Thermodynamic derivations of conditions for chemical equilibrium and of Onsager reciprocal relations for chemical reactors

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1 JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 6 8 AUGUST 2004 Themodynami deivations of onditions fo hemial equilibium and of Onsage eipoal elations fo hemial eatos Gian Paolo Beetta a) Univesità di Besia, via Banze 38, Besia, I-25123, Italy Elias P. Gyftopoulos b) Massahusetts Institute of Tehnology, Cambidge, Massahusetts Reeived 2 Mah 2004; aepted 6 Apil 2004 Fo an isolated hemial eato, we deive the onditions fo hemial equilibium in tems of eithe enegy, volume, and amounts of onstituents o tempeatue, pessue, and omposition, with speial emphasis on what is meant by tempeatue and hemial potentials as the system poeeds though nonequilibium states towads stable hemial equilibium. Fo nonequilibium states, we give both analytial expessions and pitoial epesentations of the assumptions and impliations undelying hemial dynamis models. In the viinity of the hemial equilibium state, we expess the affinities of the hemial eations, the eation ates, and the ate of entopy geneation as funtions of the eation oodinates and deive Onsage eipoal elations without eouse to statistial flutuations, time evesal, and the piniple of miosopi evesibility Ameian Institute of Physis. DOI: / I. INTRODUCTION In authoitative disussions 1 3 of hemial equilibium among onstituents of a system A, the ondition of equilibium in the pesene of one hemial eation mehanism is pesumably shown to be i1 i i 1,n 2,...,n 0, whee i, fo i1,2,...,, is the ith stoihiometi oeffiient of the hemial eation mehanism i1 i A i 0. A i denotes the ith onstituent, i the hemial potential of the ith onstituent, U the enegy, V the volume, n 1,n 2,...,n the amounts of onstituents given by the elations 1 2 n i n ia i fo i1,2,...,, 3 and n ia, fo i1,2,...,, the amount of the ith onstituent fo whih the value of the eation oodinate is equal to zeo. Fo given values of U, V, n 1a,n 2a,...,n a, 1, 2,...,, ondition 1 yields the value 0 fo whih the system is in the hemial equilibium o stable equilibium state. Thus, at hemial equilibium, the amounts of onstituents ae given by the elations n i0 n ia i 0 fo i1,2,..., 4 and the oesponding mole fations o omposition by the elations a Eletoni mail: beetta@unibs.it b Eletoni mail: epgyft@aol.om y i0 n ia i 0 n a 0 fo i1,2,...,, 5 whee n a i1 n ia and i1 i. In the disussions just ited, 1 3 it is also stated that ondition 1 esults fom the equiement that, fo an isolated system, the value of the sum i1 i ()dn i at the hemial equilibium state must be zeo fo any vaiations of the amounts of onstituents ompatible with the stoihiomety of the eation mehanism, whee nn 1,n 2,...,n denotes the amounts of the onstituents. Even though expeiene shows that ondition 1 leads to esults onsistent with obsevations, its deivation and meaning ae poblemati. Aoding to the seond law of themodynamis, 4 an isolated system with one o moe hemial eations, and given values of U, V, n 1a,n 2a,...,n a, admits one and only one stable equilibium state. To that state oesponds a unique omposition. Any omposition that deviates fom that of the stable equilibium state oesponds to a state that is not stable equilibium and, theefoe, no hemial potentials an be defined. So what funtions i ( 1,n 2,...,n ) should be used in ondition 1 in ode to find 0 and the hemial equilibium state? We investigate this question and find a satisfatoy answe fo any system A that satisfies the model assumption of what we all a simple system. Fo suh a system, the hemial potentials appeaing in ondition 1 ae those of a suogate simple system B onsisting of the same onstituents as A, with the diffeene, howeve, that all eation mehanisms ae tuned. We disuss biefly the definition of a simple system in Se. II, the deivation of onditions fo hemial equilibium of an isolated hemial eato with onstituents and hemial eations in tems of enegy, volume, and hemial /2004/121(6)/2718/11/$ Ameian Institute of Physis

2 J. Chem. Phys., Vol. 121, No. 6, 8 August 2004 Conditions fo hemial equilibium 2719 potentials in Se. III, the deivation of the same onditions in tems of tempeatue, pessue, and mole fations in Se. IV, the ate of entopy geneation in the eato in Se. V, eipoal elations in Se. VI, and ou onlusions in Se. VII. II. SIMPLE SYSTEMS We define a system as simple 4 if it satisfies the following thee onditions: i it has volume as one of the paametes; ii in any of its stable equilibium states, if it is patitioned into a set of ontiguous subsystems in mutual stable equilibium, the effets of the patitions on the values of all popeties ae negligible; 5 and iii in any of its stable equilibium states, the instantaneous swithing o on of one o moe intenal eation mehanisms, suh as a hemial eation, auses negligible instantaneous hanges in the values of enegy, entopy, volume, and amounts of onstituents. In geneal, eithe the intodution of patitions o the instantaneous swithing on o of hemial eation mehanisms o both have definite effets on a system. Fo example, using the tools of quantum theoy, 6,7 we an show that the swithing on of a eation mehanism equies the swithing on of an additional tem in the Hamiltonian opeato of the system, whih affets the funtional fom of the fundamental elation fo stable equilibium states. Again, using the tools of quantum theoy, we an show that the swithing of a eation mehanism equies the destution of oelations among onstituents and, in geneal, esults in a edution of the value of the entopy. Nevetheless, we an also show that these effets beome less and less impotant, and negligible fo all patial puposes, if the value of the amount of eah onstituent is lage than 10 Refs. 6 and 7. Hene the simple system model is appliable to most patial systems, inluding the nanovolume and miovolume sale, with suffiiently lage amounts of onstituents. n i0 n ia j i j0 fo i1,2,...,. 7 j1 Moeove, the values U, V, n a n 1a,n 2a,...,n a, and the stoihiometi oeffiients i ( j) fo i1,2,..., and j 1,2,..., detemine uniquely the values of all the popeties and quantities that haateize the hemial equilibium state, inluding the values of the entopy S, eah j0, and eah n i0. We wite the dependenes of the latte in the fom SS a ;, 8 j0 j0 a ; fo j1,2,...,, 9 n i0 n i0 a ; fo i1,2,...,. 10 In geneal, we annot find the expliit funtional foms of Eqs Fo simple systems, howeve, the poblem is somewhat less ompliated beause we an expess hemial equilibium popeties in tems of stable equilibium popeties of a multionstituent system in whih all the hemial eation mehanisms ae inhibited swithed. To see how this is done, we poeed as follows. Fist, we onside a simple system B onsisting of the same types of onstituents as system A but with all the hemial eation mehanisms inhibited swithed. Of ouse, A and B ae diffeent systems beause they ae subjet to diffeent intenal foes and onstaints. We assume that B is in a stable equilibium state with values U of the enegy, V of the volume, and nn 1,n 2,...,n of the amounts of the onstituents. We denote the entopy at this stable equilibium state by the fundamental elation III. DERIVATION OF CONDITIONS FOR CHEMICAL EQUILIBRIUM S S, 11 We onside a simple system A having enegy U, volume V, and onstituents A 1,A 2,...,A with initial amounts n 1a,n 2a,...,n a, subjet to hemial eation mehanisms i1 i j A i 0 fo j1,2,...,, 6 and deive the onditions fo A to be in a stable equilibium ( j) o, synonymously, in a hemial equilibium state, whee i ae the stoihiometi oeffiients of the jth hemial eation. In geneal, the hemial eato just defined, fo eah given set of values of U, V, n a, and, admits an infinite numbe of states. Howeve, the laws of themodynamis equie that among these states one and only one be a hemial equilibium state, and this state has the lagest value of the entopy. 4 We all the latte equiement the highest o lagest entopy piniple. At the hemial equilibium state, the values of the amounts of onstituents n 10,n 20,...,n 0 and the oesponding eation oodinates 10, 20,..., 0 satisfy the ompatibility elations whee we use the subsipt to emphasize that all the eation mehanisms ae swithed. Next, we assume that the hemial eation mehanisms ae instantly swithed on, that is, all the eations defined by the stoihiometi oeffiients ae no longe inhibited. As a esult, we obtain again system A. Beause in ou disussion of hemial eatos we go bak and foth between systems A and B by swithing and swithing on the hemial eation mehanisms, we all system B the suogate system of A. Beause the suogate system B is simple and initially in a stable equilibium state, immediately afte swithing on the eation mehanisms the state of system A has the same values of S, U, V, n 1,n 2,...,n as the oesponding values of the stable equilibium state of B. In geneal, howeve, this state of A is not stable equilibium. Fo example, if B is a quiesent mixtue of gasoline vapo and ai at oom tempeatue and we ativate the eation mehanisms by a minute spak, we instantly podue a nonequilibium state of system A in whih the eations ae no longe inhibited the buning

3 2720 J. Chem. Phys., Vol. 121, No. 6, 8 August 2004 G. P. Beetta and E. P. Gyftopoulos of the gasoline is poeeding even though the instantaneous petubations of the values of S, U, V, n 1,n 2,...,n intodued by the spak ae negligible. In othe wods, it is by vitue of the impotant key assumption of the simple system model by whih eations an be swithed on and without signifiantly alteing the value of any popety that we sueed in expessing popeties of stable equilibium states as well as of a lass of nonequilibium states of a eating system in tems of the known stable equilibium popeties of noneating multiomponent systems. Among all the states of A that may be obtained fom B in the manne just ited, we onside the subset that has given values U and V of the enegy and volume, and amounts of onstituents that ae ompatible with given values n 1a,n 2a,...,n a. We denote eah of these states by A and eognize that it oesponds to a set of values of the eation oodinates 1, 2,..., suh that n i n ia j i j fo i1,2,...,, 12 j1 whee all the n i s ae non-negative. Among all the states A, the one with the lagest entopy is the unique hemial equilibium state with enegy U, volume V, and amounts of onstituents ompatible with n a, that is, n 0 n a 0. We denote the hemial equilibium state by A 0. To pove that indeed state A 0 oesponds to the lagest entopy, we assume that anothe state A 0 A 0 with entopy S 0, not belonging to the family of states A, is the hemial equilibium state that oesponds to the given values U, V, n a,. Then S 0 S 0 beause A 0 has the lagest entopy. Now, stating fom A 0, we swith the hemial eation mehanisms. Beause system A is simple and A 0 is a stable equilibium state, the esulting state B 0 of suogate system B has the same values U, V, and n 0 as A 0 and, in patiula, its entopy is S 0. State B 0 annot be stable equilibium beause, if it wee, then upon swithing the hemial eation mehanisms bak on we would obtain again state A 0 and onlude that it belongs to the family A ontaditing the fat that A 0 has the lagest entopy. On the othe hand, if state B 0 is not stable equilibium, then the stable equilibium state of B with values U, V, and n 0 would have entopy SS 0, and swithing on the eations beginning with this state would yield a state in the family A that has entopy SS 0 S 0, again ontaditing ou stipulation that A 0 has the lagest entopy. Theefoe, if A 0 is the hemial equilibium state, it must oinide with state A 0 beause unde the speified onditions thee is one and only one stable equilibium state. Beause we an expess the entopy S of a state A in tems of the entopy S () of the state of the suogate system to whih A oesponds, we an detemine the hemial equilibium entopy S( a ;) Eq. 8 by finding the lagest value of S (). To find the lagest value just ited, we fist wite the entopy S of a state A in the fom S S a, 13 whee in the fundamental elation S S () we use the shothand notation n a fo the set n ( n 1a j1 j) ( 1 j, n 2a j1 j) ( 2 j,...,n a j1 j) j. Then we note that in ode fo A 0 to be the state of lagest entopy among all the states A with given U, V, and n a, the values of 0 must be suh that S 0 fo j1,2,...,, 14 j a,, whee the subsipts n a,, and denote, espetively, that eah of the amounts n ia, eah of the stoihiometi oeffiients i ( j), and eah of the eation oodinates i that do not appea in the deivative ae kept fixed. Fo j1,2,...,, fom Eq. 13 we find that S j a,, i1 i1 S n i i, j T i n i j n a,, Y j, a i1 i, i j, 15a 15b 15 15d whee T is the tempeatue and i, the hemial potential of onstituent i of the stable equilibium state of the suogate system B that oesponds to A 0, 1/T, and in witing Eqs. 15b and 15d we use the elations (S/n i ) i /T i and Eq. 12, and in witing Eq. 15 we ( define Y j, A j, /T whee A j, i1 j) i i, is the so-alled affinity of the jth eation, whih is lealy a stable equilibium popety of suogate system B. Fo finite values of T, we see fom Eqs. 14 and 15 that a set of neessay onditions that elate U, V, n a,,, and 0 at hemial equilibium ae i1 i j i, a 0 0 fo j1,2,..., 16 o, equivalently, Y j, ( a 0 )0. In the next setion we show that T and i fo i1,2,..., ae also equal to the tempeatue and hemial potentials of the hemial equilibium state of system A. Eah of Eqs. 16 is the hemial equilibium equation fo the oesponding eation mehanism. Fo eah given set of values U, V, n a, and, Eqs. 16 ae neessay onditions fo hemial equilibium. They may be solved to yield Eqs. 8 to 10 and, theefoe, all popeties of the hemial equilibium state. They onfim the statement made ealie to the effet that popeties of hemial equilibium may be expessed in tems of popeties of a multionstituent system with all hemial eation mehanisms swithed. Fo the extemum oesponding to Eqs. 14 to be a elative maximum, it is also neessay that the seond-ode

4 J. Chem. Phys., Vol. 121, No. 6, 8 August 2004 Conditions fo hemial equilibium 2721 patial diffeential of S with espet to the eation oodinates 1, 2,..., be negative. To show that indeed this is the ase, we stat with Eq. 13 and find that k Y j, a,,y j Y k, a,,y 22 d 2 S a, 1 2 j1 1 2 j1 1 2 p1 1 2 p1 2 S k1 j k d j d k a, 2 S k1 p1 q1 2 S q1 n p n q j1 2 S q1 n p n q j p k q d j d k n p n q dn p dn q 0, j p d j k q d k k1 17 ( whee we use dn i j1 j) i d j fo i1,2,...,. The inequality is always satisfied beause S is the fundamental elation of the suogate system B and, as suh, it is onave with espet to evey n i fo i1,2,..., Ref. 4. Eah of the neessay onditions fo hemial equilibium eah of Eqs. 16 is expessed as a funtion of enegy, volume, and amounts of onstituents of the hemial equilibium state. In the next setion we ewite these onditions in tems of tempeatue and pessue athe than enegy and volume. Fo fixed values of U, V, n a, and, fom Eqs. 13 and 15 we see that eah of the funtions S and Y j, fo j 1,2,..., depends solely on the eation oodinates k fo k1,2,...,. Aodingly, fo j,k1,2,...,, we an wite 2 S a j k o, equivalently, Y k, j a,, 2 S a k j Y j, k 18, 19 a,, that is, the matix with elements a jk (Y j, / k ) a,, is symmeti. Moeove, if we invet the elations Y k, Y k, a fo k1,2,...,, 20 with espet to the vaiables 1, 2,...,, we obtain the elations j j a,,y fo j1,2,...,, 21 and using the popeties of Jaobians we an easily show that the matix with elements b jk ( j /Y k, ) a,,y is also symmeti; that is, fo j,k1,2,...,, fo both zeo and nonzeo values of Y, that is, not only at the hemial equilibium state of the eato A, but also fo any nonequilibium state A that we model with the oesponding stable equilibium state of the suogate system B. Relations 19 and 22 ae among the many Maxwell elations that an be established fo stable equilibium states of a multionstituent system, both fo the suogate system B and the hemial equilibium state of eato A. Relation 22 implies that fo the state of the suogate system to emain in a stable equilibium state and, hene, fo the state of system A to emain within the family A, of the fou hanges d j, d k, dy j,, and dy k,, we an speify only thee abitaily and independently. IV. CONDITIONS FOR CHEMICAL EQUILIBRIUM IN TERMS OF TEMPERATURE AND PRESSURE Rathe than using enegy, volume, and amounts of onstituents as independent vaiables, it is often moe onvenient to expess eah hemial equilibium equation Eqs. 16 in tems of tempeatue, pessue, and mole fations. To this end, we note that the stable equilibium state of the suogate system B obtained by swithing the eation mehanisms at a hemial equilibium state of system A has not only the same values of enegy, entopy, volume, and amounts of onstituents as the hemial equilibium state, but also the same values of tempeatue, pessue, and hemial potentials 1, 2,...,. Fo i1,2,...,, to pove the last assetion, we eall the definitions of tempeatue, pessue, and hemial potentials 4 as given, espetively, by the elations T a ;1/S/U V,na,, p a ;S/V U,na, /S/U V,na,, i a ;S/n ia a, /S/U V,na,, whee S( a ;) is the fundamental elation fo the hemial equilibium states Eq. 8. Next, we expess the fundamental elation S of system A in tems of that of the suogate system B by evaluating S Eq. 13 at 0 as given by Eq. 9, so that SS a ;S a 0 a ;. 23 Thus, fo the invese tempeatue of a hemial equilibium state, we find that

5 2722 J. Chem. Phys., Vol. 121, No. 6, 8 August 2004 G. P. Beetta and E. P. Gyftopoulos 1 T 1 T a ; U S V,n a, S U V,n i1 24a 24b S n i i j1 j j0 U, V,n a 24 S U V,n 1 j1 j0 U j i V,n a, i1 i, 24d i i a ; T S n ia a, T S n i j1 k j k1 j0 n ia a, T S n i j1 S n k 26a 26b 26 j0 n ia j k k, a, k1 26d S U V,n 24e T S n i 26e 1 T a 0 a ; 1, T 24f whee in witing Eqs. 24 and 24e we use Eq. 23 and the hemial equilibium equations Eqs. 16, espetively. So the tempeatue T of a hemial equilibium state equals the tempeatue T of the oesponding state of the suogate system B. Fo the pessue of the hemial equilibium state we find that pp a ; T V S U,n a, T S V U,n i1 25a 25b S n i i j1 j j0 V, U,n a T S V U,n j1 j0 V j i i, U,n a, i d i, a 0 a ; i,, 26f whee in witing Eq. 26e we use Eqs. 16 and 24. Sothe hemial potential i of the ith onstituent of a hemial equilibium state of system A equals the hemial potential i, of the ith onstituent of the oesponding state of the suogate system B. It is notewothy that the identity of values of tempeatue, pessue, and hemial potentials of a hemial equilibium state with the values of the espetive popeties of a stable equilibium state of the suogate system obtains only at hemial equilibium, beause then and only then ae the hemial equilibium equations Eqs. 16 satisfied. Away fom hemial equilibium states, tempeatue, pessue, and hemial potentials ae not defined fo system A beause all suh states ae not stable equilibium. Finally, we note that Eqs indiate that, geometially, the sufaes epesented by the funtions S S( a ;) and S S ( a ) have a ontat of fist degee fo eah given set of values U, V, and n a,at 0 ( a ;), namely, at eah hemial equilibium state. As is vey well known, 4 eah hemial potential of a multionstituent system in whih all hemial eation mehanisms ae swithed may be expessed in the fom i, i, (T,p,y 1,y 2,...,y ), whee y i is the mole fation of the ith onstituent. Using the stoihiometi elations and the hemial equilibium equations, we find T S V U,n p a 0 a ; p, 25e 25f whee in witing Eq. 25e we use Eqs. 16 and 24. Sothe pessue p of a hemial equilibium state equals the pessue p of the oesponding state of the suogate system B. Fo eah hemial potential of the hemial equilibium state we find y i y i n a n ia j1 j i j n a j1 j j and, fo j1,2,...,, i1 j i i, T,p, n 1a j1 j 1 j0 n a j1 j,..., j0..., n a j1 j j0 n a j1 j j0 TY j, T,p,y 0 0, 27 28

6 J. Chem. Phys., Vol. 121, No. 6, 8 August 2004 Conditions fo hemial equilibium whee n a i1 n ia, ( j) ( i1 j) i, and y i0 y i (n a 0 ). Equations 28 epesent the hemial equilibium equations as funtions of T, p, and the mole fations of the hemial equilibium state. The funtional dependenes of the hemial potentials on T, p, and y ae those of suogate system B. Fo given values of T, p, and n a, we an solve the equations 28 fo the unknowns 10, 20,..., 0 and hene detemine the hemial equilibium omposition y 10,y 20,...,y 0 and the values of the oesponding amounts of onstituents, n 10,n 20,...,n 0. Convesely, if the values of T, p, y 1,y 2,...,y ae given but do not satisfy Eqs. 28, we would onlude that the state is not hemial equilibium. Then, of ouse, the values of T and p efe to a state of the suogate system whih beomes a nonequilibium state of eato A when the eations ae tuned on. If the hemial potentials ae witten as i, T,p,y ii T,pRT ln a i, T,p,y, 29 G j T,p,n a,, i1 i1 G n i T,p,n n i j n a,, i, i j TY j, T,p,n a 0, whee we use the equations (G/n i ) T,p,n i fo i 1,2,...,. These onditions ae satisfied if the stable equilibium state of the suogate system oesponds to the hemial equilibium state of A that is, if the hemial potentials satisfy the hemial equilibium equations Eqs. 16. The exteme value of G is a elative minimum beause it an be shown that the seond ode patial diffeential of G with espet to eah of the j s is positive. By equating seond ode deivatives of G with espet to the j s, fo j, k1,2,...,, whee ii (T,p) is the pue onstituent hemial potential at T and p, R the gas onstant, and a i, a i, (T,p,y) the ativity of the ith onstituent of the suogate system, it is easy to see that we an ewite the affinities as A j, TY j, i i j ii T,pRT ln i1 a i, j i, 30 and, defining the equilibium onstant at tempeatue T and pessue p fo the jth eation as K j T,pexp 1 RT i we an wite i1 i j ii T,p, a i, j i K j T,pexpY j, /R and ewite the hemial equilibium equations 28 in the well-known mass-ation-law fom i1 a i, T,p,y 0 j i K j T,p. 33 Anothe inteesting esult is that the lowest value of the Gibbs fee enegy of the suogate system B obtains at the state of B that oesponds to the hemial equilibium state of A. Indeed, the Gibbs fee enegy of the suogate system B is G G T,p,n 1,n 2,...,n. 34 If the amounts of onstituents ae ompatible with n a and the hemial eations onfom to Eqs. 12, we an ewite G in the fom G (T,p,n a ). Fo given T, p, n a, and, an exteme value of G obtains povided that, fo eah j1,2,...,, 2 G T,p,n a j k we obtain the Maxwell elations Y k, j Y j, T,p,n k T,p,n a,, 2 G T,p,n a k j, a,, Moeove, inveting the set of elations Y k, Y k, (T,p,n a ) fo k1,2,...,, with espet to the vaiables 1, 2,...,, we obtain the set of elations j j (T,p,n a,,y ) fo j1,2,...,, and using the popeties of Jaobians, we obtain the Maxwell elations j 38 k Y j, T,p,n a,,y Y k, T,p,n a,,y fo both zeo and nonzeo values of Y, that is, not only at the hemial equilibium state of the eato A, but also fo any nonequilibium state A that we model with the oesponding stable equilibium state of the suogate system B. Relations 37 and 38 an also be viewed as diet onsequenes of the Maxwell elations fo the suogate system obtained by equating seond ode deivatives of G with espet to the n j s, k, n j j,, 39a n k T,p,n o, equivalently, ln a k, n j T,p,n T,p,n ln a j, n k T,p,n. 39b We onlude ou deivation by summaizing the esults pitoially with the help of the enegy vesus entopy gaphs intodued in Ref. 4. Fo given values of V and n a, thee pojetions of states ae supeimposed on the single U vesus S diagam shown in Fig. 1: 1 the pojetion of the states A 1 of system A that oinide with the stable equilibium states of the suogate system B fo the given volume V of the hemial eato and fixed values n 1 n a 1, 2 the

7 2724 J. Chem. Phys., Vol. 121, No. 6, 8 August 2004 G. P. Beetta and E. P. Gyftopoulos FIG. 1. Enegy vesus entopy gaph of the states of simple system A with given values of V, n a, and. The uve labeled 1 epesents the states A 1 and oinides with the uve of the stable equilibium states of suogate system B fo the given value V of the volume and fixed values n 1 n a 1 of the amounts of onstituents that is, a fixed 1. The uve labeled 2 epesents the states A 2 and oinides with the uve of the stable equilibium states of suogate system B fo the given value of V of the volume and fixed values n 2 n a 2 of the amounts of onstituents that is, a fixed 2. The uve labeled 0 epesents the hemial equilibium states of system A fo the given values of V, n a, and. pojetion of the states A 2 of system A that oinide with the stable equilibium states of the suogate equation B fo the same given volume V and fixed values n 2 n a 2, and 3 the pojetion of the hemial equilibium states of system A fo the same given values of V, n a, and. The set of values 1 is hosen so that 1 0 (U 1,V,n a ;) and, theefoe, at the enegy U 1 the lous of states A 1 is tangent to the uve of the hemial equilibium states of A. Similaly, the value 2 is suh that 2 0 (U 2,V,n a ;) and, theefoe, the lous of states A 2 is tangent to the uve of the hemial equilibium states of A at the enegy U 2 of the hemial eato. We see that the uve of the hemial equilibium states is the envelope of the loi of states A fo all possible values of. We also see that the states A 1 epesent states of system A that ae not stable equilibium, exept at the enegy U 1, and similaly the states A 2, exept at enegy U 2. In geneal, they ae eithe nonequilibium o nonstable equilibium states and yet an be desibed using the stable equilibium popeties of the suogate system. V. RATE OF ENTROPY GENERATION In the ouse of hemial eations in an isolated system A with onstituents and hemial eation mehanisms, the system passes though a sequene of nonequilibium states, and entopy is geneated until the system eahes hemial equilibium. At hemial equilibium, all hanges ease the ate of hange of eah eation oodinate is zeo and theeafte the system emains in the stable equilibium state. The igoous and omplete evaluation of the evolutions of the popeties of the system as funtions of time fom any state that is not stable equilibium towads the oesponding hemial o stable equilibium state, and theefoe the ate of entopy geneation in a geneal nonequilibium state is outside the sope of this atile, and we ae not disussing it. Instead, nevetheless, we deive an estimate of the ate of entopy geneation in tems of the affinities of the suogate system B of A and the ates of hange of the eation oodinates of the hemial eation mehanisms. We will see that this estimate is infomative both about what might be onsideed as the diving foes of the eations and about whethe the so-alled piniple of miosopi evesibility plays any ole in entopy geneation. To deive this estimate, we poeed as follows. Fo given values U, V, n a, and, the values of the amounts of onstituents nn(t)n 1,n 2,...,n, and the value of the entopy S(t) ae funtions of time. Speifially, the value of S(t) is smalle than the value of S (t) of the suogate system B; that is, StS t S a t, 40 whee the equal sign applies only at the hemial equilibium state. The justifiation of Inequality 40 is that, by definition, S (t) oesponds to the entopy of a stable equilibium state whih, by vitue of the lagest entopy piniple, is highe than the entopy of any othe state with the same values of U, V, and n(t). Beause at hemial equilibium both S(t) and S (t) assume the same lagest value, an estimate of the ate of entopy geneation Ṡ i the ate of entopy geneation by ievesibility in the isolated system A is obtained by assuming that A is always in one of the states A defined in Se. III, so that the value of eah popety is equal to the oesponding stable equilibium state value of the suogate system B. This assumption oesponds to the following two-step oneptual model. We stat at time t with the suogate system B in a stable equilibium state B t,eq oesponding to the state A (t) of the eato with eation oodinates (t). We then tun the eations on fo an infinitesimal lapse of time dt at the end of whih we immediately tun them. As a esult, the eato is in state A tdt, the eations have hanged the omposition, and, in geneal, the suogate system at time tdt is in a nonequilibium state B tdt. We allow the additional lapse of time t that is neessay fo the suogate system to eah the stable equilibium state B tdtt,eq oesponding to the state A (tdtt ) of the eato with the new omposition. Now we tun on the eations again, and so on. The values of the entopy ae as follows: S B A t,eq S (t) and B A B A S tdt S tdt S tdtt,eqs (tdtt). The entopy geneated by ievesibility in the fist time inteval is A A A S i,(t,tdt) S tdt S (t) and in the seond B A A S i,(tdt,tdtt) S (tdtt) S tdt. An impotant simplifiation is obtained if tdt, that is, if the time t

8 J. Chem. Phys., Vol. 121, No. 6, 8 August 2004 Conditions fo hemial equilibium 2725 FIG. 2. Enegy vs entopy gaphs of the states of simple system A with given values of V, n a,, and. State A (t),0 (t) evolves in time while emaining on the dashed uve labeled (t), 0 (t) oesponding to the lagest entopy patial hemial equilibium state ompatible with the given values U, V, n a,, and the instantaneous values of eation oodinates (t) of the slow ate-ontolling eations. The dotted uves epesent the families of states A (t1 ), a, A (t 1 ),, A b (t 2 ),, A (t 2 ), d fo abitay values a, b,, d. The solid uve labeled 0, 0 epesents the hemial equilibium states of system A fo the given values of U, V, n a,, and, whee 0 0 ( 0 ). taken by the suogate system to eah stable equilibium afte a hange in omposition is muh shote than the time dt taken by the eato to affet suh hange in omposition. In suh a ase, the seond step of ou oneptual model an be negleted, and the stable equilibium states of the suogate system ae suffiient to desibe the states of the eato along the entie poess. This is tantamount to assuming that among all the intenal inteations and dynamial mehanisms that dive the nonequilibium state towads stable equilibium, the hemial eations with stoihiometi oeffiients ae the only mehanisms apable of hanging the omposition of the system and, on the time sale hosen fo the desiption, they ae the slow, ate-ontolling mehanisms, while all the othe mehanisms ae assumed to be muh faste, so as to dive the system in negligible time to its lagest entopy state ompatible with the instantaneous values of n(t) and theefoe maintain the state of the eato along the family of states A (t) suh that if at any instant in time we tun the eations we obtain the suogate system at o vey lose to the stable equilibium state with entopy S a (t). Unde this assumption, Ṡ i ds a t dt i1 i1 j1 S n i i, T n i t n a, j1 i j j j1 i1 i j i, T j jy j, Y t Y a t, whee is the ow veto of the eation ates 1, 2,..., and Y the olumn veto of the atios Y j, A j, /T ( fo j1,2,...,, whee A j, i1 j) i i, is the affinity of the jth eation evaluated at the stable equilibium state of suogate system B with values U, V, n(t)n a (t). A futhe simplifiation is obtained if thee is a subset of hemial eation mehanisms that ae muh faste than the othes. Then, only the slow eations ae ate ontolling, wheeas the fast eations dive the system in negligible time to its lagest entopy state ompatible with the instantaneous omposition n(t), whih vaies slowly as a esult of the ates of the slow eations. Denoting the stoihiometi oeffiients and the oodinates of the slow eations by and and of the fast eations by and, Eqs. 6 and 12 ae eplaed by and i1 i1 i j A i 0 fo j1,2,..., 42a i k A i 0 fo k1,2,...,, 42b n i tn ia j1 i j j t k1 i k k t. 43 In Eq. 43 we assume that the k s adjust instantly to hanges in omposition indued by the slowly vaying j s so as to maintain system A at the lagest entopy patial hemial equilibium state A,0 ompatible with the given values of U, V, n a,, and the instantaneous values of the j (t) s see Fig. 2, that is, fo k1,2,...,, 41 k t k0 a t;, 44

9 2726 J. Chem. Phys., Vol. 121, No. 6, 8 August 2004 G. P. Beetta and E. P. Gyftopoulos whee the k0 s ae the values that maximize S, S ( a ) fo the given values of U, V, n a,,, and (t), o equivalently, fo k1,2,...,: S, k a,, Y k, a t As a esult of these assumptions, system A evolves though the family of states A (t),0 (t), the entopy S(t) is appoximated in tems of S of the suogate system B, StS a t 0 a t;, and the ate of entopy geneation is given by the elation Ṡ i j1 jy j, k1 k0 Y k, 46 and note that nowhee in ou deivation we make efeene to the onept of miosopi evesibility. Next we make use of the ondition that, at evey state A, Ṡ i () must be non-negative. In patiula, we apply this ondition in the viinity of the hemial equilibium state by expanding Ṡ i () into a Taylo seies about 0 up to seond ode and using the fat that Y j, ( 0 )0 Eq. 35, j( 0 )0 Eq. 49, and, of ouse, Ṡ i ( 0 )0. Thus, fo j,k1,2,...,, we find Ṡ i j Ṡ i j k 1 0 j 0 Y, 0 0 Y, j 0 0, 51 j 0 Y, k 0 j1 jty j, a t 1 k 0 Y, j a t;, 47 beause at eah instant of time Y k, 0 Eq. 45. It is lea that the model we pesent hee is valid fo homogeneous states of the eato. In addition, it povides the oneptual bakgound also fo the so-alled loal equilibium assumption upon whih the ontinuum fluid dynamis teatment of nonhomogeneous states is based. VI. RECIPROCAL RELATIONS Aoding to the foegoing disussion, we poeed unde the assumption appoximation that the homogeneous state of the isolated eato A belongs at eah instant in time to the family of states A. Fo fixed values of U, V, n a, and, the only independent vaiables of the family of A states ae the eation oodinates j s. We futhe assume that eah eation ate k is a funtion of the state A and, hene, of the j s, that is, with 0 0, whee, fo simpliity, fom hee on we do not wite expliitly the dependenes on the given values of U, V, n a,, and. Condition 49 is neessay beause at hemial equilibium all eation ates ae zeo. Realling that Y Y () Eqs. 15, we an wite Eqs. 41 and 47 fo the ate of entopy geneation in the fom Ṡ i Ṡ i 1 Y, Y 50 and, theefoe, Ṡ i 1 2 j1 k1 j1 k1 1 j1 k1 1 m1 2 Ṡ i j k 0 j j0 k k0 Y, j j j0 k k0 k Y m, Y m, j 0 j j0 k k0 1 m1 1 m1 Y, k Y m, 0 Y, Y m, L m, Y, Y m,, whee we define the matix L with elements L m, 8 Y m, 0. 53a 53b 53 53d 53e 54 In Eq. 53 we use the tunated Taylo seies expansion about 0 of the elations PY Y, 55 whih follow 8 fom substituting into Eq. 48 the elations (Y ) obtained fom the invesion at fixed U, V, n a,, and ) of the set of elations Y Y (), that is, fo j, 1,2,...,,

10 J. Chem. Phys., Vol. 121, No. 6, 8 August 2004 Conditions fo hemial equilibium Y m1 j 0 m1 8 Y m, 0 Y m, m1 L m, Y m,, 56a 56b 8 Y m, 0 Y m, j In Eq. 53d we use the tunated expansion about 0 of the elations Y Y (), that is, fo 1,2,...,, Y m, j1 Y m, j 0 j j0. 58 Ou esults an be intepeted in the ustomay manne of the so-alled Onsage s linea themodynami theoy of ievesible poesses Eah affinity A m, o, bette, eah Y m, A m, /T fo m1,2,...,, an be egaded as a diving foe and eah eation ate j, fo j 1,2,...,, as a flux that depends on all the diving foes: that is, diving foes and fluxes ae oupled. If the fluxes ae expessed as funtions of the diving foes o vie vesa, the oeffiients of the linea appoximation Eqs. 56 of these funtions in the viinity of the hemial equilibium state an be egaded as genealized ondutivities. In view of the elations (Y ), with (0) 0,we ewite Eq. 50 in the fom Ṡ i Ṡ i Y 1 Y Y, PY Y, 59 whee P(Y ) (Y ) with P(0) ( 0 )0. The expansion of this fom in the viinity of the hemial equilibium state yields Ṡ i Y 1 2 j1 k1 2 Ṡ i Y Y Y j, Y j, Y k,. k,0 60 Diet ompaison of Eq. 60 with Eq. 53e shows that, fo j,k1,2,...,, L jk, 1 2 Ṡ i Y 1 2 Ṡ i Y L 2 Y j, Y k,0 2 Y k, Y jk,, j,0 61 and theefoe the matix L is symmeti, that is, its elements obey the Onsage 10 eipoal elations. Equation 53a shows that the leading tem in the expansion of Ṡ i aound the hemial equilibium state is a quadati fom in the distanes ( j j0 ) fom the hemial equilibium state. Beause in the viinity of the hemial equilibium state evey j j0 an take both negative and positive values, the ondition that Ṡ i be always nonnegative implies that the oeffiients of the quadati fom ae elements of a nonnegative definite matix, that is, S 0, 62 whee the elements of the matix S ae given by the elations S jk, 1 2 Ṡ i. 2 j k 0 63 Similaly, Eq. 53e shows that the leading tem in the expansion of Ṡ i aound the hemial equilibium state is also a quadati fom in the diving foes Y j, (). Beause in the viinity of the hemial equilibium state evey Y j, () an take both negative and positive values, the ondition that Ṡ i be always non-negative implies that also the oeffiients of this quadati fom ae elements of a nonnegative definite matix L 0 0; 64 that is, the symmeti matix of eipoity o Onsage oeffiients is non-negative definite. We finally emphasize that the main esults of this setion deive fom the stutue of the leading tems of Taylo expansions valid only in the viinity of the hemial equilibium state. By ontast, the model developed in the pevious setions in tems of the A family of states and the stable equilibium popeties of the suogate system is valid both fa fom and nea the hemial equilibium state. As it is well known, 11 the elation P(Y ) between eation ates and diving foes is in geneal nonlinea. Nevetheless, even fo states that ae vey fa fom hemial equilibium, elations 50 and 59 ae valid, togethe with the ondition that Ṡ i 0, and povide impotant guidane in the development of hemial kinetis models. VII. CONCLUSIONS The themodynami deivations of onditions fo hemial equilibium, of Onsage eipoity elations, and of the popeties of a patially impotant family of nonequilibium states pesented hee diffe fom the deivations pesented in patially all teatises of themodynamis applied to hemial eatos. Ou motivation fo developing this deivation is based on a evolutionay 13 oneption of themodynamis as a nonstatistial 14 physial theoy 4,15 that applies to all systems both maosopi and miosopi, to all states both themodynami equilibium and not themodynami equilibium, and that disloses that entopy is an inheent intinsi nondestutible popety of matte well defined fo all systems and all states, in the same sense as inetial mass is an inheent popety of matte. Ou nonstatistial deivation of Onsage elations fo an isolated hemial eato shows that the aguments based on statistial flutuations, time evesal, and the piniple of miosopi evesibility, 9 12 whih ae invaiably used in all taditional deivations, ae not essential and, theefoe, play no fundamental ole in the themodynami theoy of ievesible poesses.

11 2728 J. Chem. Phys., Vol. 121, No. 6, 8 August 2004 G. P. Beetta and E. P. Gyftopoulos 1 E. A. Guggenheim, Themodynamis, 5th evised ed. Noth-Holland, Amstedam, K. Denbigh, The Piniples of Chemial Equilibium, 2nd ed. Cambidge Univesity Pess, London, M. Modell and R. C. Reid, Themodynamis and Its Appliations Pentie Hall, Englewood Cliffs, NJ, E. P. Gyftopoulos and G. P. Beetta, Themodynamis: Foundations and Appliations Mamillan, New Yok, The simple system model extends to heteogeneous stable equilibium states also in the pesene of extenal fields and sufae effets, povided a ontinuous phase model is adopted. See, e.g., J. W. Cahn and J. E. Hilliad, J. Chem. Phys. 28, ; L. Mistua, ibid. 83, ; and Ref J. C. Slate, Quantum Theoy of Moleules and Solids MGaw-Hill, New Yok, 1965, Vol.2. 7 G. N. Hatsopoulos and E. P. Gyftopoulos, Themioni Enegy Convesion MIT Pess, Cambidge, MA, 1979, Vol Fo the sake of peision, we intodue the diffeent haate P to emphasize that, as defined by Eq. 55, the funtional dependene implied by witing P(Y ) is diffeent fom that implied by witing (). 9 R. C. Tolman, The Piniples of Statistial Mehanis Oxfod Univesity Pess, Oxfod, L. Onsage, Phys. Rev. 37, ; 38, S. R. degoot and P. Mazu, Nonequilibium Themodynamis Noth- Holland, Amstedam, A. Kathalsky and P. F. Cuan, Nonequilibium Themodynamis in Biophysis Havad Univesity Pess, Cambidge, MA, We use the tem evolutionay in the sense intodued by T. S. Kuhn, The Stutue of Sientifi Revolutions The Univesity of Chiago Pess, Chiago, We use the tem nonstatistial to emphasize that entopy is defined see Refs. 4 and 15 fo all systems, maosopi and miosopi, and all states, themodynami equilibium and not themodynami equilibium, without any neessity fo eithe statistial o infomation theoeti aguments. 15 G. N. Hatsopoulos and E. P. Gyftopoulos, Found. Phys. 6, ; 6, ; 6, ; 6, ; G. P. Beetta, ibid. 17,