A Counterexample to the Second Law of Thermodynamics

Transcription

1 Applied Pysis Resear; Vol. 6, No. 3; 2014 ISSN E-ISSN Publised by Canadian Center of Siene and Eduation A Counterexample to te Seond Law of ermodynamis 1 Douglas, AZ, USA José C Íñiguez 1 Correspondene: José C Íñiguez, st Street, Douglas, AZ 85607, USA. Tel: iniguez.jose@gmail.om Reeived: Mar 7, 2014 Aepted: April 17, 2014 Online Publised: April 29, 2014 doi: /apr.v6n3p100 URL: ttp://dx.doi.org/ /apr.v6n3p100 Abstrat e onatenation of one yle in te operation of a reversible eat engine wit tat proess in wi te engine s work output is irreversibly degraded into eat at te temperature of te old reservoir, transforms te effets of te former into a ouple of irreversible eat transfers. One of tese eat transfers is found to be redited, via termodynami-santioned proedures, wit two different entropy anges. A logial analysis entered on te priniple of ontradition rejets one of tese entropy anges as well as te termodynami notion leading to it. e rejeted notion, te one assigning a zero entropy ange to te eat-to-work transformation, brings to ligt a ounterexample to te seond law of termodynamis in te form of an isotermal and reversible proess wit a different from zero total entropy ange. Keywords: law of inreasing entropy, onstant entropy reversibility riterion, ounterexample, eat to work transformation, entropy, negentropy 1. Introdution No one as expressed te true rea of te law of inreasing entropy in simpler terms tan Roger Caillois: Clausius and Darwin annot bot be rigt (Prigogine, 1980). Even now, as te following quotes attest, te need is felt for a new teoretial onstrution apable of doing wat Prigogine s Dissipative Strutures paradigm was supposed to ave aomplised: solving te paradox represented by te seond law s future of deay, and nature s inessant suess in turning aos into order. us wile Farmer states: I d like to believe (life) is desribed by some ounterpart of te seond law of termodynamis some law tat would desribe te tendeny of matter to organize itself, and tat would predit te general properties of organization we d expet to see in te universe (Waldrop, 1992); Kauffman, on is part, inquires Could tere possibly be a fourt law of termodynamis for open termodynami systems, some law tat governs biosperes anywere in te osmos or te osmos itself (Kauffman, 2004)? As will ere be sown, no fourt law is needed to bring self-organizing penomena into te realm of termodynamis, but a simple elimination of a teoretial flaw idden among te notions sustaining te edifie of wat is known as te law of inreasing entropy. e onstrution arising from tis orretion manages to bring togeter, in a simple equation, te opposites at play in natural proesses: te entropy inreasing effets assoiated to te irreversible degradation of work into eat and te entropy dereasing effets of work reation via te upgrading of eat into work. 2. Bakground 2.1 Reversible Cylial Proesses Aording to Clausius in a Simple Cylial Proess two variations in respet to eat take plae, viz. tat a ertain quantity of eat is onverted into work (or generated out of work), and anoter quantity of eat passes from a otter into a older body (or vie versa)...e relation between tese two transformations is...tat wi is to be expressed by te Seond Main Priniple (Clausius, 1879a). For te ase of a reversible eat engine absorbing a ylial amount eat Q from a ot reservoir of temperature T, tese two transformations an be represented via te following self evident notation: [ Q ( T ) W ] rev, [ Q ( ) Q ( )] rev. Here Q stands for tat portion of Q tat ends up as work (W ) in an appropriate meanial reservoir, and Q for te remaining portion of Q wi is reversibly transferred to a old reservoir of temperature T, wit T T. From te previous onsiderations it is understood tat 100

2 Applied Pysis Resear Vol. 6, No. 3; 2014 Q Q (1) Q And also tat Q W (2) e amount of work produed by tis reversible engine per yle of operation is quantified by te following equation W Q ( T T ) / T (3) Here te term ( T )/ T represents te effiieny of te reversible engine. It quantifies te fration of every unit of eat oming out of te ot reservoir tat ends up transformed into te superior form of energy we all work. It was in terms of tese two transformations tat Clausius originally expressed wat was to be known as te seond law of termodynamis...te algebrai sum of all te transformations wi our in a ylial proess must always be positive or in te limit equal to zero... (Clausius, 1879b). Later on tis statement was generalized in te following terms...e teorem wi... was enuniated in referene to irular proesses only...as tus assumed a more general form, and may be enuniated tus:- e algebrai sum of all te transformations ourring in any alteration of ondition watever an only be positive, or as an extreme ase, equal to noting... (Clausius, 1862). For te reversible eat engine being onsidered and in terms of te previously introdued notation, tese statements find te following matematial representation S [ ) W] rev S[ Q ( ) Q ( )] rev 0 (4) Equation (4) embodies te zero total entropy priniple i.e. te form taken by te seond law of termodynamis wen applied to reversible proesses. Serves to note ere tat at differene wit urrent termodynamis wi desribes te total or universe entropy ange of a given proess in terms of te ombined entropy anges of te bodies involved in it, Clausius approa does it in terms of wat migt be alled elementary proesses or transformations su as te transformation of eat into work, te transformation of eat between two bodies, te transformation of reatants into produts in a emial reation, te degradation of eat into work, et. e entropy ange of a given transformation is tis way te resultant of te entropy ontributions of ea and every one of te bodies in it partiipating. If a body takes part in a given proess ten its ontribution, eiter individual or as part of a olletive, an be identified eiter as a omponent of a transformation or as a transformation itself. e summation of te entropy anges of all te transformations subsumed by a given proess determines, on its part, its total entropy ange. e identifiation of te individual entropy anges for te two transformations appearing in Equation (4) an be aomplised via te termodynami notion tat assoiates a zero entropy ange to te prodution of work out of eat. us, wen performing te entropy balane for a reversible eat engine Pitzer and Brewer tells us tat in it te work term does not appear...sine it involves no entropy... (Pitzer & Brewer, 1961a). e justifiation for tis position appears to be linked to te fat tat te effet of [ Q ( T ) W ] rev, as well as tat of its opposite [ W T )] rev finds manifestation in te ange of ondition of a purely-meanial body (be it te rising or lowering of weigts, te elongation/ompression of a spring, or any oter equivalent meanism), restrited by its very nature to reversible ergo isentropi proesses. is rationale takes te form S wt. 0 in te entropy balanes performed by Bent (1965a) for a number of proesses, and is also found in te following quote by Barrow we are at liberty, you sould reognize, to asribe any features to tis new entropy funtion tat we like, te requirement being tat we onstrut a funtion tat is self onsistent and allows us to form a useful expression for te seond law. In tis way we furter speify tat, for all proesses, ds me res 0 (Barrow, 1973). Here ds me res 0 refers to te entropy ange for te meanial reservoir. e previous statements an all be found subsumed one way or anoter in te termodynami definition of a meanial reservoir (or reversible work soure) as A system enlosed by adiabati impermeable walls in wi all proesses of interest are quasistati (Tsoegl, 2000). An equivalent definition an be found in Callen (2007). Aording to te previous onsiderations, te entropy ange for te eat-to-work transformation taking plae in a reversible eat engine an be written as 101

3 Applied Pysis Resear Vol. 6, No. 3; 2014 S [ T ) W] 0 (5) e ombination of Equations (4) and (5) leads, in turn, to S [ Q ( T ) Q ( T )] 0 (6) Note tat tese entropy anges onform to te zero total entropy ange demanded by te seond law for reversible proesses. e fat tat ea of tese transformations takes into onsideration te entropy ontribution of all te bodies in it partiipating makes ea of tem a universe in itself (and a sub-universe of te eat engine), and teir respetive entropy anges total entropy anges. 2.2 e Irreversible Transfer of Heat e entropy ange for te diret, irreversible transfer of an amount of eat Q from a ot reservoir of temperature T to a old reservoir of temperature T is known to be (Bevan-Ott & Boerio-Goates, 2000) S[ T ) T )] In orrespondene wit te fat tat tis irreversible proess takes plae wit te sole involvement of te said reservoirs, we find its entropy ange determined by te summation of te entropy anges of tese two bodies. us, Q / and Q / respetively represent te entropy anges of te ot and old reservoirs due to te release, in te former, and te absorption, in te latter, of te amount of eat Q. e validity of tis result, it needs to be pointed out, stems from te validity of te notions on wi it stands: te first law of termodynamis ( du Q W ) and te definition of te entropy, ds Qrev / T. e fat tat a eat bat or eat reservoir is a onstant temperature body apable of exanging eat but not work ( W 0 ) allows any of its anges to be desribed by du dq. is equation makes evident te fat tat any eat exanged by a eat reservoir is equal to te ange of a point funtion. Being tis so te ange of state of te eat bat is tus entirely determined by te eat transfer dq and ds is terefore te same weter tis transfer is reversible or not. (Denbig, 1968). Consequently, te entropy ange of a eat reservoir of temperature T beomes simply S Q / T. It is tis equation te one produing te entropy anges of Equation (7). It sould also be noted from Equation (7) tat wit te exeption of te zero entropy ange arising at tat limit ase of reservoirs of te same temperature, in any oter situation in wi as assumed T T, te entropy ange will be positive and as su in agreement wit te law of inreasing entropy, te form adopted by te seond law of termodynamis in te ase of irreversible proesses su as te transfer of eat ere being onsidered. e re-expression of te rigt and side of Equation (7) as ( Q ( ) / ) / permits, wit te aid of Equation (3), te identifiation of ( ) / ) wit te work Q ould ave produed ad it been fed to a reversible engine working between te said two reservoirs. On tis perspetive it an be properly alled te work lost by te irreversible transit of Q. Its representation as [ W lost ( Q)] permits giving Equation (7) te following form 102 irr irr rev rev Q T lost Q T (7) S [ T ) T )] [ W ( Q)] / T (8) e term [ W lost ( Q)] was osen to distinguis Equation (8) from tat oter equation of similar form found in seond law termodynamis wi quantifies te entropi effet of te degradation of atual work into eat If an amount of work W is degraded to eat at temperature T, te inrease in entropy is d S irr W / T (Pitzer & Brewer, 1961b). e work degrading proess to wi te previous quote makes referene and te entropy ange by it produed, for te purpose of te disussions tat follow, will be represented as follows S [ W )] irr W / T (9) 2.3 Work Prodution in Cylial Proesses As known, te ylial evolution of a reversible eat engine is produed by te onatenation of a reversible and isotermal expansion at te temperature of te ot reservoir; an adiabati and reversible expansion taking te working substane, wi will eretofore be taken as an ideal gas, from te temperature of te ot reservoir to tat of te old reservoir; an isotermal and reversible ompression at te temperature of te old reservoir, and finally, losing te yle, an adiabati and reversible ompression taking te gas bak to its initial ondition.

4 Applied Pysis Resear Vol. 6, No. 3; 2014 On reason of te fat tat in ylial proesses te eat-to-work and work-to-eat transformations take plae along te isotermal and reversible proesses is tat a loser look to tese proesses will be taken in wat follows. Besides te ideal gas appropriately ontained in a ylinder fitted wit a fritionless and weigtless piston, an isotermal and reversible proess demands te onourse of a eat reservoir of te same temperature of te gas, and also of a meanial reservoir to store/supply te work produed/required along te expansion/ompression. In te ase of an isotermal and reversible expansion (IRE), te ideal gas manages to quantitatively transform into work all of te eat absorbed from te reservoir. Now, before any amount of eat Q an be transformed into work, it as first to flow from te reservoir to te gas. is allows us to reognize tat two onomitant proesses are taking plae along te expansion: e transfer of eat between two bodies of te same temperature, and te atual transformation of eat into work by te expanding gas. Aording to te notation above introdued, te total entropy ange for tis proess an be written as follows S[ IRE] S[ T ) T )] rev S[ T ) W ] rev 0 (10) at Equation (10) is equal to zero finds explanation in te fat tat te entropy anges of te two transformations in it appearing are also zero: tat of te eat transfer on reason of it taking plae between two bodies of te same temperature; and tat of te eat-to-work transformation on reason of Equation (5). is equation is te matematial representation of te seond law of termodynamis in te form it applies to isotermal and reversible expansions. e fat tat S [ T ) W ] rev 0 is inompatible wit tis priniple will allow te onstrution of a ounterexample to it. Serves to note tat in an (IRE) taken by itself, i.e. not as part of a eat engine, full onversion of eat into work is aieved at te prie of leaving te gas in an expanded ondition. In order to ave a seond work-produing round te work produed will ave to be expended in full in taking te gas bak to its ompressed original position, so no work will remain at te end of tis peuliar yle. e ontinuous transformation of eat into work needs to make use of a eat engine and tis, in turn demands a ylial proess su as te one desribed at te beginning of tis setion. Any of tese engines, even if reversible, and exluding reservoirs of infinite or absolute zero temperatures, will never be able to aieve te full onversion of eat into work. 2.4 e Priniple of Contradition Aristotle s priniple of ontradition will be te riterion used in tis work to identify wi of two ontraditory statements is at fault. For te purpose of tis task tis priniple will be expressed as follows: X annot be bot X and not X at te same time and in te same respet. Violation of tis priniple implies a ontradition. e ontradition onsists in tat even if being X exludes te possibility of not being X at te same time and in te same respet, bot of tese positions are taken as true wen in reality tey divide te true and te false between tem i.e., one member of te pair must be true and te oter false (Horn, 2010). Any set of statements involving ontraditions beomes, tis way, logially inonsistent. 3. e Contraditions of Seond Law ermodynamis 3.1 From One Cyle of a Reversible Heat Engine to a Couple of Irreversible Heat Transfers e effets of one yle in te operation of te reversible eat engine disussed in Setion 2.1 ave been depited in Figure 1(a). ere, te (impliit) engine is sown transferring te amount of eat Q from te ot to te old reservoir, as well as transforming into work (W ) te amount of eat Q of temperature T. As before stated, Q Q Q. If tis yle is now onatenated wit tat proess sown in 1(b) in wi W is irreversibly degraded, say via a fritional meanism, into an equivalent amount of eat ending up in te old reservoir, we will see te original effets of te reversible engine redued to te two irreversible eat transfers sown in 1(). In order to understand te transit from 1(a)+1(b) to 1() let us onsider te following two fats. 1) at te onatenation of te eat-to-work transformation sown in 1(a), wit te work-to-eat transformation sown in 1(b), produes as its sole effet te irreversible transfer of Q from te ot to te old reservoir sown in 1() as Q irr : Having left te ot reservoir of temperature T to be transformed into W in 1(a), Q reappears in 1(b) entering te old reservoir upon te transformation of W bak into eat taking plae tere. erefore [ Q ( ) W ] rev [ W )] irr [ ) )] irr (11) As sould be noted from Figure 1, te oter transformation taking plae in 1(a) [ Q ( ) Q ( )] rev reappears in 1() as Q, irr, sortand for [ Q ( ) Q ( )] irr. 2) at wile te availability of W in 1(a) is te reason beind te reversible ondition of te two 103

5 Applied Pysis Resear Vol. 6, No. 3; 2014 transformations tere taking plae te simple feeding of W to te inverse yle as te effet of restoring te initial state witout anges being left elsewere; It is te unavailability of work in 1() wat explains te irreversible ondition of te two eat transfers tere depited. e reason is simple. e transfer bak in 1() of Q or Q, or bot, from te old to te ot reservoir demands te expenditure of work. e fat tat none is ere available means tat su a reversion an only take plae at te prie of leaving a permanent ange in tat body alled to supply te required amount of work. Q, rev Q W W Q, irr Qirr Q T (a) (b) () Figure 1. Out of te amount of eat Q released by te ot reservoir of temperature T, te reversible eat engine impliit in (a) manages to transfer te portion Q to te old reservoir of temperature T, and to transform te differene Q Q Q into te equivalent amount of work W. e furter degradation of W into eat at te temperature of te old reservoir sown in (b) as te effet of reduing te original engine effets to tose of te two irreversible eat transfers sown in (). Here Q, rev Q, irr and Q irr stand, respetively, for [ Q ( ) Q ( )] rev, [ Q ( ) Q ( )] irr, and [ Q ( ) T )] irr 3.2 e First Entropy Cange for [ Q ( ) T )] irr e irreversible nature of bot of te eat transfers sown in 1() makes tem idential in nature to tat desribed in Setion 2.2. If so, teir respetive entropy anges an, at te ligt of tat expressed by Equation (7), be written as follows Q Q S ) )] irr Q Q S Q ( ) Q ( ] irr [ (12) [ (13) e addition of Equations (12) and (13) leads, wit te onourse of Equation (3), te fats tat Q Q Q, and Q W, as well as some algebrai rearrangements, to te following equivalent expressions for te total entropy ange of proess 1(): Q Q Q Q Q Q Q[( ) / ] Q W S[ 1( )] irr ( ) ( ) (14) 3.3 e Seond Entropy Cange for [ Q ( ) T )] irr e entropy ange for te irreversible transfer of Q sown in 1() an, in attention to Equation (11), be written as follows S [ ) )] irr S[ ) W ] rev S[ W )] irr (15) Furter substitution in te previous expression of Equations (5) and (9) produes S [ ) )] irr 0 W / (16) e ombination of Equation (16) wit te appropriate equivalent expressions given by Equation (14) for W / T leads to Q Q Q Q Q Q S[ ) )] irr 0 [( ) ( )] 0 [ ] (17) 104

6 Applied Pysis Resear Vol. 6, No. 3; 2014 A simple omparison of Equation (12) wit Equation (17) makes evident te fat tat in te latter [ Q ( ) T )] irr is redited wit te wole of te entropy ange for proess 1(); in oter words, ere te irreversible transfer of Q appears responsible not only for its own entropy ange, but also for tat of te irreversible transfer of Q. e fat tat Equations (12) and (17) define te entropy ange of te very same proess in terms of ontraditing notions, leads to te onlusion tat one of tese equations is at fault. e validation of Equation (7) and by extension tat of Equation (12) arried on in Setion 2.2 in terms of te first law and te definition of te entropy is evidene enoug to put in doubt te result given in Equation (17). To prove tat as suspeted, su a result is inorret, is te matter of te following argument wi starts reognizing tat te entropy ange for [ Q ( ) T )] irr appearing in Equation (17) is not given, as sould be expeted, in terms Q, as is done by Equation (12), but in terms of bot Q and Q ombined, or equivalently, in terms of Q. In oter words, even if te eat transferred is Q, te effets orrespond to Q. e notion embedded in Equation (17) tat Q an be bot: Q wen transferred from te ot to te old reservoir and not Q in regard to te entropy ange produed by tis transfer violates te priniple of ontradition. Note tat bot statements refer to tat moment in te proess in wi te transfer of Q takes plae, and are given in respet to te onatenation of proesses depited in Figure 1. Notwitstanding te fat tat being Q exludes te possibility of not being Q at te same time in te same respet, Equation (17) takes bot of tese statements as true. It is on reason of tis tat te entropy ange given by Equation (17) for [ Q ( ) T )] irr is rejeted as false. It needs to be mentioned ere tat tere exists anoter perspetive, independent from te one just disussed, from wi te logial inonsisteny of Equation (17) an be approaed (Íñiguez, 2011, 2012). It will be briefly mentioned in Setion e Logial Imbroglio of Seond Law ermodynamis e rejetion on logial grounds of Equation (17), and by extension of Equation (16) on wi it originates, eliminates te ontradition of aving two different entropy anges for te same proess. is result simply onfirms te trutfulness of Equation (12), previously validated in terms of te notions sustaining it. Now, even if Equations (16) and (17) are out of te way, it is imperative to identify te reason of teir logial inonsisteny. is will be done below. Let us start tis disussion by writing te following expression resulting from te appropriate ombination of Equations (12) and (15) Q Q S ) )] irr S[ ) W ] rev S[ W )] irr [ (18) A term by term omparison between te two rigtmost expressions of te previous equation allows us, on reason of te independene of [ Q ( T ) W ] rev and Q / on T ; as well as of te independene of [ W T )] irr and Q / on T, to make te following identifiations S [ T ) W ] Q / T (19) rev S [ W )] irr Q / (20) e substitution of Equation (20) in Equation (15) permits writing te following expression Q S ) )] irr S[ ) W ] rev [ (21) Let us now agree tat any laim of orretness for Equation (19) is in opposition not only to te aepted termodynami notion embodied by Equation (5) giving a zero entropy ange to [ Q ( T ) W ] rev but, as already mentioned in regard to Equation (10), also to te seond law in its zero total entropy version. Note now tat if in aeptane of Equation (5) we replae S [ T ) W ] rev in Equation (21) wit a zero, we will be regenerating Equation (16) and troug it Equation (17); bot of tem inorret on logial grounds. Note ten tat it is not te W / term (or equivalently, Q / ) of Equation (16) wat makes tese equations inorret. is term, written as Q /, belongs in te orret equation for [ Q ( ) T )] irr as attested by Equation (18). It is te zero aompanying W / in Equation (16) wat allows tis term to morp its identity into tat of te terms sown in te rigt and side of Equation (17). e fat tat tis zero omes from Equation (5) allows us first of all to identify tis equation as te origin of te logial onundrum previously unveiled and as su te true objet of logial rejetion by te argument of te previous setion; and also to onfirm te validity of Equation (12) and troug it tat of Equations (19) and (20). ese results provide us, via te substitution of Equation (19) in Equation (10), wit a logially onsistent 105

7 Applied Pysis Resear Vol. 6, No. 3; 2014 ounterexample to te seond law in te form of a reversible and isotermal ideal gas expansion wit a non-zero, atually negative total entropy ange of magnitude Q / T, i.e. S[ IRE] S[ T) T)] rev S[ T) W ] rev Q / T (22) Note tat te negative total entropy for te isotermal and reversible expansion makes tis proess negentropi. Serves well in tis regard to reall ere tat te entropy ange given by Equation (19) an also found in Clausius work on te seond law (Clausius, 1879). 3.5 An Isotermal and Reversible Compression e fat tat te oupling of a reversible proess wit its preise inverse brings te universe bak to its original ondition means tat ea and every effet of te forward proess is reverted by its inverse, and if tis is so, te entropy ange of te latter must be of equal magnitude but opposite sign to tat of te former so tat teir ombination, as orresponds to a point funtion, produe a net ange of zero for tis yle. Wen applied to te inverse of te (IRE) introdued in Setion 2.3 and wose orret entropy ange appears quoted in Equation (22) i.e. to an isotermal and reversible ompression (IRC), we will ave tat te following is true S[ IRC] S[ T) T)] rev S[ W T)] rev Q / T (23) S[ W T )] rev Q / T (24) e different from zero total entropy anges of te isotermal and reversible ideal gas proesses desribed by Equations (22) and (23) provides a glimpse at te paradigmati anges subsumed by te replaement of Equation (5) by Equation (19). In order to keep expanding tis new view is tat te following disussions are presented 3.6 A Partially Reversible Isotermal Expansion Let us onsider te isotermal and irreversible expansion depited in Figure 2. In it a portion Q out of te amount of eat Q released by te ot reservoir of temperature T finds a diret, irreversible pat to te old reservoir of temperature T ; wit te oter portion, Q, being transformed into an equivalent amount of work, Q W. Note tat te work produing proess is reversible on reason of te fat tat Q an be returned to te ot reservoir, witout any additional ange taking plae, via an isotermal and reversible ompression propelled wit W. e fat tat no work was generated in te transfer of Q allows us to realize tat its transfer bak to te ot reservoir annot be aomplised witout leaving a ange in tat body alled to supply te missing work. ese onsiderations permit writing te following equation for te total entropy ange of tis proess S[ proess 2] S irr S rev (25) e reversible and irreversible omponents of te universe entropy ange an, on reason of Equation (7) and Equation (19) be written, respetively, as follows Sirr Q ( ) /( ) (26) S rev Q / W / (27) In terms of wi Equation (25) beomes S[ proess 2] [ Q ( ) /( )] ( Q / ) (28) 106

8 Applied Pysis Resear Vol. 6, No. 3; 2014 T Q W Q, irr T Figure 2. e isotermal expansion depited is irreversible on reason of te fat tat only te portion Q out of te total amount of eat Q, Q Q Q, released by te ot reservoir of temperature T is transformed into work. e remaining portion Q bypasses te work prodution proess by flowing diretly to te old reservoir of temperature T. e irr sub index as been added to Q to empasize te irreversible nature of tis eat transfer Note now tat tis equation orretly reprodues te entropy anges for te two extremes tere involved. If Q 0 ten no irreversible leaking of eat takes plae and Q Q. In tis situation all of te eat released by ot reservoir ends up transformed into work and te total entropy ange beomes S [ proess2] Q / T (29) If, on te oter and, all of te eat released by te ot reservoir finds its irreversible way to te old reservoir ten Q Q. In tis situation Q 0 and no work is at all produed. erefore S proess 2] Q ( T T ) /( T T ) (30) [ It is important realize tat work is a ommon link to te opposite effets sown in Equation (28). us wile te negentropi ontribution of te reversible term given by Equation (27) appears in terms of te work (W ) output of te proess i.e. in terms of te work gained : (31) S rev W gained / e entropi ontribution of te irreversible term given in Equation (26), as previously disussed in referene to Equation (8), an be written in terms of te wasted work produing potential tat Q arries wit it in its irreversible transit to te old reservoir, i.e. in terms of te lost work arried on by Q S [ W ( Q )] / T (32) irr lost ese last two equations allows us to write te following re-expression of Equation (25) S[ proess 2] [ Wlost ( Q ) / ] ( W gained / ) (33) is equation, ommon to oter (non-reversible) termodynami proesses involving te transformation of eat into work (Íñiguez, 2012), allows us to see tat in tis new perspetive te entropy inrease assoiated to te lost work is opposed by te negentropy assoiated to te work gained. e predominane of one of tese opposite over te oter depends on te effiieny of te operation. If te effiieny for tis isotermal expansion is defined as Q / Q ten its value at te irreversible extreme ( Q W 0 ) is zero, wile at te reversible end ( Q W Q ) is one. It sould be noted tat in te zero-effiient irreversible realm in wi no work is produed (no work produed, no negentropy) Equation (33) will lose its rigtmost term to beome, as previously noted in referene to Equation (7), te following re-expression of te law of inreasing entropy: S[ proess 2] [ Wlost ( Q ) / ]. e previous argument allows us to see tat it is at te zero-effiient irreversible limit were te rule of te law of inreasing entropy is absolute. In tis perspetive te seond law as we know it orretly desribes te domain of irreversible proesses. As soon as eat starts finding its way into te superior form of energy we all work, te desription of te situation demands te use of Equation (33) or its equivalent re-expressions. It omes apparent from Equation (33) tat on reason of being T T, every unit of work wasted produes a larger amount of entropy tan te negentropy brougt forward by every unit of work gained. In tis perspetive te punisment for wasting work exeeds te reward for gaining it. Inerent to te previous disussion is te fat tat te total entropy of te irreversible isotermal expansion being onsidered transits from a positive value at 0 to a negative value at 1. But if tis is so, tere must be 107

9 Applied Pysis Resear Vol. 6, No. 3; 2014 an effiieny o at wi te total entropy, as a refletion of idential magnitude but opposite sign ontributions from te entropi and negentropi proesses tere taking plae, beomes zero. is effiieny an be identified for te proess under study by setting Equation (28) equal to zero and solving for o. Wen tis done we get (34) o ( T ) / Any of tese proesses taking plae wit o will be entropi. All tose for wi o will be negentropi, and tose omplying wit o, isentropi. e previous disussion unveils te fat tat te entropy is more tan a state funtion; It is also te indiator of te transition point from entropi to negentropi universes (or vie versa). e separation of te total entropy ange of termodynami proesses in entropi and negentropi ontributions allows us to give preise meaning to te notions of termodynami aos and termodynami order. e latter finding expression in te negentropi region were te effet of te ordered energy we all work overomes tat of te disordered energy we all eat. In te opposite ase it is te former termodynami aos- te one finding expression. On tis perspetive te transition from entropi to negentropi universes an be likened wit te transition from aos to order and vie versa. If we are apable of developing expressions for te work gained appearing in self-organizing penomena as spatial-funtional order i.e. for te work involved in te onstrution and sustaining of te strutures evident in tose proesses v. g. e rolls or ells appearing in Bénard Convetion (Íñiguez, 2010), as well as for te work lost in tose proesses, we will be able to test via Equation (33) if as posited ere, tese strutures materialize te transitions of tese systems from entropi to negentropi, as well as to determine and ompare wit experiment te transition point at wi disorder beomes order. 3.7 Heat as Meanial Energy Two elements of judgment are required for te argument to be presented below. e first one makes use of te ontrast made by Smit and Van Ness between eat and work, as follows One of te learest interpretations of te seond-law priniple may be aieved by onsidering te differenes between te two forms of energy, eat and work. In tis regard experiene teaes tat tere is a differene in quality between eat and work. e experimental fats exibited by tese autors in support of tis notion are, on te one and, te pratially unrestrited onvertibility from one form of work to anoter, and in te oter, te severely restrited onvertibility of eat into any form of work. e effiienies for te latter kind of proesses are so low in omparison wit tose obtained for te transformation of work from one form to anoter tat tere an be no esape from te onlusion tat tere is an intrinsi differene between eat and work. is differene may be summarized by saying tat eat is a less versatile or more degraded form of energy tan work and tat work migt be termed energy of a iger quality tan eat (Smit & Van Ness, 1965). Note tat te previous onsiderations imply a ommon quality for te different forms of meanial energy, or equivalently, for te different forms of work. ese notions find omplement in te ommon interpretation of te total entropy inrease in irreversible proesses as a quantitative measure, or index, of te degradation of energy as work to energy as eat (Weber & Meissner, 1957). e seond element refers to te fat tat any proess taking plae in a onservative meanial system, su as te inter-onversions of one form of meanial energy to anoter, takes plae at onstant entropy. Let us now transribe ere te total entropy expression for an isotermal and reversible expansion as written by Bent in aord wit urrent termodynami wisdom (Bent, 1965b). S [ IRE] ( Q / T) ( Q / T) SWt. 0 (35) In te previous equation S Wt. refers to te entropy ange of te onservative weigt-in pulley system used by Bent as meanial reservoir. In it te weigt s position anges in response to te eat-to-work transformation represented as [ Q ( T ) W ] rev. Bent s original expression makes use of te internal energy ange of te eat reservoir for te quantifiation of its own entropy ange as well as tat of te gas. e fat tat for a eat reservoir, as previously noted, E Q, allows re-expressing tese entropy anges in terms of Q as sown in Equation (35). From a omparison between Equations (35) and (10) two tings beome evident: tat te term S [ T ) T )] rev of te latter orretly orresponds wit te first two terms of te former, and tat for seond law termodynamis, as we know it, te entropy ange assoiated to te weigt s displaement ( S Wt. ) is idential wit tat of te termodynami proess upgrading eat into work, i.e. 108

10 Applied Pysis Resear Vol. 6, No. 3; 2014 S Wt. S[ T ) W ] rev 0 (36) For te purposes of te oming disussion te two rigtmost terms of te previous equation will be written as follows S [ T ) E Wt. ] rev 0 (37) In it te work term as been substituted by its equivalent, te energy of te weigt. Let us now onsider te situation in wi te lifting of te weigt in Bent s meanial reservoir is produed via te unoiling of a nonfritional spring previously oupled to it. e fat tat tis proess involves no oter ange but te transformation of te elasti energy of te spring into te gravitational-potential energy of te weigt makes it, as previously noted, an isentropi proess. ese onsiderations are refleted in te following expression S [ E spring E Wt ] rev 0 (38) As made evident by a omparison between Equations (37) and (38), for urrent termodynamis te transformation of eat into meanial energy in Equation (37) is entropially indistinguisable from te transformation of elasti energy into gravitational-potential energy represented in Equation (38); in oter words, eat as an energy form is, in regard to Equation (37), indistinguisable from any oter form of meanial energy, as it is apable of transiting, of morping into te gravitational-potential energy of te weigt at onstant entropy. is onlusion is, owever, in opposition to experiene. As previously noted, eat is intrinsially different from meanial energy. In negating tis experiene supported fat, te seond law negates itself, as its foundation rests preisely in te notion tat tese two energy forms are intrinsially different. It needs to be stressed tat wat as been faulted is not te isentropi nature of te meanial reservoir, but its identifiation sown in Equation (36) wit te termodynami proess responsible for te eat-to-work transformation. e transformation of eat into work and te entropy ange to it assoiated is not, as urrent termodynamis posits, an attribute of te meanial reservoir but of te olletive of bodies in it involved. In te ase of te (IRE) it is an attribute of te eat reservoir, te gas, and te meanial reservoir. Take any of tese bodies out and te eat-to-work transformation annot take plae. e extreme redutionist position of urrent termodynamis in terms of te entropies of bodies, finds its limit in te impossibility to assign te entropy ange of te eat-to-work-transformation to any of te bodies tere involved. Wen Bent s total entropy ange for te isotermal and reversible ideal gas expansion given in Equation (35) is looked upon at te ligt of te fat tat transformation [ Q ( T) W ] rev involves all te bodies taking part in it meanial reservoir inluded- it beomes evident tat it needs to be orreted via te replaement of S Wt. by S [ T ) W ] rev. is simple replaement embodies te oneptual sift separating urrent seond law termodynamis from te formulation ere being introdued, i.e. ( Q / T ) ( Q / T ) SWt. ( Q / T) ( Q / T ) S[ T) W ] rev (39) Materializing in te identifiation of S [ T ) W ] rev wit Q/ T tis ange produes te orret result for te total entropy ange of an isotermal and reversible ideal gas expansion: S [ IRE] Q / T (40) 3.8 e Spontaneous Flow of Heat From a Colder to a Hotter Body A different perspetive to te ontraditions inurred by seond law termodynamis, as we know it, in santioning two ontraditory entropy anges for [ Q ( ) T )] irr, an be unveiled troug te following argument. Let us ten assume tat tat te following equation, a ombination of Equations (14) and (17), is true S [ 1( )] irr S[ ) )] irr (41) e problem wit tis equations starts one we reognize te fat, evident from Figure 1() tat S [1( )] irr is also given by te following equation (42) S [ 1( )] irr S[ ) )] irr S[ Q ( ) Q ( )] irr A omparison between te last two equations igligts te point tat te validity of urrent seond law 109

11 Applied Pysis Resear Vol. 6, No. 3; 2014 termodynamis position embodied in Equation (41) presupposes te validity of te following relation S [ Q ( T ) Q ( T )] 0 (43) It is troug a simple analysis of te previous equation tat two important flaws subsumed by tis position ome to ligt. 1) e fat tat te proess to wi Equation (43) makes referene is reversible on reason of its entropy ange and irreversible on reason of te fat tat in 1() it is impossible to transfer Q bak to te ot reservoir witout anges remaining elsewere, attest to te non-equivalene between te onstant total entropy reversibility riterion and te possibility of restoring te preise initial ondition of te universe. Here we ave a reversible proess tat an t be reverted. 2) e only possibility open for te zero total entropy reversibility riterion to remain valid in regard to wat Equation (43) expresses, i.e. te only possibility open for us to trade irr for rev in Equation (43) demands from Q te ability to flow of itself, unassisted, from te old to te ot reservoir. In oter words, te validity of te zero total entropy reversibility riterion, and by extension tat of te total entropy inreasing quality of irreversible proesses, demands te non-validity of tat priniple to wi te seond law is supposedly equivalent, namely, tat eat annot, of itself, pass from a older to a otter body (Clausius, 1879d). 3.9 e Reversible to Irreversible Transition of Q ( T ) Q ( T )] 110 irr [ Wit te problem regarding te entropy ange for [ Q ( ) T )] irr finally settled, tere is still te matter of te ange experiened by [ Q ( ) Q ( )] from 1(a) to 1() onsisting in te transition from a reversible ondition and a zero entropy ange to an irreversible ondition and an entropy ange of magnitude ( Q / ) ( Q / ). Let us agree tat in 1(a) no lost work is produed by te reversible transit of Q to te old reservoir. If tis were not so ten, ontrary to experiene, proesses apable of outputting larger amounts of work tan te reversible work would be possible. e previous onsideration explains, via te notion subsumed by Equation (8), te zero entropy ange assoiated to te reversible transfer of Q (no lost work, no entropy prodution). is situation anges radially, owever, one proess 1(b) takes effet. Here, subsumed in te two irreversible transfers depited in 1() we find W reappearing as te dissipated work produing potential (te lost work) arried by tese two eat transfers in teir irreversible transit to te old reservoir. e fat tat wile in 1(a) te transfer of Q, rev takes plae witout lost work and onsequently at onstant entropy, and tat in 1() te transfer of Q, irr is aompanied wit a wasted work produing potential in te amount of ( Q ( ) / ) and an entropy ange of ( Q ( ) / ) / means tat te entropy ange for te transit from te former to te latter taking plae on reason of 1(b) an be written as S [ Q, irr ] S[ Q, rev ] Q ( ) / (44) If so, ten in agreement wit our previous result sown in Equation (13) we get 4. Conlusions S [ Q, irr ] S[ Q, rev ] ( Q ( ) / ) Q ( ) / Hailed as te supreme law of nature by Eddington (Nikulov & Seean, 2004), te law of inreasing entropy as been ere sown to be a logially inonsistent onstrution. Its orretion materializes in an equation taking into aount te opposing tendenies at te ore of natural penomena, order and aos. Even if te evidene is lear and appears unassailable, te final verdit belongs, as Plank as said in tis ontext, to experiene no more effetive weapon an be used by bot ampions and opponents of te seond law tan indefatigable endeavor to follow te real purport of tis law to te utmost onsequenes, taking te latter one by one to te igest ourt of appeal -experiene. Watever te deision may be, lasting gain will arue to us from su a proeeding, sine tereby we serve te ief end of natural siene te enlargement of our stok of knowledge (Plank, 1990). Referenes Barrow, G. M. (1973). Pysial Cemistry (3rd ed., p. 168). New York, NY: MGraw-Hill Book Company. Bent, H. A. (1965a). e Seond Law (pp , 77-81). New York, NY: Oxford University press. Bent, H. A. (1965b). e Seond Law (pp ). New York, NY: Oxford University press. Bevan-Ott, J., & Boerio-Goates, J. (2000). Cemial ermodynamis, Priniples and Appliations (pp ). London, UK: Aademi Press. Callen, H. B. (2007). ermodynamis and an Introdution to ermostatis (2nd ed., p. 103). New York, NY: (45)

12 Applied Pysis Resear Vol. 6, No. 3; 2014 Jon Wiley and Sons. Clausius, R. (1862). On te Appliation of te eorem of Equivalene of Transformations to te Internal Work of a Mass of Matter. Pilosopial Magazine Series 4, 24(159), 211. ttp://dx.doi.org/ / Clausius, R. (1879a). e Meanial eory of Heat (pp ). London, UK: Mamillan and Co. Retrieved from ttps://arive.org/details/u Clausius, R. (1879b). e Meanial eory of Heat (p. 213). London, UK: Mamillan and Co. Retrieved from ttps://arive.org/details/u Clausius, R. (1879). e Meanial eory of Heat (pp ). London, UK: Mamillan and Co. Retrieved from ttps://arive.org/details/u Clausius, R. (1879d). e Meanial eory of Heat (p. 78). London, UK: Mamillan and Co. Retrieved from ttps://arive.org/details/u Denbig, K. (1968). e priniples of Cemial Equilibrium (2nd ed., p. 40). London, UK: Cambridge University Press. Horn, L. R. (2010). Contradition. Stanford Enylopedia of Pilosopy. Retrieved from ttp://plato.stanford.edu/entries/ontradition Íñiguez, J. C. (2010). An unusually effiient oupling of Carnot engines. Apeiron, 17(2), Retrieved from ttp://redsift.vif.om/journalfiles/v17no2pdf/v17n2ini.pdf Íñiguez, J. C. (2011). A ermodynami Impasse: A onstant entropy irreversible proess. Pysial Cemistry: An Indian Journal, 6(3), Retrieved from ttps:// Íñiguez, J. C. (2012). A Critial Study of te Law of Inreasing Entropy. Consooken, PA: Infinity Publising Co. Kauffman, S. (2004). Investigations (p. 2). New York, NY: Oxford University Press. Nikulov, A., & Seean, D. (2004). e Seond Law Mystique. Entropy, 6, Pitzer, K. S., & Brewer, L. (1961a). ermodynamis (2nd ed., p. 95). (Revision of Lewis and Randall). New York, NY: MGraw-Hill International Student Edition. Pitzer, K. S., & Brewer, L. (1961b). ermodynamis (2nd ed., pp ). (Revision of Lewis and Randall). New York, NY: MGraw-Hill International Student Edition. Plank, M. (1990). Treatise on ermodynamis (p. 107). New York, NY: Dover. Prigogine, I. (1980). From Being to Beoming: Time and omplexity in te pysial sienes (p. 84). New York, NY: W. H. Freeman and Company. Smit, J. M., & Van Ness, H. C. (1965). Introdution to Cemial Engineering ermodynamis (pp ). Mexio, DF: MGraw-Hill Book Co., International Student Edition. Tsoegl, N. W. (2000). Fundamentals of Equilibrium and Steady-State ermodynamis (pp ). Amsterdam, Holland: Elsevier. ttp://dx.doi.org/ /b / Waldrop, M. M. (1992). Complexity: e emerging siene at te edge of order and aos (p. 288). New York, NY: Toustone. Weber, H. C., & Meissner, H. P. (1957). ermodynamis for Cemial Engineers (p.168). New York, NY: Jon Wiley and Sons. Copyrigts Copyrigt for tis artile is retained by te autor(s), wit first publiation rigts granted to te journal. is is an open-aess artile distributed under te terms and onditions of te Creative Commons Attribution liense (ttp://reativeommons.org/lienses/by/3.0/). 111