Mesoscopic Nonequilibrium Thermodynamics Gives the Same Thermodynamic Basis to Butler-Volmer and Nernst Equations

Transcription

1 J. Phy. Chem. B 2003, 107, Meocopic Nonequilibrium Thermodynamic Give the Same Thermodynamic Bai to Butler-Volmer and Nernt Equation J. M. Rubi and S. Kjeltrup* Department of Chemitry, Faculty of Natural Science and Technology, Norwegian UniVerity of Science and Technology, Trondheim, 7491-Norway ReceiVed: May 5, 2003; In Final Form: September 16, 2003 Meocopic nonequilibrium thermodynamic i ued to derive the Butler-Volmer equation, or the tationary tate nonlinear relation between the electric current denity and the overpotential of an electrode urface. The equation i derived from a linear flux-force relationhip at the meocopic level for the oxidation of a reactant to it charged component. The urface wa defined with exce variable (a Gibb urface). The Butler- Volmer equation wa derived uing the aumption of local electrochemical equilibrium in the urface on the meocopic level. The reult wa valid for an iothermal electrode with reaction-controlled charge tranfer and with equilibrium for the reactant between the adjacent bulk phae and the urface. The formulation that i ued for the meocopic level i conitent with nonequilibrium thermodynamic for urface and, thu, with the econd law of thermodynamic. The Nernt equation i recovered in the reverible limit. The reverible/ diipative nature of the charge-tranfer proce i dicued on thi bai. 1. Introduction The theory of reaction kinetic 1 ha been ued uccefully to give alo electrode kinetic a broader bai. By introducing the law of ma action, the concept of unidirectional rate, and the tranition tate, the Butler-Volmer equation i derived, and it ha been hown to contain alo the Nernt equation, ee, for example, ref 2-4. Alo, the Marcu theory ue the concept of unidirectional rate and the tranition tate. 5 The Butler-Volmer equation, the central equation in electrode kinetic, give the current denity j a a nonlinear function of the overpotential η: j ) j 0 [e (1-R)ηF/RT - e -RηF/RT ] (1) Here j 0 i the equilibrium exchange current denity, the rate of formation of product or reactant in equilibrium, and R i the tranfer coefficient, that decribe how the electric potential alter the activation energy barrier of the reaction. The ymbol F, R, and T are Faraday contant, the ga contant, and the abolute temperature. The exchange current denity i a function of the activation energy E a and the height of the barrier j 0 ) Be -E a/rt where B i a pre-exponential factor. At equilibrium, one can derive the Nernt equation from thi equation in a imilar way a one derive the equilibrium contant in reaction kinetic theory. The Nernt equation decribe the reverible converion of the reaction Gibb energy r G into electric energy φ, when one mole of charge i * To whom correpondence hould be addreed. On leave from Department of Fundamental Phyic, Univerity of Barcelona, Spain. (2) ( φ) jf0 )- r G/F (3) tranferred in the outer circuit. For large overpotential, one of the exponential term in the Butler-Volmer equation can be neglected, and one obtain the Tafel equation. The ytem i then conidered to be far from equilibrium. More complicated expreion than eq 1 are available; ee, for example, ref 3. The Nernt equation ha it firm bai alo in thermodynamic, but the Butler-Volmer equation ha o far a kinetic bai only. Far from chemical and thermodynamic equilibrium, it i relevant to ak: I there really a bai for uing thermodynamic function to decribe energy converion at the electrode urface? In thi work, we how that a thermodynamic bai can be etablihed for the Butler-Volmer equation and that the Nernt equation i contained in the ame formulation. The aim of thi work i to etablih a well defined thermodynamic bai for the decription of energy converion during electrode reaction. Such a bai can be found from a branch of thermodynamic that ha not yet been applied to electrochemitry, namely meocopic nonequilibrium thermodynamic. 6 The theory ha o far been applied to chemical reaction 7 and particle motion with inertial effect. 8 Through the application of meocopic nonequilibrium thermodynamic to energy converion procee at the electrode, we hall derive eq 1, but with a different et of aumption than uual. The urpriing finding i that we hall need the aumption of local electrochemical equilibrium in the reaction coordinate pace. In reaction kinetic, one normally aume equilibrium between the reactant and the activated pecie, but thi aumption i not needed here. We hall alo ee that the rate equation in eq 1 originate from an expreion that tart with the net rate of reaction, imilar to well-known rate equation uch a Fourier and Fick law. A thermodynamic bai for the equation can thu be given, uing nonequilibrium thermodynamic theory for the meocopic level. We hall accomplih thi by putting together well-known concept in a new way. We hall firt define the electrode urface following Gibb, 9 uing hi exce variable (ection /jp030572g CCC: $ American Chemical Society Publihed on Web 11/08/2003

2 13472 J. Phy. Chem. B, Vol. 107, No. 48, 2003 Rubi and Kjeltrup 2). In ection 3 we give the entropy production rate for the urface. Thi wa done already for everal cae and alo for a polarized electrode. The internal coordinate from meocopic nonequilibrium thermodynamic i in thi context the reaction coordinate. The reaction coordinate i introduced in ection 4, and the particular of the meocopic nonequilibrium thermodynamic are given. By uing thi theory, we can arrive at an expreion for the entropy production rate at the meocopic level under certain condition. The overpotential of an electrode i defined from it meaurement 14 in ection 5. Under iothermal, tationary tate condition, with equilibrium for the reactant at the urface and with local electrochemical equilibrium along the reaction coordinate, we obtain the Butler-Volmer equation in ection 6 without conidering the unidirectional rate in the electrode reaction. The flux-force relationhip on the meocopic level i linear and conform completely to nonequilibrium thermodynamic theory. However, at the integrated level, the level that i related to meaurement, the relationhip become nonlinear. 2. The Sytem The electrode urface i here regarded a an open, autonomou thermodynamic ytem, fully decribed by it exce variable. It decription i tandard in equilibrium 3 and originate from Gibb. 9 A an example we chooe the electrochemical (anodic) oxidation of the metal A at an anode: A f A + + e - (4) The urface ha an extenion in pace that contain A, A +, and e -. The urface contain A in the amount Γ A and the charged particle in the amount Γ A + and Γe -. We chooe the extenion of the urface o that it i electroneutral. For the exce denitie, thi give Γ A + ) Γ e - (5) Since the electron are predominantly preent in the conduction band of the metal and the cation are predominantly facing the electrolyte double layer, the urface i polarized. Mot likely there i therefore alo a pecific adorption of electrolyte from the olution to the polarized urface. The urface exce denitie are determined according to the precription of Gibb 9 and are given for intance in unit of mol/m 2. The charged pecie give the urface a polarization P (in C/m). The additional extenive variable of the urface are the entropy denity and the internal energy denity u. Ma conervation for A give r A ) dγ A+ )- dγ A dt dt where r A i the rate of the anodic reaction. Chemical energy i tranformed into electrical energy in the electrochemical reaction in eq 4. In order for A to eparate into charge, A + and e -, A goe through an activated tate. The beginning and final tate in thi proce are normally, and alo here, thought of a being connected by a path. The ditance along the path i meaured by a reaction coordinate γ. The coordinate i in tate pace and i called an internal variable in (6) meocopic nonequilibrium thermodynamic. There i no poible external control for thi type of variable. The derivation that follow can be carried out for other cae than our example (eq 4), following the procedure that i outlined below. 3. Entropy Production Rate for an Electrode Surface The entropy balance for the urface i where i the urface exce entropy and J i the entropy flux; ee, for example, ref 15 The upercript i mean into the urface, and o mean out of the area element. The area element i now called the thermodynamic urface or imply the urface. The entropy flux contain the meaurable heat flux, J q, and the thermodynamic entropie time their correponding ma flux, here for A only. The tranported entropy of A +, S / A +, i included in J q in thi choice of fluxe. The exce entropy production rate of a polarized electrode urface, σ, wa given by Bedeaux and Kjeltrup Ratkje. 10 It i found in each cae from the Gibb equation and the firt law for the urface. In thi ytem, the Gibb equation i T d ) du - (µ A + where ɛ 0 i the dielectric contant of the vacuum, D eq i the diplacement field of the adjacent phae, and µ i i the chemical potential. The time rate of change in the exce variable in the Gibb equation give T d dt ) du with the reaction Gibb energy for the reaction that take place in the urface The firt law for the urface i d dt ) J i - J o + σ (7) J ) 1 T J q + J A S A (8) + µ e - - µ A )dγ A + - D eq dp (9) ɛ 0 dt - G r A - D eq G ) (µ A + The fluxe are determined by their value in the homogeneou phae adjacent to the urface. There i a heat flux into the urface, J q i, and out of the urface, J q o, and a flux of A into the urface, J A i, that carrie enthalpy H A. The electrical potential jump φ time the electric current denity j give the electric energy delivered by the urface, and D i the diplacement field when the ytem i out of equilibrium. The firt law decribe how upplied electric energy i tranformed into internal energy (enthalpy) and heat, and how the tate of polarization i changed or, vice vera, how fluxe of enthalpy, heat, and polarization can give electric energy. ɛ 0 dp dt (10) + µ e - - µ A ) (11) du dt ) J i q + J i A H A - J o q - j φ + D dp ɛ 0 dt (12)

3 Meocopic Nonequilibrium Thermodynamic J. Phy. Chem. B, Vol. 107, No. 48, By introducing the firt law into eq 10, rearranging the term, and comparing the reult with the entropy balance (eq 7 and 8), one find the exce entropy production rate (ee ref 10 for more detail): σ ) J q i( 1 T - 1 i) T qo( + J 1 T - 1 ) o T - 1 T [µ A,T - µ m A,T ]J o A - The chemical potential, µ A,T r A - 1 T G T j φ + 1 (D - D ) eq dp T ɛ 0 dt and µ m A,T, are taken at the ame temperature, here the temperature of the electrolyte. The upercript m refer to the electrolyte. To compare with experiment, we hall introduce the condition of an iothermal urface, with the ame chemical potential of A a that of it adjacent bulk phae (µ A,T ) µ m A,T ). The deviation of the diplacement field from it equilibrium value i equal and oppoite to the electric potential difference acro the urface (D - D eq ) ɛ 0 (13) )- φ d (14) with d being the urface thickne. The expreion 1/d dp /dt i the diplacement current j dipl, and j tot ) j + j dipl (15) By introducing thee relation, we are left with two term in the exce entropy production rate σ )- 1 T j tot φ - r A T G (16) Thi entropy production rate give two fluxe in the ytem, linearly related to their force. j tot )-L 11 φ - L 12 G (17) r A )-L 21 φ - L 22 G (18) Relation uch a thee were ued to find the urface impedance 13 and Randle equivalent circuit. The urface impedance i found by applying an ocillation potential to the urface and recording the correponding ocillating electric current denity. The chemical force in thi experiment i independent of the electrical force if the urface can accumulate reactant or product. For a urface with contant polarization, like here, the term containing dp /dt diappear, and j ) j tot. Under contant polarization, the two driving force are no longer independent and r A ) j tot /F ) j/f (19) Charge conervation in the urface mean that the electric current denity j i contant through the urface and equal to the flux of electron or the flux of poitive ion. All derivation from now on are made for thi condition. The entropy production rate i reduced to σ )- 1 T j G [ φ + F ] (20) The well-known limiting value of thi expreion for zero electric current can now be found: lim(σ ) jf0 )- j G [ φ + T F ] ) 0 (21) The Nernt equation reult in ( φ) jf0 )- G jf0 F Here refer to the tate of the product and the tate of the reactant in the urface. With repect to the reactant and product, we peak of a two-tate decription on the macrocopic thermodynamic level. Equation 22 expree that the chemical energy i completely tranformed into electrical energy. The potential jump acro the urface i defined by a reverible tranformation of (all) chemical energy in the urface into electrical energy. Under a reverible tranformation, there i a balance between the electrical force and the chemical force. The urface i not in chemical equilibrium a long a G i different from zero, but it may be regarded a being in electrochemical equilibrium, when eq 22 applie. Thi i in agreement with the common perception of electrochemical equilibrium. According to nonequilibrium thermodynamic theory, 15 the exce entropy production rate (eq 20) give the following fluxforce relationhip: The problem to be olved i now how we can reconcile thi eemingly linear relation with the common Butler-Volmer equation. We hall ee that thi can be done by introducing the variable that change continuouly between the two chemical tate, the reaction coordinate or the internal coordinate, mentioned in the preceding ection. 4. A Continuou Reaction Path with Local Electrochemical Equilibrium (22) G j )-L[ φ + F ] (23) The entropy production rate given above refer to the Gibb urface. The electric potential wa the integral acro the urface, and the chemical reaction wa given a the difference between two chemical tate, the tate of the product and the tate of the reactant. It i poible to find another expreion, if we introduce the reaction coordinate γ from reaction kinetic theory 1 and replace the difference by integral. The different value of the γ coordinate correpond to the different configuration that the ytem adopt during it evolution from the inital to the final tate. By doing thi, we change to a meocopic level in the decription. The variable at thi level cannot be controlled from the outide of the ytem. The thermodynamic theory dealing with thi level i therefore called meocopic nonequilibrium thermodynamic. Since j and r A are contant, we are eeking a variation in φ(γ) and G(γ) that obey σ )- j γ T γ1 2[ φ(γ) γ + 1 G (γ) F γ ] dγ (24)

4 13474 J. Phy. Chem. B, Vol. 107, No. 48, 2003 Rubi and Kjeltrup Figure 1. Effective chemical potential a a function of the internal coordinate γ. Thi variation give a poitive driving force and a poitive entropy production at any location in γ pace. The electric potential difference i thu φ ) γ1 γ 2 φ(γ) γ dγ ) φ(γ 2 ) - φ(γ 1 ) (25) and the reaction Gibb energy i likewie recovered by G ) γ1 γ 2 G (γ) γ dγ ) µ A+(γ 2 ) + µ e -(γ 2 ) - µ A (γ 1 ) (26) The difference ign in the above two equation i now referring to difference in tate pace. We can define the effective chemical potential of the reactant in tate 1 µ A(γ 1 ) ) µ A (γ 1 ) + Fφ(γ 1 ) (27) and the effective chemical potential for the product in tate 2 µ A+e-(γ 2 ) ) µ A +(γ 2 ) + µ e -(γ 2 ) + Fφ(γ 2 ) (28) Thi effective chemical potential i equal to the electrochemical potential of A +, when the chemical potential of electron i mall. To make poible a ditinction from the electrochemical potential, we introduce the name effective chemical potential. In the next tep, we aume that thee expreion are valid at any location in the internal coordinate pace, that i, that the normal thermodynamic relation apply in γ-pace: µ A(γ) ) µ A (γ) + Fφ(γ) µ A+e-(γ) ) µ A +(γ) + µ e -(γ) + Fφ(γ) (29) We then move one tep further and introduce the aumption of local electrochemical equilibrium in the internal coordinate pace. With local equilibrium along the coordinate γ, we have µ A +(γ) + µ e -(γ) + Fφ(γ) ) µ A (γ) + Fφ(γ) ) µ (γ) (30) The effective chemical potential of the reactant i alway equal to the effective chemical potential of the product. In other word, there i local equilibrium at any coordinate by thi criterion for equilibrium. The phyical meaning of thi condition i a follow: At each coordinate, along the continuou path between the product and the reactant tate, there i an enemble of particle of a particular energy and tate of polarization. The energy of any tate along the reaction path i now expreed imply by µ (γ). The effective chemical potential µ (γ) i illutrated in Figure 1. The difference between the product and the reactant tate of the effective chemical potential i negative, ince the anode Figure 2. Intrinic barrier C(γ) (lower part of the figure). A linear potential profile i drawn in the upper part of the figure, and the combination of C(γ) and φ(γ) i alo hown. The urface potential drop i indicated to the right in the figure. reaction i pontaneou. In global electrochemical equilibrium, the two tate value are the ame, and µ ) 0 (31) Thi i yet another formulation of the Nernt equation (eq 22). By introducing eq 30 into the entropy production rate (eq 24), we can now write σ )- 1 γ 2 j T γ1 F[ µ (γ) γ ] dγ (32) Thi expreion for the entropy production rate i equivalent to the expreion in eq 20. It mean that all equation above apply when there i local chemical equilibrium along the internal coordinate pace. The effective driving force time it flux, the electric current denity, give the diipation of energy of the electrode urface through eq 32. Only the difference between the chemical force and the electric force, the effective driving force of the reaction, lead to diipation of energy. Chemical reaction that take place in homogeneou phae are purely diipative phenomena. At the electrode urface, however, the reaction can be driven backward or forward, depending on the direction of the electric current. There i thu an element of control or of reveribility in the electrochemical reaction that ditinguihe it from the imple chemical reaction. According to the econd law, σ > 0. We hall make the aumption, a de Groot and Mazur alo did, 15 that the entropy production i poitive, alo on the meocopic level: σ(γ) )- j T F[ µ (γ) γ ] dγ > 0 (33) The effective chemical potential in Figure 1 agree with thi aumption. To comply with the condition in eq 33, we mut have a monotone decreaing function. We now pecify the chemical contribution of eq 30 further: µ (γ) ) C(γ) + RT ln Γ(γ) + Fφ(γ) (34) The econd term on the right in thi equation i the normal configurational contribution to the chemical potential. The exce urface adorption Γ, or the denity of A or A +,ia function of γ. The term C(γ) i the intrinic barrier that the reactant ha to pa on the way to product; ee Figure 2. The

5 Meocopic Nonequilibrium Thermodynamic J. Phy. Chem. B, Vol. 107, No. 48, urface potential drop that appear in the equation of the preceding ection i then equal to φ j in eq 37. The reaction Gibb energy of the electrode i G j.wecan add and ubtract thi value in eq 37. Thi give, with the definition of the effective driving force, η ) µ + ( G j - G jf0 ) 1 F (38) When, furthermore, G j ) G jf0 )- φ jf0, we have from thi equation that the overpotential can be identified with the effective driving force defined by eq 32. η ) µ 1 F (39) Figure 3. Illutration of the configurational part of the chemical potential and the electric potential plu the barrier at equilibrium. There i a ymmetry around the horizontal line of contant effective chemical potential. variation in the electric potential i here illutrated by a traight line, labeled φ(γ). The effective chemical potential can be further rewritten a µ ) RT ln fγ (35) where f(γ) i the activity coefficient, defined a f(γ) ) e (C+Fφ)/RT (36) The activity coefficient expree the variation in the barrier plu the electrical potential in γ pace. The ituation at global electrochemical equilibrium i illutrated in Figure 3. We ee the variation in the um of the barrier and the electric potential, C + φ, a a function of γ. With F φ )-RT ln Γ, the configurational part of the chemical potential i everywhere oppoite and equal to thi function. The equilibrium value of the effective chemical potential i contant and appear a a line of ymmetry in the figure. 5. The Overpotential Meaurement The overpotential i meaured under tationary tate condition with the help of three electrode; ee, for example, ref 14. A net current i paed between the working electrode (the electrode of interet) and an auxiliary electrode. The current denity at the working electrode i j. The potential between the working electrode and a reference electrode of the ame kind that i kept at zero current denity i φ j. When the electric current denity of the working electrode i zero, the meaured potential i φ jf0. The overpotential i then the difference between the potential drop at a given current denity and that at zero current denity: η ) φ j - φ jf0 (37) The overpotential i defined a poitive. In the preent cae of an anodic overpotential, φ j > φ jf0. By definition, when j ) 0, η ) 0. Suppoe now that the overpotential i due to a rate-limiting reaction at the electrode urface and that we have been able to correct the meaurement of φ j for concentration gradient contribution and ohmic reitance drop in the electrolyte. The The aumption G j ) G jf0 i true when the chemical potential are little affected by the current denity. The overpotential time the electric current i then a meaure of the diipation of energy at the electrode urface. We have alo aumed that T ) 0 and µ i ) 0 applie to the urface. 6. The Butler-Volmer Equation We are now in a poition to find the condition for which the Butler-Volmer equation appear. From the entropy production (eq 33) we can define the thermodynamic force, conjugate to the flux j: The force appear a the derivative of the denity. Thi i normal for force on the macrocopic level (for intance in Fick law). The force refer to the urface temperature, T,byeq24. The central aumption in nonequilibrium thermodynamic i that of linear flux-force relationhip. The aumption i normally ued on the macrocopic level. We hall now ue it for the meocopic level. The linear law that follow from the local entropy production rate in γ pace i then We ued a a condition for the derivation in the preceding ection that the electric current i contant. Since the force varie with γ, o will L(γ). The linear law i now rewritten with the coefficient of tranport Thi coefficient can be interpreted a the electric mobility for motion of the charge carrier A + in the complex A + e - in the polarized electrode (the motion of the electron i not ratelimiting). Since the current ha been taken contant along the internal coordinate, we obtain by integration where l ) R/I and X(γ) )- 1 µ FT γ )- R fγ FfΓ γ j )-L(γ) R 1 F fγ fγ γ (40) (41) j )-u + e (-(C+Fφ)/RT) R dγ eµ /RT (42) u + ) L(γ) FΓ (43) j )-l[e µ 2/RT - e µ 1/RT ] (44) I ) 1 2 u+ -1 e (C+Fφ)/RT dγ (45)

6 13476 J. Phy. Chem. B, Vol. 107, No. 48, 2003 Rubi and Kjeltrup Figure 4. Effective chemical potential and the overpotential. i the integral of the invere mobility weighted with the proper Boltzmann factor. The effective chemical potential at equilibrium µ eq mut have a value between µ 1 and µ 2. We ubtract and add the factor µ eq in the exponential of eq 44 and obtain j )-le µ eq/rt [e (µ 2-µ eq )/RT - e (µ 1-µ eq )/RT ] (46) So far we have only aumed a tationary tate, and eq 46 i the mot general outcome of the derivation. We have een that a proce that i linear in the internal variable lead to a nonlinear proce in the external variable pace. To find the Butler-Volmer equation, we mut relate the effective chemical potential to the overpotential. We make the following identification with the ditance a and b in Figure 4: µ 1 - µ eq ) a ) (1 -R)Fη (47) µ 2 - µ eq )-b )-RFη (48) The difference between the equation make the difference in effective chemical potential equal to -Fη. The fraction R i defined by thee identitie. When thee expreion are introduced into eq 44, we obtain the Butler-Volmer equation j ) j 0 [e (1-R)ηF/RT - e -RηF/RT ] (49) Thi complete the derivation of the Butler-Volmer equation from meocopic nonequilibrium thermodynamic. The equation ha thu been given a thermodynamic bai. Slight difference appear in the interpretation of the variable. The exchange current denity depend on the equilibrium tate through j 0 ) le µ eq (50) In the Butler-Volmer equation there i a imilar dependence on the activation energy for the barrier. The value of j 0 depend here on the barrier C through the variable l. Our variable R i imilar to the ymmetry factor R in the Butler-Volmer equation, eq 1. The value of R in our derivation can be related to the ymmetry of the problem: It meaure the difference of the firt tate and the final tate from the equilibrium tate. The ucce of the Butler-Volmer equation for decription of meaurement i beyond doubt. We have derived thi equation for tationary tate condition, making the aumption that the effective driving force can be replaced by η. When thi aumption i not true, we can ue the more general reult, eq Dicuion The Butler-Volmer equation i uually derived by analyi of unidirectional procee acro an activation barrier, that i, by kinetic theory; ee, for example, ref 4. While there are imilaritie between our thermodynamic approach to the problem and the uual kinetic theory, there are alo difference. The imilaritie and difference hall now be dicued. Both derivation make ue of the reaction coordinate in a decription of the activated proce. Common i that the reaction coordinate meaure the progre of the chemical reaction. In meocopic nonequilibrium thermodynamic the reaction coordinate ha alo the particular tatu of being the internal variable that i ued to decribe the tate of the ytem. The chemical reaction i not in global equilibrium, nor in reaction kinetic, nor in meocopic nonequilibrium thermodynamic. In reaction kinetic, 1 one aume equilibrium between the reactant tate and the activated tate. Our baic aumption i different from thi. We ue the aumption of local chemical equilibrium along the whole reaction path, and we find the Butler-Volmer equation on thi bai. Therefore, we can tate that procee that can be decribed by the Butler-Volmer equation can be regarded a happening in local equilibrium. And, there i no need to quetion the ue of thermodynamic variable in the decription of the urface reaction, in the nonlinear current-potential regime. It i of coure jutified to ak whether the aumption of local equilibrium i good. Some evidence i emerging that it i a good aumption in electroneutral ytem and alo with urface. 16,17 The ue of linear flux-force relation and local equilibrium i very different from the uual premie that are ued to derive the Butler-Volmer equation. In kinetic theory, the difference between the exponential in the Butler-Volmer equation tem from a picture of unidirectional fluxe acro the barrier. Clearly, thi picture decribe a ituation far from equilibrium. One may then rightly wonder whether, for example, the Gibb equation hold. Thi doubt i eliminated by our procedure. We derive the nonlinear flux equation, by conidering only the net rate, and condition are compatible with nonequilibrium thermodynamic. We tart with a linear relation in a proper variable pace. By integration to the macrocopic level, the nonlinear relation i found. Our derivation reolve the problem of the temperature of the reaction, poed by Eyring and Eyring. 1 It i clear from our derivation that the temperature of the reaction i the urface temperature. The urface temperature i defined by Gibb exce variable for the urface. Reaction kinetic theory ha no precription for the temperature. The Butler-Volmer equation and the Nernt equation have for the firt time been given the ame thermodynamic bai by the preent work. The bai i the entropy production rate of the urface. We aw firt that the reverible limit of the entropy production rate gave the well-known Nernt equation. We next ued the condition of local electrochemical equilibrium to find the Butler-Volmer equation from the ame entropy production rate. The entropy production rate of the electrode urface at the tationary tate wa determined by the effective driving force and the electric current denity. The effective driving force wa the um of the chemical and electrical force. The overpotential, a it wa defined by experiment, i not necearily equal to thi effective driving force. Since the overpotential i not necearily equal to the effective driving force, it may not only reflect diipative term only. Thi topic deerve further attention and hould be teted experimentally.

7 Meocopic Nonequilibrium Thermodynamic J. Phy. Chem. B, Vol. 107, No. 48, Meocopic nonequilibrium thermodynamic i a part of claical nonequilibrium thermodynamic. 15 The aumption of a linear flux-force relation and of local chemical equilibrium are central in nonequilibrium thermodynamic. The tarting point in nonequilibrium thermodynamic i alway fluxe that are linear function of their conjugate force. Nonequilibrium thermodynamic become a nonlinear theory, in particular by it incluion of variable on the meocopic level. Meocopic nonequilibrium thermodynamic ue the ame ytematic procedure a claical nonequilibrium thermodynamic. The crucial point, that i olved by the meocopic verion of thi theory, i that a proper definition can be found for the driving force. In our derivation, we have ued a omewhat different procedure than the one originally given by de Groot and Mazur. 15 Thee author tarted with the entropy change of an adiabatic ytem. Since the preent ytem i not adiabatic, it i neceary to find the entropy production on the meocopic level from it counterpart on the macrocopic level. The proper driving force i then more eaily defined. The importance of the preent work lie in the ytematic procedure it can give for further derivation and in the undertanding it can offer. We found that certain premie mut be fulfilled for the Butler-Volmer equation to apply. The urface mut be iothermal, there mut be equilibrium at the urface for adorbed pecie, and one mut be able to identify the overpotential with the effective driving force. Experiment, done for other condition than thee, may give new inight and help develop the theory further. A derivation, tarting with the econd law of thermodynamic, offer inight into the diipative nature of the proce. Chemical energy i alway converted into electrical energy during electrode reaction. The converion i mot clearly decribed in the macrocopic form in eq 23 and 22. The central quetion i alway: Which power i lot in the converion; or how much energy i diipated? In the analyi above, the effective driving force time the electric current gave the complete power lo. The effective driving force wa only equal to the overpotential of the electrode for certain condition, a defined by meaurement. The reaction Gibb energy may be a function of the tate of polarization, G (P ). It i then not correct to replace µ by η. It will be intereting to purue thi point in the future, to find all diipative part of the overpotential. Nonequilibrium thermodynamic give a ytematic bai for including contribution alo from nonzero heat fluxe and thermal force. Bedeaux and Kjeltrup tudied the effect of temperature gradient 10 and lack of equilibrium between the electrolyte and the urface. 11 While it i not practical to tudy all effect combined, it i ueful to have information of the limiting cae and how they all derive from a common bai. Nonequilibrium thermodynamic for the meocopic level expand the poibilitie further to exploit the ue of internal variable and their relation with experiment. Meocopic nonequilibrium themodynamic provide a ytematic way to find kinetic law alo for other condition than the one ued here. 8. Concluion We have hown that the equation ued for reaction controlled charge tranfer, the Butler-Volmer equation, ha a general bai in nonequilibrium thermodynamic. In the derivation we took advantage of the ytematic procedure offered by meocopic nonequilibrium thermodynamic. We arrived at the Butler-Volmer equation uing aumption of local electrochemical equilibrium along the reaction coordinate pace, and a linear flux-force relation for thi coordinate pace. The derivation require a certain identification of the overpotential with the effective driving force to be true. By etting a more general bai for thi important electrochemical equation, we open poibilitie for other tudie when the aumption do not hold, that i, when there are gradient between the urface and it cloe urrounding, and when the tationary tate condition doe not apply. Acknowledgment. Thi work wa completed while J.M.R. wa Onager profeor at NTNU. Nork Hydro ASA i thanked for ponoring the profeorhip. Reference and Note (1) Eyring, H.; Eyring, E. Modern Chemical Kinetic; Chapman and Hall: London, (2) Crow, D. R. Principle and application of electrochemitry, 4th ed.; Blackie Academic & Profeional: London, (3) Bockri, J. O. M.; Khan, S. U. M. Surface Electrochemitry. A molecular level approach; Plenum: (4) Dwayne Miller, R. J.; McLendon, G. L.; Nozik, A. J.; Scmickler, W.; Willig, F. Surface Electron-Tranfer Procee; VCH Publiher: New York, (5) Marcu, R. Can. J. Chem. 1959, 37, 155. (6) Vilar, J. M.; Rubi, J. M. Proc. Natl. Acad. Sci. 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