Thermodynamics of Stability of on equilibrium Steady States R. P. Rastogi and Ram Shabd Gorakhpur University, Gorakhpur-273001, India Thermodynamic theory of the stability of equilibrium states is well estahlished and good accounts are available in standard textbooks (1.2). , , . The stabilitv of noneauilibrium states is of considerable current interest in view of its application to oscillatory processes, limit cycle behavior, and bifurcation phenomena ( 3 , 4 ) .The thermodynamic theory of stability is well advanced ( 5 ) .The purpose of this article is to present a concise and critical account of this important development in nonequilihrium thermodynamics. The criterion for the stability of nunequilibrium steady states has been critically examined for consecutive and monomolecular triangular reactions, autocatalytic reactions, auto-inhibited reactions, and the Lotka-Volterra model. ~
~
Stability of Equilibrium States According to Lyapounov, "a system is stable, if upon application of a disturbance, it returns to its original state when the disturhance is removed." Stability conditions for equilibrium are well known. Since entropy is maximum a t equilibrium, virtual fluctuations, 6S (Fig. 11, in entropy, S, would obey the following condition (5)
Stote parameter Figure 1. Fluctuation in entropy because of fluctuation in one of the state parameters.
i.e., rate of virtual variation of entropy would be less than zero. E and V denote the energy and volume of the system. Similarily, as seen from Figure 2, we obtain This means that the rate of virtual variation of internal energy would he greater than zero. We have used the concept of virtual displacement from equilibrium because of fluctuations in parameters describing the thermodynamic state of the system. It should be noted that such displacements would be non-spontaneous. Since entropy is maximum in the equilibrium state, and since the displacement 6S would result in lowering of entropy, we get condition (1).For stability of the equilibrium state, it is also necessary that (5) which is the condition for maxima noting that the curve in Figure 1is convex, and similarily (6'E)s.v > 0
(4)
which is the condition for minima noting that the curve in Figure 2, is concave. It can be shown by standard thermodynamic arguments that
State parameter Figure 2. Fluctuation in energy because of fluctuation in one of the state parameters
where p, is chemical potential of the component y.From eqn. ( 5 ) ,the following conditions can be ohtained, (a) For thermal stability
(b) For mechanical stability
(c) For stability of chemical equilibrium where s is the local value of entropy, C, is heat capacity a t constant volume, p is density and x is isothermal compressibility. p,,, is defined as
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Journal of Chemical Education
&,,4N76N,,
>0
(8)
From eqn. (6) it follows that on giving energy to the system, the temperature must increase at constant volume. Similarly, a t constant temperature, the volume must decrease as pressure is increased due to eqn. (7). Several criteria for the stability of nonequilibrium steady
states have been suggested by Prigogine and Glansdorff ( 5 ) . However, the thermodynamic study of stability of nonequilihrium steady states has not received sufficient attention. The purpose of the present communication is to review these criteria and investigate the stability of a few typical nonequilihrium states. Nonequilibrium Steady States Near Equilibrium
It has been shown (6) that in a nonequilihrium state having n-flows, J I , Jz,. . . , J k ,Jk+l,. . . ,Jnand n-independent forces X1...X,,then
when a certain number of forces X1. . .Xh are kept constant and r denotes entropy production due to irreversible processes. J k + , . . J"
=o
(10) Equation (9) is obtained when (1) linear phenomenological relations are assumed; (2) Onsager reciprocity relation is satisfied; and (3) phenomenological coefficients are assumed to he constant. Thus, in the steady state when the fluxes, J k + l . . J, are zero, entropy production is minimum, according to eqn. (9). In other words, if we plot o against X,, we would get a curve of the type shown in Figure 3, and the minimum would correspond to the steady state. Let us examine the minimum of entropy production curve in Figure 3. If the steady state is perturbed by small perturbation in X,, and 6na0 (11) then the system would return back to the steady state since entropy production is a t a minimum in this state. Accordingly, inequality condition (11) represents the stahility condition of the steady state. Now since a is given by o = ZJhXa
(12)
we have, upon small perturbation, bo = 2JabXa
+ ZXaWa
(13)
Further,
Stability of Nonequiiibrium Steady States
The most useful function characterizing such states that are not necessarily in the linear region is 62S, which is a homogeneous function of degree two. I t has a quadratic form with constant values of the associated coefficients. I t can be easily shown that the Eulerian derivative of 6"sis just its time derivative (7). Since PS is negative, the condition of stability of the nonequilibrium steady state according to second method of Liapounov is given by
i.e., the time derivative of the second differential of entropy corresponding to virtual change is greater than zero. This second differential of entropy arises due to virtual changes in the variables defining 6 s . Evidently, 6% is a Liapounov function (see Appendix 1). Inequality (16) would he valid as long as the Gihhs equation remains valid outside equilibrium. In the case of chemical reactions occurring in an isothermalisobaric system, eqn. (16) would always be true so long as the system is continuously stirred and concentration and temperature gradients are not generated. I t is obvious that ( a 2 G ) ~ ,and p ( a 2 A ) ~ ,would v also be Liapounov functions, and since both are positive and greater than zero, the corresponding stahility conditions would he as follows
In other words, the time derivative of the second differential of Gihhs function, G, or Helmholtz function, A, corresponding to virtual changes is always negative for stability of nonequilibrium states (Appendix 2). It can be shown by thermodynamic arguments that
Accordingly, it follows that
f
Z6JabxkdV > 0
(19)
where 6Jk and 6Xh are the deviations of the flux J k and force Xk from the reference state defined ( 8 )hy (20)
dJk. = J 4 - J 0 k
so that ZJkbXa = ZXibJt
=
'1&~
(15)
I t follows from eqn. (14) that ZJkbXa = ZXi6Ji
0
This means that the sum of the product of fluxes and virtual variation in forces, or the sum of the product of forces and the virtual variation in fluxes, would be greater than or equal to zero for stable noneqnilihrium states. We shall consider the physical significance of eqn. (15) in a later section.
In this case, the reference state is the nonequilibrium steady state, and J o k and Xoh refer to the flux and force in this situation. If we perturb the steady state by perturbing only one of the forces X,, it follows from inequality (19), that bJ,6X,
>0
(22)
which shows that the perturhation 6J, and the perturbation 6X, always have the same sign. Theorem of Minimum Entropy Production and Stability Condition
We note that eqns. (9) and (15) are obtained when the condition of minimum entropy production is taken into account. In addition, eqn. (15) corresponds to stability condition (19). This can he proved as follows. Jkis the flux when the system is perturbed from the steady state by perturbing Xk. According to arguments in the section on stability of equilibrium states, Jokis zero a t the steady state, which is our reference state, so that J h = 6Jk and eqn. (15) reduces to condition (19). However, it is clear that eqn. (15) is more restrictive than condition (19). Nonequilibrium Steady States Involving Chemical Reaction Figure 3. Fluctuation in n due to fluctuation in X,.
Condition (19) is particularly useful in the case of chemical reactions when we consider isolated systems. As long as conVolume 60
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541
vection does not start in a chemically reacting system, the Gihhs equation is valid for the entire nonequilihrium range. This may he achieved provided the system is well stirred. The analysis of chemical reactions under isothermal conditions is particularly instructive. By perturhing the steady state concentration of one of the reacting species, all the independent fluxes and forces would also be perturhed. For this purpose, we shall examine the following class of reactions.
The steady state in the monomolecular triangular reaction will he stable. This would he true whether [A]., [B]., or [C]. is perturhed. Stability of Steady States in an Autoinhibitory Reaction
Let us consider the following reaction kl
X+Y+Z
(30)
h-i
1) Monomolecular triangular reaction 2) Autoinhihited reaction 3) Autocatalytic reaction 4) Lotka-Volterra Model
We consider the situation when the reverse reaction is insignificant and the forward reaction is autoinhihitory. Accordingly, the reaction rate for the forward reaction would he expressed as,
Stability in Monomolecuiar Triangular Reactions
Some of the examples of monomolecular triangular reactions are as follows: 1) Isomerization of hutenes (9)
cis-butene = trans- butene 2) Triangular isomerization reaction of cymene ( 1 0 ) 3) Isomerization reaction of 2-pentenoic acid (111.
~
~
where k l is rate constant for the forward reaction. Affinity, A , of the reaction would he written as
If a small perturhation is made in the concentration of X in the steady state, the corresponding perturbation in w and A would he given by
We consider the stability problem for the following monomolecular triangular reaction
Accordingly, we find that B
R
'
C
h: (2)
where k l , k z , and k:, are the specific reaction rate constants for the forward reactions while k'l, h'z, and k'3 are those for the reverse reactions. Further, let Al, A2, and A s he the affinities of the reactions (I),(2), and (3), respectively. These are related to chemica! potentials of the various species in the following manner
when the system is far removed from equilihrium. The stability criterion predicts that for the above reaction the steady state will he unstable. An example of an autoinhihitory reaction (12) is 209
-
302
which oheys the rate law
A i = PA - PB; A2 = I*B - PC; A S = P C - P A
where subscripts A, B, and C denote the corresponding quantity for the various species. The chemical reaction rates J 1 , J2, and Jg for reactions (I),(2), and (3) and their corresponding affinities are given by
The decomposition of ammonia on platinum surface (13) oheys the rate law. ~ P N H ~
dt
PNH~ PHs
where K is the rate constant, P N H ~and P H denote ~ the partial pressure of NH3 and Hz. Similarly, oxidation of SO2 a t a platinum surface is inhihited by the trioxide formed in the contact process. Many enzymatic reactions are known to he inhihited by the reaction products. Out of these, only two fluxes and two affinities are independentsinceJ1+J2+J3=0andAl+A2+A~=O. In order to test the stability of steady states, we introduce a small perturhation in steady state concentration [XA]..The corresponding fluctuation in Jiand Ai would he given by
Stabiiity of Steady States in an Autocatalytic' Reaction
Let us consider the following reaction which is autocatalytic with respect to X. kl
X+Y+2X
(36)
k-1
The affinity of the reaction is expressed as
where K is the equilihrium constant and R is the gas constant,
Hence
' Thermal decomposition of ammonium perchlorate is known to be
p
p
-
~
~
~
autocatalytics ( 14). In the Belousev-Zhabotinskiioscillatory reaction. it is believed that production of HBrO, is autocatalytic ( 15). It is known that conversion of trypsinogen into trypsin is catalyzed by trypsin itself ( 76).
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Journal of Chemical Education
and I is the temperature of the system. The reaction rate, w, is given by
k%
X+Y&2Y;
kl
K'=-
k-2
k-,
. f + H & Ekl;
where k, is the rate of the forward reaction and k-1 is the rate of the reverse reaction. In order to check the steady state stahility, a small perturbation close to the steady state of X concentration, say 61x1, is made. The corresponding perturbation in affinity and rate of reaction would he written as
1-3
k-s
143) (44)
This scheme is applicable to population oscillations in ecosystems ( l a ) , where X and Y denote the population of prey and predator, and A being the source of food to X. Affinities and reaction rates for reaction (421, (431, and (44) are given by
Applying the stahility criteria, it follows that the nonequilihrium steady state would he unstable provided R 6(AIT)6w = - -lk~[Y] [XI.
- 2k-1[X.J16[X]/~< 0
In other words, the system would he unstable provided, k11Yl > ZkdXl.
follows i . , , when the rate of forward reaction is greater than twice the rate of the reverse reaction during equilibrium. When Y = [Y]., the above condition cannot be satisfied and the steady state would always he stable. However, if the reaction is takine: p l ~ wI I I 311 open qstetn, TI),.~ w ~ ~ e n t r a ,,i t i Y~ can m he maintained ,uc h !hill the abuw ~ ( a d i t i mis ilhlisiivd 'I'his twinl is further examined below. Let us consider the reaction system in contact with two large reservoirs of Y and X so that, in effect, the system hecomes an open2 one
The entropy production, a, in the overall system would he due to 1) transport of Y from the reservoir I to subsystem 11, 2) chemical reaction in subsystem 11, 3) transport of X from the subsystem I1 to reservoir I11 and would be given by where J y and J x denotes the mass flux associated with transfer of Y from I to I1 and the transfer of X from I1 to 111 while A y and A x are the corresponding forces. w, denotes the rate of chemical reaction in I1 while A, denotes the corresponding affinity. We maintain the concentration [Y] > 2 [Y]. in chamber I and X in chamber 111 while in chamber 11 we maintain the concentration of X equal to [XI. and the concentration of Y equal to [Y]. Now, if we perturb the concentration of X, the system would be unstable provided
The subscript o refers to corresponding concentrations in the steady state when wl = 0; wz = 0; 0 3 = 0. Now,
Noting t.hat the steady state concentrations [XI. and [Y]. are given by
so that
When equilibrium is far to the right-hand side i.e., when the rates of the reverse reactions are negligible as compared to the rates of the corresponding forward reactions,
Putting the value of [Y]. from eqn. (57) in eqn. (54) and noting that kl[X]. 0 we find
--
bAxbJx + 6A,6w, < 0 Now, if dAxdJx > 0 but 6A,6w, < 0, then the system has the potential of being destabilized. Stability of Steady State in Lotka-VoMerra Model
We consider the following scheme of homogeneous chemical reactions,
-
which is the condition for marginal stahility as obtained by Nicolis (19). Thus, for marginal stability, the sum of the product of virtual changes in the reaction rate and the corresponding force is equal to zero. Let us examine the stability of a reaction system with respect to perturhation in [Y].. The affinities and rates would he perturbed as follows
-
Belousov-Zhabotinsky oscillatory reaction has been studied in a continuous flow stirred tank reactor (17). Volume 60
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543
which means that 6%is a homogeneous function of degree two. T h e Eulerian derivative of function can be written as
so that
.. Substituting the value of (621, we g e t
[XI. a n d [Y]. f r o m eqn. (55) into eqn.
T h e time derivative of function f i s given by 1 (If --
2 ill
The s t e a d y state c a n be unstable, p r o v i d e d klkzks[A][B] > k-l(Zk-zk-dE1
+ k32[B]2)
(63)
However, w h e n w e neglect t h e b a c k w a r d r e a c t i o n i.e., w h e n we consider t h e situation far away f r o m equilibrium t o the left side, eqn. (62) o n using eqn. (56) yields t h e marginal stability condition given b y e q n . (58). Acknowledgment
= (ax
iir + by) ax - (bx + cy) at at
(7)
Comparison of eqns. (6) and (7) shows that the Eulerian derivative of 6%is just its time derivative. Now, we may evaluate s'6 explicitly, differentiating both sides of eqn. (3) with respect to time. We get
Setting Ae = Ar, Au = 60 and using relations (2),we obtain
+
~/ii~i'6.)~ = 6T-'ar6e 6(pT-')at6u Far multicomponent system, s ' 6 can be written as
(9)
O n e of us (R.P.R.) is t h a n k f u l t o Prof. I. Prigogine a n d Prof. G. Nicolis for constructive discussions d u r i n g h i s s t a y at t h e Universite L i b r e d e Bruxelles.
Thus, the time derivative of 6's for a multicomponent system can be expressed as folluws
Appendix I Here we intend tu show that 6?S is a Lyapounov function. We first pruceed to show that, 1) I t is a negative definite function. 2) Its Eulerian derivative is just the time derivative of the functiun. We consider, for simplicity, a one component system. Assuming I c d equilibrium, the specific entropy would be written as s = s(e, u )
(1)
where e and u are specific energy and specific volume. Furthermore, we have the following relations from equilibrium theory
where p is the pressure. T h e second order differential of s, obtained by using aTaylor expansion of the increment As, would be given by
Using eqn. (3) and the standard thermodynamic relations, we obtain
where all the terms have been defined earher. Since all the terms in the bracket are positive definite, d's is a negative definite quantity.:' We now proceed to prove that 62s is a homogeneous function of degree two. According to Euler's theorem, a function fix, y) will be a homogeneous function of degree n if it satisfies the following conditiun,
'har6% = 6T-'at6e
+ 6(pT-L)ar6u- 2: 6(fi.,T-')&6NY
(11)
7
Deducing the stability condition is straightforward using the Lyapounov stability criteria. It follows from Lyapounov theorem that the nonequilibrium state will be stable if
a, (6zs)3 o (12) since 6?s is negative. Equation (12) expresses the stability criteria in a local form which is no longer appmpriate for cases where boundary conditions are taken into account. T h e general stability criteria in a global form can be ubtained by integratingeqn. (12)over the whole volume of the system. The final stability condition in the global I'urm is given by r ~ , < c t h e r n i t h6-.5 < t u h e n 5 d,nt.tr> t h c g l ~ ~ Lm. ~t ln e p ~ . w w r . :-i I. n r ; ~ t ~ ~ r J ~ ~ it: ~k:~i AIVHL I)?
Appendix 2 We shall obtain here a n explicit relationship between at6Zsand virtual variation in fluxes and forces for a simple case when diffusion and chemical reactions are absent. In other words, we consider thermal stability and keep 6" and 6N, equal to zero. Thus eqn. (11) reduces to
liza162~ = 6T-'at6e
(13)
Far the entire volume V, at62Sis given by
J I ~ = ~lii.rpat62~ ~ ~ d~v s
Equation (2) can be written in a simplified manner as follows f = ax2
+ 2bxy + cy2
where
pa,& = - div. 6Jc where J,, is heat flow.
For an isolated system entropy has to be maximum at equilibrium. AS a result the first-order differential of entropy would vanish and the second-order differential, 62s.would be negative definite.
and
Hence, it follows that
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Journal of Chemical Education
6s = 0 and 6% < 0
From eqns. (14)and (151, we obtain 'har6%S= - J ' d T 1 div. 6 J c d V
(16)
Equation (16) can be transibrmed intc the fblhwing form4 '/zdt62S = -JdV div. 6 T - l 6 J ~+ J d V b J ~ g r a d .(6T-')
a,6p, = d i v . 6J,
(17)
Equation (17) can be further simplified using the Gailbs divergence theorem which relates the integral . I I;., Kerhgue. 1%.H.. J. ('hem S i r . 3 3 , I:lfi'l 1IY40). 1121 Mwre. W. .I., "l'hyiical Chemirtry," Lungman8 Gmen and Co.. Ltd., London, 1962, P. 258.
(1:Il Stevenr, H..Chapman and Hall LLd. and Science I'aperliaekr TOPPBD Company. L t d , T ~ k y o 1971, , yp. 8:i-84. 1141 .lari,hr. 1' W. M..snd Whitehead. H. M.. C h ~ mR e u , 69,551 119691. 1151 Hrleiencr lxl,p.:l45. (161 Ehat,A. A , m d I'aannn, R. (:."Kinei,rzand Mechanirm,"dohn WileyandSms. I n c . NewYs&, 1961, p. 1%K u n i t ~ M., . and Ni,rthrc,p..J. H.,d 1;m Physinliim 19,991 I1Y:IBI. 1171 Relerence 141, p. ,!i". 1181 Reference id), p. 451. 1191 Niaiiir.G., Ado. in Chem Phys., 19, 25:!ilY71l.
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