production and lost for some irreversible processes Francesco Di Liberto

To cite this version:

Francesco Di Liberto. and lost work for some irreversible processes. Philosophical Magazine, Taylor & Francis, 2007, 87 (3-5), pp.569-579. ￿10.1080/14786430600909006￿. ￿hal-00513741￿

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Entropy production and lost work for some irreversible processes

Journal: Philosophical Magazine & Philosophical Magazine Letters

Manuscript ID: TPHM-06-Apr-0109.R1

Journal Selection: Philosophical Magazine

Date Submitted by the 09-Jun-2006 Author:

Complete List of Authors: di Liberto, Francesco; Università di Napoli, INFN.CNR-CNISM, Dipartimento di Scienze fisiche

Keywords: statistical physics, , transformations

Keywords (user supplied): irreversibility, entropy production, Clausius inequality

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1 1 2 3 Entropy production and lost work for some irreversible processes 4 5 6 Francesco di Liberto 7 8 Dipartimento di Scienze Fisiche 9 10 Università di Napoli “Federico II” 11 Complesso universitario Monte S. Angelo 12 Via Cintia - 80126 Napoli (Italy) 13 [email protected] 14 tel. + 39 081 676486 - fax + 39 081 676346 15 16 For Peer Review Only 17 In this paper we analyse in depth the Lost Work in an (i.e. 18 WLost = WRe v −WIrrev ) . This quantity is also called ‘degraded ’ or ‘Energy 19 unavailable to do work’. Usually in textbooks one can find the relation W ≡ TS , 20 Lost U 21 which, for many processes , is not suitable to evaluate the Lost Work. Here we find for 22 W a more general relation in terms of internal and external Entropy production, π 23 Lost int 24 and π ext , quantities which enable also to write down in a simple way the Clausius 25 inequality. Examples are given for elementary processes. 26 27 28 Keywords : irreversibility, entropy production, 29 30 1. Introduction 31 32 33 Entropy production, a fascinating subject, has attracted many physics researches even in cosmological physics 34 35 [1], moreover in the past ten years there has been renewed interest in thermodynamics of engines; many 36 papers address issues of maximum , maximum efficiency and minimum Entropy production both from 37 38 practical and theoretical point of view [2-6]. 39 40 One of the main points in this field is the analysis of Available Energy and of the Lost Work. Here we 41 42 give a general relation between Lost Work and Entropy production, merging together the pioneering papers of 43 44 Sommerfeld (1964), Prigogine (1967), Leff (1975) and Marcella (1992), which contain many examples of 45 such relation, and the substance-like approach to the Entropy of the Karlsruhe Physics Course due mainly to 46 47 Job (1972), Falk, Hermann and Schmid (1983) and Fuchs (1987). 48 49 It is well known [7-13] that for some elementary irreversible process, like the irreversible isothermal 50 51 expansion of an in contact with an heat source T , the work performed by the gas in such process W 52 53 is related to the reversible work WRe v (i.e. the work performed by the gas in the corresponding reversible 54 55 process) by means of the relation 56 (1) 57 W = WRe v − T SU 58 where S is the total entropy change of the universe (system + external heat sources). The degraded energy 59 U 60 TSU is usually called WLost ‘the Lost work’, i. e.:

WLost = WRe v − W

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2 1 2 3 4 the work that could have been performed in the related reversible process (here the reversible expansion); it is 5 also called ‘energy unavailable to do work’. 6 7 By the energy balance, the same relation holds for the amount of extracted from the source T 8 9 = − 10 Q QRe v T SU 11 12 13 Therefore TSU is also called the ‘Lost heat QLost ’, i.e. the additional heat that could have been drawn from 14 ‡ 15 the source in the related reversible process . 16 For Peer Review Only 17 The total variation of Entropy , SU , is usually called ‘Entropy production’. The second Law claims that 18 19 SU ≥ 0 20

21 The relation between Entropy production and WLost (or QLost ) is the main subject of this paper. In Sec.3 22 23 we will find a relation more general than relation (1) . To introduce the subject let us remind the steps that 24 25 lead to the relation (1).[9] 26 27 For a process (A—>B) in which the system (for example, the ideal gas) absorbs a given amount of 28 heat Q from the heat source at T and performs some work W , the entropy production of the 29 ext 30 31 Universe, i. e. the variation of Entropy of the system+variation of Entropy of the external source, , is 32 Q 33 SU ≡ S sys + Sext = S sys − (2) 34 Text 35 36 From the energy balance U sys =Q −W it follows 37 38 Text SU = Text S sys − U sys −W (3) 39 40 If the process is reversible then S = 0 and W ≡ W = T S − U , 41 U Re v ext sys sys 42 = − = 43 Therefore Text SU WRe v W WLost (4) 44 45 which defines the Lost Work and proves relation (1). There are however some irreversible processes for 46 which relation (4) is not suitable to evaluate the Lost Work, for example the irreversible adiabatic processes, 47 48 [11] in which there is some W , some Entropy production S , but no external source T . 49 Lost U ext 50 In general for an irreversible process, S > 0 , . from relation (3), it follows 51 U 52 W < W (5) 53 Re v 54 55 i.e. the Reversible Work is the maximum amount of work that can be performed in the given process 56 57 58 59 ‡ For an irreversible compression TSU is sometime called WExtra or QExtra [10] i.e. the excess of work performed on the 60 system in the irreversible process with respect to the reversible one (or the excess of heat given to the source

in the irreversible process). In a forthcoming paper we will show that WExtra is related to the environment temperature and to the entropy productions

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2 In Sec. 3 we evaluate WLost for some simple irreversible processes, refine relation (4) taking account of 3 4 internal and external irreversibility and give a general procedure to evaluate the Lost Work. Such procedure 5 6 follows from the analysis of Sec.2 where it has been shown that often the total Entropy production is due to 7 8 the entropy production of the sub-systems. When the subsystems are the system and the external source, their 9 entropy productions has been called respectively internal and external, i.e. S = π + π 10 U int ext 11 In Sec.2 the entropy balance and the entropy productions for irreversible processes are analyzed by 12 13 means of the substance-like approach. In the following the heat quantities Q ’s are positive unless explicitly 14 15 stated and the system is almost always the ideal gas. 16 For Peer Review Only 17 2. Entropy production for irreversible processes 18 19 In this Section are given some examples of Entropy production for elementary processes. First we analyse the 20 21 reversible isothermal expansion (A-->B) of one mole of monatomic ideal gas at temperature T which 22 23 receives the heat Q from a source at temperature T . For the ideal gas we have 24 25 S gas = Sin − Sout (6) 26 27 QRe v VB 28 where Sout = 0 and the Entropy which comes into the system is S In = = R ln , since 29 T VA 30 B B 31 V Q = δQ = PdV = RT ln B . The heat which flows from the heat source into the gas is Q , which 32 Re v ∫ Re v ∫ V Re v 33 A A A 34 is also the work performed by the system in the reversible isothermal expansion. The increase in Entropy for 35 36 B δQ V 37 = = B the gas is S gas ∫ R ln ; R=8.314 J/mol. K° is the universal constant for the gases. 38 A T VA 39 40 For the heat Source it holds 41 ext ext ext 42 S = Sin − Sout (7) 43 44 − Q Q where S ext = rev , S ext = 0 and S ext = Re v . 45 T in out T 46 47 For the Universe S = S + S ext = 0. In this example (a reversible process) the Entropy is conserved. 48 U gas 49 50 Let us turn to the irreversibility and take a look at the irreversible isothermal expansion at temperature T of 51 one mole of monatomic ideal gas from the state A to the state B (let, for example, P = 4P ). This can be done 52 A B 53 by means of thermal contact with a source at temperature T or at temperature greater than T . 54 55 I) Thermal contact with a source at temperature T = T 56 ext 57 58 59 60

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4 1 2 3 4 m P ext ≡ P 5 B 6 7 P 8 A 9 10 11 T 12 13 14 15 Figure 1. Ideal gas in thermal contact with the heat source T 16 For Peer Review Only 17 18 The ideal gas, in contact with the source T , is at P = 4P = 4P ext by means of some mass m on the 19 A B 20 mass-less piston of area Σ . Let V be its . The mass is removed from the piston and the ideal gas 21 A 22 = ext 23 performs an isothermal irreversible expansion and reaches the volume VB at pressure PB P . In the 24 (V −V ) 3 25 expansion the gas has performed the work W = P Σ B A = RT (8) 26 ext Σ 4 27 28 By means of the Energy Balance we can see that the heat which lives the source and goes into the system is 29 30 Q = W (9) 31 32 The increase of the Entropy in the ideal gas is the same as for the reversible process i.e. 33 B δQ V 34 S = Re v = R ln B = R ln 4 35 gas ∫ T V 36 A A 37 We can verify that now relation (6) is not fulfilled, in fact S = 0, since no Entropy goes out from the gas , 38 out 39 Q 3 40 and S In = = R , so we can see that S gas ≠ Sin − Sout . 41 T 4 42 43 To restore the balance one must add to the right-hand side a quantity π int , the Entropy production due to the 44 45 internal irreversibility 46 = − + π 47 S gas Sin Sout int (10) 48 3 49 We see that π is π = R ln 4 − R . 50 int int 4 51 52 On the same footing, to take in account the external irreversibility, we introduce the quantity π ext which is 53 54 defined by the general relation 55 56 ext ext ext S = Sin − Sout + π ext (11) 57 58 Whith the constraint that S = S + S ext = π + π (12) 59 U gas in ext 60 − Q Q In this irreversible process, since S ext = , S ext = 0 and S ext = it is easy to verify that π = 0 T in out T ext

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5 1 2 There is no external irreversibility, there is no external Entropy production. It is well known indeed that the 3 4 isothermal exchanges of heat between heat reservoirs are reversible. 5 π 6 In the following , as consequence of definitions (11) and (12) we define U , the entropy production due to 7 8 the internal and external entropy productions : 9 = + ext = π = π + π 10 SU S syst S U int ext (13) 11 12 The entropic balance (10) is reported in Fig. 2, where the circle is the system (i.e. the ideal gas) 13 14 15 16 For Peer Review Only 17 18 π = S (↑) − S 19 int syst In 20 21 22 23 24 25 26 SIn 27 28 29 30 Figure 2. The entropic balance for the ideal gas. 31 32 33 The entropic balance for the heat source (11) is reported in Fig. 3, where the square is the heat source T 34 35 36 37

38 ext 39 S out 40 41 42 43 44 π = S ext (↓) + S ext = 0 45 ext out 46 47 48 Figure 3. The entropic balance for the heat source T 49 50 51 52 4 II ) Thermal contact with an heat source at T >T (for instance T = T ). 53 ext ext 3 54 55 In this case we want to make the previous irreversible expansion of the gas ,from the state ( P ,V ,T) to the 56 A A 57 state (P ,V ,T ) by means of the heat Q coming from the heat source T > T . The gas which is in the state 58 B B ext 59 60 ()PA ,VA ,T is brought in thermal contact with the heat source Text and the mass on the piston is removed. The thermal contact with the source is now shorter than before in order to not increase the temperature of the gas. It is clear that in such new process there is some external irreversibility, some external Entropy

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2 production π ext , because the heat Q flows from a hotter ( Text ) to a colder source ( T ) . For such irreversible 3 4 Q Q 5 flow of the heat Q it is well known that the change in entropy is S = − . This quantity will be our 6 T Text 7 Q Q 8 external entropy production i.e. π = − ; therefore 9 ext T Text 10 11 ext QRe v Q Q Q 12 π U = S syst + S = − = π int + π ext = π int + − (14) 13 T Text T Text 14 15 which gives for the internal entropy production the result 16 For Peer Review Only Q Q 17 π = Re v − (15) 18 int T T 19 20 More examples of Entropy productions in irreversible processes are given in ref [11] . 21 22 To conclude this Section we remark that the global Entropy change is related the local Entropy productions by 23 24 means of the following relation : 25 26 π ≡ S = S + S ext = π + π 27 U U sys int ext 28 29 The second law of the thermodynamics claims that the global Entropy production is greater or equal zero 30 31 i.e.SU ≥ 0 , but from these examples we see that also π int ≥ 0 and π ext ≥ 0 , this suggests that in each 32 33 subsystem the Entropy cannot be destroyed. On the other hand, from the substance-like approach of 34 35 Karlsruhe, i.e. from the local Entropy balance, (that we can write for each subsystem) this condition is 36 completely natural § [7],[12]. The proof that for each subsystem π ≥ 0 has been given in Sommerfeld (1964) 37 38 [7]. Moreover as a consequence of this formulation of the Second law of the Thermodynamics we have the 39 40 following formulation of Clausius inequality: if the system makes a whatsoever cycle, relation (10) implies: 41 42 43 δQ δQ 44 = = + π ⇒ ≤ 0 ∫ dS syst ∫ int ∫ 0 (16) 45 Tsyst Tsyst 46 47 where δQ > 0 , i.e. it is positive, when it comes into the system and T is the system’s temperature in each 48 syst 49 step of the cycle. This formulation of the Clausius inequality seems more simple and elegant than the 50 51 traditional one. Similarly for the external source, when it makes a whatsoever cycle, from relation (11), it 52 53 follows 54 55 56 57 58 59 60 § This local formulation of the second law was also given by Prigogine [7]. In a recent paper [14] it has been pointed out that those irreversible processes for which some local entropy production π is negative (if any) are more efficient than the corresponding reversible processes. Here we will find always π ≥ 0 .

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7 1 2 δQ δQ 0 = dS ext = + π ext ⇒ ≤ 0 ** (17) 3 ∫ ∫ T ∫ T 4 ext ext 5 6 3-Lost Work : examples and general expression 7 In this section we evaluate the Lost Work for the processes of the Sec.2. For each process we can easily 8 9 evaluate the work available in the related totally- reversible process, from this we subtract the effective work 10 11 performed in the irreversible process and this difference gives the Lost Work. This enables us to check 12 13 whether relation (5) is suitable to give the Lost Work in terms of the Entropy production. As already pointed 14 15 out, for adiabatic processes [11] we need a more general relation than (4). In this section such general link 16 between Entropy productionFor and PeerLost Work is finallyReview given. Only 17 18 I) For the irreversible isotherm expansion at temperature T , as has been already shown, the Lost Work is 19 20 3 W = W −W = RT ln 4 − RT 21 Lost Re v 4 22 23 On the other hand relation (4) gives the same result : 24 25 3 26 WLost = Tπ U = Tπ int = T (R ln 4 − R) 27 4 28 29 II) For the irreversible isotherm expansion at temperature T , by means of a source at Text ≥ T , the Total 30 31 Reversible Work is the Reversible work of the gas + the work of an auxiliary reversible engine working 32 33 between Text and T . For the gas WRe v (gas ) = Q Re v = RT ln 4 34 35 The auxiliary reversible engine brings the heat Q Re v to the ideal gas at temperature T and takes from the 36 37 Max Max Q QRe v 38 heat source Text the heat Q which is related to Q Re v by the relation = : it therefore does the 39 Text T

40 Max 41 work WRe v (engine ) = Q − QRe v . The total reversible work is 42 Max 43 WRe vTot = WRe v (gas ) +WRe v (engine ) = Q (18) 44 45 3 46 On the other hand the work performed by the gas in the irreversible expansion is W = Q = RT , therefore 47 4 48 = − = Max − 49 WLost WRe vTot W Q Q (19a) 50 51 The same result is given by the relation (4) 52 Q Q 53 W = T π = T ( Re v − irrev ) = Q Max − Q . (!9b) 54 Lost ext U ext T Text 55 56 Relation (4) is therefore suitable to evaluate the Lost Work in this process. On the other hand we are aware of 57 58 the fact that the Lost Work is due to internal and external irreversibility and we expect that 59 60

** A similar remark is due to Marcella[9]

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8 1 2 3 1) W (int) = Q − Q = RT ln 4 − RT (20) 3 Lost Re v 4 4 5 2) W (ext ) = Q Max − Q (21) 6 Lost Re v 7 But relation (4) does not give this deep insight. 8 9 Therefore we want to write down a more intuitive and general expression of the Lost Work in terms of the 10 11 internal and external Entropy production, which can be suitable also for irreversible adiabatic processes[11]. 12 13 Let us outline the way to find it. We simply replicate, for each subsystem, the argument reminded in the 14 15 introduction. Looking at the System at temperature Tsys , in the process some heat Q comes in and some work 16 For Peer Review Only 17 W comes out, therefore from relation (10) and the First Law 18 19 Q S syst = + π int 20 Tsys 21 22 T π = S − Q = S − U −W 23 sys int sys sys sys 24 25 26 If the process is Endo-reversible (π = 0), we have 27 int 28 0 = S − U −W Endo 29 sys sys Re v

30 Endo 31 Which defines WRe v , therefore 32 Endo 33 Tsys π int = WRe v −W = QRe v − Q (23) 34 35 i.e. Tsys π = WLost (int) is the lost work due to the internal irreversibility, i.e. the lost work with respect to the 36 int 37 Endo-reversible process, the process in which the gas performs the reversible isothermal expansion A B 38 39 It remains to evaluate the external Lost Work with respect to the Endo-reversible process. In the 40 41 Endo-reversible process, from relation (11) 42 Q Q Q 43 π Endo = Re v + S = Re v − Re v 44 ext T ext T T 45 ext 46 And by the definition of QMax 47 48 π Endo = Max − 49 Text ext Q QRe v (24) 50 51 As expected! Relations (23) and (24) are obtained by simply replication ,for each subsystem, of the argument 52 outlined in the introduction. 53 54 Therefore, in general 55 56 Endo WLost = Tsys π int + Text π ext (25a) 57 58 or if the system has variable temperature (See Appendix A) 59 60 B = δπ + π Endo WLost ∫ Tsys int Text ext (25b) A

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2 Or for both variable, and for Text ranging between TC and TD (Seee Appendix A) 3 4 B D = δπ + δπ Endo 5 WLost ∫ Tsys int ∫ Text ext (25c) 6 A C 7 8 In ref [11] applications of relations (25) are given for other irreversible processes : isobaric, adiabatic etc. 9 10 11 12 4- Conclusion 13 π = π + π 14 We have shown that the relation U int ext is suitable to give in a short way the Clausius inequality 15 16 and mainly to give a generalFor expression Peer of the LostReview Work in terms of Only the entropy production. We believe 17 that the relation π = π + π will be also useful to make an analysis of the Extra Work ( W ) i.e. the 18 U int ext Extra 19 20 excess of work that is performed on the system in some irreversible process. The excess will be evaluated 21 with respect to work performed in the reversible one. That analysis is in progress. 22 23 Acknowledgments : This work is mainly due to useful discussion with Marco Zannetti, Caterina Gizzi 24 Fissore, Michele D’Anna, Corrado Agnes and with my friend Franco Siringo to whose memory this paper is 25 especially dedicated. 26 27 This version of the paper is due to the many encouraging remarks of the referee, which are welcomed and 28 acknowledged. 29 APPENDIX A 30 Here we prove relations (25b) and (25c) 31 When T and T are variable, we must consider infinitesimal steps, i.e. the related quasi-static process. 32 sys ext 33 Looking at the System at temperature Tsys , in the infinitesimal process some heat δQ comes in the system and 34 35 some work δW leaves it, therefore from relation (10) and the First Law δQ 36 dS = + δπ q s.. 37 syst T int 38 sys q s.. 39 Where δπ int is the infinitesimal entropy production in the related quasi-static irreversible process. 40 δπ q s.. = − δ = − − δ 41 Tsys int Tsys dS sys Q Tsys dS sys dU sys W 42 43 44 45 46 δW 47 48 49 Tsys 50 δQ 51 52 53 54 55 Fig. 4 Some heat comes in the system and some work leaves the system in the infinitesimal step. 56 57 q s.. Endo 58 If the infinitesimal process is Endo-Reversible ( δπ int = 0 ) it holds 0 = Tsys dS sys − dU sys − δWRe v 59 Therefore 60 q s.. Endo Tsys δπ int = δWRe v − δW = δQRe v − δQ (A1)

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10 1 2 q s.. i.e.Tsys δπ int = δWLost (int) is the infinitesimal Lost work due to the Internal irreversibility, i. e. the 3 4 infinitesimal Lost work with respect to the Endo-reversible process. 5 6 Therefore 7 8 B B = δπ q s.. = δ Endo − 9 WLost (int) ∫ Tsys int ∫ WRe v W 10 A A 11 12 Remark that for the adiabatic process, many Endo-reversible paths are possible[11] 13 Finally we evaluate for each infinitesimal step the External Lost Work with respect to the Endo-reversible 14 15 process. In each step of the Endo-reversible process 16 For Peer Review Only 17 δQ δQ δQ δπ Endo = Re v − dS = Re v − Re v 18 ext T ext T T 19 sys sys ext 20 δQ 21 T δπ Endo = T Re v − δQ = δQ Max − δQ (A2) 22 ext ext ext Re v Re v Tsys 23 24 Therefore W (ext ) = T π Endo (A3a) 25 Lost ext ext 26 Or for T ranging between T and T 27 ext C D

28 D 29 W (ext ) = T δπ Endo (A3b) 30 Lost ∫ ext ext 31 C 32 In conclusion for both temperatures variable 33 34 B D 35 = δπ q s.. + δπ Endo WLost ∫ Tsys int ∫ Text ext (A4) 36 A C 37 38 References 39 40 [1] W.H. Zurek, Entropy Evaporated by a Black Hole, Phys.Rev. Lett 49 1683 (1982); 41 42 P. Kanti, Evaporating black holes and extra-dimensions, Int. J. Mod. Phys. A19 4899 (2004). 43 [2] F. Angulo-Brown, An ecological optimization criterion for finite- heat engines, J. Appl. Phys . 69 7465 44 45 (1991). 46 47 [3] Z. Yan and L. Chen, The fundamental optimal relation and the bounds of power output efficiency for an 48 49 irreversible Carnot engine, J. Phys . A: Math . and Gen 28 , 6167 (1995). 50 [4] A. Bejan, Entropy generation minimization: the new thermodynamics of finite size devices and finite-time 51 52 processes, J. Appl. Phys . 79 1191 (1996) and References therein. 53 54 [5] L.G. Chen, C. Wu and F.R. Sun, Finite time thermodynamics or entropy generation minimization of 55 56 energy systems, J. Non- Equil. Thermodyn. 25 327 (1999) and References therein. 57 58 [6] A.M. Tsirlin and V. Kazakov, Maximal work problem in finite-time Thermodynamics, Phys. Rev . E 62 59 307 (2000); 60 E. Allahverdyan and T. M Nieuwenhuizen, Optimizing the Classical , Phys. Rev. Letters 85 232 (2000);

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11 1 2 F. di Liberto, Complexity in step-wise ideal gas , Physica A 314 331-344 (2002). 3 4 [7] H.L. Callendar, The of Heat and Carnot's Principle , Proc. Phys. Soc . 23 153 (1911); 5 A. Sommerfeld, Thermodynamics and , in Lectures in Theoretical Physics –Vol. V - 6 7 Chp.II, Sec.21, pp. 152-155 (Academic Press, 1964); 8 9 I. Prigogine, Thermodynamics of irreversible Processes (Interscience Publishers, New York, 1967). 10 11 G. Job, Neudarstellung der Warmlehre, die Entropie als Warme (Berlin, 1972). 12 13 G. Falk, F. Hermann and G.B. Schmid, Energy Forms or Energy Careers , Am. J. Phys. 51 1074 (1983). 14 [8] H.S. Leff and L. Jones Gerald, Irreversibility, entropy production and , Am. J. Phys. 43 15 16 973 (1975); For Peer Review Only 17 18 H.S. Leff, Heat engine and the performance of the external work, Am. J. Phys. 46 218 (1978); 19 20 H.S. Leff, Thermal efficiency at maximum work output: new results for old heat engines, Am .J. Phys 55 602 21 (1987); 22 23 P.T. Landsberg and H.S. Leff, Thermodynamic cycles with nearly universal maximum-work efficiencies, J. 24 25 Phys. A: Math . and Gen 22 4019 (1989). 26 27 [9] V.T. Marcella, Entropy production and the second law of thermodynamics: an introduction to second law 28 analysis, Am . J. Phys. 60 888-895 (1992). 29 30 [10] R.E. Reynolds, Comment on ‘Entropy production and the second law of thermodynamics: an 31 32 introduction to second law analysis’, Am. J. Phys. 62 92 (1994). 33 34 [11] F. di Liberto. Entropy production and lost work for irreversible processes (2006) 35 http://www.fedoa.unina.it/345/ 36 37 38 [12] M. Vicentini Missoni, Dal calore all’Entropia (La Nuova Italia Scientifica, Roma, 1992); 39 40 H.U. Fuchs, The dynamics of heat (Springer, New York 1996). 41 Eur. J. Phys. 21 42 F. Hermann, The Karlsruhe Physics Course, 49 (2000) ; 43 C. Agnes, M. D’Anna, F. Hermann and P. Pianezz, L’Entropia Giocosa, Atti XLI Congresso AIF, 34 (2002). 44 45 M. D'Anna, U. Kocher, P. Lubini, S. Sciorini”L'equazione di bilancio dell'energia e dell'entropia” La fisica 46 nella Scuola Vol.XXXVIII (2005,)290 47 [13] O. Kafri., Y.B. Band and R.D. Levine, Is work output optimized in a reversible operation?, Chem. Phys. 48 49 Lett. 77 441 (1981). 50 51 [14] J.I. Belandria, Positive and negative entropy production in an ideal-gas expansion, Europhys. Lett. 70 446 52 53 (2005) . 54 55 56 57 58 59 60

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1 2 3 4 5 m ext ≡ 6 P PB 7 8 9 PA 10 11 12 T 13 Figure 1. Ideal gas in thermal contact with the heat source T 14 15 16 ------For Peer Review Only 17 18 19 20 21 22 π = ↑ − 23 int Ssyst ( ) SIn 24 25 26 27 28 29 30 31 Figure 2. The entropic balance for the ideal gas 32 ------33 ext 34 S out 35 36 37 38 39 π = ext ↓ + ext = 40 ext S ( ) Sout 0 41 42 43 44 Figure 3. The entropic balance for the heat source T 45 46 ------47 48 49 δ 50 W 51 52 53 Tsys 54 δQ 55 56 57 58 59 Fig. 4 Some heat comes in the system and some work leaves the system in the infinitesimal step. 60

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