Irreversible thermodynamic analysis and application for molecular heat engines. Umberto Lucia 1, Emin Açıkkalp 2

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1 Ivsibl thmodynamic analysis and application fo molcula hat ngins Umbto Lucia, min Açıkkalp Dipatimnto ngia "Galilo Fais", Politcnico di Toino, Coso Duca dgli Abuzzi 4, 09 Toino, Italy; Dpatmnt of Mchanical and Manufactuing ngining, ngining Faculty, Bilcik S.. Univsity, Bilcik 0, Tuky; Abstact Th aim of this pap is to dtmin lost woks in a molcula ngin and compa sults with maco (classical) hat ngins. Fistly, ivsibl thmodynamics a viwd fo maco and molcula cycls. Scondly, ivsibl thmodynamics appoachs a applid fo a quantum hat ngin with -/ spin systm. Finally, lost woks a dtmind fo considd systm and sults show that maco and molcula hat ngins oby sam limitations. Moov, a quantum thmodynamic appoach is suggstd in od to xplain th sults pviously obtaind fom an atomic viwpoint. Kywods: Molcula hat ngins, Ivsibility, Lost woks, Quantum hat ngin, Quantum constuctal law, Non-quilibium thmodynamics. Intoduction Applications of thmodynamics ang fom fficintly valuation of ngy soucs, lationships among poptis of matt, and vn living matt. ngy is a thmodynamic popty, and it is always constant in th univs; it dosnt dstoy but only changs fom a fom to anoth on, lik wok to hat, wok to lcticity, tc. []. Th usful wok is obtaind by valuating th vaiations of th ngy, which mans that any chang in a systm is always latd to a tansition btwn, at last, two diffnt systm stats.

2 ngy changs, convsion on fom to anoth fom, and intactions with nvionmnt a fully xplaind by th laws of thmodynamics. Th fist law of thmodynamics xpsss th consvation of ngy, whil th scond law stats that ntopy continuously incass fo th systm and its nvionmnt []. Th scond law givs us infomation about th quality of th ngy as wll as allows us to valuat th ivsibility of any al systm []. Scintists and ngins hav always bn tying to obtain th maximum usful wok and to dcas losss o ivsibility to th minimum lvl latd to th spcific constaints [3-6]. Duing th last dcads, a continuous intst on mico- and nano-thmodynamic cycls is gowing, with dvlopmnts in nano-tchnology. Thmodynamic assssmnts fo quantum- and nano-systms, hnc quantum thmodynamics, has bn focusd on [7-76]. In cnt yas, thmodynamic dvlopmnts in th fild of nanotchnology hav aisd novl qustions about thmodynamics away fom th thmodynamic limit. Indd, quantum hat ngins a dvics that convt hat into wok dscibd by th laws of quantum and statistical thmodynamics. Thy hav bn a subjct of intns sach du to thi gat pactical applications, as, fo xampl:. Diffnt thmomty appoach with th aim to duc th dimnsions of th pob and in pushing away fom th thmalisation timscal to obtain a tmpatu masumnt spons in th mgnc of small thmomts fo nano-scal us;. Th us in opto-mchanical systms fo th alization of nano-scal quantum thmal machins with paticula intst fo nano-mchanical sonatos and quantum optomchanical ngins, which should convt incohnt thmal ngy into cohnt mchanical wok fo pow applications in photovoltaic systms; 3. Th bactia convsion of light ngy into biofuls;

3 and many oths. Unlik a classical hat ngin, in quantum hat ngins, th ngy xchang btwn th systm and thmal svois occus in quantizd fashion. Thfo th quantum hat ngins a modlld as sts having disct ngy lvls unlik classical ngins. But, on of th opn poblms in this topic is th link btwn classical and quantum thmodynamics fo th analysis and dsigning of th quantum hat ngins. In 803, Laza Canot dvlopd a mathmatical analysis of th fficincy of pullys and inclind plans [77] in a gnal discussion on th consvation of mchanical ngy. H highlightd that, in any movmnt, th always xists a loss of "momnt of activity". In 84, his son Sadi Canot [3] intoducd th concpt of th idal ngin. It is a systm which opats on a compltly vsibl cycl without any dissipation. But, this sult sms nonsnsical bcaus, appantly, this systm has no ivsibility and, consquntly, it could convt all th absobd hat into wok, without any ngy loss. On th contay, Canot povd that [3,4]:. All idal ngins opating btwn th sam two thmal baths of tmpatu T and T, with T > T, has th sam fficincy C = T /T ;. Any oth ngin has an fficincy such that always < C. Consquntly, th fficincy of a vsibl Canot cycl is th upp bound of thmal fficincy fo any hat ngin woking btwn th sam tmpatu limits [3,4,7,77]. Canot s gnal conclusion on hat ngins is no mo than th poof of th xistnc of natual limit fo th convsion at of th hat into th mchanical ngy [4]. A gat numb of attmpts hav bn dvlopd to impov th calculation of th fficincy of th al machin [5,7,78-86] bcaus all th thmodynamic pocsss occu in finit-siz dvics and in finit-tim, in psnc of ivsibility. Th Canots limit is invitabl fo any natual systm [4], and it has always bn xpimntally vifid. 3

4 In this pap, it is aimd to dtmin ivsibilitis in a quantum hat ngin. Ivsibilitis a classifid xtnal, intnal and total ivsibilitis that is sum of intnal and xtnal ivsibilitis. A mthod is psntd to dscib ivsibilitis fo a quantum hat ngin opating -/ spin systm and thn som numical sult a submittd. Ivsibility fom a quantum point of viw In This Sction w consid th continuous intaction btwn atomic lctons and th nvionmnt photons. Fo simplicity, but without any loss of gnality, w consid th Hydogn-lik atoms in intaction with th lctomagntic wavs psnt in thi nvionmnt. Th lctomagntic wav is a flow of photons, which incoms into th atoms, a absobd by th atomic lctons if th lctomagntic wav fquncy is sonant, and outcoms fom thm. At atomic lvl, th photons can b absobd by th lctons of th atoms, and an lctonic ngy tansition occus btwn ngy lvls of two atomic stationay stats. Thn, th photons a mittd by th xcitd lctons, whn thy jump down into th ngy lvl of th oiginal stationay stat. Appantly, th a no changs in th ngy of th atom, but only in th lctonic tansition. But, in ality th xists a chang in th kintic ngy of th cnt of mass of th atom, which is usually ngligibl in lation to th to th ngy chang in lctonic tansition. Moov, th tim of occunc of th ngy vaiation of th atomic cnt of mass (0-3 s) is gat than th tim of lctonic tansition (0-5 s). H, w stss that an ngy vaiation of th atomic cnt of mass xists and it cannot b nglctd if w consid a gat numb of intaction as it happns at macoscopic lvl [87]. 4

5 Any atomic stationay stat has a wll dfind ngy lvl, idntifid by th pincipal quantum numb n [88-95]. An lctonic tansition btwn two ngy lvls can occu following th quantum slction ul n = n f n i = [88-95], wh th subscipt f mans final stat and th subscipt i mans initial stat. Th atom has an atomic numb Z and only on lcton in th last obital. This lcton movs in its obital, fo which, following th appoach usd in spctoscopy [93-96], w can intoducd [87]:. Th appant atomic adius: 4 0 n n () mz. Th ngy of th atomic lvl: n m Z () n 3. th Sommfld-Wilson ul stats that [38-4]: p dn p n mv n (3) wh p is th angula momntum of th lcton, bing n dfind by th lation (), n =,, 3,... is th pincipal quantum numb, always intg, and is th Diac constant, p = m v is th lctonic momntum, wh m is th mass of th lcton and v its vlocity insid th atom, is th lmntay chag, and 0 is th lctic pmittivity. Considing an Hydognlik atom, at initial stat, th gomtic fnc systm can b fixd in th cnt of mass of th nuclus, so that th atom is at st with null momntum p atm. Its Schödings quation is [88-96]: m N V N m N N tot N,, (4) wh is th Diac constant, m N is th mass of th nuclus, m is th mass of th lcton, N is th nuclus coodinat, is th lcton coodinat, V is th lctostatic potntial, N 5

6 tot is th total ngy, and, is th wav function. Now, by using th lativ N coodinats N, th coodinats of th cnt of mass R m m / m m, N N N th total mass M m N m, th ducd mass m N m, th momntum of th cnt of mass P M R i R, and momntum of th ducd mass paticl p i, th quation (4) bcoms [88-96]: M V R tot,, R R (5) Th wav function R R (5) in th following two quations:, is usually intoducd to spaat th quation R M V R R CM (6) wh CM = P /M is th ngy of th f paticl cnt of mass, and is th ngy of th bound paticl of ducd mass, such that tot, and V Z /. CM Now, w consid th Hydogn-lik atom in intaction with xtnal lctomagntic wavs. Th lctomagntic adiation is a flux of photons, with [96,97]:. Th ngy : h (7) wh h is th Plancks constant ( J s), and is th fquncy of th lctomagntic wav;. Th momntum p : h p c (8) 6

7 W dfin th thmodynamic contol volum as th sph with cnt in th cnt of th atomic nuclus and adius dfind by th lation () with n + instad of n. Consquntly, th intaction btwn th lctomagntic adiation and th Hydogn-lik atom can b analysd as th intaction btwn th flux of photons with an opn systm (th atom of pincipal quantum numb n), though th bod of th contol volum dfind by th sph of adius: 4 n m Z 0 (9) with cnt in th cnt of th atomic nuclus. Th atomic lcton absobs th incoming photon whn its fquncy is th sonant fquncy, quid by th tansition btwn th initial i and final f ngy lvls [88-97], cosponding to th quantizd ngy: f i (0) h wh h is th Plancks constant. mission of th this photon sults in th vs pocss. Th momntum of th incoming photon is hu c /c, wh u c is th vso of popagation of th lctomagntic wav, and c is th vlocity of light in vacuum. Whn an lcton absobs th incoming photon, th atomic momntum bcoms [88-97]: p atm h u c () c and th lcton undgos an ngy lvls tansition, fom th stationay stat of ngy i to th stationay stat of ngy f, which sults [87]: f patm h i h i h () M M c wh p atm/m is th kintic ngy gaind by th atom, and M is th mass of th atom. So, w can obtain [87-97]: 7

8 8 M c h h i f (3) In a simila way, fo th mission of a photon, w can obtain [87-97]: M c h h f i (4) As a consqunc of th absoption of th photon, th laws of consvation of momntum and ngy du to th absoption of th photon, hold to [87-97]: M M m m M m m m m M N N N N N N N N p p R p p p p p R P (5) wh p N is th momntum of th nuclus and p is th momntum of th lcton, N m m and N m m M. Consquntly, th Schödings quation bcoms [87]: R R R V M CM (6) Whn th photon is mittd, following th sam appoach, w can obtain [87]: R R R V M CM (7) with th wav function givn as, R R. W can valuat th ngy footpint of th pocss as: h M m H h CM ftp,, R R (8)

9 wh H is th Hamiltonian of th intaction. In this way, w hav povn that a micoscopic ivsibility xists. Indd, th lation (8) psnts th coction tm h/mc latd to th ivsibility occus duing th photon absoption-mission pocss by th lcton of a Hydogn-lik atom. This tm can b valuatd considing that th ngy of an lctonic tansition is of th od of 0-3 J, whil th ngy, Mc, latd to th mass of an atom M is of th od of 0-8 J. So, th coction tm h/mc is of th od of J, ngligibl compad to in th dnominato, obtaining th wll known lation (0) usd in atomic spctoscopy. But, w must highlight that fo a singl atom w may not consid this coction bcaus it is vy small in lation to th tansition ngy, but w stss that this ngy coction xists, and it is th ngy footpint of th pocss. Ths sults allow us to xplain th Canots sults. Indd, as a consqunc of th continuous intaction btwn lctomagntic wavs and matt, any systm loss ngy fo micoscopic ivsibility and, consquntly, any systm cannot convt th whol ngy absobd into wok. Indd, ou sult consists in pointing out that th intaction btwn a photon and an lcton in an atom affcts th ngy lvl both of th lcton and of th cnt of mass of th atom. Whn w consid a macoscopic systm, w must consid th global ffct of an Avogados numb of atoms (0 3 atoms), and th macoscopic ffct of th atomic ngy footpint fo lctomagntic intaction btwn atoms/molculs and photons sults of th od of J mol -. This ngy is lost by matt fo thmal disquilibium, which causs a continuum lctomagntic intaction [73,75,76]. But, this macoscopic ivsibility is no mo than a consqunc of th micoscopic ivsibility. Now, w must consid th macoscopic ffct of this micoscopic considations. Ivsibl thmodynamics analysis 9

10 ntopy may b calld as thmal ngy cannot b tund into usful wok o somtims, it is calld as disod in th molcula stuctu. ntopy is a stat function and ntopy chang fo a vsibl systm is: S S Q S S dt T g v 0 n ds Qi Ginsin G dt i T 0 i in out out s out dt (9) wh S is th ntopy vaiation that could b obtaind though a vsibl path on which th systm xchangs th sam fluxs acoss its boundais, S g is th ntopy gnation at, i.. th ntopy vaiation du to ivsibility and it psnts how considd systm [78,98], τ is th liftim of th pocss und considation, which can b dfind as th ang of tim in which th pocss occus [9, 78, 99], and Q is th hat xchangd, T is th tmpatu of th thmal souc, s is th spcific ntopy and G is th mass flow. ntopy gnation and obtaind wok fom a systm a two thmodynamic quantitis latd on anoth, bcaus, vsibl wok is th maximum wok that can b povidd any systm, whil actual systms includ ivsibility xpssd by th ntopy gnation, that causs ducing at vsibl wok. Wok intactions of a considd systm by using of kintic ngy thom can b wittn as [7,00,0]: W s W W (0) f i k wh W s is th wok don by th nvionmnt on th systm, i.. th wok don by th xtnal focs to th bod of th systm, W f is th wok lost du to xtnal ivsibility, k is th kintic ngy of th systm, W i is th intnal wok, such that [7]: W i W W () v i fi wh v Wi vsibl intnal wok and W fi lost wok sultd fom intnal ivsibility. W s is th wok don by th systm on th nvionmnt and it can b dscibd as [7,00,0]: 0

11 W s W W () s f Consquntly, lations obtaind fom th fist law of thmodynamics [00,0]: Q W s Q W U i U k (3) wh U th intnal ngy of th systm. k can b wittn as following [75]: 0 k dt 0 Q W U dt Q W U s s (4) Accoding to Annila and Salth, th Noth appoach [0] : 0 k dt nh with n (5) wh n multiplis of quanta h is Planck constant. Using qs. (4) and (5), on can gt: h Q Ws U n n (6) wh is ducd Planck constant. q. (6) shows lationship Annila and Salth sults and ivsibl thmodynamics. Accoding to Gouy-Stodola thom, total lost wok is [0-4]: iv W W fi W f T0 S (7) g wh T 0 is th nvionmntal tmpatu and S g is th ntopy gnation. Manipulating pvious quations, on can gt: v W i Ws n T0 S g Considing an idal systm (without ivsibilitis): (8) W v i n Ws (9) If lctonic tansition in an atom is considd fo which th diffnc of th ngy in a cycl, i.. th absoption and mission of a photon: v v W W W W n n i s i s (30)

12 Fo an ivsibl pocss th quation (30) bcoms: v v n n W i Ws Wi Ws T0 S g T0 S g T0 S g (3) An atom, its gound stat of ngy is 0, absobs a photon of fquncy ν and it passs stat of ngy.lctonic tansition is: 0 h (3) Thotically th atom can hav th vs tansition to th gound stat 0 h (33) without any footpint of th pocss; indd, considing th cycl of absoption and mission of th photon th footpint sults: h h 0 (34) 0 0 But, in an atom, a bound lcton intacts (lctostatic foc) with th atomic nuclus and this intaction must hav a consqunc in th pocss considd, as w hav highlightd in th pvious sction. Whn a photon is absobd by a bound lcton, th following tansition occus: 0 h ka (35) with ka th kintic ngy acquid by th atom as a consqunc of th ngy and momntum consvation. Consquntly, it is ncssay a gat quantity of ngy, spct to th idal cas, to obtain th sam tansition btwn th two stationay stats. Thn, whn th photon is mittd th following tansition occus: 0 h ka (36) with ka th kintic ngy in th final stat. Th footpint of th pocss can b valuatd as: 0 0 h h ka ka ka (37) ka and, considing th lation (3):

13 S g ka ka (38) T 0 with T 0 nvionmntal tmpatu. In this cas th nvionmntal ngy can b latd to an ngy fnc stat, fo xampl th tmpatu of th atomic nuclus. Th sult obtaind ags with th pvious obtaind by using a quantum appoach. Now, w show an xampl on a molcula hat ngin, in od to confim numically ou sults. Application to molcula hat ngin A hat ngin is a machin convting hat ngy into wok and thy a cyclic ngins. Fist law of th thmodynamics, which is about th convsion of ngy, says that diffnc of ngy inlt to th systm and ngy outlt fom th systm fo hat poducing ngins is qual to ngy chang in th systm. It is xpssd by th diffntial fom of th quation (3); Q W du (39) Fo a cyclic systm at th stady stat conditions. Fist law of th thmodynamics is wittn as follows: Q W (40) Fo quantum thmodynamic systm hat and wok changs and thy sult [6]: Q ds (4) W Sd (4) wh, ω is th ngy-lvl gap and S is th spin. xpctation valu of spin opato is [6] : S Tanh (43) 3

14 Th tmpatu (thoughout this pap tmpatu will f top ath than T fo simplicity), if not statd othwis ( = l / k B T, wh T is th absolut tmpatu) [6]. In calculations, k B and h a assumd as fo simplicity. Ral ngins opating with thmodynamic cycl always includ ivsibilitis basd on ntopy gnation and lost wok sultd fom ivsibilitis. Dfining th lost wok can b accomplishd by compaing th hat ngin with totally vsibl (o vsibl) cycl calld as Canot ngin. Canot hat ngin is not only xtnal vsibl but intnal ivsibl. Anoth thotical cycl is calld as Cuzon-Ahlbon ngin that is intnal vsibl (ndovsibl) but xtnal ivsibl. On can dtmind xtnal ivsibilitis by moving th wok poducd by ndovsibl cycl fom th wok poducd by vsibl cycl. Rvsibl (Canot) cycl, ndovsibl and ivsibl (actual hat ngin) cycls (actual hat ngin) fo a quantum hat ngin a shown in Figu. Using quations hat tansf quations [59], wok output of vsibl cycl of spin systm can b wittn as follows: H Cosh v H H Q ln H Tanh Tanh H H Cosh (44) Q Tanh Tanh L 4 Cosh v 4 L 4 3 L 3 L ln L L 3 Cosh (45) W Q Q v v v H L (46) as follows Simila to vsibl cycl, wok gnatd by ndovsibl cycl can b obtaind by 4

15 h Cosh ndo h h Q ln H Tanh Tanh h h Cosh (47) l 4 Cosh ndo 4 l 3 l Q ln L Tanh Tanh l l 3 Cosh (48) W Q Q ndo ndo ndo H L (49) wh,,, 3 4 a dfind in th following lation [6]: Tanh S Tanh S Tanh S Tanh S,, 3, 4 h h l l (35) W invstigat a two-lvl systm and th woking mdium is non-intacting spin -/ systms. Considd hat ngin cycl involv two isothmal banchs connctd by two ivsibl adiabatic banchs. Hat ngin opats btwn hat svois at β H and β L, which a thmal phonon systms. Th svois a infinitly lag and thi intnal laxations a vy stong. In addition, tim dpndnt xtnal magntic fild is applid to th systm. Actual (ivsibl) wok output can b calculatd by using th following quations: h Cosh iv h h Q ln H Tanh Tanh h h Cosh (50) * l 4 * * * * Cosh iv 4 l4 3 l3 Q ln L Tanh Tanh * l l 3 Cosh (5) W Q Q iv iv iv H L (5) 5

16 Accoding to th quantum adiabatic thom, apid chang in th xtnal magntic fild causs a quantum non-adiabatic phnomnon. Th ffct of th quantum non-adiabatic phnomnon on th pfomanc chaactistics of th hat ngin cycl is simila to intnally dissipativ fiction in th classical analysis. x is a paamt sultd fom th intnal ivsibility in adiabatic banch -3 and 4-.Sinc chang of th S to S 3 and S 4 to S is lina, x is a fist gad paamt. Accoding to this dfinition, S 3 and S 4 can b wittn as follows [59, 60]: S S x 3 (53) * H h 3 Tanh Tanh x (54) S S x 4 (55) * L l 4 Tanh x Tanh (56) Lost wok sultd fom xtnal ivsibility is: xt v nd W W W (57) Lost wok bcaus of xtnal and intnal ivsibilitis (total ivsibilitis) is calculatd by moving wok output of th actual hat ngin fom th wok output obtaind vsibl hat ngin: tot v iv W W W (58) Finally intnal ivsibilitis a calculatd by moving th last wok basd on th xtnal fom ivsibilitis th lost wok basd on total ivsibility: int tot xt W W W (59) Accoding to Rfnc [76] ntopy gnation can b xpssd as: W T tot o ka T o ka (60) 6

17 W tot ka ka (6) wh, ka is th kintic ngy acquid by th atom as a consqunc of th ngy and momntum consvations and ka is th kintic ngy of th atom in th final stat, cosponding to th initial gound stat of th lcton involvd in th tansition. xtnal, intnal and total ivsibilitis a calculatd spctivly by using th pvious lations h obtaind: W xt CoshH CoshH CoshH H l L ln h H L h l L ln ln Cosh H CoshH CoshH Hl Cosh L hh l ln H H H hh l hh l LTanh H ( h l ) L Actanh Tanh Tanh L Hl Cosh L Hl H H H hh ltanh hh l LTanh H ( h l ) L ActanhTanh Tanh L H l h H Tanh l L (6) W int H Tanh x CoshH H H ln ln Actanh Tanh Tanh H Cosh Tanh x H l H H H H AcTanhx Tanh x Ta nh AcTanh Tanh Tanh H H AcTanh x Tanh Tanh x (63) 7

18 W tot H Hl H Cosh Tanh x Cosh Cosh( H ) L ln ln ln ln H Hl H Cosh Cosh Tanh Cosh( H ) x L H h L l H H H H ActanhTanh Tanh Actanh x Tanh x Tanh H Tanh h l Hl H H HTanh Actanh Tanh Tanh L H Tanh L h H H Hl Actanh x Tanh Tanh x HTanh L l L (64) In Tabl th paamts usd in calculations a listd. Aft calculations wok outputs v fo vsibl ( W ), ndovsibl ( W ndo ) and ivsibl cycls( iv W ) a found and shown in Tabl. Calculatd sults fo th quantum hat ngin a matching with sults of a maco scal hat ngin; vsibl wok has th biggst, ndovsibl is th scond and wok output of th ivsibl cycl is th smallst on. This povs that quantum thmodynamics and classical thmodynamics a accodanc with ach oth. Using wok outputs of vsibl, ndovsibl and ivsibl cycls, total, xtnal and intnal ivsibilitis a dtmind. Rsults show that xtnal ivsibilitis of th quantum cycl a much mo than (mo than th tims) intnal ivsibilitis. It can b said that ivsibilitis sultd fom th finit tmpatu hat tansf mo ffctiv than ivsibilitis basd on quantum fiction. In addition to that, w dfin total ivsibilitis in tms of kintic ngis dscibd in [75], which is footpint of th pocss. 4. Conclusions 8

19 Famous Canot ngin is a totally vsibl, which psnts upp limits fo hat ngins. This mans it has maximum fficincy and maximum wok output. Howv, Canot hat ngin is thotical and,unfotunatly, it is impossibl that hat ngins opats without losss in ality. In oth wods, th a always losss in actual thmal cycls and ason of ths losss a sultd fom th ntopy gnation. Ths losss, which caus to dcas in wok output and fficincy, a calld as lost wok, ivsibilitis o xgy dstuction. Ths ivsibilitis may b dividd into intnally and xtnally basd. As it fd bfo, Canot hat ngin is totally vsibl, th is no ivsibility o lost wok in it, ndovsibl hat ngin is dscibd as intnally vsibl (howv, xtnally ivsibl). In classical thmodynamics, lationship btwn ths hat ngins can b dfind and valus of ivsibilitis might b calculatd. In this study, lost woks a dtmind fo molcula hat ngin that is -/ spin systm and it is sn that sults cospond to maco hat ngins. In addition, ou sults a simila with f [98-0], which a about mio/molcula hat ngins and quantum thmodynamics. It is commndd that nano/quantum thmodynamics should b focusd on bcaus of advancs in nano tchnolgy. Th lation obtaind show that duing any pocss, th always xist a typ of ivsibility, du to intaction of th matt with th lctomagntic wavs of th nvionmnt, vn if fictions dont xist. This ivsibility is latd to th natu itslf of th matt, and it is du to th xistnc of th spontanous flows btwn opn systms and nvionmnt du to thmal disquilibium. Consquntly, pat of th ngy absobd by th systm is convtd in atomic ivsibility and cannot b usd to convt th absobd hat into wok. H w hav suggstd a nw appoach to xplain th macoscopic ivsibility in thmodynamics by intoducing an ngy footpint in quantum mchanics and diving a chang in th Schödings quation fo a Hydogn-lik atom. W hav followd th 9

20 instins, Schödings and Gibbs considations on th intaction btwn paticls and thmal adiation (photons), which lads to consid th atom as an opn systm in intaction with an xtnal flows of photons. In conclusion, w stat that th quantum mchanical analysis shows that paticl path infomation isnt psvd bcaus th paticl intactions with photons in th thmal adiation fild chang th intnal stats of th paticls thmslvs with a micoscopically ivsibility, in accodanc with th ivsibl masumnts that John von Numann showd incas th ntopy [03]. Th sults h obtaind is a confimation of th hypothsis that an ngy footpint du to ivsibility xists also in atomic lvl tansition [04-08]. Ivsibility is th sult of th intaction btwn opn systm and its nvionmnt. Rfncs. Dinc, I.; Cngl, Y.A. ngy, ntopy and xgy concpts and thi ols in thmal ngining. ntopy 3 (00) Dolv, S.; litzu, A.C. Biology and Thmodynamics: Smingly-Opposit Phnomna in Sach of a Unifid Paadigm. Th instin Quatly: Jounal of Biology and Mdicin 5 (998) Canot, S. Rèflxion su la puissanc motic du fu su l machin a dèvlopp ctt puissanc. Bachli Libai, Pais, Lucia, U. Canot fficincy: Why?. Physica A 39 (03) Clausius, R. Mchanical Thoy of Hat with its Applications to th Stam ngin and to Physical Poptis of Bodis. John van Voost, London, Lavnda, B.H. Thmodynamics of ivsibl pocsss. Dov, Minola, USA, Tusdll, C. Rational Thmodynamics. Sping-Vlag, Blin, Gmany,

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30 Tabl. Paamts usd in calculations Paamt Unit Valu H J L J h J l J x J 5 J 3 30

31 Tabl. Calculatd valus fo th cycl Paamt Unit Valu v W J 0.04 ndo W J 0.06 iv W J xt W J int W J tot W J

32 Figu. Canot quantum hat ngin, ndovsibl quantum hat ngin and ivsibl quantum hat ngin 3