THEPRINCIPLE OF MINIMUM ENTROPY PRODUCTION
Dec., 1960
and, as S is the symbol for a substrate which does not change rapidly ‘Rf.M+-’ aRf,,w+/dtl
< A+-’ < A--’
dA+/dt (ki/k-z)dA+/dt A- --I dA+ /dt !Rf,M---IdRr,\z-/bt/ < A-‘ dA- ldt l R f , M + - ’ dRf,M-/bti < A+ -‘(k-n/ki)dA-/dt A+-’ tlA-/dt bRd,M+/dt dRd.M-/bt = 0 IRf.N--’
dRf.M+/all
1857
Since no branching chains are involved the steady state is stable.
(7.3)
Conclusions
N
(7.4) (7.5)
We have illustrated the application of the general set of inequalities derived in Section 2 by a sufficient number of examples to demonstrate its (7.6) power in indicating the conditions necessary for a (7.7) steady state. Usually it is found that the requirement for a steady state reduces to the necessity also for the concentrations of reactive intermediates IdARx+/dM+/ = k-1 A (7.8) to be small compared to those of other species. ldARN+/dM-I = ] ~ A R M - / ~ M =+XI (7.9) However, even in rather unusual situations, it is laAR~~-/02V-l = kz X (7.10) found that the inequalities, if properly applied, are Typically, we must show that adequate t o disentangle the various factors involved, and can serve as a guide to the conditions XA- > > idL4+/dti (7.11) under which a steady state can occur, without the But necessity of integrating kinetic equations. When IdA+/dtl = lk-IM+ - klSA+l (7.12) many reactions are involved the process may beAt a steady state, from (7.1) come rather tedious, but it resolves by a systematic k - i M + - k#A+ = X[M- - M + ] (7.13) procedure all questions concerning the existence If M- and M+ are intermediates which are present and stability of a steady state. in small concentrations this is clearly much smaller I am indebted to Dr. W. Forst for reading and than the left-hand side of (7.11), and again we see commenting upon portions of this paper and for a that the more nearly equilibrium is established translation of the paper by Frank-Kamenetsky, between &I+and A+ the more valid is the inequal- and t o the referee for a criticism which resulted ity. in an improvement in the discussion of (2.9).
-
-7
+
+
THE PRINCIPLE OF MINI31UM ENTROPY PRODUCTION AND THE KINETJC APPROACH TO IRREVERSIBLE THERMODYNA1\IICS1 BY 0. K. RICE Department of Chemistry, University of North Carolina, Chapel Hill,North Carolina Rececued Apral I, 1960
A distinction is made between a steady state in detail, in which certain reactive intermediates come t o a steady state, and a steady state in the large, in which substances in larger concentrations come to a steady state under certain restraints. It is shown very generally that in either case a state is reached in which the rate of entropy production is a minimum consistent with the constraints imposed, provided the system is not far removed from equilibrium. A mechanism is considered which may be applied to the consideration of flow processes. The relation of these ideas to the general principles of irreversible thermodynamics is considered, and it shows that the laws of irreversible thermodynamics follow from the minimum-entropy principle. The article closes with a discussion of the assumptions involved in this deduction and their significance.
It is well known that when a steady state becomes established in a system which is only slightly displaced from equilibrium the rate of production of entropy is a minimum. This condition as it applies t o chemical reactions has been examined in some detail by Prigogine2and by Bak.3 These considerations involve the usual methods of irreversible thermodynamics in which the entropy production is expressed in terms of forces and fluxes. While Prigogine has considered some cases which do not directly involve Onsager’s reciprocal relations, the particular types of steady state considered were not too close to those of interest to chemists, and the principal interest seems t o have been in the relation between the reciprocal relations and the law of microscopic reversibility. It is, (1) Work assisted b y the Katlonal Science Foundation. (2) See I. Prigogine, “Introduction t o Thermodynamics of Irreversible Processes,” Charles C . ‘Ihomas, Springfield, Ill., 1955, Chap.
VI. (3) T. A . 13ak Thwis, University of Copenhagen, l % Y ,
Chap. 2.
however, possible to introduce this subject by a direct consideration of steady states in which short-lived intermediates are involved, as, indeed, they are, in one way or another, in most reactions. The principle of minimum entropy production can be established in a very general fashion on the basis of the assumptions usual in chemical kinetics, including, indeed, the law of microscopic reversibility, without any direct reference to the reciprocal relations. It is then possible to obtain the formalism of irreversible thermodynamics using the minimum entropy principle as a basis. A number of interesting points emerge when this exercise is carried out. A short-lived intermediate comes quickly to a steady state14and the concentrations of the substances which are present in larger amounts change slowly. Under certain circumstances these (4) See 0. K. Rice, THISJOURNAL, 66, 1851 (1961) (precedingsrticle) and D. .4.Frank-Kamenetsky Zhur. Fis. Khim., 14, 605 (1940).
0. K. RICE
1858
concentrations can also reach a steady state. Thus we have two types of steady state, the first type involving the intermediates which we may call the steady state in detail, and the second type involving the substances present in measurable concentrations, which we call the steady state in the large. 1. Minimurn Entropy Production and the Steady State.-An intermediate, which we shall call M, may be involved in a series of reactions of varying complexity, which may be written in the generalized form (the subscript i characterizing a particular reaction). aiAi biBi ... miM niNi 4- . . .
+
+
+
IC i
qiQi
k-
+
+ riRi + . . . mi’M + piPi +
. . ,
VOl. f34 dASi/aM
= (mi’
- mi)A&
(1.3)
where ASM is a factor which is common to all the reactions. Also we can express A S i by taking the first term in a Taylor’s expansion about the concentration Me,;of M which is necessary to bring the particular reaction into equilibrium. Thus AS1 = (fM - hf,,i)(dASi/bM) = (M - Me,i)(mi’ - tni)AS~ (1.4) Using eq. 1.2 and 1.4, we see that d@/bM = 0 gives Zi[miki(AiaiBibi mi’k-i(&iqiRiri
mi)ASu k-i(&iqiRi’i
. . Mmi-lNini ,
. . . Mmit-1PiPi
+ Zi[ki(AiGiBibi
. . .) . . . ) ] ( M - Me,i )(mi’ -
.. . . . . M m i ’ P i p i . . .)](mi’ - m i ) A s ~= 0 (1.5)
i
Here the large 1et)tersrepresent chemical substances, Now we can write . . . Qi,Ril . . . present in large quantities, . . . P i , . . . being other intermediates, and the small letters representing stoichiometric const,ants. The subscript i is used since different substances by the definition of M e , i . Hence the ith bracketed quantity under the second summation of may be involved in the various reaction^.^ (1.5) can he written A steady st’atefor M will occur when Ai,Bi, Xi,
. . Nini . . . ) ( i l l m i - M , , i m i ) L i(QiqiRi+i . . . P i p i . . . ) (Mmi‘ - M e , i m i ’ ) the italic letters representing concentrations (or which, since M is close to Me,i, is equivalent to activities), and the summation being over all reac- nziki(AiaiBibi . . . A:ini . . ) X m i - l ( M - M , , i ) tions in which M is involved. We now let ASi be mi’ki(QiqiRiri . . . P i p i . . ( M - Me,i) the entropy when the reaction proceeds by one Zi(mi‘
- mi)[12,(AiaiBibi . . . M m i N n i . . .)
-
)] = 0
ki(AiaiBibi
,
(1.1)
,
,)M”’1’-‘
unit in a large isoEated system. The rate of entropy production dX/dt = @ will be given by = ~ , [ k i ( AiaiB,bi
k-
. . . iTfmiNini . . .) . . . $fmi’PiPi
i(&iP1h!iri
.. .)]A S i
(1.2)
We could have followed Prigogine and others in expressing the rate of entropy production in terms of the affinities and degree of advancement of the various reactions, using the ideas of de Donder, instead of expressing the rate in terms of entropy production in an isolated system, but the latter method seems somewhat simplier for our purposes. For the entropy production to be a minimum a necessary condition is d@/dM = 0. To evaluate this we must first find out something about the ASi. We note in the first plme that we assume that none of the reactions are far displaced from equilibrium. If any given reaction i is in equilibrium A X i = 0. Furthermore, we see that a change of M will change all the A S i in related ways. Thus we may write5a ( 5 ) I t may be noted t h a t b y putting the substance M on both sides of the equation, and allowing mi’ and mi t o take on non-integral values, and by allowing the possibility t h a t some of the other substances appear on both sides ( e . ~ . , Ai and Qi might be identical) we allow suflicient generality t o deal with any situation t h a t may arise. Because t.he equilibrium const,ant must be expressible a s a simple quotient in mi must be an integer. terms of activities, a difference like mi’ We might, in general, expect the rate constants in either direction to be rather complex functions of all the activities, h u t a function which would be the same for ki a n d k i and which would thus divide out of the equilibrium constant to leave it a combination of simple powers. For very small shifts from a given state of equilibrium such a function can be sufficiently well approximated b y a product of single powers of the various variables, the power in each case being determined to give the correct value of the first partial derivative. (5a) A ~ might M be differentfor different reactions if concentrat.ions are wed, since the thermodynamic functions for one substance might be changed by changing theconcontralion of another. hot all difficultiw
-
and hence is equal to the corresponding quantity in the first summation sign. Thus a+,ldM = 22i[ki(AiaiBibi . . . M m i n ’ i n i . . . ) k-l(&igiRiri . . . M m i ‘ P i p i . . .](mi‘ - V Z ~ ) A S M(1.7)
Comparing eq. 1.1and 1.7 we see that the condition of minimum entropy production is the same as the condition for the steady state. It remains to be verified that the rate of entropy production is a minimum rather than a maximum. From eq. 1.7 bQ/dM2 = 2Zi[kimi(AiaiBibi . . . MmiNni . . . ) k-,rni‘(&i’JiRiri
,
. , M m i ’ P i P i . , . ) ] fi!f-l
(mi’
- mi)bSu (1.8)
Since we are never far from equilibrium, eq. 1.6 will be nearly fulfulled for the value of M actually existing. Hence we may see that the right-hand side of eq. 1.8 will have a sign opposite to that of ASM. Since eq. 1.3exhibits ASM as the change with concentration of the entropy increment per unit M introduced by a process into the system, it is clear that ASM will be negative, for the entropy increment per unit M is less the greater M is. Therefore, d2(dS/dt)/bM2 is positive, which shows that dS/dt is a minimum. One can carry out similar differentiations for any one of the short lived intermediates. Indeed, it is also possible to carry them out for the other substances, also. The lowest minimum will be obtained when the derivative of @ with respect to the concentrations of all possible substances is zero. However, this lowest minimum will naturally occur of this sort are avoided by replacing concentrationh by activities, (which is, strictly, also necessary in eq. 1.1 and 1.2).
Dec., 1960
THEPRINCIPLE OF MINIMUM ENTROPY PRODUCTION
1859
at = 0, when all subgtances are in mutual Also, the effective temperatures for the unit and its equilibrium. It is possible to set up conditions inverse will be slightly different. We can write a such that the concentrations of some of the species representative reaction scheme for this situation as Ai,Bi, . . . &i,R1,. . . , i.e., the substances present kl x ki in large concentration, are held fixed by some artiM+ S M - Z S + AS + A+ ficial means, by introducing or removing some of k- , x k- z this material. The system will then come to a S represents the rest! of the substrate which, in the steady state with respect to all of the unconstrained of the above description, since in a simple case concentrations, the steady-state equations for light the transference unit transfers only A, will be the any particular species being of the general type same for M+ and M-. The rate constant for (l.l),regardless of whether it is a short-lived inter- M+ + M- , which is designated as A, will, as already mediate or not, and this will be the situation of noted, be t,he same as that for ILI- -+ AI+. Since minimum entropy production under the constraints different temperatures are involved, me will have applied. lcl # k-2 and k-I # k2, and, in fact, kllIc-1 # The difference between the short-lived inter- Ic-Zlkz. Because an energy level of M+ can go mediates and the other substances lies in the fact over into a corresponding one of M- and vice versa, that the former come to their steady state very for the reaction M+ --c M- we can mite AS1 = quickly; that is, the steady state in detail is 0 if M+ = M-. In general rapidly attained. The other concentrations then A S A = (M- - A!f+)AS, (2.1) change slowly, and the system eventually drifts where toward the steady state in the large, which is, if the system is close to equilibrium, a state of A S M = dASx/hM- = -hLiSh/dJf+ (2.2) minimum entropy production. Of course, if the and Mare, respectively, capable of Since M+ system is close to equilibrium, this drift time may S A+ and S Acoming into equilibrium with be small, but it will in any case be long compared to the time required to establish the steady state we can write AS1 = -(M+ - M+,JASM (2.3) in detail. 2. Application to Row Processes.-The reac- and tions which we have considered can be of a rather AS2 = (M- - M - . , ) A S M (2.4) general t j p e . We can imagine, for example, that we have constraints on the system, so that some of The assumption here, of course, is that one can get a the substances involved could be a t a different true equilibrium even when there are temperature pressure from others. Pressure differences and and concentration gradients. As I have predifferences in concentration may be maintained by viously remarked,6 this seems inherent in the assemipermeable membranes and pistons, in a way sumption that such a system has definable thermowhich requires no gain or loss of entropy. But if dynamic functions. It is equivalent to the Kelvin temperature gradients are present we are in im- principle, applied to elementary processes. It is now seen that by the device of introducing mediate trouble, because transfer of material cannot occur without transfer of heat, so the tem- the transference units we have reduced the mechperature difference cannot be maintained in an anism to one in which all the reactions involved can isolated system. Thus one cannot be certain that come to an equilibrium in which the A8 vanishes. equality of the rates of forward and back reactions So the same calculations can be made, and the really implies a state where AS = 0, a condition steady state will be a state of minimum entropy which is quite essential for the deductions we have production. This will be true for the steady state given. If semipermeable membranes are not al- for M+ and M- if S, A+ and A- are held fixed. lowed, similar situations may occur with concen- It will also be true for the steady state for A+ if Ais held fixed. The first situation, of course, repretration gradients. In situations of this sort, I have proposed6 that sents the steady state in detail. If, say, A- is we consider a (‘transference unit,” or “transfer held fixed we will eventually approach the steady complex.” A transference unit, which we will state in the large, in which A+ is determined by now designate as M+, is a group of molecules, the A - . Of course, if k+ were equal to k-, then by outside regions of which blend in with the surround- obvious symmetry A+ = A - . However, in genings, but which contains some portion which is in a eral, k+ and k- are not equal on account of the special configuration which can go over to temperature gradient, and A+ and A - come to a another state in which transfer of material and steady relationship, with a difference in A+ and Aheat has occurred. The latter state is the in- which corresponds roughly to the Soret effect. By verse unit, which we will call M-. It can go back considering this somewhat oversimplified example, to M+, with the same probability that holds for the we see the relationship between the steady state in direct reaction. In a simple case, a transference detail and the steady state in the large in the usual complex IS striving to come to equilibrium with problems of irreversible thermodynamics. 3. Irreversible Thermodynamics.-In irreverssome pari icular substance A (the substance transferred) a t some particular point in space, where ible thermodynamics, certain “fluxes” Ji are the activity of A is, say, L4+. This inverse unit expressed in terms of “forces” Xi by a linear rewill be trying to come to equilibrium a t a slightly lationship displaced posilion where the activity is A - . Ji = ZjLijXj (3.1) (6) 0.K. Rice, TEIBJOURNAL, 61, 622 (1957). and the entropy production is given by
+
+
0. K. RICE
1860
*
= =
ZiJlXi 2,.,L,,X,X,
(3.2)
If certain of the X, are assigned specific values then the system comes eventually to a steady state in which the other J i are equal to zero if none of the component involved is added to or removed from the system. The Ji corresponding to a fixed Xi cannot be zero, for material of type i must be added at one end of the system and removed at the other in order to keep the X , constant. In general the number of indices i or j involved will be very much less than the number of reactions involved in our discussion in Sections 1and 2. The implication is that most of the reactions are already in a steady state (essentially the steady state in detail is already established) even when the Ji are not in a steady state. When the final steady state is reached, the steady state is established in detail and in the large. This must be a situation of minimum entropy production under whatever restraints are assumed to exist. The reciprocal relations follow from the minimum entropy production for a steady state, as well as the reverse. Suppose, for example, we fix all the X , a t zero except XI, which is fixed at a definite non-zero value, and X s , which is free. Then for the entropy production to be a minimum with respect to variation of X2 2Lzzxz
+ (LlZ + LZdXI = 0
(3.3)
But for J z to be equal to zero, Le., for a steady state to exist under these circumstances, we see from (3.1) that L21X1
+ L22XZ = 0
(3.4)
It then follows from eq. 3.3 and 3.4 that Ll2 = Ln1
(3.5)
and so for the other pairs. If Xa were fixed and XI free, the same thing could have been proved, but it is not necessary to prove it twice. This is fortunate, for one of the forces can be associated with the temperature gradient. As we have noted in Section 2, the case of a fixed temperature gradient can be satisfactorily handled. On the other hand, while it is possible to leave the temperature free and reach a steady state with other gradients fixed, it may not be possible to describe such a steady state in terms of chemical equations. So such a steady state would have a somewhat different basis. But once we have the reciprocal relations it can also be shown to be a state of minimum entropy production. From the above discussion, it can be seen that the laws of irreversible thermodynamics follow
Vol. 64
from the assumptions we have made, through the vehicle of the law of minimum entropy production. 4. Analysis of the Assumptions Made.-As we have remarked, a detailed consideration of the system involves a great many more variables than are ordinarily included in equations like (3.1) and (3.2). We can, however, as in Section 1, imagine the entropy production expressed in terms of all the chemical reactions, plus a term due to that part of the conduction of heat which does not involve reaction or transfer of material. All these processes occur independently of each other, if we use activities in our equations and consider the generalization noted in footnote 5. Thus we have avoided cross terms like Lij(i # j ) in our formulation of the entropy production. If we assume that the reciprocal relations hold, it will always be possible to diagonalize the matrix of the coefficients, Lij, and so arrive a t a formulation of the type we have considered.’ This point was noted by Bak,3 who stated that it formed the basis for the use of transference units. He also stated that the reciprocal relations cannot be proved from transference units. Whether one states that the reciprocal relations cannot be proved by the use of transference units is certainly to a large extent a question of semantics. One cannot prove them without making certain assumptions, the most important one being, in t’he present formulation, that AS1 and AS2 of eq. 2.3 and 2.4 are zero when M+ and M- are in “equilibrium” with the substrate. It may be that one would prefer the assumption made by Onsager, namely that the rate of regression of fluctuations is given by the macroscopic equations, but whichever assumption one prefers, it is of interest that they lead to the same results. Of course, either treatment involves the law of microscopic reversibility. This is involved in the present treatment because of the consideration of the direct and reverse reactions in the individual equilibria. It was further stated by Bak that the chief interest in the use of transference units lies in the fact that one may sometimes imagine what their nature might be. This is certainly a point of some importance, but a more profound interest lies in the possibility of interpreting the macroscopic phenomena in terms of independent element,ary processes in a rather direct and pictorial manner. In order to do this, it is not necessary to specify the nature of a transference unit in anything but the broadest outline. (7) bctually, of course. the steady-state conditions of Sections 1 and 2 are somewhat involved and cannot be formulated in terms of having certain of the J ’ s constant. Indeed this would give nothing of interest in the diagonal representation.
