Study of the possibility of eliminating the Gibbs paradox within the framework of classical thermodynamics *

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1 tudy of the possblty of elnatng the Gbbs paradox wthn the fraework of classcal therodynacs * V. Ihnatovych Departent of Phlosophy, Natonal echncal Unversty of Ukrane Kyv Polytechnc Insttute, Kyv, Ukrane e-al: V.Ihnatovych@kp.ua Abstract he forulas for the entropy of deal gases xture and the entropy change n xng of deal gases on the bass of the thrd law of therodynacs were obtaned. It s shown that when usng these forulas, the Gbbs paradox wthn the fraework of classcal therodynacs does not arse. 1. Introducton he Gbbs paradox arose n the theoretcal consderng a queston of a change n the entropy n xng of two deal gases. It conssts n that the value of the change of entropy durng the xng of two deal gases (entropy of xng), wth the sae ntal teperature and pressure does not dependent on the propertes of the xed gases (untl they are dfferent), and unevenly becoes zero n the transton fro xng of dfferent gases to xng of dentcal gases. A jup of the entropy of xng s paradoxcal, when paraeter of gases dfference goes to zero wthout a jup [1-10]. he search of ths paradox explanatons lasts for ore than 100 years and s descrbed n detal n the onograph [10], where about ffty orgnal explanatons for ths paradox are gven and a concluson that there s no generally accepted explanaton s ade. In consderng ths paradox, the varous authors dscussed, n partcular (and dd not coe to coon opnon), as to whether the concluson of the jup of entropy of xng s connected wth the assupton of the exstence of dscrete dfferences between the paraeters of the xed gases or not, and can the concluson of ths jup be elnated, f we accept the prese of the possblty of a sooth transton fro one gas to another [1-10]. * centfc Notes of the Acadey of cences of Hgher chool of Ukrane V.III. pp (n Ukranan). Represented by an acadecan of the Acadey of cences of Hgher chool of Ukrane prof. N.Vrchenko. 1

2 In our opnon, any questons about ths paradox are not solved because n consderng ths paradox ts logcal and atheatcal aspects were not taken nto account: the pont that the concluson about the paradoxcal jup was ade n certan dscourses and regards to atheatcal behavor of the functons of any varables the entropy of deal gases xng, for whch we can derve the forula was not taken nto account. If these crcustances are taken nto account, t ust be sad that the characterstcs of peculartes of behavor of the functon for whch there s a forula, are deterned by the peculartes of ths forula and the peculartes of behavor of varables and paraeters that are ncluded n the forula. he forula for the entropy of xng s obtaned by the developent on the bass of certan ntal forulas. It s understood that the peculartes of behavor of the entropy of xng are caused by the peculartes of the forulas on whch bass the forula for the entropy of xng s developed. If we develop a forula for the entropy of xng, usng other ntal forulas, we can get the other peculartes of the behavor of the entropy of xng. In ths paper the author ntends to show that the correspondng forulas can be obtaned n classcal therodynacs and thus elnate the Gbbs paradox n classcal therodynacs. ones: 2. Prelnares In the papers concernng the Gbbs paradox, the followng forulas are used as the ntal Δ =, g j g =, (2) =, (3) = n( c ln Rln p + ) p 0p where Δ the entropy of xng, the entropy of the xture, the entropy of the g syste consstng of subsystes, separated by pereable parttons, j the entropy of j-th subsyste, the entropy of -th deal gas a xture coponent, n the nuber of oles of the -th gas, c p ts olar theral capacty at constant pressure, therodynac teperature, 0p ntegraton constant, whch depends on the nature of the gas and does not depend on n, c, V, p,. Forulas (2) (4) express the absolute value of the entropy of therodynac systes. In classcal therodynacs, the absolute value of the entropy s deterned on the bass of the second and thrd laws of therodynacs. (1) (4) 2

3 Accordng to the second law of therodynacs, δq d =, (5) where δ Q the aount of heat absorbed by the syste n eleentary equlbru process. Accordng to the thrd law of therodynacs, the entropy of a pure substance an deal crystal when = 0 equals to zero. Accordngly, as to the second and thrd laws of therodynacs, the absolute value of the entropy of any pure substance, whch when = 0 s an deal crystal, equals to: δq p dh С p = = d. + = + Т (6) Т where syste; transton, Q p the heat that s absorbed by the syste at constant pressure; H the enthalpy of the Δ H the enthalpy change n k-th phase transton, the teperature of k-th phase C p the theral capactes of substance at constant pressure. he forula (6) can be used to deterne the entropy of substance, whch s an deal gas at a gven pressure p and teperature 1. Of course, when the teperature lows at constant pressure such substance becoes not deal gas (real gas), then lqud and sold. At 1 the entropy of such substance can be deterned by the forulas (4) and (6). Accordngly, f С d С ln Rln p. p = 0 + Т 0 p then the forula (6) agrees wth the forula (4). he entropy of the ultcoponent syste, whch s at teperature of a xture of deal gases, s equal to: δq С d. (7) p dh p = = + + = Т Т where H the enthalpy of the syste; C the theral capactes of the syste at constant p pressure, Δ H the enthalpy change n k-th phase transton n the syste; the teperature of k-th phase transton n the syste, the entropy of the syste when = 0. 0 In the forula (7) the fact that the entropy of the ultcoponent syste at = 0 can dffer fro zero s taken nto account. 3

4 For a syste, whch s coposed of gases wth equal teperature, separated by parttons, Q = Q. (8) Fro (5) and (8) follows (2). 3. Defnton of the entropy of deal gases xng usng the thrd law of therodynacs In order not to be dstracted by nor detals, we consder the case of xng of two deal gases wth equal ntal teperatures and pressures, prevously separated by an pereable partton. We also assue that the syste contans one ole of each gas. For ths case of (1), (6) (8) follows: C ( c + c ) Δ = d p p1 p Т (9) Т Т If we use the forulas (2) (4), for a xture of dfferent gases we obtan Δ = 2Rln2, (10) and for a xture of dentcal gases Δ = 0. (11) Accordng to (9), the value of the entropy of deal gases xng (at 1 ) depends on the dfference n the behavor of functons C, c p p 1 + c, p 2, Т teperature range 0 1, whch coplcated depends on the propertes the gases, as at low n the Т Т 1 2 teperatures, gases and ther xtures are not deal and at certan teperatures are ovng to a condensed state. It s known that the closer the xture coponents are by ther propertes, the closer are the values of functons C and p c p 1 + c, p and + Т and Т Т 1 2 accordng to (9), the closer to zero s the value of Δ. Consequently, n deternng the entropy of gases and xtures on the bass of the second and thrd laws of therodynacs there s no queston of the ndependence of the entropy of xng of the knd of gases, whch, accordng.d. Khatun [1, p. 24], s an ntegral part of the Gbbs paradox. he xtures of optcal soers are deal. herefore, for dfferent gases optcal soers the value deterned by the forula (9) s equal to zero. When deternng the entropy accordng to the equaton (9) there s also a jup of entropy of xng n the transton fro the xng of dfferent to the xng of dentcal gases. 4

5 If the value Δ has any peculartes of behavor when approachng the propertes of xed c gases, then, accordng to (9), they ay be due only to the peculartes of behavor of values C, p c, p Т, Т. In partcular, the jup of the entropy of xng ay take place only n the event when any the ndcated values changes unevenly. Respectvely, the entropy of xng, deterned usng the thrd law of therodynacs, cannot have any peculartes of the behavor, not connected wth the peculartes of behavor of heat, enthalpy or nternal energy of deal gases. hus, f we start fro the thrd law of therodynacs, t s possble to obtan conclusons regardng the behavor of the entropy of deal gases xng that ake up the content of the Gbbs paradox. Due to the fact that the results, obtaned usng the thrd law of therodynacs, contradct to the results obtaned on the bass of forulas (2) (4), at least one of the forulas (2) (4) contradcts the thrd law of therodynacs. It s shown above that the forula (6) does not contradct to the forula (4), and that the forula (2) can be used to deterne the entropy of deal gases xng usng the thrd law of therodynacs. herefore, the dfferences of behavor of the value Δ, obtaned on the bass of the forulas (2) (4) and the value Δ, obtaned on the bass of the forulas (5) (8), are condtoned by the dfferences of behavor of values, deterned by the forulas (3) and (7). herefore, we can assue that the forula (3), whch expresses the Gbbs theore of entropy of deal gases xng, cannot be reconcled wth the thrd law of therodynacs. Let us note that n the classcal therodynacs, Gbbs theore s proven on the bass of Dalton law about the pressure of deal gases xture (see, for exaple [1 3, 8]). It s easy to ake sure that we can strctly get only the forula for the entropy change when the state of an deal gas xture changes fro the forula of the entropy of an deal gas and Dalton law: Δ = Δ, (12) and the forula (3) does not follow t. 4. Conclusons Gbbs paradox n the forulaton of the peculartes of atheatcal behavor of the entropy of deal gases xng n the transton fro xng of dfferent gases to xng of 5

6 dentcal gases n the classcal therodynacs can be elnated fro ths theory, f for defnton of the entropy of xng we use the thrd law of therodynacs nstead of the Gbbs theore about the entropy of an deal gases xng. It can be assued that the Gbbs theore about the entropy of deal gases xng contradcts to forulas, based on the second and thrd laws of therodynacs. References 1. I. P. Bazarov, Gbbs paradox and ts soluton // Zhurnal Fzchesko Kh (7). P [n Russan]. 2. I.P. Bazarov, Paradoxes of gases xng // Uspekh Fzcheskkh Nauk (3). pp [n Russan]. 3. I. P.Bazarov, herodynacs. Moscow: Vysshk, 1991 [n Russan] 4. L.A.Blyuenfeld, A.Yu. Grosberg, Gbbs paradox and the concept of syste constructon n therodynacs and statstcal physcs // Bophyscs (3). pp [n Russan]. 5. J.. Warsawskj, A. V. chenn, On the entropy of systes contanng coponents dffcult to dstngush / / Doklady Akade Nauk R (5). p [n Russan] 6. J. M. Gelfer, V. L. Ljuboshts, M. I. Podgoretskj, Gbbs Paradox and dentty of partcles n quantu echancs. Moscow. Nauka Publshers, 1975 [n Russan] 7. V. B. Gubn. About physcs, atheatcs and ethodology. Moscow: PAIM, p. [n Russan] 8. B. M. Kedrov hree aspects of the atoc theory. Gbbs paradox. he logcal aspect. Moscow: Nauka Publshers [n Russan] 9. V. L. Lyuboshts, M. I. Podgoretsky, Entropy of polarzed gases and Gbbs paradox // Doklady Akade Nauk R (3). P [n Russan] 10.. D. Khaytun, he Hstory of the Gbbs' Paradox; Moscow: Nauka Publshers, 1986 [n Russan] 6