps to exactitude) E&R pp 86-91, Raff pp."

Kory Morris
5 years ago
Views:

1 statements of the Second Law of hermodynamics Chemistry 163B Refrigerators and Generalization of Ideal Gas Carnot (four steps to exactitude) E&R pp 86-91, Raff pp Macroscopic properties of an isolated system eventually assume constant values (e.g. pressure in two bulbs of gas becomes constant; two block of metal reach same ) [Andrews. p37] 2. It is impossible to construct a device that operates in cycles and that converts heat into work without producing some other change in the surroundings. Kelvin s Statement [Raff p 157]; Carnot Cycle 3. 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, refrigerator 4. In the neighborhood of any prescribed initial state there are states which cannot be reached by any adiabatic process ~ Caratheodory s statement [Andrews p. 58] 1 2 roadmap for second law goals for lecture[s] 1. Phenomenological statements (what is ALWAYS observed) 2. Ideal gas Carnot [reversible] cycle efficiency of heat work (Carnot cycle transfers heat only at and L ) 3. Any cyclic engine operating between and L must have an equal or lower efficiency than Carnot OR VIOLAE one of the phenomenological statements (observations) 4. Generalize Carnot to any reversible cycle (E&R fig 5.4) 5. Show that for this REVERSIBLE cycle 0 ( inexact differential) but 0 ( something special about ) L Carnot in reverse: refrigerators and heat pumps Show that ε ideal gas rev Carnot ε any other machine otherwise one of the phenomenological statements violated Not only for ideal gas but ε ideal gas rev Carnot = ε any other rev Carnot ersible ersible ds ds 0 and S is SAE FN CION 6. S, entropy and spontaneous changes 3 4 Carnot engine arithmetic (for below table see handout) remember Carnot engine does net work on surroundings and net heat q (I) + q L(III) = total 5 6 1

perpetual motion of the first kind (produce work with no heat input) Escher Waterfall remember Carnot engine cyclic machine (water flows downhill to pillar on left; round and round) no energy

the second kind (produce work extracting heat from cooler source to run machine at warmer temperature) refrigerators and heat pumps: running the Carnot cycle in REVERSE Brownian Ratchet it only

> 2 http://wapedia.mobi/en/brownian_ratchet#1. 9 however the Carnot cycle is a combination of reversible processes 10 heat pumps and refrigerators (fig. 5.

2 perpetual motion of the first kind (produce work with no heat input) Escher Waterfall remember Carnot engine cyclic machine (water flows downhill to pillar on left; round and round) no energy input work on surroundings does net work on surroundings but extracts heat from surroundings at q (I) < 0 VIOLAES 1 st LAW gives off heat to surroundings at L q L(III) > perpetual motion of the second kind (produce work extracting heat from cooler source to run machine at warmer temperature) refrigerators and heat pumps: running the Carnot cycle in REVERSE Brownian Ratchet it only makes sense to talk about running a process in REVERSE for reversible processes (on a PV diagram there is no reverse process for irreversible expansion against constant P ext ) get work only if 1 > however the Carnot cycle is a combination of reversible processes 10 heat pumps and refrigerators (fig E&R) 3rd reverse Carnot cycle: a refrigerator q < 0 heat out at w sys = surr > 0 work done ON system 11 q L >0 heat in at L cyclic process IV adiabatic expansion III isothermal expansion II adiabatic compression I isothermal compression 12 2

Carnot refrigerator arithmetic (just negatives of Carnot engine) for below table see handout reverse Carnot cycle: a heat pump q < 0 heat out at w sys = surr > 0 work done ON system Coefficient of

3 Carnot refrigerator arithmetic (just negatives of Carnot engine) for below table see handout reverse Carnot cycle: a heat pump q < 0 heat out at w sys = surr > 0 work done ON system Coefficient of performance of refrigerator: q surr _ III q III L r CR w w total total L q L >0 heat in at L cyclic process IV adiabatic expansion III isothermal expansion II adiabatic compression I isothermal compression E&R eqn goodness of performance one more important FACOID about Carnot machine Since refrigerators and heat pumps are just Carnot machines in reverse, the relationship for û= total /q I = ( - L) / holds for these cycles. However only for engines does it represent the efficiency or goodness of performance (work total out /heat in ). For a refrigerator performance is (heat in at L /work total ) qfrom freezer qiii L r CR wtotal wtotal L E&R eqn 5.45 HW#6 P28 (E&RP5.33) For a heat pump performance is (heat out at /work total ) q q E&R eqn 5.44 given off to room I hp CHP wtotal wtotal L ( > 1) for complete reversible Carnot cycle: " d= =? 0 " dh= H =? 0 " rev = q =? π0 " dw rev = w =? π0 B ALSO LE S LOOK A: cycle rev one more important FACOID about Carnot machine Generalization of Ideal Gas Carnot Cycle P1 nr ln P2 I 0 (adiabatic) II P1 nrl ln P2 III L 0 (adiabatic) IV rev? 0 cycle have we uncovered a new state function entropy (S)?? S ds rev does it just apply to ideal gas Carnot cycle??? Our work on the (reversible) Carnot Cycle using an ideal gas as the working substance lead to the relationship total - L = = However, the second law applies to machines with any working substance and general cycles. he following presentation shows that a (reversible) Carnot machine with any working substance cannot have rev any ws > carnot ideal gas. REVERSING the directions of the ideal gas and any rev Carnot (e.g. Raff, pp ) shows that rev any ws = carnot ideal gas One can also show (E&R Fig 5.4, Raff ) that any Carnot machine has " rev / =0 and that any reversible machine cycle can be expressed as a sum of Carnot cycles. 18 3

4 Carnot machine Carnot machine: cyclic, reversible, machine exchanging heat with environment at only can x > c for any Carnot machine X vs ideal gas Carnot machine C??? (generalizing ε rev =1- L / that was derived for reversible ideal gas Carnot cycle) machine X: any working substance (not necessarily ideal gas) exchanges heat at only two temperatures and L two temperatures, L A Carnot engine: forward Carnot cycle engine heat pump Carnot heat pump: reverse Carnot cycle L wtotal 1 = applies to both forward and q reverse Carnot cycles the machine X operates as a Carnot engine doing work, while the machine C operates as a reverse Carnot ideal gas engine acting as a heat pump. In this example the work done by X, (w t ) x is totally used by C: (w t ) c = (w t ) x CAN ε x > ε Carnot : worksheet for numerical example (all w x goes into w Carnot ) CAN ε x > ε Carnot : worksheet for numerical example (all w x goes into w Carnot ) RY: ε x =0.6 > ε carnot =0.5 ; specify heat into x (q u ) x =100 J and all (w out ) x =(w in ) carnot RY: ε x =0.6 > ε carnot =0.5 ; specify heat into x (q u ) x =100 J and all (w out ) x =(w in ) carnot û x= J (wt ) x =-60J -120J +100J -120J engine violation 0.6 of 2 nd Law 0.5 heat pump J -60J (q u ) C =-120J û C=0.5-40J -60J t t q = ε= ε =q +q L+w t=0 q w=t ε for each of the cyclic processes w total =0 Clausius (q ) total = -20J (into u ) (q L ) total = +20J (out of L ) 21 ε= t q w total =0 t t q = ε =q +q L+w =0 (q ) total = -20J (into u ) w=t ε (q L ) total = +20J (out of L ) 22 for each of the cyclic processes violation of 2 nd Law Clausius Statement of Second Law (numerical example) general proof: can x > c for Carnot machine X vs ideal gas Carnot machine C??? we assumed an ε x > ε C for ideal gas Carnot and calculated w total =0 (q L ) total =+20J (into combo engine out of cold reservoir at lower ) (q ) total =-20J (out of combo engine into heat reservoir at upper ) B: 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 if second law holds x > c cannot be true 23 (q ) < 0 (q ) > 0 c x -(w ) (w ) -(w ) ε = = t c t x t x c (q ) (q ) x (q ) c c x (w ) = (q ) ε c (w ) = -(q ) ε t x c t x x x (q ) + (q ) + (w ) = 0 (q ) + (q ) + (w ) = 0 L c c t c L x x t x ) x -(q ε ε t o g iv e - ( w ) = ( w ),, a n d ( q ) ( q ) C x t x t C C x (q ) ε ε C x C ε 1 (q ) (q ) (q ) (q ) w h ic h w ill b e < 0, fo r ε > ε x ε C (q L ) to ta l (q L ) x (q L ) C -(q ) x - (w t ) x + -(q ) C + (w t ) x (q ) to ta l to ta l x C x x C 24 4

5 general proof: can x > c for Carnot machine X vs ideal gas Carnot machine C??? HW#4 Prob. 23 So (q ) total < 0 and (q L ) total = - (q ) total > 0 with w total =(w t ) x + (w t ) c =0 so what s the problem??? q transferred L with no net work violates Clausius statement of second law so what s the solution??? x > c is not a condition consistent with the Second Law setup for (HW prob #23) ε x efficiency greater than Carnot RY: ε x =0.6 > ε carnot =0.5 ; specify heat into x (q u) x=100 J and all (2/3)(w out) x=(w in) carnot ε x > ε C (no way!!; slides #20-#24); but can ε x < ε C (ε x =0.5 < ε c = 0.6)?? 100 J -83⅓ J engine -50 J J 50 J heat pump 33⅓J û X=0.6 û C=0.5 (q ) X =100J (w total ) x =60J (w total ) c =+40J (q ) c =-80J (w sys to surr ) x =20J (w internal to C ) X =-40J calc: (q L ) x, (q L ) c, (q L ) total, (q ) total, (w) total ANOHER CONSEQENCE OF ASSMING ε x > ε C (2nd Law Statements) 27 derivation a la slide #21 with input of (q u) x=100j gives (q u) total=+16⅔ (out of u) (q L) total=-16 ⅔ (into L) heat from hotter to cooler ε x =0.5 < ε c = 0.6 is OK (no violation) if X is machine and C is heat pump (real machine can be less efficient than reversible Carnot heat pump) Clausius 28 CAN ε x =0.5 < ε c = 0.6 if X is a some cyclic REVERSIBLE machine moral of the story for machines exchanging heat at and L NO: for reversible X and reversible Carnot, WHY: can now run X as heat pump ε x =0.5 heat pump and Carnot as machine ε c = 0.6 engine -120J J same calc as slides J -40J same violation as ε x > ε c ε Machine must be less than or equal to ε HeatPump 29 ε reversible Carnot ideal gas = (1- L / ) ε reversible any working substance cannot be > ε reversible Carnot ideal gas (slides 21-22) ε reversible any working substance cannot be < ε reversible Carnot ideal gas (slide 29) ε reversible IN GENERAL = ε reversible Carnot ideal gas = (1- L / ) (ε irreversible ) engine < ε maximum = (1- L / ) (slide 28) ε maximum = ε reversible = (1- L / ) general 30 5

generalization to arbitrary cycle (E&R Fig. 5.4, Raff Fig. 4.

sum of Carnot cycles 31 have shown generally for any reversible CYCLIC engine operating between and L : wtotal wtotal L 1 so now 1 thus L 1 1

reversible cycle 1. Phenomenological statements (what is ALWAYS observed) 2.

Any cyclic engine operating between and L must have an equal or lower efficiency than Carnot OR VIOLAE one of the phenomenological statements

6 generalization to arbitrary cycle (E&R Fig. 5.4, Raff Fig. 4.11) a REALLY BIG RESL: connecting ε and entropy WHA about reversible cycle in general (maybe not only two s)? sum of Carnot cycles 31 have shown generally for any reversible CYCLIC engine operating between and L : wtotal wtotal L 1 so now 1 thus L 1 1 L 0 FOR HE CYCLE (ANY reversible cyclic process operating between and L ) 32 the BOOM LINE roadmap for second law so generally for this reversible cycle 1. Phenomenological statements (what is ALWAYS observed) 2. Ideal gas Carnot [reversible] cycle efficiency of heat work (Carnot cycle transfers heat only at and L ) DEFINE: ds reversible 3. Any cyclic engine operating between and L must have an equal or lower efficiency than Carnot OR VIOLAE one of the phenomenological statements (observations) 4. Generalize Carnot to any reversible cycle (E&R fig 5.4) ds reversible 0 and S is SAE FNCION Show that for this REVERSIBLE cycle 0 ( inexact differential) but 0 ( something special about ) L 6. SAE FNCION S, entropy and spontaneous changes (more to come) 34 did Sadi imagine his ideal gas Carnot Cycle?

7 37 38 Carnot vs Einstein??? (Milivoje Kostic) 39 7

Chemistry 163B Winter 2013 Clausius Inequality and ΔS ideal gas

Chemistry 163B Winter 2013 Clausius Inequality and ΔS ideal gas Chemistry 163B q rev, Clausius Inequality and calculating ΔS for ideal gas,, changes (HW#5) Challenged enmanship Notes 1 statements of the Second Law of hermodynamics 1. Macroscopic properties of an isolated