Proposal for Generalized Exergy and Entropy Properties Based on Stable Equilibrium of Composite System-Reservoir

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1 Joural of oder Physcs, 2013, 4, Publshed Ole July 2013 ( Proposal for Geeralzed Exergy ad Etropy Propertes Based o Stable Equlbrum of Composte System-eservor Perfracesco Palazzo echp, ome, Italy Emal: ppalazzo@techp.com eceved Aprl 20, 2013; revsed ay 25, 2013; accepted Jue 27, 2013 Copyrght 2013 Perfracesco Palazzo. hs s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. ABSAC he preset theoretcal study represets a proposal amed at vestgatg about the possblty of geeralzg the caocal etropy-exergy relatoshp ad the reservor cocept. he method adopted assumes the equalty of pressure ad chemcal potetal as ecessary codtos of mutual stable equlbrum betwee a system ad a reservor addto to the equalty of temperature that costtutes the bass for defg etropy as dervg from eergy ad exergy cocepts. A attempt s made to defe mechacal ad chemcal etropy as a addtoal ad addtve compoet of geeralzed etropy formulated from geeralzed exergy property. he mplcatos exergy method ad the possble egeerg applcatos of ths approach are outled as future developmets amog the domas volved. Keywords: Adabatc Avalablty; Avalable Eergy; Geeralzed Exergy; Geeralzed Etropy; eservor; Stable Equlbrum; Exergy ethod 1. Itroducto 2. heoretcal odel ad ethod he formulato of the Secod Law s based o the exstece ad uqueess of the state of stable equlbrum of a system teractg wth a reservor amog all the states of a system that have a gve value of the eergy ad are compatble wth a gve set of values of the amouts of costtuets ad of the parameters [1]. hs geeral statemet requres complace wth the ecessary codtos of equalty of (absolute) temperature, pressure ad chemcal potetal betwee system ad reservor. he Secod Law mples the defto of etropy property relato to temperature. he etropy of system A ca also be derved as the dfferece of eergy E ad exergy EX assocated wth the geeralzed avalable eergy wth respect to a reservor at costat temperature [1]. he Secod Law, expressed as dcated above, has to accout for pressure ad chemcal potetal addto to temperature, hece the purpose of ths preset study s to coceve a geeralzed formulato of exergy, ad cosequetly of etropy, based o equalty of pressure ad chemcal potetal, addto to temperature, betwee a system A ad a reservor at costat temperature, costat pressure P ad costat chemcal potetal. he followg assumptos are posted: 1) the aalyss s focused o smple systems accordg to the termalogy ad deftos adopted by Gyftopoulos ad Beretta [1]; 2) the exteral referece system behaves as a reservor as defed the lterature [1]; 3) the system may be large or small, eve at molecular level, ad may experece states of equlbrum ad oequlbrum [1]; 4) physcal ad chemcal exergy are accouted for; 5) ketc eergy ad the potetal eergy of the system as a whole are eglected. hese assumptos are posted order to costtute the requremets complyg wth the fudametal framework to whch the addressed lterature refers. he method adopted assumes the equalty of pressure ad chemcal potetal of the composte system-reservor A as addtoal ecessary codtos for mutual stable equlbrum, other tha equalty of temperature. he formulato of the etropy property s derved from exergy, related to a reservor at costat temperature, costat pressure ad costat chemcal potetal. hs set of geeralzed potetals equalty costtutes a ecessary ad suffcet codto for the stable equlbrum of a composte system-reservor cosstet wth the Secod Law Copyrght 2013 Sces. JP

2 P. PALAZZO 53 statemet. 3. hermal Etropy Derved from hermal Exergy he behavour of the exergy property s characterzed by addtvty because t s defed cosderg a exteral referece system or a teral part of the system tself that behaves as a reservor [1,2]. Exergy s derved from the geeralzed avalable eergy that s a cosequece of the cocept of adabatc avalablty whe the system teracts wth a reservor [1]; the deftos of avalable eergy ad exergy are based o the mutual stable equlbrum of a system A wth a reservor at costat temperature ad accout for work teracto by meas of a weght process; ths mples that, as defed these terms, (thermal) exergy expresses the maxmum et A AX 10 useful work W coectg two states 0 ad 1 obtaed by meas of a weght process resultg from the avalable eergy betwee varable temperatures of system A ad costat temperature of reservor as expressed by the followg equatos adoptg the symbology used by Gyftopoulos ad Beretta [1]: A AX W U1 U0 S1 S0 EX AX P V V he defto of etropy s derved from the dfferece betwee eergy ad avalable eergy [1] ad s here defed as thermal etropy S by vrtue of the cosderatos dscussed above: 1 S S E E (1) (2) hese equatos, reported the addressed lterature, costtute the outset of the preset study uderped by the cosderato that equalty of temperature s a ecessary codto that s ot suffcet to prove that a system s stable equlbrum wth the reservor. Ideed, eve though two teractg systems are thermal stable equlbrum owg to equalty of temperatures, the two systems may experece oequlbrum states due to the o-ull dfferece betwee pressures or chemcal potetals. he equalty of total potetal ad pressure betwee system ad reservor should therefore costtute the set of addtoal ecessary codtos to esure the stable equlbrum so that the equalty of the complete set of geeralzed potetals costtutes a ecessary ad suffcet codto for mutual stable equlbrum complace wth the Secod Law as worded by Gyftopoulos ad Beretta [1]. As a cosequece of the procedure adopted for ts defto by the sad Authors [1], etropy s a addtve property that ca be geeralzed to clude the cotrbuto of addtoal compoets fulfllg to the codtos of equalty of pressure ad chemcal potetal that guaratee the mutual stable equlbrum of the composte system-reservor. A tutve ratoale of the procedure here adopted to defe etropy may be explaed cosderg that teral eergy s characterzed by a hybrdzato of ordered ad dsordered (due to dstrbuto of molecule posto ad velocty of system s partcles) eergy status. Etropy may be regarded as the measure of the amout of dsordered eergy released to the reservor resultg from the dfferece of hybrd eergy ordered ad dsordered ad avalable eergy ordered eergy trasferred, as useful teracto, to the exteral system. 4. echacal Etropy Derved from echacal Exergy he formulato of thermal exergy Equato (1) expresses the weght process as the result (ad the measure) of the maxmum et useful work that ca be extracted from a system teractg wth a thermal reservor. he weght process ca also be regarded as the mmum et useful work delvered to the system ad hece ca be assocated wth the maxmum et useful heat accordg to the cocepts of equvalece ad tercovertblty [3-5]. I ths case, the weght process s calculated as the result of the teracto of the system wth a mechacal reservor at costat pressure P. he mechacal reservor s here assumed to possess the same propertes as a thermal reservor [1,2,6] that s characterzed ad defed wthout explct referece to ay specfc form of geeralzed potetal. he equalty of pressure betwee system ad reservor s assumed to be a addtoal ecessary codto of mutual stable equlbrum. hs codto should therefore be compled wth equalty of temperature to esure that the status of composte system-reservor s oe of stable equlbrum. If the cocept of geeralzed avalable eergy [1] s ow referred to, the the formulato of mechacal exergy should be traslated to the followg expresso where the superscrpt stads for echacal reservor sce the composte system-reservor udergoes a work teracto ad the physcal meag becomes the maxmum et useful heat of the system: AX 10 A EX Q (3) he defto of mechacal exergy s the bass for dervg the expresso of mechacal etropy usg the procedure adopted for thermal exergy ad thermal etropy: S1 S0 E1 E0 PV (4) Copyrght 2013 Sces. JP

3 54 P. PALAZZO hs expresso of mechacal etropy ca be demostrated cosderg that the procedure coceved by Gyftopoulos ad Beretta provg the formulato of (thermal) etropy [1], does ot mpose ay restrcto wth respect to the form of eergy ad geeralzed avalable eergy that costtute the expresso of etropy. herefore the same procedure ca be cosdered vald regardless of the physcal ature of the propertes volved. he mmum amout of weght process correspods to the maxmum amout of heat teracto: E E 1 W A 10 IN PV lv lv 0 whch expresses the (mmum) amout of work teracto absorbed from mechacal reservor at costat pressure P ad costat V (total volume cocdes wth specfc volume the partcular case of a reservor). Equato (5), substtuted the former relato, expresses the mechacal etropy: S (5) S lv lv (6) 5. Chemcal Etropy Derved from Chemcal Exergy he chemcal potetal geerated by teracto amog the molecules of a ope system costtutes a compoet of teral eergy ad s defed as a form of dsordered eergy, as thermal eergy s [7]. he maxmum work of a reversble (terally ad exterally) chemcal reacto s expressed by the Gbbs NE fucto so that WEV G. he Va t Hoff equlb- rum ope system at costat temperature ad costat pressure s a sutable devce for reproducg a typcal chemcal reacto where A, B are reactats ad C, D are products [7]: G 0 l KP (7) 0 AA BB CC DD s adopted to express maxmum work takg to accout the teral mechasm of chemcal reacto by meas of the expresso depedg o the equlbrum costat K P of the reacto. he Gbbs relato, obtaed from mass ad teral eergy balace, s as follows: where du ds PdV d U U ds S V V, S, VS,, U dv d VS,, (8) U represets the chemcal potetal of the th - costtuet. Chemcal exergy s defed by Kotas [7] as the maxmum work obtaable from a substace whe t s brought from the evrometal state to the dead state by meas of processes volvg teracto oly wth the evromet. he chemcal reservor ca be characterzed accordg to the defto proposed by Gyftopoulos ad Beretta [1] as a reservor wth varable amouts of costtuets. he whole chemcal ad physcal process s modelled by a schema subdvded two typcal steps [7]: the tal molecular system form s rearragemet (molecular structure) ad the fal molecular system s dmesoal chage (geometry-kematcs or pressure ad temperature). he two represetatve processes are: chemcal reacto ope system ad physcal operato ope system; the two processes seres provde a expresso of maxmum et useful work wthdraw from the system expressed by Equato [7]: AX C A P1 EX W10 l (9a) P0 ass teracto s the characterstcs of the chemcal eergy trasfer ad s moved by the dfferece of chemcal potetal betwee the system ad the reservor. I the more geeral case of a mxture of chemcal costtuets: AX C A EX W10 xl x AX 10 EX W C A C (9b) he equalty of total potetals s accouted for as a addtoal ecessary codto of mutual stable equlbrum betwee the system ad the reservor other tha the equalty of temperature [1]. hs mples a defto of chemcal etropy derved from chemcal exergy ad chemcal eergy le wth the methodology prevously adopted. If the cocept of geeralzed avalable eergy s ow aga cosdered, the formulato of chemcal exergy should be traslated to the followg expresso: (10) where the superscrpt C stads for Chemcal reservor sce the composte of system ad reservor udergoes a mass teracto. Now that chemcal exergy s defed, ad cosderg that etropy s a addtve property, the expresso used for etropy assocated wth heat teracto ca be exteded to chemcal potetal depedg o mass teracto: C 1 C S S E E (11) hs costtutes the expresso of the chemcal etropy derved from chemcal exergy based o the equalty of Copyrght 2013 Sces. JP

4 P. PALAZZO 55 chemcal potetal whch costtutes a ecessary codto for mutual stable equlbrum betwee the system ad the chemcal reservor. Sce etropy s a heret property of all systems [8-11], chemcal etropy would be characterzed by the chemcal potetal of all atoms ad sub-molecules that costtute all compouds ad determe the supra-molecular archtecture ad cofguratos of all molecular systems. he dmesos ad shapes of molecular structures play, ths perspectve, a fudametal role determg the mmum etropy level that esures the stablty of matter ad ts capablty to react wth other co-reactats, as well as to udergo edogeous or exogeous processes. ypcal s orgac chemstry ad bochemstry whch same chemcal formula are capable of descrbg materals ad compouds that dsplay dfferet physcal ad chemcal characterstcs ad propertes. he vbrato degree of freedom s a addtoal aspect correlated to molecular structure complexty. 6. Geeralzed Exergy ad Etropy he defto of a thermo-chemcal-mechacal reservor characterzed by costat temperature, pressure ad chemcal potetal mples the addtvty [1,2] of the compoets costtutg the geeralzed exergy: EX G EX EX C EX (12) he teral eergy balace of the composte systemreservor, adoptg the symbology [1], provdes the amout of weght process due to thermal, mechacal ad chemcal cotrbutos: A G A A EX W Q SYSE ESEVOI U U U U U U SYSE, Q SYSE, W Q W SYSE C, UC U (13) Q, where: U Q s the mmum heat teracto C, wth the thermal reservor; U s the mmum mass teracto wth the chemcal reservor ad W, U W s the mmum work teracto wth the mechacal reservor. he sothermal process realzes eergy coverso ad at the same tme a etropy coverso from thermal etropy to mechacal etropy that occurs due to smultaeous heat teracto ad work teracto. Both coversos are accouted for the physcal exergy expresso: G SYSE ESEVOI EX U U OAL, U U0 S, C, S S P V (14) OAL, where the term S represets the cotrbuto to etropy coverso oly occurrg sde the res-, C, ervor ad the terms PV S S represet the cotrbuto trasferred from the system to the reservor. It s oteworthy that etropy coverso s heret eergy coverso ad that etropy coverso requres the addtoal term that cotrbutes to exergy balace expressed the above formulato whch therefore cosders the effect of both eergy ad etropy coverso processes. Cosderg that etropy s a addtve property costtuted by compoets deduced from the correspodg geeralzed exergy s compoets, the geeralzed etropy may be defed as the sum of thermal, mechacal ad chemcal terms: G C S S S S. 7. emarks upo the Gbbs-Duhem elato he Gbbs-Duhem relato [1] costtutes a codto amog all tesve propertes temperature pressure ad chemcal potetal that defe the state of a heterogeeous system. If the system s homogeeous ad costtuted by oe costtuet oly, there are o phase chages or chemcal reacto mechasms sde the system mplyg the system tself s at chemcal equlbrum ad the Gbbs-Duhem relato s: Sd VdPd 0 (15) Chemcal potetal s defed as the compoet of teral eergy geerated by the teractos of ter-partcle postos ad relatve dstace, except for ketc potetal (due to ter-partcle relatve velocty). Cosderg these assumptos, the system model characterstcs ca be assmlated to those adopted the Ketc heory of Gas [12] whch, partcular, cosders molecules udergog elastc repulsve teracto forces (Leard- Joes) o collso wth other molecules ad wth the wall of the cotaer but otherwse exert o attractve teracto forces (Va der Waals) o each other or o the cotaer wall [13]. he cotaer walls represet a geometrcal volume costrat codto mposed o the system. I the specal case of the system s udergog a sothermal process, the pressure chages are due to the chage of volume that determes the frequecy of partcle collsos [12]; f the system s characterzed as assumed, the the dfferetal of chemcal potetal amog all atoms or molecules s due to the temperature that s the oly ter-partcle ketc eergy trasformed to ter-partcle potetal eergy due to repulsve collso teractos (attractve teractos are eglgble by assumpto). Cosderg that o chemcal reactos occur sde the system as assumed, the d 0, as reported by Kotas [6] for systems wth a fxed chemcal compos- Copyrght 2013 Sces. JP

5 56 P. PALAZZO to, ad the Gbbs-Duhem relatos becomes: Sd VdP (16) where S ad V are ot ull both beg heret propertes of ay system [8]. If the system udergoes a sothermal reversble process, the the temperature remas costat by defto, so that d 0; o the other sde, dp s ot ull the same sothermal process ad therefore a cosstecy appears. he same cosstecy s dsplayed f a sobarc process s accouted for usg the Gbbs-Duhem relato where pressure remas costat, so that dp 0, ad temperature does ot. hs cosstecy volves the tesve propertes temperature ad pressure whch determe the thermo-mechacal coversos volved the cocept of avalable eergy ad exergy. he alteratve approach set forth the preset study resolves ths cosstecy. he chemcal potetal expressed by the Gbbs fucto reveals that etropy property s varable the sothermal process ad cosequetly the chemcal potetal s ot costat, whch s cotradcto wth the assumpto set forth. he chemcal potetal that appears assumed system model s due exclusvely to the repulsve teractos o collsos ad depeds solely o molecules velocty, so that temperature costtutes the frst cotrbuto to pressure. he secod cotrbuto s due to the (specfc) volume determg the frequecy of collsos betwee molecules ad the exteral system whch, o the other sde, does ot determe the chemcal potetal due to attractve teractos amog the molecules whch does ot exst the model assumed. If d 0 ad d 0 ad cosderg attractve teractos as eglgble, repulsve teracto potetal s equal to ketc potetal (trasformato durg collso oly). he Gbbs relato: du dsp dv QW (17) ca be reformulated dfferet terms adoptg thermal etropy ad mechacal etropy as defed Equato (6): d d PV U S ds QW (18) ad trasformed, by usg the state equato PV vald ths specal case, ad cosstet wth the Ketc heory of Gases [12], the followg form: du ds ds ds ds QW (19) hs, assocated wth the temperature, takes to accout ether heat or work teractos cotrbutg to varatos teral eergy. It s oteworthy that, the case of work teracto, the work put to the system, whch s cosdered postve, correspods to a decrease of mechacal etropy as per Equato (6). hs s the opposte of heat teracto that s postve f thermal etropy creases. I other terms, heat put causes thermal etropy to crease, work put causes mechacal etropy to decrease, therefore work depeds o pressure the opposte maer that heat depeds o temperature. If the system releases work, t creases mechacal etropy because mechacal eergy, dstrbuted amog the partcles that costtute the system, progressvely becomes smlar to thermal eergy.e. eergy dstrbuted by the velocty of the same partcles. Icreasg volume meas that pressure s progressvely determed by partcles ketc eergy (temperature) wth respect to the cotrbuto of the frequecy of collsos amog partcles ad wth the exteral system surface determed by the volume. Pressure s thus progressvely the more lke temperature as volume creases. he balace of etropy, alog the sothermal expaso reversble process d U 0, s deduced from the OAL equato S S S 0 where S s the thermal etropy put due to heat flowg to the system ad S s the mechacal etropy output from the system whch s the opposte of the mechacal etropy put flow heret the expaso work output from the system tself. he physcal meag s that coverso from a ketc form to a geometrc form, requred to covert heat to work, ecesstates a crease mechacal etropy. Due to the fact that mechacal etropy must eter the system because of work output, the the mechacal etropy drecto s verted compared to the thermal etropy drecto the system s etropy balace. otal etropy s cosequetly costat f ether thermal ad mechacal etropy eter to the system. he ratoale of ths statemet s that these two etropy compoets have a opposte org ad elde each other. OAL If the term S s expressed by meas of OAL S S S 0, the total etropy, resultg from the addto of thermal ad mechacal compoets of etropy, mples that thermal etropy would be costat a sothermal reversble process that requres heat teracto by meas of thermal etropy exchage. O the other sde, pressure does ot derve from ter-partcle chemcal potetal ad s just the mechacal effect produced by the temperature tself (apparet potetal). he Gbbs relato, expressed terms of the Equato (19), resolves the apparet cosstecy hghlghted the Gbbs-Duhem relato. I fact, ths ca be reformulated as follows: Copyrght 2013 Sces. JP

6 P. PALAZZO 57 du S ds d ds d ds d (20) he Euler relato s obtaed from the Gbbs relato by tegrato at costat temperature ad costat chemcal potetal [1], so that: U S OAL S S S S (21) If compared wth the classcal expresso of the Gbbs relato U S PV, the term PV corre- spods to the term S. oreover, the case of a sothermal process (ad absece of chemcal reactos so that chemcal potetal s costat) t requres that total etropy s costat also mplyg that thermal etropy varato s equal to mechacal etropy varato. It s oteworthy that, the case of a deal system as assumed, teral eergy U s assocated wth the ketc eergy of the molecules, ad thus to temperature oly; however, teral eergy compoets deped o the terms ad PV whch both deped o volume as well. hs depedece esures that real systems whch thermodyamc codtos are affected by teractos amog molecules that determe the potetal eergy, are characterzed by volume. I dfferetal terms, the Euler relatos are: du OAL OAL ds S d dd (22) ds ds S S d dd ds ds S d S d dd O combg the Gbbs relato ad Euler relato expressed the terms set forth, the Gbbs-Duhem relato Sd VdP d0 assumes the form: S d S d d S S d d (23) OAL S d d 0 where: costtutes the ter-partcle ketc potetal compoet of teral eergy resultg the PV macroscopc work teracto trasferred by meas of a weght process; costtutes the ter-partcle chemcal potetal compoet of teral eergy resultg the PV macroscopc work teracto trasferred by meas of a weght process. he ketc ad chemcal costtute the two fudametal potetals at mcroscopc ter-partcle level teractg at macroscopc level that costtute the herarchcal geometrc ad kematc structure. he dualsm of ketc potetal ad chemcal potetal costtutes the heret structure of potetals eve the specal case of a deal system for whch ter-partcle potetal eergy ull. I ths case, fact, potetal eergy stll exst the form of repulsve reacto potetal eergy that s due to ketc eergy trasformed o collso oly, wthout macroscopc effects o the etre system. hs dfferet form of the Gbbs-Duhem relato resolves the apparet cosstecy the specal case of the sothermal deal process. I fact, d 0 because the system model s deal ad d 0 remas the oly codto to be satsfed sce dp o loger appears the Gbbs-Duhem relato as expressed Equato (23). he ratoale of these statemets ca also be foud the behavour of elemets, molecules ad atoms, costtutg the system as a whole. I fact, the sothermal process, the temperature ad subsequet termolecular repulsve teractos o each collso are costat ad the varatos of ketc potetal ad chemcal potetal (due to termolecular repulsve teractos) are therefore ull: d 0 ad cosequetly d 0. I the case of the sobarc process, temperature ad pressure are varable or ketc potetal ad chemcal potetal (due to repulsve teractos at each ter-partcle collso) both chage alog the sobarc process: d 0 ad cosequetly d 0. Eve the case of a deal system, there s dualsm ad symmetry of ketc eergy ad potetal eergy amog the molecules so that d 0 ad d 0. Pressure s the mechacal effect of the cotrbuto related to ketc teracto ad related potetal ad chemcal teracto ad related potetal. I ths perspectve, pressure ca vewed as the outcome of the temperature ad chemcal potetal of a complex mult-partcle system, coverted to work teracto wth the exteral reservor ad wth the exteral weght process. Fally, otwthstadg the restrctos assumed for the model adopted, the behavour of the system s coheret wth expectatos terms of pheomea ad tedecy of the propertes the geeral case of real systems ad processes where each partcle expereces attractve teracto wth all others, ad does ot cotradct the fudametals reported the lterature. 8. Cocluso he three addtve compoets of exergy dscussed ths study costtute the compoets of geeralzed exergy that depeds o temperature, pressure ad chemcal potetal ad, at mcroscopc level, o the ketc eergy ad potetal eergy geerated by teractos amog the molecules of the system. oreover, three compoets of etropy property have bee ferred from the correspodg exergy compoets. I partcular, chemcal exergy ad etropy are correlated to the molecular structure of matter due to the composte of molecules geome- Copyrght 2013 Sces. JP

7 58 P. PALAZZO try ad chemcal bods characterstcs. he am of seekg a property related to molecular or supra-molecular archtecture s to obta a method able to predct a-pror stablty as well as capablty self-assemblg processes ad the related termedate phases of chemcal compouds that are ot avalable the evromet ad whch could udergo a buldg process by meas of ao-sceces techologes. Such a method would make t possble to desg materals characterzed by propertes that could be evaluated pror to beg realzed ad to cofrm predctos by meas of expermets ad laboratory tests. Fally, the method of Etropy Geerato mzato (EG) [14-16] assocated wth the Exteded Exergy Accoutg (EEA) [17,18] could be further geeralzed to provde a overarchg paradgm for aalysg the whole process. EFEENCES [1] E. Gyftopoulos ad G. P. Beretta, hermodyamcs: Foudatos ad Applcatos, Dover Publcatos, New York, [2]. A. Gaggol, Iteratoal Joural of Appled hermodyamcs, Vol. 1, 1998, pp [3] W.. Dubar, N. Lor ad. A. Gaggol, Joural of Eergy esources echology, Vol. 114, [4]. A. Gaggol, D. H. chardso ad A. J. Bowma, Joural of Eergy esources echology, Vol. 124, 2002, pp do: / [5]. A. Gaggol ad D.. Paulus Jr. Avalable Eergy Part II: Gbbs Exteded, rasacto of the ASE 2002, [6]. A. Gaggol, he Dead State, Proceedgs of 25th ECOS, Peruga, [7]. J. Kotas, he Exergy ethod of hermal Plat Aalyss, eprt Edto, Kreger Publshg Compay, Kreger, [8] E. P. Gyftopoulos, Iteratoal Joural of hermodyamcs, Vol. 9, 2006, pp [9] G. P. Beretta, Iteratoal Joural of hermodyamcs, [10] E. Zach, Iteratoal Joural of hermodyamcs, [11] E. Zach ad G. P. Beretta, Iteratoal Joural of hermodyamcs, Vol. 13, 2010, pp [12] H. A. Jakobse, Chemcal eactor odelg, Sprger- Verlag, Berl, Hedelberg, do: / _2 [13] B. H. Brasde ad C. J. Joacha, Physcs of Atoms ad olecules, 2d Edto, Pretce Hall, Upper Saddle ver, [14]. L. Hll, Statstcal echacs. Prcples ad Selected Applcatos, Dover, New York, [15] E. Scubba, Etropy, Vol. 12, 2010, pp do: /e [16] E. Scubba, Iteratoal Joural of hermodyamcs, Vol. 14, 2011, pp [17]. Bell ad E. Scubba, Iteratoal Joural of Exergy, Vol. 5, [18] E. Scubba, Eergy, Vol. 28, 2003, pp do: /s (03) Copyrght 2013 Sces. JP