Thermodynamic derivations of conditions for chemical equilibrium and of Onsager reciprocal relations for chemical reactors
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(2) J. Chem. Phys., Vol. 121, No. 6, 8 August 2004. Conditions for chemical equilibrium. potentials in Sec. III, the derivation of the same conditions in terms of temperature, pressure, and mole fractions in Sec. IV, the rate of entropy generation in the reactor in Sec. V, reciprocal relations in Sec. VI, and our conclusions in Sec. VII. II. SIMPLE SYSTEMS. We define a system as simple4 if it satisfies the following three conditions: 共i兲 it has volume as one of the parameters; 共ii兲 in any of its stable equilibrium states, if it is partitioned into a set of contiguous subsystems in mutual stable equilibrium, the effects of the partitions on the values of all properties are negligible;5 and 共iii兲 in any of its stable equilibrium states, the instantaneous switching off or on of one or more internal reaction mechanisms, such as a chemical reaction, causes negligible instantaneous changes in the values of energy, entropy, volume, and amounts of constituents. In general, either the introduction of partitions or the instantaneous switching on or off of chemical reaction mechanisms or both have definite effects on a system. For example, using the tools of quantum theory,6,7 we can show that the switching on of a reaction mechanism requires the switching on of an additional term in the Hamiltonian operator of the system, which affects the functional form of the fundamental relation for stable equilibrium states. Again, using the tools of quantum theory, we can show that the switching off of a reaction mechanism requires the ‘‘destruction’’ of correlations among constituents and, in general, results in a reduction of the value of the entropy. Nevertheless, we can also show that these effects become less and less important, and negligible for all practical purposes, if the value of the amount of each constituent is larger than 10 共Refs. 6 and 7兲. Hence the simple system model is applicable to most practical systems, including the nanovolume and microvolume scale, with sufficiently large amounts of constituents. III. DERIVATION OF CONDITIONS FOR CHEMICAL EQUILIBRIUM. r. 兺 共i j 兲A i ⫽0. i⫽1. for j⫽1,2,...,c,. c. n i0 ⫽n ia ⫹. 兺. j⫽1. 共i j 兲 j0. 共6兲. and derive the conditions for A to be in a stable equilibrium or, synonymously, in a chemical equilibrium state, where (i j) are the stoichiometric coefficients of the jth chemical reaction. In general, the chemical reactor just defined, for each given set of values of U, V, na , and , admits an infinite number of states. However, the laws of thermodynamics require that among these states one and only one be a chemical equilibrium state, and this state has the largest value of the entropy.4 We call the latter requirement the highest or largest entropy principle. At the chemical equilibrium state, the values of the amounts of constituents n 10 ,n 20 ,...,n r0 and the corresponding reaction coordinates 10 , 20 ,..., r0 satisfy the compatibility relations. for i⫽1,2,...,r.. 共7兲. Moreover, the values U, V, na ⫽ 兵 n 1a ,n 2a ,...,n ra 其 , and the stoichiometric coefficients ⫽ 兵 (i j) for i⫽1,2,...,r and j ⫽1,2,...,c 其 determine uniquely the values of all the properties and quantities that characterize the chemical equilibrium state, including the values of the entropy S, each j0 , and each n i0 . We write the dependences of the latter in the form S⫽S 共 U,V,na ; 兲 ,. 共8兲. j0 ⫽ j0 共 U,V,na ; 兲. for j⫽1,2,...,c,. 共9兲. n i0 ⫽n i0 共 U,V,na ; 兲. for i⫽1,2,...,r.. 共10兲. In general, we cannot find the explicit functional forms of Eqs. 共8兲–共10兲. For simple systems, however, the problem is somewhat less complicated because we can express chemical equilibrium properties in terms of stable equilibrium properties of a multiconstituent system in which all the chemical reaction mechanisms are inhibited—switched off. To see how this is done, we proceed as follows. First, we consider a simple system B consisting of the same r types of constituents as system A but with all the chemical reaction mechanisms inhibited—switched off. Of course, A and B are different systems because they are subject to different internal forces and constraints. We assume that B is in a stable equilibrium state with values U of the energy, V of the volume, and n⫽ 兵 n 1 ,n 2 ,...,n r 其 of the amounts of the r constituents. We denote the entropy at this stable equilibrium state by the fundamental relation S off⫽S off共 U,V,n兲 ,. We consider a simple system A having energy U, volume V, and constituents A 1 ,A 2 ,...,A r with initial amounts n 1a ,n 2a ,...,n ra , subject to c chemical reaction mechanisms. 2719. 共11兲. where we use the subscript ‘‘off’’ to emphasize that all the reaction mechanisms are switched off. Next, we assume that the chemical reaction mechanisms are instantly switched on, that is, all the reactions defined by the stoichiometric coefficients are no longer inhibited. As a result, we obtain again system A. Because in our discussion of chemical reactors we go back and forth between systems A and B by switching off and switching on the chemical reaction mechanisms, we call system B the surrogate system of A. Because the surrogate system B is simple and initially in a stable equilibrium state, immediately after switching on the reaction mechanisms the state of system A has the same values of S, U, V, n 1 ,n 2 ,...,n r as the corresponding values of the stable equilibrium state of B. In general, however, this state of A is not stable equilibrium. For example, if B is a quiescent mixture of gasoline vapor and air at room temperature and we activate the reaction mechanisms by a minute spark, we instantly produce a nonequilibrium state of system A in which the reactions are no longer inhibited—the burning. Downloaded 02 Sep 2004 to 18.95.1.27. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp.
(3) 2720. J. Chem. Phys., Vol. 121, No. 6, 8 August 2004. G. P. Beretta and E. P. Gyftopoulos. of the gasoline is proceeding—even though the instantaneous perturbations of the values of S, U, V, n 1 ,n 2 ,...,n r introduced by the spark are negligible. In other words, it is by virtue of the important key assumption of the simple system model by which reactions can be switched on and off without significantly altering the value of any property that we succeed in expressing properties of stable equilibrium states as well as of a class of nonequilibrium states of a reacting system in terms of the known stable equilibrium properties of nonreacting multicomponent systems. Among all the states of A that may be obtained from B in the manner just cited, we consider the subset that has given values U and V of the energy and volume, and amounts of constituents that are compatible with given values n 1a ,n 2a ,...,n ra . We denote each of these states by A and recognize that it corresponds to a set of values of the c reaction coordinates ⫽ 兵 1 , 2 ,..., c 其 such that. where in the fundamental relation S off⫽S off(U,V,n) we use the shorthand notation na ⫹ • for the set n ⫽ 兵 n 1a ⫹ 兺 cj⫽1 (1j) j , n 2a ⫹ 兺 cj⫽1 (2j) j ,..., n ra ⫹ 兺 cj⫽1 r( j) j 其 . Then we note that in order for A 0 to be the state of largest entropy among all the states A with given U, V, and na , the values of 0 must be such that. 冉 冊 S j. ⫽0. 共14兲. for j⫽1,2,...,c,. U,V,na , , . where the subscripts na , , and denote, respectively, that each of the amounts n ia , each of the stoichiometric coefficients (i j) , and each of the reaction coordinates i that do not appear in the derivative are kept fixed. For j⫽1,2,...,c, from Eq. 共13兲 we find that. 冉 冊 S j. r. ⫽ U,V,na , , . 兺 i⫽1. 冉 冊 冉 冊 S off ni. U,V,n. ni j. 共15a兲 na , , . c. n i ⫽n ia ⫹. 兺. j⫽1. 共i j 兲 j. for i⫽1,2,...,r,. 共12兲. where all the n i ’s are non-negative. Among all the states A , the one with the largest entropy is the unique chemical equilibrium state with energy U, volume V, and amounts of constituents compatible with na , that is, n0 ⫽na ⫹ • 0 . We denote the chemical equilibrium state by A 0 . To prove that indeed state A 0 corresponds to the largest entropy, we assume that another state A 0 ⫽A 0 with entropy S 0 , not belonging to the family of states A , is the chemical equilibrium state that corresponds to the given values U, V, na , . Then S 0 ⬍S 0 because A 0 has the largest entropy. Now, starting from A 0 , we switch off the chemical reaction mechanisms. Because system A is simple and A 0 is a stable equilibrium state, the resulting state B 0 of surrogate system B has the same values U, V, and n0 as A 0 and, in particular, its entropy is S 0 . State B 0 cannot be stable equilibrium because, if it were, then upon switching the chemical reaction mechanisms back on we would obtain again state A 0 and conclude that it belongs to the family A contradicting the fact that A 0 has the largest entropy. On the other hand, if state B 0 is not stable equilibrium, then the stable equilibrium state of B with values U, V, and n0 would have entropy S⬎S 0 , and switching on the reactions beginning with this state would yield a state in the family A that has entropy S⬎S 0 ⬎S 0 , again contradicting our stipulation that A 0 has the largest entropy. Therefore, if A 0 is the chemical equilibrium state, it must coincide with state A 0 because under the specified conditions there is one and only one stable equilibrium state. Because we can express the entropy S of a state A in terms of the entropy S off(U,V,n) of the state of the surrogate system to which A corresponds, we can determine the chemical equilibrium entropy S(U,V,na ; ) 关Eq. 共8兲兴 by finding the largest value of S off(U,V,n). To find the largest value just cited, we first write the entropy S of a state A in the form S ⫽S off共 U,V,na ⫹ • 兲 ,. 共13兲. r. ⫽⫺. 兺 i⫽1. i,off 共 j 兲 T off i. ⫽Y j,off共 U,V,na ⫹ • 兲. 共15b兲 共15c兲. r. ⫽. 兺 共 ⫺ i,off off兲 共i j 兲 ,. i⫽1. 共15d兲. where T off is the temperature and i,off the chemical potential of constituent i of the stable equilibrium state of the surrogate system B that corresponds to A 0 , ⫽1/T, and in writing Eqs. 共15b兲 and 共15d兲 we use the relations ( S/ n i ) U,V,n ⫽⫺ i /T⫽⫺ i and Eq. 共12兲, and in writing Eq. 共15c兲 we r define Y j,off⫽A j,off /T off where A j,off⫽⫺ 兺 i⫽1 (i j) i,off is the so-called affinity of the jth reaction, which is clearly a stable equilibrium property of surrogate system B. For finite values of T off , we see from Eqs. 共14兲 and 共15兲 that a set of necessary conditions that relate U, V, na , off , , and 0 at chemical equilibrium are r. 兺 共i j 兲 i,off共 U,V,na ⫹ • 0 兲 ⫽0. i⫽1. for j⫽1,2,...,c. 共16兲. or, equivalently, Y j,off(U,V,na ⫹ • 0 )⫽0. In the next section we show that T and i for i⫽1,2,...,r are also equal to the temperature and chemical potentials of the chemical equilibrium state of system A. Each of Eqs. 共16兲 is the chemical equilibrium equation for the corresponding reaction mechanism. For each given set of values U, V, na , and , Eqs. 共16兲 are c necessary conditions for chemical equilibrium. They may be solved to yield Eqs. 共8兲 to 共10兲 and, therefore, all properties of the chemical equilibrium state. They confirm the statement made earlier to the effect that properties of chemical equilibrium may be expressed in terms of properties of a multiconstituent system with all chemical reaction mechanisms switched off. For the extremum corresponding to Eqs. 共14兲 to be a relative maximum, it is also necessary that the second-order. Downloaded 02 Sep 2004 to 18.95.1.27. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp.
(4) J. Chem. Phys., Vol. 121, No. 6, 8 August 2004. Conditions for chemical equilibrium. partial differential of S with respect to the c reaction coordinates 1 , 2 ,..., c be negative. To show that indeed this is the case, we start with Eq. 共13兲 and find that 共 d 2 S 兲 U,V,na , . 1 ⫽ 2 1 ⫽ 2. c. c. 兺兺. j⫽1 k⫽1 c. 冉. c. 2S jk r. r. 兺兺 兺 兺 j⫽1 k⫽1 p⫽1 q⫽1. 冉 兺 兺冉 r. r. ⫽. 1 2 p⫽1. ⫽. 1 2 p⫽1. 兺 q⫽1 兺 r. r. q⫽1. 冊. d j d k. 2 S off n p n q. 冊. 共pj 兲 共qk 兲 d j d k U,V,n c. c. 2 S off n p n q. U,V,n j⫽1. 2 S off n p n q. U,V,n. 兺. 共pj 兲 d j. 兺. k⫽1. 共qk 兲 d k. dn p dn q ⬍0,. ⫽ U,V,na , ,Yoff. 冉 冊 j Y k,off. 共22兲 U,V,na , ,Yoff. 共17兲. where we use dn i ⫽ 兺 cj⫽1 (i j) d j for i⫽1,2,...,r. The inequality is always satisfied because S off is the fundamental relation of the surrogate system B and, as such, it is concave with respect to every n i for i⫽1,2,...,r 共Ref. 4兲. Each of the necessary conditions for chemical equilibrium—each of Eqs. 共16兲—is expressed as a function of energy, volume, and amounts of constituents of the chemical equilibrium state. In the next section we rewrite these conditions in terms of temperature and pressure rather than energy and volume. For fixed values of U, V, na , and , from Eqs. 共13兲 and 共15兲 we see that each of the functions S off and Y j,off for j ⫽1,2,...,c depends solely on the reaction coordinates k for k⫽1,2,...,c. Accordingly, for j,k⫽1,2,...,c, we can write. 2 S off共 U,V,na ⫹ • 兲 2 S off共 U,V,na ⫹ • 兲 ⫽ jk k j. k Y j,off. for both zero and nonzero values of Yoff , that is, not only at the chemical equilibrium state of the reactor A, but also for any nonequilibrium state A that we model with the corresponding stable equilibrium state of the surrogate system B. Relations 共19兲 and 共22兲 are among the many Maxwell relations that can be established for stable equilibrium states of a multiconstituent system, both for the surrogate system B and the chemical equilibrium state of reactor A. Relation 共22兲 implies that for the state of the surrogate system to remain in a stable equilibrium state and, hence, for the state of system A to remain within the family A , of the four changes d j , d k , dY j,off , and dY k,off , we can specify only three arbitrarily and independently.. U,V,na , . 冉 冊 冊. 冉 冊. 2721. 共18兲. IV. CONDITIONS FOR CHEMICAL EQUILIBRIUM IN TERMS OF TEMPERATURE AND PRESSURE. Rather than using energy, volume, and amounts of constituents as independent variables, it is often more convenient to express each chemical equilibrium equation 关Eqs. 共16兲兴 in terms of temperature, pressure, and mole fractions. To this end, we note that the stable equilibrium state of the surrogate system B obtained by switching off the reaction mechanisms at a chemical equilibrium state of system A has not only the same values of energy, entropy, volume, and amounts of constituents as the chemical equilibrium state, but also the same values of temperature, pressure, and chemical potentials 1 , 2 ,..., r . For i⫽1,2,...,r, to prove the last assertion, we recall the definitions of temperature, pressure, and chemical potentials4 as given, respectively, by the relations. or, equivalently,. 冉 冊 Y k,off j. ⫽ U,V,na , , . 冉 冊 Y j,off k. T 共 U,V,na ; 兲 ⫽1/共 S/ U 兲 V,na , , ,. 共19兲. U,V,na , , . that is, the c⫻c matrix with elements a jk ⫽( Y j,off / k ) U,V,na , , is symmetric. Moreover, if we invert the relations Y k,off⫽Y k,off共 U,V,na ⫹ • 兲. for k⫽1,2,...,c,. 共20兲. with respect to the variables 1 , 2 ,..., c , we obtain the relations j ⫽ j 共 U,V,na , ,Yoff兲. for j⫽1,2,...,c,. 共21兲. and using the properties of Jacobians we can easily show that the matrix with elements b jk ⫽( j / Y k,off) U,V,na , ,Yoff is also symmetric; that is, for j,k⫽1,2,...,c,. p 共 U,V,na ; 兲 ⫽ 共 S/ V 兲 U,na , / 共 S/ U 兲 V,na , ,. i 共 U,V,na ; 兲 ⫽⫺ 共 S/ n ia 兲 U,V,na , / 共 S/ U 兲 V,na , , where S(U,V,na ; ) is the fundamental relation for the chemical equilibrium states 关Eq. 共8兲兴. Next, we express the fundamental relation S of system A in terms of that of the surrogate system B by evaluating S off 关Eq. 共13兲兴 at 0 as given by Eq. 共9兲, so that S⫽S 共 U,V,na ; 兲 ⫽S off„U,V,na ⫹ • 0 共 U,V,na ; 兲 …. 共23兲 Thus, for the inverse temperature of a chemical equilibrium state, we find that. Downloaded 02 Sep 2004 to 18.95.1.27. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp.
(5) 2722. J. Chem. Phys., Vol. 121, No. 6, 8 August 2004. G. P. Beretta and E. P. Gyftopoulos. 1 1 ⫽ T T 共 U,V,na ; 兲 ⫽. ⫽. 冉 冊 S U. ⫽⫺T 共24b兲 V,na , . 冋冉 冊 S off U. 冉 冊. S off ⫽ U. ⫽. ⫽. 冉 冊 S off U. r. ⫹ V,n. 兺. i⫽1. 冉 冊. c. S off ni. 兺. U,V,n j⫽1. 冋 兺冉 冊 c. 1⫺. j⫽1. V,n. 共i j 兲. j0 U. 冉 冊 j0 U. r. 兺. V,na , i⫽1. 共i j 兲 i,off. V,na , . 册. ⫽⫺T. 册. 共24f兲. 冉 冊. 共25b兲 U,na , . 冋冉 冊 S off V. 冉 冊 S off V. ⫽T off. r. ⫹ U,n. 兺 i⫽1. 兺冉 冊 c. U,n. S off V. S off ni. c. U,V,n. 兺 共i j 兲 j⫽1. 冉 冊 j0 V. U,na , . 册. 共25c兲 ⫺. 冉 冊. 冉 冊. j⫽1. j0 V. r. 兺 共i j 兲 i,off. U,na , i⫽1. 共25d兲. ⫽⫺T off. U,V,n. j0 n ia. 兺. k⫽1. ⫹ U,V,n. S off ni. S off nk. U,V,na , c. 冉 冊. 冉 冊. 兺. j⫽1. 册. 共26c兲. 冉 冊 j0 n ia. U,V,n. r. 兺. U,V,na , k⫽1. 共kj 兲 k,off 共26d兲 共26e兲. U,V,n. 共26f兲. where in writing Eq. 共26e兲 we use Eqs. 共16兲 and 共24兲. So the chemical potential i of the ith constituent of a chemical equilibrium state of system A equals the chemical potential i,off of the ith constituent of the corresponding state of the surrogate system B. It is noteworthy that the identity of values of temperature, pressure, and chemical potentials of a chemical equilibrium state with the values of the respective properties of a stable equilibrium state of the surrogate system obtains only at chemical equilibrium, because then and only then are the chemical equilibrium equations 关Eqs. 共16兲兴 satisfied. Away from chemical equilibrium states, temperature, pressure, and chemical potentials are not defined for system A because all such states are not stable equilibrium. Finally, we note that Eqs. 共24兲–共26兲 indicate that, geometrically, the surfaces represented by the functions S ⫽S(U,V,na ; ) and S off⫽S off(U,V,na ⫹ • ) have a contact of first degree for each given set of values U, V, and na , at ⫽ 0 (U,V,na ; ), namely, at each chemical equilibrium state. As is very well known,4 each chemical potential of a multiconstituent system in which all chemical reaction mechanisms are switched off may be expressed in the form i,off⫽ i,off(T,p,y 1 ,y 2 ,...,y r ), where y i is the mole fraction of the ith constituent. Using the stoichiometric relations and the chemical equilibrium equations, we find y i ⫽y i 共 na ⫹ • 兲 ⫽. n ia ⫹ 兺 cj⫽1 共i j 兲 j n a ⫹ 兺 cj⫽1 共 j 兲 j. 共27兲. and, for j⫽1,2,...,c, r. where in writing Eq. 共25e兲 we use Eqs. 共16兲 and 共24兲. So the pressure p of a chemical equilibrium state equals the pressure p off of the corresponding state of the surrogate system B. For each chemical potential of the chemical equilibrium state we find. ⫹. ⫽ i,off„U,V,na ⫹ • 0 共 U,V,na ; 兲 …⫽ i,off ,. U,n. 共25f兲. r. 冉 冊. 冉 冊. 共25e兲. ⫽p off„U,V,na ⫹ • 0 共 U,V,na ; 兲 …⫽p off ,. 共26b兲. 冋冉 冊. 兺 共kj 兲 j⫽1. 共26a兲. U,V,na , . S off ni. V,n. 共25a兲. ⫽T. S n ia. S off ⫽⫺T ni. 共24d兲. p⫽p 共 U,V,na ; 兲. ⫽T. ⫻. 共24c兲. where in writing Eqs. 共24c兲 and 共24e兲 we use Eq. 共23兲 and the chemical equilibrium equations 关Eqs. 共16兲兴, respectively. So the temperature T of a chemical equilibrium state equals the temperature T off of the corresponding state of the surrogate system B. For the pressure of the chemical equilibrium state we find that. S V. 冉 冊. c. 共24e兲. 1 1 , ⫽ T off„U,V,na ⫹ • 0 共 U,V,na ; 兲 … T off. ⫽T. i ⫽ i 共 U,V,na ; 兲. 共24a兲. 兺. i⫽1. 共i j 兲 i,off ...,. 冉. T,p,. n 1a ⫹ 兺 cj⫽1 共1j 兲 j0 n a ⫹ 兺 cj⫽1 共 j 兲 j0. n ra ⫹ 兺 cj⫽1 共r j 兲 j0 n a ⫹ 兺 cj⫽1 共 j 兲 j0. 冊. ,...,. ⫽⫺TY j,off共 T,p,y0 兲 ⫽0,. 共28兲. Downloaded 02 Sep 2004 to 18.95.1.27. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp.
(6) J. Chem. Phys., Vol. 121, No. 6, 8 August 2004. Conditions for chemical equilibrium. r r where n a⫽兺i⫽1 n ia , ( j) ⫽兺i⫽1 (i j) , and y i0 ⫽y i (na ⫹• 0 ). Equations 共28兲 represent the chemical equilibrium equations as functions of T, p, and the mole fractions of the chemical equilibrium state. The functional dependences of the chemical potentials on T, p, and y are those of surrogate system B. For given values of T, p, and n a , we can solve the c equations 共28兲 for the c unknowns 10 , 20 ,..., c0 and hence determine the chemical equilibrium composition y 10 ,y 20 ,...,y r0 and the values of the corresponding amounts of constituents, n 10 ,n 20 ,...,n r0 . Conversely, if the values of T, p, y 1 ,y 2 ,...,y r are given but do not satisfy Eqs. 共28兲, we would conclude that the state is not chemical equilibrium. Then, of course, the values of T and p refer to a state of the surrogate system which becomes a nonequilibrium state of reactor A when the reactions are turned on. If the chemical potentials are written as. i,off共 T,p,y兲 ⫽ ii 共 T, p 兲 ⫹RT ln a i,off共 T, p,y兲 ,. 共29兲. where ii (T,p) is the pure constituent chemical potential at T and p, R the gas constant, and a i,off⫽a i,off(T, p,y) the activity of the ith constituent of the surrogate system, it is easy to see that we can rewrite the affinities as. r. r. 兺i 共i j 兲 ii共 T, p 兲 ⫺RT ln i⫽1 兿 关 a i,off兴 . 共 j兲 i. ,. 共30兲. and, defining the equilibrium ‘‘constant’’ at temperature T and pressure p for the jth reaction as. 冉. 1 K j 共 T,p 兲 ⫽exp ⫺ RT. r. 兺i 共i j 兲 ii共 T, p 兲. 冊. ,. 共31兲. we can write r. 兿 关 a i,off兴 . i⫽1. 共 j兲 i. ⫽K j 共 T, p 兲 exp共 ⫺Y j,off /R 兲. 共32兲. and rewrite the chemical equilibrium equations 共28兲 in the well-known mass-action-law form r. 兿 关 a i,off共 T,p,y0 兲兴 . i⫽1. G off j. r. ⫽ T,p,na , , . 共 j兲 i. ⫽K j 共 T, p 兲 .. 共33兲. Another interesting result is that the lowest value of the Gibbs free energy of the surrogate system B obtains at the state of B that corresponds to the chemical equilibrium state of A. Indeed, the Gibbs free energy of the surrogate system B is G off⫽G off共 T,p,n 1 ,n 2 ,...,n r 兲 .. 共34兲. If the amounts of constituents are compatible with na and the c chemical reactions conform to Eqs. 共12兲, we can rewrite G off in the form G off(T, p,na ⫹ • ). For given T, p, na , and , an extreme value of G off obtains provided that, for each j⫽1,2,...,c,. 兺 i⫽1. 冉 冊 冉 冊 G off ni. T,p,n. ni j. na , , . r. ⫽. 兺 i,off 共i j 兲. i⫽1. ⫽⫺TY j,off共 T,p,na ⫹ • 兲 ⫽0,. 共35兲. where we use the equations ( G/ n i ) T,p,n⫽ i for i ⫽1,2,...,r. These conditions are satisfied if the stable equilibrium state of the surrogate system corresponds to the chemical equilibrium state of A—that is, if the chemical potentials satisfy the chemical equilibrium equations 关Eqs. 共16兲兴. The extreme value of G off is a relative minimum because it can be shown that the second order partial differential of G off with respect to each of the j ’s is positive. By equating second order derivatives of G off with respect to the j ’s, for j, k⫽1,2,...,c,. 2 G off共 T,p,na ⫹ • 兲 2 G off共 T,p,na ⫹ • 兲 ⫽ , jk k j. 共36兲. we obtain the Maxwell relations. 冉 冊 Y k,off j. A j,off⫽TY j,off ⫽⫺. 冉 冊. 2723. ⫽ T,p,na , , . 冉 冊 Y j,off k. .. 共37兲. T,p,na , , . Moreover, inverting the set of relations Y k,off ⫽Y k,off(T,p,na ⫹ • ) for k⫽1,2,...,c, with respect to the variables 1 , 2 ,..., c , we obtain the set of relations j ⫽ j (T,p,na , ,Yoff) for j⫽1,2,...,c, and using the properties of Jacobians, we obtain the Maxwell relations. 冉 冊 k Y j,off. ⫽ T,p,na , ,Yoff. 冉 冊 j Y k,off. 共38兲 T,p,na , ,Yoff. for both zero and nonzero values of Yoff , that is, not only at the chemical equilibrium state of the reactor A, but also for any nonequilibrium state A that we model with the corresponding stable equilibrium state of the surrogate system B. Relations 共37兲 and 共38兲 can also be viewed as direct consequences of the Maxwell relations for the surrogate system obtained by equating second order derivatives of G off with respect to the n j ’s,. 冉. k,off n j. 冊 冉 冊 j,off nk. ⫽. T,p,n. 共39a兲. , T,p,n. or, equivalently,. 冉. ln a k,off n j. 冊 冉 ⫽. T,p,n. ln a j,off nk. 冊. .. 共39b兲. T,p,n. We conclude our derivation by summarizing the results pictorially with the help of the energy versus entropy graphs introduced in Ref. 4. For given values of V and na , three projections of states are superimposed on the single U versus S diagram shown in Fig. 1: 共1兲 the projection of the states A 1 of system A that coincide with the stable equilibrium states of the surrogate system B for the given volume V of the chemical reactor and fixed values n1 ⫽na ⫹ • 1 , 共2兲 the. Downloaded 02 Sep 2004 to 18.95.1.27. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp.
(7) 2724. J. Chem. Phys., Vol. 121, No. 6, 8 August 2004. FIG. 1. Energy versus entropy graph of the states of simple system A with given values of V, na , and . The curve labeled 1 represents the states A 1 and coincides with the curve of the stable equilibrium states of surrogate system B for the given value V of the volume and fixed values n1 ⫽na ⫹ • 1 of the amounts of constituents—that is, a fixed 1 . The curve labeled 2 represents the states A 2 and coincides with the curve of the stable equilibrium states of surrogate system B for the given value of V of the volume and fixed values n2 ⫽na ⫹ • 2 of the amounts of constituents— that is, a fixed 2 . The curve labeled 0 represents the chemical equilibrium states of system A for the given values of V, na , and .. projection of the states A 2 of system A that coincide with the stable equilibrium states of the surrogate equation B for the same given volume V and fixed values n2 ⫽na ⫹ • 2 , and 共3兲 the projection of the chemical equilibrium states of system A for the same given values of V, na , and . The set of values 1 is chosen so that 1 ⫽ 0 (U 1 ,V,na ; ) and, therefore, at the energy U 1 the locus of states A 1 is tangent to the curve of the chemical equilibrium states of A. Similarly, the value 2 is such that 2 ⫽ 0 (U 2 ,V,na ; ) and, therefore, the locus of states A 2 is tangent to the curve of the chemical equilibrium states of A at the energy U 2 of the chemical reactor. We see that the curve of the chemical equilibrium states is the envelope of the loci of states A for all possible values of . We also see that the states A 1 represent states of system A that are not stable equilibrium, except at the energy U 1 , and similarly the states A 2 , except at energy U 2 . In general, they are either nonequilibrium or nonstable equilibrium states and yet can be described using the stable equilibrium properties of the surrogate system.. V. RATE OF ENTROPY GENERATION. In the course of chemical reactions in an isolated system A with r constituents and c chemical reaction mechanisms, the system passes through a sequence of nonequilibrium states, and entropy is generated until the system reaches chemical equilibrium. At chemical equilibrium, all changes. G. P. Beretta and E. P. Gyftopoulos. cease—the rate of change of each reaction coordinate is zero—and thereafter the system remains in the stable equilibrium state. The rigorous and complete evaluation of the evolutions of the properties of the system as functions of time from any state that is not stable equilibrium towards the corresponding chemical or stable equilibrium state, and therefore the rate of entropy generation in a general nonequilibrium state is outside the scope of this article, and we are not discussing it. Instead, nevertheless, we derive an estimate of the rate of entropy generation in terms of the c affinities of the surrogate system B of A and the rates of change of the reaction coordinates of the c chemical reaction mechanisms. We will see that this estimate is informative both about what might be considered as the driving forces of the reactions and about whether the so-called principle of microscopic reversibility plays any role in entropy generation. To derive this estimate, we proceed as follows. For given values U, V, na , and , the values of the amounts of constituents n⫽n(t)⫽ 兵 n 1 ,n 2 ,...,n r 其 , and the value of the entropy S(t) are functions of time. Specifically, the value of S(t) is smaller than the value of S off„U,V,n(t)… of the surrogate system B; that is, S 共 t 兲 ⭐S off„U,V,n共 t 兲 …⫽S off„U,V,na ⫹ • 共 t 兲 …,. 共40兲. where the equal sign applies only at the chemical equilibrium state. The justification of Inequality 共40兲 is that, by definition, S off„U,V,n(t)… corresponds to the entropy of a stable equilibrium state which, by virtue of the largest entropy principle, is higher than the entropy of any other state with the same values of U, V, and n(t). Because at chemical equilibrium both S(t) and S off„U,V,n(t)… assume the same largest value, an estimate of the rate of entropy generation S˙ irr—the rate of entropy generation by irreversibility in the isolated system A—is obtained by assuming that A is always in one of the states A defined in Sec. III, so that the value of each property is equal to the corresponding stable equilibrium state value of the surrogate system B. This assumption corresponds to the following two-step conceptual model. We start at time t with the surrogate system B in a stable equilibrium state B t,eq corresponding to the state A (t) of the reactor with reaction coordinates (t). We then turn the reactions on for an infinitesimal lapse of time dt at the end of which we immediately turn them off. As a result, the reactor is in state A t⫹dt , the reactions have changed the composition, and, in general, the surrogate system at time t⫹dt is in a nonequilibrium state B t⫹dt . We allow the additional lapse of time ␦ t ⬘ that is necessary for the surrogate system to reach the stable equilibrium state B t⫹dt⫹ ␦ t ⬘ ,eq corresponding to the state A (t⫹dt⫹ ␦ t ⬘ ) of the reactor with the new composition. Now we turn on the reactions again, and so on. B ⫽S A(t) and The values of the entropy are as follows: S t,eq A B B A S t⫹dt ⫽S t⫹dt ⬍S t⫹dt⫹ ␦ t ⬘ ,eq⫽S (t⫹dt⫹ ␦ t ⬘ ) . The entropy generated by irreversibility in the first time interval is A A ⫽S t⫹dt ⫺S A(t) and in the second S irr,(t,t⫹dt) B. A. A S irr,(t⫹dt,t⫹dt⫹ ␦ t ⬘ ) ⫽S (t⫹dt⫹ ␦ t ⬘ ) ⫺S t⫹dt . An important simplification is obtained if ␦ t ⬘ Ⰶdt , that is, if the time ␦ t ⬘. Downloaded 02 Sep 2004 to 18.95.1.27. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp.
(8) J. Chem. Phys., Vol. 121, No. 6, 8 August 2004. Conditions for chemical equilibrium. 2725. FIG. 2. Energy vs entropy graphs of the states of simple system A with given values of V, na , , and ⬘ . State A (t), ⬘ „(t)… evolves in time while remaining 0 on the dashed curve labeled (t), 0⬘ „(t)… corresponding to the largest entropy 共partial chemical equilibrium兲 state compatible with the given values U, V, na , , ⬘ and the instantaneous values of reaction coordinates (t) of the slow rate-controlling reactions. The dotted curves represent the families of states A (t 1 ), ⬘ , A (t 1 ), ⬘ , A (t 2 ), ⬘ , A (t 2 ), ⬘ for arbitrary values a⬘ , ⬘b , c⬘ , ⬘d . The solid curve labeled 0 , ⬘0 represents the chemical equilibrium states of system a b c d A for the given values of U, V, na , , and ⬘ , where 0⬘ ⫽ 0⬘ ( 0 ).. taken by the surrogate system to reach stable equilibrium after a change in composition is much shorter than the time dt taken by the reactor to affect such change in composition. In such a case, the second step of our conceptual model can be neglected, and the stable equilibrium states of the surrogate system are sufficient to describe the states of the reactor along the entire process. This is tantamount to assuming that among all the internal interactions and dynamical mechanisms that drive the nonequilibrium state towards stable equilibrium, the chemical reactions with stoichiometric coefficients are the only mechanisms capable of changing the composition of the system and, on the time scale chosen for the description, they are the slow, rate-controlling mechanisms, while all the other mechanisms are assumed to be much faster, so as to drive the system in negligible time to its largest entropy state compatible with the instantaneous values of n(t) and therefore maintain the state of the reactor along the family of states A (t) such that if at any instant in time we turn off the reactions we obtain the surrogate system at 共or very close to兲 the stable equilibrium state with entropy S off„U,V,na ⫹ • (t)…. Under this assumption, S˙ irr⫽ ⫽. 兺. i⫽1. 冉 冊 冉 冊 S off ni. r. ⫽⫺. 兺 i⫽1. U,V,n c. ni t. 兺. j⫽1. 兺 共i j 兲A i ⫽0. and r. 兺 i⬘共 k 兲A i ⫽0. for k⫽1,2,...,c ⬘ ,. c. na , c. i,off 兺 共i j 兲˙ j ⫽ j⫽1 兺 T off j⫽1. r. 兺. i⫽1. 冉. ⫺. 共i j 兲 i,off T off. 冊. ˙ j. ˙ j Y j,off⫽ ˙ •Yoff⫽ ˙ 共 t 兲 •Yoff„U,V,na ⫹ • 共 t 兲 …, 共41兲. 共42a兲. for j⫽1,2,...,c. i⫽1. n i 共 t 兲 ⫽n ia ⫹. c. ⫽. r. i⫽1. dS off„U,V,na ⫹ • 共 t 兲 … dt r. where ˙ is the row vector of the c reaction rates ˙ 1 ,˙ 2 ,...,˙ c and Y off the column vector of the c ratios Y j,off⫽A j,off /T off r for j⫽1,2,...,c, where A j,off⫽⫺ 兺 i⫽1 (i j) i,off is the affinity of the jth reaction evaluated at the stable equilibrium state of surrogate system B with values U, V, n(t)⫽na ⫹ • (t). A further simplification is obtained if there is a subset of chemical reaction mechanisms that are much faster than the others. Then, only the slow reactions are rate controlling, whereas the fast reactions drive the system in negligible time to its largest entropy state compatible with the instantaneous composition n(t), which varies slowly as a result of the rates of the slow reactions. Denoting the stoichiometric coefficients and the coordinates of the slow reactions by and and of the fast reactions by ⬘ and ⬘ , Eqs. 共6兲 and 共12兲 are replaced by. 兺. j⫽1. 共i j 兲 j 共 t 兲 ⫹. c⬘. 兺. k⫽1. ⬘i 共 k 兲 k⬘ 共 t 兲 .. 共42b兲. 共43兲. In Eq. 共43兲 we assume that the ⬘k ’s adjust instantly to changes in composition induced by the slowly varying j ’s so as to maintain system A at the largest entropy 共partial chemical equilibrium兲 state A , ⬘ compatible with the given 0 values of U, V, na , , ⬘ and the instantaneous values of the j (t)’s 共see Fig. 2兲, that is, for k⫽1,2,...,c ⬘ ,. ⬘ „U,V,na ⫹ • 共 t 兲 ; ⬘ …, ⬘k 共 t 兲 ⫽ k0. 共44兲. Downloaded 02 Sep 2004 to 18.95.1.27. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp.
(9) 2726. J. Chem. Phys., Vol. 121, No. 6, 8 August 2004. G. P. Beretta and E. P. Gyftopoulos. ⬘ ’s are the values that maximize S , ⬘ where the k0 ⫽S off(U,V,na ⫹ • ⫹ ⬘ • ⬘ ) for the given values of U, V, na , , ⬘ , and (t), or equivalently, for k⫽1,2,...,c ⬘ :. 冉 冊 S , ⬘ k⬘. U,V,na ⫹ • , ⬘ , ⬘. ⬘ „U,V,na ⫹ • 共 t 兲 ⫹ ⬘ • ⬘0 …⫽0. ⫽Y k,off. 共45兲. As a result of these assumptions, system A evolves through the family of states A (t), ⬘ „(t)… , the entropy S(t) is approxi0 mated in terms of S off of the surrogate system B, S 共 t 兲 ⫽S off共 U,V,na ⫹ • 共 t 兲 ⫹ ⬘ • 0⬘ „U,V,na ⫹ • 共 t 兲 ; ⬘ …兲 ,. and note that nowhere in our derivation we make reference to the concept of microscopic reversibility. Next we make use of the condition that, at every state A , S˙ irr( ) must be non-negative. In particular, we apply this condition in the vicinity of the chemical equilibrium state by expanding S˙ irr( ) into a Taylor series about 0 up to second order and using the fact that Y j,off( 0 )⫽0 关Eq. 共35兲兴, ˙ j ( 0 )⫽0 关Eq. 共49兲兴, and, of course, S˙ irr( 0 )⫽0. Thus, for j,k⫽1,2,...,c, we find. 冏. S˙ irr⫽. ⫽ 0. ⫹. 冏. 2 S˙ irr共 兲 jk. c⬘. ⬘ 兺 ˙ j Y j,off⫹ k⫽1 兺 ˙ k0⬘ Y k,off j⫽1. ⫽ 0. 兺 ˙ j 共 t 兲 Y j,off共 U,V,na ⫹ • 共 t 兲 j⫽1. ˙ ᐉ j. 兺. ᐉ⫽1. ⫹ 共47兲. ⬘ ⫽0 关Eq. 共45兲兴. because at each instant of time Y k,off It is clear that the model we present here is valid for homogeneous states of the reactor. In addition, it provides the conceptual background also for the so-called local equilibrium assumption upon which the continuum fluid dynamics treatment of nonhomogeneous states is based.. 兺. ᐉ⫽1. 0. 冏. ˙ ᐉ k. c. S˙ irr共 兲 ⫽. 2 S˙ irr共 兲 jk. 兺 k⫽1 兺. c. ⫽. c. 1 2 j⫽1. c. c. 兺兺 兺 j⫽1 k⫽1 ᐉ⫽1. 冏. ˙ ᐉ j. 0. 冏. 冏. Y ᐉ,off j. 冏. Y ᐉ,off k. ˙ 0. ⫽0,. 共51兲. 0. 0. 冏. Y ᐉ,off j. 共52兲 0. 共 j ⫺ j0 兲共 k ⫺ k0 兲 ⫹¯ 0. 共53a兲. 冏. Y ᐉ,off k. 共 j ⫺ j0 兲共 k ⫺ k0 兲 0. ⫹¯ c. According to the foregoing discussion, we proceed under the assumption–approximation that the 共homogeneous兲 state of the isolated reactor A belongs at each instant in time to the family of states A . For fixed values of U, V, na , and , the only independent variables of the family of A states are the reaction coordinates j ’s. We further assume that each reaction rate ˙ k is a function of the state A and, hence, of the j ’s, that is,. ˙ ⫽ ˙ 共 兲. Y ᐉ,off共 0 兲. and, therefore,. VI. RECIPROCAL RELATIONS. 共48兲. with. ⫽. 共53b兲 c. c. c. 兺兺 兺 兺 j⫽1 k⫽1 ᐉ⫽1 m⫽1. 冏. 8 ᐉ Y m,off. 0. 冏. Y m,off j. 0. ⫻ 共 j ⫺ j0 兲共 k ⫺ k0 兲 ⫹¯ c. ⫽. c. 兺 兺 ᐉ⫽1 m⫽1 c. ⫽. 冏. 8 ᐉ Y m,off. 0. 冏. Y ᐉ,off k. Y ᐉ,off共 兲 Y m,off共 兲 ⫹¯. 0. 共53c兲 共53d兲. c. 兺 兺 L ᐉm,offY ᐉ,off共 兲 Y m,off共 兲 ⫹¯ , ᐉ⫽1 m⫽1. 共53e兲. where we define the c⫻c matrix L with elements. ˙ 共 0 兲 ⫽0,. 共49兲. where, for simplicity, from here on we do not write explicitly the dependences on the given values of U, V, na , , and ⬘ . Condition 共49兲 is necessary because at chemical equilibrium all reaction rates are zero. Recalling that Yoff⫽Yoff( ) 关Eqs. 共15兲兴, we can write Eqs. 共41兲 and 共47兲 for the rate of entropy generation in the form c. S˙ irr⫽S˙ irr共 兲 ⫽. 冏. c. c. ⫹ ⬘ • 0⬘ „U,V,na ⫹ • 共 t 兲 ; ⬘ …兲 ,. 0. 兺 ˙ ᐉ共 0 兲 ᐉ⫽1. c. ⫽. ˙ ᐉ j. 兺 ᐉ⫽1 c. 共46兲. and the rate of entropy generation is given by the relation c. 冏. c. S˙ irr共 兲 j. 兺. ᐉ⫽1. ˙ ᐉ 共 兲 Y ᐉ,off共 兲 ⫽ ˙ 共 兲 •Yoff共 兲. 共50兲. L ᐉm,off⫽. 冏. 8 ᐉ Y m,off. .. 共54兲. 0. In Eq. 共53c兲 we use the truncated Taylor series expansion about 0 of the relations. ˙ ⫽ P 共 Yoff兲 ⫽ ˙ „共 Yoff兲 …,. 共55兲. which follow from substituting into Eq. 共48兲 the relations ⫽ (Yoff) obtained from the inversion 共at fixed U, V, na , , and ⬘ ) of the set of relations Yoff⫽Yoff( ), that is, for j,ᐉ ⫽1,2,...,c, 8. Downloaded 02 Sep 2004 to 18.95.1.27. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp.
(10) J. Chem. Phys., Vol. 121, No. 6, 8 August 2004. 冏. c. 8 ᐉ „Yoff共 兲 …⫽. 8 ᐉ Y m,off. 兺 m⫽1. Conditions for chemical equilibrium. Y m,off共 兲 ⫹¯. 共56a兲. 0. ⬙ ⭓0, Soff. 共62兲. ⬙ are given by the where the elements of the c⫻c matrix Soff relations. c. ⫽. 冏. ˙ ᐉ j. 0. L ᐉm,offY m,off共 兲 ⫹¯ ,. 冏. c. ⫽. 兺. m⫽1. 8 ᐉ Y m,off. 兺 m⫽1. 冏. Y m,off j. 0. 共56b兲. ⫹¯ .. 共57兲. 0. In Eq. 共53d兲 we use the truncated expansion about 0 of the relations Yoff⫽Yoff( ), that is, for ᐉ⫽1,2,...,c,. 冏. c. Y m,off共 兲 ⫽. Y m,off j. 兺 j⫽1. 共 j ⫺ j0 兲 ⫹¯ .. 共58兲. 0. Our results can be interpreted in the customary manner of the so-called ‘‘Onsager’s linear thermodynamic theory of irreversible processes.’’ 9–12 Each affinity Am,off or, better, each Y m,off⫽Am,off /T off for m⫽1,2,...,c, can be regarded as a ‘‘driving force’’ and each reaction rate ˙ j , for j ⫽1,2,...,c, as a ‘‘flux’’ that depends on all the driving forces: that is, ‘‘driving forces’’ and ‘‘fluxes’’ are coupled. If the fluxes are expressed as functions of the driving forces or vice versa, the coefficients of the linear approximation 关Eqs. 共56兲兴 of these functions in the vicinity of the chemical equilibrium state can be regarded as ‘‘generalized conductivities.’’ In view of the relations ⫽ (Yoff), with (0)⫽ 0 , we rewrite Eq. 共50兲 in the form c. S˙ irr⫽S˙ irr共 Yoff兲 ⫽. 兺. ᐉ⫽1. ˙ ᐉ „共 Yoff兲 …Y ᐉ,off⫽ P 共 Yoff兲 •Yoff , 共59兲. where ˙ ⫽ P (Yoff)⫽ ˙ „(Yoff)… with P (0)⫽ ˙ ( 0 )⫽0. The expansion of this form in the vicinity of the chemical equilibrium state yields c. 1 S˙ irr共 Yoff兲 ⫽ 2 j⫽1. c. 兺 k⫽1 兺. 冏. 2 S˙ irr共 Yoff兲 Y Y ⫹¯ . Y j,off Y k,off 0 j,off k,off 共60兲. Direct comparison of Eq. 共60兲 with Eq. 共53e兲 shows that, for j,k⫽1,2,...,c, L jk,off⫽. 冏. 冏. 1 2 S˙ irr共 Yoff兲 1 2 S˙ irr共 Yoff兲 ⫽ ⫽L jk,off , 2 Y j,off Y k,off 0 2 Y k,off Y j,off 0 共61兲. and therefore the matrix L is symmetric, that is, its elements obey the Onsager10 reciprocal relations. Equation 共53a兲 shows that the leading term in the expansion of S˙ irr around the chemical equilibrium state is a quadratic form in the ‘‘distances’’ ( j ⫺ j0 ) from the chemical equilibrium state. Because in the vicinity of the chemical equilibrium state every j ⫺ j0 can take both negative and positive values, the condition that S˙ irr be always nonnegative implies that the coefficients of the quadratic form are elements of a nonnegative definite matrix, that is,. 2727. S ⬙jk,off⫽. 冏. 1 2 S˙ irr共 兲 2 jk. .. 共63兲. 0. Similarly, Eq. 共53e兲 shows that the leading term in the expansion of S˙ irr around the chemical equilibrium state is also a quadratic form in the ‘‘driving forces’’ Y j,off( ). Because in the vicinity of the chemical equilibrium state every Y j,off( ) can take both negative and positive values, the condition that S˙ irr be always non-negative implies that also the coefficients of this quadratic form are elements of a nonnegative definite matrix L共 0 兲 ⭓0;. 共64兲. that is, the symmetric matrix of reciprocity or Onsager coefficients is non-negative definite. We finally emphasize that the main results of this section derive from the structure of the leading terms of Taylor expansions valid only in the vicinity of the chemical equilibrium state. By contrast, the model developed in the previous sections in terms of the A family of states and the stable equilibrium properties of the surrogate system is valid both far from and near the chemical equilibrium state. As it is well known,11 the relation ˙ ⫽ P (Yoff) between reaction rates and driving forces is in general nonlinear. Nevertheless, even for states that are very far from chemical equilibrium, relations 共50兲 and 共59兲 are valid, together with the condition that S˙ irr⭓0, and provide important guidance in the development of chemical kinetics models. VII. CONCLUSIONS. The thermodynamic derivations of conditions for chemical equilibrium, of Onsager reciprocity relations, and of the properties of a practically important family of nonequilibrium states presented here differ from the derivations presented in practically all treatises of thermodynamics applied to chemical reactors. Our motivation for developing this derivation is based on a ‘‘revolutionary’’ 13 conception of thermodynamics as a ‘‘nonstatistical’’ 14 physical theory4,15 that applies to all systems 共both macroscopic and microscopic兲, to all states 共both thermodynamic equilibrium and not thermodynamic equilibrium兲, and that discloses that entropy is an inherent 共intrinsic兲 nondestructible property of matter 共well defined for all systems and all states兲, in the same sense as inertial mass is an inherent property of matter. Our nonstatistical derivation of Onsager relations for an isolated chemical reactor shows that the arguments based on statistical fluctuations, time reversal, and the principle of microscopic reversibility,9–12 which are invariably used in all traditional derivations, are not essential and, therefore, play no fundamental role in the thermodynamic theory of irreversible processes.. Downloaded 02 Sep 2004 to 18.95.1.27. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp.
(11) 2728. J. Chem. Phys., Vol. 121, No. 6, 8 August 2004. E. A. Guggenheim, Thermodynamics, 5th revised ed. 共North-Holland, Amsterdam, 1967兲. 2 K. Denbigh, The Principles of Chemical Equilibrium, 2nd ed. 共Cambridge University Press, London, 1966兲. 3 M. Modell and R. C. Reid, Thermodynamics and Its Applications 共Prentice Hall, Englewood Cliffs, NJ, 1974兲. 4 E. P. Gyftopoulos and G. P. Beretta, Thermodynamics: Foundations and Applications 共Macmillan, New York, 1991兲. 5 The simple system model extends to heterogeneous stable equilibrium states also in the presence of external fields and surface effects, provided a continuous phase model is adopted. See, e.g., J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 共1958兲; L. Mistura, ibid. 83, 3633 共1985兲; and Ref. 7. 6 J. C. Slater, Quantum Theory of Molecules and Solids 共McGraw-Hill, New York, 1965兲, Vol. 2. 7 G. N. Hatsopoulos and E. P. Gyftopoulos, Thermionic Energy Conversion 共MIT Press, Cambridge, MA, 1979兲, Vol. 2. 8 For the sake of precision, we introduce the different character P to em1. G. P. Beretta and E. P. Gyftopoulos phasize that, as defined by Eq. 共55兲, the functional dependence implied by writing ˙ ⫽ P (Yoff) is different from that implied by writing ˙ ⫽ ˙ ( ). 9 R. C. Tolman, The Principles of Statistical Mechanics 共Oxford University Press, Oxford, 1938兲. 10 L. Onsager, Phys. Rev. 37, 405 共1931兲; 38, 2265 共1931兲. 11 S. R. deGroot and P. Mazur, Nonequilibrium Thermodynamics 共NorthHolland, Amsterdam, 1962兲. 12 A. Katchalsky and P. F. Curran, Nonequilibrium Thermodynamics in Biophysics 共Harvard University Press, Cambridge, MA, 1967兲. 13 We use the term ‘‘revolutionary’’ in the sense introduced by T. S. Kuhn, The Structure of Scientific Revolutions 共The University of Chicago Press, Chicago, 1970兲. 14 We use the term ‘‘nonstatistical’’ to emphasize that entropy is defined 共see Refs. 4 and 15兲 for all systems, macroscopic and microscopic, and all states, thermodynamic equilibrium and not thermodynamic equilibrium, without any necessity for either statistical or information theoretic arguments. 15 G. N. Hatsopoulos and E. P. Gyftopoulos, Found. Phys. 6, 15 共1976兲; 6, 127 共1976兲; 6, 439 共1976兲; 6, 561 共1976兲; G. P. Beretta, ibid. 17, 365 共1987兲.. Downloaded 02 Sep 2004 to 18.95.1.27. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp.
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