Thermodynamic stability of elementary chemical reactions proceeding at finite rates revisited using Lyapunov function analysis

Energy 30 (2005) 897–913 www.elsevier.com/locate/energy

Thermodynamic stability of elementary chemical reactions proceeding at finite rates revisited using Lyapunov function analysis Chandrakant S. Burande, Anil A. Bhalekar
Department of Chemistry, Nagpur University, Amravati Road Campus, Nagpur 440 033, India

Abstract The thermodynamic stability of a few representative elementary chemical reactions proceeding at finite rates has been investigated using the recently proposed thermodynamic Lyapunov function and following the steps of Lyapunov’s second method (also termed as the direct method) of stability of motion. The thermodynamic Lyapunov function; Ls, used herein is the excess rate of entropy production in the thermodynamic perturbation space, which thereby inherits the dictates of the second law of thermodynamics. This Lyapunov function is not the same as the excess entropy rate that one encounters in thermodynamic (irreversible) literature. The model chemical conversions studied in this presentation are A þ B ! vx X and A þ B Ð mx X. For the sake of simplicity, the thermal effects of chemical reactions have been considered as not adding to the perturbation as our main aim was to demonstrate how one should use systematically the proposed thermodynamic Lyapunov function following the steps of Lyapunov’s second method of stability of motion. The domains of thermodynamic stability under the constantly acting small disturbances, thermodynamic asymptotic stability and thermodynamic instability in these model systems get established. # 2004 Elsevier Ltd. All rights reserved.

1. Introduction Lyapunov’s second method of stability of motion [1–3] has a generality element, which is very much akin to that of thermodynamics. That is, in the former method one identifies a sign definite Lyapunov function and then the differential equations of motion under investigation are used to compute the time rate of change of Lyapunov function in the perturbation space. The


Corresponding author. Tel.: +91-712-2500429; fax: +91-712-2224362. E-mail address: [email protected] (A.A. Bhalekar).

0360-5442/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2004.04.004

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behavior of the time rate of change of the Lyapunov function so obtained then provides the desired information, namely whether the motion (mechanical) under investigation is stable [1,2] or asymptotically stable [3] and/or stable under constantly acting small disturbances [3] or unstable [1–3]. Thus, in this direct method, there is no need to rigorously solve the differential equations of motion. Indeed, this latter course is adopted in Lyapunov’s fist method [1–3], wherein one tackles every case individually. This then allows the investigator to map the regions of stability and instability if any. In essence, Lyapunov’s second method involves effecting of a sufficiently small perturbation (in principle, beyond the ones those originate on account of natural fluctuations) and then observing the response in the corresponding perturbation space. In spite of such an elegance of Lyapunov’s second (direct) method, one hardly finds in the stability considerations of thermodynamics [4–14] the incorporation of the systematic steps of the former. This existing gap then prompted us to attend to this subject matter afresh. The first and the obvious task that pondered our minds was to see whether a single thermodynamic Lyapunov function that can be used for determining the stability of all thermodynamic (equilibrium or nonequilibrium) states could be identified or not. Precisely this very demand, in our opinion, remained a big detrimental in the development of a unified thermodynamic stability theory though, for example, the second differential of entropy and Gibbs free energy or De Donderian chemical affinity have been proposed as Lyapunov functions [6,7,10–14], of course, based on the dictates of the second law of thermodynamics. However, their utility as thermodynamic Lyapunov functions turned out to be considerably limited as they brought long with them an element of close to equilibrium in the case of spatially non-uniform systems or the so-called local equilibrium assumption remained a prerequisite. In order to remove these restrictions we have defined a thermodynamic Lyapunov function, Ls, in an appropriate thermodynamic space [15–17] using the entropy source strength function, rs, which is a positive definite quantity as per the second law of thermodynamics, appearing in Clausius–Duhem inequality [4–6] that reads as, q

ds þ div J s ¼ rs  0 dt

(1)

where q is the mass density, s is the per unit mass entropy, Js is the entropy flux density, and t is time. The terms involved in Eq. (1) are position and time dependent but for the sake of brevity this dependence has not been shown. We define Ls as,

Ls ¼
rs  r0s
> 0 (2) where r0s is the entropy source strength along the real trajectory which is under investigation and rs is that along the perturbed one. Notice that the definition of Ls of Eq. (2) accommodates equally well the cases in which by perturbation the entropy source strength increases or decreases (rs ?r0s ). Before proceeding further let us comprehend that the use of rs as an objective function to construct Ls has an inherent advantage. As has been recalled above rs is a guaranteed strict positive definite function by the second law of thermodynamics. Moreover, the thermal energy of the chemical reaction does not influence this property of rs. Therefore, there is no need to know at all whether the total entropy of the system is increasing or decreasing function during the course of the real trajectory, in the act of perturbation and during the evol-

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ution of the system after perturbation. That is, one is totally spared from worrying about the involved energetics of the chemical reaction and the behavior of the entropy function.1 Further, the coordinates of the perturbation space, ai ¼ jyi  y0i j, where yi and y0i are the coordinates of the motion along perturbed and real trajectories, respectively (in Lyapunov’s method, ais are defined as absolute values to deal with the situations beyond natural fluctuations), have their differential equation, dai ¼ fi ðt; a1 ; a2 ; . . . ; an Þ dt

ði ¼ 1; 2; . . . ; nÞ

(3)

and vanish only on the real trajectory. Notice that the definition of ais adopted herein includes both positive and negative perturbations of the variables and also saves us from considering by mistake the tangential space of the real trajectory as the perturbation space. However, this care has not been taken in earlier proposals [7,10–13]. Thus, our analysis of stability pertains strictly to the motion within the perturbation space. Other advantages of the above definition of ais are that we are led to the autonomous equations of motion (that is, one remains well within the perturbation space) and the question of bistable stability gets included as per the design of this apparatus if in given nonequilibrium situation both the positive and negative perturbations in coordinates of real trajectory are feasible. Thus, the trivial solution of Eq. (3) is, ai  0

ði ¼ 1; 2; . . . ; nÞ

(4)

which is the equation of unperturbed motion. Thus, by virtue of the second law of thermodynamics we have [5], rs ¼ rs ðt; a1 ; a2 ; . . . ; an Þ > 0

(5)

r0s ¼ rs ðt; 0; 0; . . . ; 0Þ > 0

(6)

and correspondingly, function Ls gets expressed as, Ls ¼ Ls ðt; a1 ; a2 ; . . . ; an Þ > 0

(7)

L0s ¼ Ls ðt; 0; 0; . . . ; 0Þ  0

(8)

that is, by definition Ls has a strict minimum at the origin. Therefore, from Eqs. (2) and (5)–(8), it is clear that the present proposal uses the excess rate of entropy production as the thermodynamic Lyapunov function, which of course is not the same as the excess entropy rate used by the Brussels school [10]. Moreover, one of the most important features of Ls is that it covers all variety of nonequilibrium situations irrespective of their nature. Now according to Lyapunov’s theory of stability of motion [1,2], if dLs @Ls X @Ls ¼ þ fi 0 dt @t @t i 1

(9)

In the case of equilibrium r0s ¼ 0 and hence by the act of perturbation the system attains a neighboring nonequilibrium state in which it is guaranteed that rs > 0 and during the evolution of the system after perturbation, we have drs =dt < 0.

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then the real trajectory, which is under scrutiny, is the stable one. On the other hand, if the lefthand side rate of Eq. (9) behaves as dLs b < 0 (10) dt where b is a positive number and vanishes only for ai ¼ 0 ði ¼ 1; 2; . . . ; nÞ then the asymptotic stability of motion is guaranteed. In addition to Eqs. (2) and (10), if the partial derivatives of Ls with respect to ai are all finite then the motion of real trajectory is stable under constantly acting small disturbances in accordance with Malkin’s theorem [3]. Our aim in this presentation is limited. We herein demonstrate somewhat in detail how the above-described apparatus deals with the question of thermodynamic stability. For this purpose, we have chosen the case of chemical reactions proceeding at finite rates in a spatially uniform system. That is, the only source of irreversibility that exits on the unperturbed trajectory is that due to the chemical conversions at finite rates. We further remain confined to the following situations, namely: 1. In the perturbation space, the only source of irreversibility that persists is that due to chemical conversions at finite rates. 2. The system in which the chemical conversion takes place remains spatially uniform even after perturbation. 3. Across the boundary of the system no temperature, pressure and concentration gradients exist and do not get generated on account of perturbations. 4. The chemical conversion may be endothermic or exothermic but the temperature of the system remains practically constant even after perturbation. The last named condition, of course, excludes the situations encountered due to the thermal effects in chemical reactors. The influence of this aspect on thermodynamic stability is under investigation and would be discussed in a future presentation. Indeed, to allow in our analysis the perturbation on account of thermal effects due to the non-zero enthalpy of reaction we need to retain corresponding perturbations in chemical affinities and the rate constants that require more involved analysis. Further, we are considering the following two representative chemical reactions, namely: k1

A þ B ! vx X k2

A þ B Ð vx X k2

(11) (12)

where vx (=1 or 2) is the stoichiometric coefficient of X, A and B are the reactants, X is the product and k1, k2 and k2 are the respective rate constants. 2. Thermodynamic space For chemically reacting systems considered herein, the traditional Gibbs relation [4] is an appropriate one. The corresponding time rate form of the traditional Gibbs relation [5,18]

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reads as, X dnk dS dU dV ¼ T 1 þ pT 1  T 1 lk dt dt dt dt k

(13)

where S is the entropy, T is the temperature, p is the pressure, U is the internal energy, V is the volume, lk is the chemical potential per mole of the component k and nks are the mole numbers. From Dalton’s law, we have [7] dnk ¼ vk dn

(14)

where n is the extent of advancement of the chemical reaction and vks are the stoichiometric coefficients taken positive for products and negative for reactants. Further, the standard expression for chemical affinity, A is [7] X (15) A ¼  lk v k k

The substitution of Eqs. (14) and (15) into Eq. (13) gives, dS dU dV dn ¼ T 1 þ pT 1 þ AT 1 dt dt dt dt

(16)

We recall that Eq. (16) is the De Donderian version of the traditional Gibbs relation in time rate form. Thus, in the absence of irreversibility in thermal and mechanical interactions of the system with its surroundings, the entropy source reads as, RS ¼

A dn >0 T dt

(17)

where di S RS ¼ ¼ dt

ð rs dV

(18)

V

Notice that Eq. (17) conforms to the dictates of the second law of thermodynamics that is the rate of entropy production is never a negative quantity [19] (cf. refer to the description below Eq. (2)). Also notice that one is using only the entropy production function in Eq. (17) that remains positive definite whether the chemical reaction happens to be exothermic or endothermic. This latter source or sink of energy in the view of constraints considered herein will determine the magnitude and sign of the rate of exchange of entropy by the system with its surroundings, which of course will add to in determining whether the total entropy of the system increases or decreases. Further, whenever it is a local level description the subscript s will be used to denote the thermodynamic Lyaponov function and also to the entropy source strength while the subscript S will be used in the case of a global level description.

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In the following sections, we present the details of our analysis of thermodynamic stability of the above chosen chemical reactions at constant T and p in various possible situations before and after the effected perturbation.

3. System not in chemical equilibrium before perturbation 3.1. Relevant aspects of stoichiometry and chemical kinetics when mole number of only one of the reactants is perturbed From stoichiometry and chemical kinetics [20], we obtain for the rates of consumption and production of reactants and product for the chemical reactions of Eqs. (11) and (12) the following relation, dni dnA dnB 1 dnx ¼ ¼ ¼ vx dt dt dt dt

ði; vx ¼ 1; 2Þ

(19)

where the subscript to n match with that for the respective forward rate constants. The reaction rates for reactions of Eqs. (11) and (12) are given by chemical kinetics for the unperturbed case (that is, on the real trajectory) as, dn01 ¼ k1 n0A n0B > 0 dt

(20)

 vx dn02 ¼ k2 n0A n0B  k2 n0X > 0 dt

(21)

where superscript 0 indicates the quantities on the unperturbed trajectory (it should not be confused with its use in chemical kinetics for initial mole numbers at the time of initiation of the chemical reaction). The positive definiteness of Eqs. (20) and (21) asserts that we are considering in particular those cases and the stages of reaction where the overall reaction proceeds in the forward direction. Now let us consider the case when only the concentration of A is perturbed by sufficiently small amount say, dnA, we then have,

dn ¼ jn  n0 j > 0; dnA ¼
nA  n0A
> 0 (22) On the perturbed trajectory, we have, dn1 ¼ k1 nA n0B > 0 dt

(23)

 vx dn2 ¼ k2 nA n0B  k2 n0X > 0 dt

(24)

As dnA by definition is sufficiently small, the positive rates of Eqs. (20) and (21) are still retained on the perturbed trajectory as shown in Eqs. (23) and (24). Of course, as such in Eqs. (23) and (24), we have excluded very close to equilibrium situations.

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Now on subtracting Eqs. (20) and (21), respectively, from Eqs. (23) and (24), we obtain,       d nA  n0A dn dn0 d n  n0 ¼ ¼ kn0B nA  n0A  ¼ (25) dt dt dt dt as the stoichiometry prescribes that, dnA ¼ dn;

dn0A ¼ dn0

(26)

Notice that we have dropped the subscripts to rate constant and n as in both the cases the form of the right-hand side expression for dðn  n0 Þ=dt obtained is the same namely k1 n0B ðnA  n0A Þ and k2 n0B ðnA  n0A Þ. The same practice will be adhered to in all the cases considered in this paper with only a few exceptions where clarity of presentation demands. Thus, we have from Eq. (25),

d
nA  n0A
¼ kn0B dnA b < 0 (27) dt


dðn  n0 Þ

¼ kn0 dnA  b > 0
(28) B

dt where b is a positive number and as evident from preceding equation it vanishes only at the origin, that is on the unperturbed trajectory. Also we do have, @n0B dn0B ¼ ; @t dt

@n0 dn0 ¼ ; etc: @t dt

(29)

as on the real trajectory ðnA  n0A Þ ¼ 0 ¼ ðn  n0 Þ. This can be appreciated as follows: dn @n @n dðdnÞ ¼ þ dt @t @ðdnÞ dt as dn ¼ 0 on the real trajectory the second term on the right-hand side of the preceding expression vanishes that is as it should be n ¼ n0 on the real trajectory. The chemical affinities, on assuming the reaction mixture as an ideal gas mixture, is given by thermodynamics on unperturbed trajectory as, n0A n0B  v þ RT ln A0j ¼ A j n0X x

! ðj ¼ 1; 2Þ

(30)

where A0j are the chemical affinities on the unperturbed trajectory, A j are the chemical affinities in the standard reference state, and R is the gas constant. As only the concentration of A is perturbed by sufficiently small amount, dnA, the chemical affinity on the perturbed trajectory in

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all the above four cases is given by,  dnA 0 A ¼ A þ RT ln 1 þ 0 nA

(31)

However, by definition, we have dnA 5 n0A the logarithmic term of Eq. (31) remains negligibly small compared to A0 and hence for all practical purposes, we have ! n0A n0B

ðj ¼ 1; 2Þ (32) Aj  Aj þ RT ln  0 vx ¼ A0j nX The local time derivative in thermodynamic perturbation space of Eq. (30) is obtained as,   0 @ðA0j =TÞ 1 1 vx @ nB j ¼R 0 þ 0 þ 0 < 0 ðj ¼ 1; 2Þ (33) @t @t nA nB nX wherein the dictates of stoichiometry of the reactions have been incorporated. Notice that,

@ðA j Þ=@t remains identically equal to zero as Aj ¼ Aj ðT; pÞ. 3.1.1. Thermodynamic stability The rate of entropy production on the perturbed trajectory in the present cases is given by Eq. (17) but in view of Eq. (32) it specifically reads as RS ¼

A0 dn >0 T dt

(34)

and that on the unperturbed trajectory we have, R0S ¼

A0 dn0 >0 T dt

(35)

As we are not working at the local level the global level thermodynamic Lyapunov function, LS , gets defined as,

(36) LS ¼
RS  R0S
> 0 Next on combining Eqs. (34) and (35) with Eq. (36), we obtain,

0
A dn A0 dn0

>0
LS ¼
 T dt T dt


(37)

which, on using Eq. (28) and in view of A0 =T being a positive quantity, further simplifies to,

A0

dðn  n0 Þ

A0 0 ¼k n dnA  b0 > 0 (38) LS ¼

dt T T B where b0 is a positive number and vanishes only on the real trajectory because dnA vanishes only

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on the unperturbed trajectory. Thus, we have LS ¼ LS ðt; dnA Þ

(39)

The total time derivative of LS reads as dLS @LS @LS dðdnA Þ ¼ þ dt dt @t @ðdnA Þ

(40)

In thermodynamic perturbation space, the local time derivative of LS is obtained from Eq. (38) as, @LS @ðA0 =TÞ 0 A0 @n0 knB dnA þ ¼ kdnA B @t @t T @t and on incorporating Eq. (33) into Eq. (41), we have,     0 @ n0B j A0j @ðLS Þ 1 1 vx @ nB j 0 ¼R 0 þ 0 þ 0 knB dnA þ kdnA <0 @t @t T @t nA nB nX

(41)

ðj; vx ¼ 1; 2Þ

(42)

whereas the gradient of LS is obtained as,  A0 0 @LS A0 @  0 ¼ knB dnA ¼ knB > 0 and finite @ðdnA Þ T @ðdnA Þ T

(43)

and on combining Eqs. (27) and (43) we have, @LS dðdnA Þ A0 2  0 2 ¼ k nB dnA < 0 dt @ðdnA Þ T

(44)

Now on substitution of Eqs. (42) and (44) into Eq. (40), we obtain,  0 " #  0 @ nB j A0j  2 A dðLS Þj 1 1 v j x dnA ¼ k n0B R 0 þ 0 þ 0 þ  k2 n0B dnA b00 < 0 dt T @t T nA nB nX ð j; vx ¼ 1; 2Þ

ð45Þ

Thus, Eq. (45) establishes that the process of chemical reactions proceedings at finite rates that studied herein are asymptotically stable (cf. Eq. (10)). Further, in view of the gradient of LS being finite as per Eq. (43) this process is also stable under constantly acting small disturbances as per Malkin’s theorem [3]. The same conclusions follow if the mole numbers of B are perturbed instead. 3.2. Thermodynamic stability when mole numbers of both the reactants are perturbed 3.2.1. The sign of both the perturbations being same In this case, the relevant expressions for perturbation are:





dn ¼
n  n0
> 0; dnA ¼
nA  n0A
> 0; dnB ¼
nB  n0B
> 0

(46)

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Correspondingly, the rates of reaction on the perturbed trajectory read as, dn1 ¼ k1 nA nB > 0 dt

(47)

 vx dn2 ¼ k2 nA nB  k2 n0x > 0 dt

(48)

The rate of change of perturbation in all the four cases is then given by,     d nA  n0A dn dn0 dðn  n0 Þ  k n0A dnB þ n0B dnA  ¼ ¼ dt dt dt dt Hence, herein the rate of change of perturbation is obtained as,
  djnA  n0A
ddnA ¼ ¼ k n0A dnB þ n0B dnA b < 0 dt dt


dðn  n0 Þ
 
¼ k n0 dnB þ n0 dnA  b > 0
A B

dt The chemical affinity on the perturbed trajectory reads as,   dnA dnB 0 A ¼ A þ RT ln 1 þ 0 þ RT ln 1 þ 0  A0 nA nB On substitution of Eq. (51) into Eq. (37), the expression for LS reads as

 A0

dðn  n0 Þ

A0  0 ¼ k nB dnA þ n0A dnB  b0 > 0 LS ¼

dt T T

(49)

(50) (51)

(52)

(53)

and hence we have, LS ¼ LS ðt; dnA ; dnB Þ

(54)

The total time derivative of LS in this case reads as dLS @LS @LS dðdnA Þ @LS dðdnB Þ þ ¼ þ dt dt dt @t @ðdnA Þ @ðdnB Þ

(55)

The local time derivative in the thermodynamic perturbation space of LS keeping dnA and dnB constant and using Eq. (33) reads as  0  @ n0  0 @ nB j A  0  @ðLS Þj 1 1 v B j j x þ 0 þ 0 ¼ kR nB dnA þ n0A dnB þ kðdnA þ dnB Þ <0 0 @t @t T @t nA nB nX ðj; vx ¼ 1; 2Þ

ð56Þ

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As we see, the right-hand side of the preceding equation @ðLS Þj =@t vanishes only at the origin. The gradients of LS in thermodynamic perturbation space are given as @LS A0 0 ¼ knB > 0 and finite @ðdnA Þ T

(57)

@LS A0 0 ¼ knA > 0 and finite @ðdnB Þ T

(58)

Therefore, in this case we have,

@LS @LS dðdnA Þ þ dt @ðdnA Þ @ðdnB Þ   A0 2  0 ¼ ð59Þ k nA þ n0B n0A dnB þ n0B dnA < 0 T which too vanishes only at the origin. Thus, we learn from Eqs. (56) and (59) that each term on the right-hand side of Eq. (55) is negative definite and vanishes only at the origin, therefore, we have @LS dðdnA Þ @LS dðdnB Þ þ ¼ dt dt @ðdnA Þ @ðdnB Þ



dLS b < 0 dt

(60)

and hence the chemical process under consideration is asymptotically stable and in view of Eqs. (57) and (58) it is also stable under the constantly acting small disturbances. 3.2.2. The two perturbations being opposite in sign 1. Let us first consider that n0A 6¼ n0B and dnA 6¼ dnB . In this case too the relevant expressions for perturbation remain the same as given in Eq. (46). However the rates of reaction on the perturbed trajectory read as    dn1 (61) ¼ k1 n0A  dnA n0B  dnB > 0 dt     vx dn2 ¼ k2 n0A  dnA n0B  dnB  k2 n0X > 0 (62) dt Herein, on using Eqs. (20), (21), (61) and (62) we obtain,       d nA  n0A d nB  n0B dn dn0 dðn  n0 Þ ¼  k n0B dnA  n0A dnB  ¼ ¼ dt dt dt dt dt

(63)

Therefore, we now have for the rate of change of perturbation the following expressions, namely:   ddnB ddnA ¼ ¼ k n0B dnA  n0A dnB 7 0 
dt 0
dt
dðn  n Þ



¼ k
n0 dnA  n0 dnB
 b > 0
B A

dt

(64) (65)

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The expression for LS is obtained as


A0

dðn  n0 Þ

A0

0 ¼ k nB dnA  n0A dnB
 b0 > 0 LS ¼

dt T T

(66)

and hence we have the functional dependence of LS as that in Eq. (54). However, the chemical affinity on the perturbed trajectory in this case too approximates to A0 as described in Eq. (52). The total time derivative of LS reads as that given in Eq. (55). For the local time derivative in thermodynamic perturbation space, on the lines of Eq. (56), we obtain the following expressions, namely,    @ n0  @ n0B j
0
1 A0j @ðLS Þj 1 v B j x 0 ¼ kR
nB dnA  nA dnB
0 þ 0 þ 0 þ kðdnA  dnB Þ 70 @t @t T @t nA nB nX ðj; vx ¼ 1; 2Þ

ð67Þ

which vanishes only at the origin. The gradients of LS in the thermodynamic perturbation space read as @LS A0 0 ¼ knB > 0 and finite @ðdnA Þ T

for n0B dnA > n0A dnB

(68)

@LS A0 0 ¼1 knB < 0 and finite @ðdnA Þ T

for n0B dnA < n0A dnB

(69)

@LS A0 0 ¼1 knA < 0 and finite @ðdnB Þ T

for n0B dnA > n0A dnB

(70)

@LS A0 0 ¼ knA > 0 and finite @ðdnB Þ T

for n0B dnA < n0A dnB

(71)

Therefore, for the case of n0B dnA > n0A dnB we have,   @LS dðdnA Þ @LS dðdnB Þ A0 2  0 þ ¼ k nA þ n0B n0B dnA  n0A dnB < 0 dt dt @ðdnA Þ @ðdnB Þ T

(72)

which vanishes only at the origin and for n0B dnA < n0A dnB , we obtain   @LS dðdnA Þ @LS dðdnB Þ A0 2  0 þ ¼ k nA þ n0B n0A dnB  n0B dnA < 0 dt dt @ðdnA Þ @ðdnB Þ T

(73)

which too vanishes only at the origin. The total time derivative of LS is then analyzed by substitution of the relevant expressions from Eqs. (67)–(73) into Eq. (55). With this exercise, we conclude that as the gradients of LS gives in Eqs. (68)–(71) are all finite the asymptotic stability and the stability under constantly acting small disturbances are guaranteed in the case when @LS =@t < 0. Further, one can easily verify from Eq. (67) that @LS =@t > 0 can be realized only if dnA 5 dnB , which in turn demands that n0B 4n0A . This then makes ðn0A dnB  n0B dnA Þ  0 and

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hence essentially we have,   @ n0B j dLS @LS A0 ðdnA  dnB Þ  k >0 dt @t T @t Thus, the process under investigation remains unstable. Thus, it is not advisable to run such processes with a large excess concentration of one of the reactants over the other. Whereas @LS =@t < 0 can be guaranteed simply by dnA > dnB which indeed is a practical reasonable demand (cf. Eq. (67)). 2. However, when n0A ¼ n0B and obviously dnA  dnB , then instead of Eq. (63) we have,     d nA  n0A d nB  n0B dn dn0 dðn  n0 Þ ¼  2kn0A dnA  ¼ ¼ (74) dt dt dt dt dt and the rate of change of perturbation now reads as, ddnB ddnA ¼ ¼ 2kn0A dnA < 0 dt dt


dðn  n0 Þ

¼ 2kn0 dnA  b > 0
A

dt 

The expression for LS is obtained as

A0

dðn  n0 Þ

A0 0 ¼2 knA dnA  b0 > 0 LS ¼

dt T T

(75) (76)

(77)

and hence we have the functional dependence of LS as that depicted in Eq. (39). However, the chemical affinity on the perturbed trajectory in this case too approximates to A0 as described in Eq. (52). The total time derivative of LS reads as that given in Eq. (40). But the local time derivative in thermodynamic perturbation space is now given as, 0  @ n0  @ðLS Þj A 2 vx A j 0 ¼ 2kdnA þ RnA 0 þ 0 < 0 ð j; vx ¼ 1; 2Þ (78) @t T @t nA nX The gradients of LS in the thermodynamic perturbation space now reads as, @LS A0 0 ¼2 knA > 0 and finite @ðdnA Þ T In view of the above, the total time derivative of LS now reads as 0  0 @nA dLS @LS @LS dðdnA Þ A 2 vx 0 ¼ 2kdnA ¼ þ þ RnA 0 þ 0 dt dt @t @ðdnA Þ T @t nA nX 0 A 2  0 2 4 k nA dnA b00 < 0 T

(79)

ð80Þ

Thus, more stringently asymptotic stability and the stability under constantly acting small disturbance are guaranteed.

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3.3. Thermodynamic stability when mole number of product is perturbed As the rate of reaction is influenced by the mole numbers of product in the case of Eq. (12) but not Eq. (11) and hence we are left to consider only the two cases of the chemical reaction of Eq. (12). Thus, following the above-described steps in this case we obtain,


dðn  n0 Þ
 
¼ 2k2 n0 p dnX  b > 0
(81) X

dt  p dðdnX Þ ¼ 2k2 n0X dnX b < 0 (82) dt  vx dnX 0 (83)  A0 A ¼ A þ RT ln 1  0 nX  @ðA0 =TÞ 1 1 vx @n0X (84) ¼ R 0 þ 0 þ 0 < 0 ðvx ¼ 1; 2Þ @t nA nB nX @t

 p A0

dðn  n0 Þ

A0 ¼2 k2 n0X ðdnX Þ  b0 > 0 LS ¼ (85)

dt T T where vx ¼ p þ 1. The preceding equations then give, 

@LS 1 1 vx A0  0 p1 @n0 0 ðdnX Þ X 7 0 ¼ 2k2 RnX 0 þ 0 þ 0  p nX @t T @t nA nB nX

(86)

 p A0 @LS ¼ 2k2 n0X > 0 and finite @ðdnX Þ T

(87)

 2p @LS dðdnX Þ A0 ¼ 4ðk2 Þ2 n0X ðdnX Þ <0 @ðdnX Þ dt T

(88)

Now the conclusions about stability follow as under. 1. For vx ¼ 1, we have p ¼ 0 and hence Eq. (86) reduces to,  @LS 1 1 vx @n0 ¼ 2k2 R 0 þ 0 þ 0 ðdnX Þ X < 0 @t @t nA nB nX

(89)

whereas from Eqs. (87) and (88), we obtain @LS A0 ¼ 2k2 > 0 and finite @ðdnX Þ T

(90)

@LS dðdnX Þ A0 ¼ 4ðk2 Þ2 ðdnX Þ <0 @ðdnX Þ dt T

(91)

Therefore, we strictly have, dLS b00 < 0 dt

(92)

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That is, asymptotic stability and stability under constantly acting small disturbances is guaranteed. 2. For vx ¼ 2, we have p ¼ 1 and hence Eq. (86) reads as 

@LS @n0X 1 1 2 A0 0 ¼ 2k2 ðdnX Þ RnX 0 þ 0 þ 0  70 (93) @t @t T nA nB nX Thus, asymptotic stability and stability under constantly acting small disturbances is guaranteed as long as the following condition is met, namely:  0 nX n0X 0 (94) A RT 0 þ 0 þ 2 nA nB Of course, the validity of this assertion depends on the instantaneous absolute value of A0 at a given stage of the progress of chemical reaction where the perturbation is effected. However, when  0 nX n0X 0 (95) A > RT 0 þ 0 þ 2 nA nB that means @LS =@t > 0 but still the stability gets guaranteed provided the following condition is followed, namely:  0 0 h  0 2 i n n0X @nX 0 0 0 þ þ 2 k n n  3k RT X  A (96) 2 A B 2 nX @t n0A n0B Thus, the possibility of instability to get encountered remains considerably diminished.

4. Final remarks This paper deals with the thorough elaboration of the systematic use of recently proposed thermodynamic Lyapunov function in investigation of thermodynamic stability of the process of chemical conversions proceeding at finite rates. To grasp it in finer details, we have chosen very elementary model chemical reactions and investigated almost all possibilities of perturbations of the real trajectories. The consistency of the framework is reflected by the conclusions that result are almost all on the expected lines. However, It is amusing to learn that there are a few situations wherein the thermodynamic stability is not guaranteed but then those situations are of little practical use. The thermodynamic stability, using the present framework, of chemical equilibrium and chemical relaxation has been already described elsewhere [21] wherein autocatalytic reactions were analyzed. Hence, we have not presented the same for the chemical reactions considered herein. However, one can easily prove for the cases of chemical equilibrium the asymptotic stability and stability under constantly acting small disturbances. On comparing it is revealed that the sufficiently small perturbations in mole numbers that are required to produce a significant perturbation from a chemical equilibrium state are relatively larger than those considered while perturbing a chemical reaction proceeding at a finite rate, in spite of this notice

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that in the former cases asymptotic stability and the stability under the constantly acting small disturbances remain guaranteed. Now it would be of interest to investigate thermodynamic stability of chemical reactions proceeding at finite rates against the perturbations in the mole numbers of the size involved in the case of realistic perturbation of chemical equilibrium. And also it is of interest to know how thermodynamic stability will behave if perturbation is effected in T (its source lies in the non-zero enthalpy of the chemical reaction coupled with the operational problems which abundantly are associated with the heat exchangers) and/or p (because of a finite volume change of the reaction in majority of cases). In the event of this the perturbations in chemical affinity and the rate constants of reaction need to be retained in thermodynamic Lyapunov function, LS or Ls. These aspects are under investigation. Recall that, we have already spelled our previously [15–17] that no theory of thermodynamic stability got evolved prior to the present proposal in which Lyapunov’s direct method has been rigorously followed. Indeed, some attempts were made by earlier workers but they had to resort to certain hypotheses (which Lavenda [9], Keizer and Fox [22], Landsberg [23] and one of us [15–17] have questioned) and they themselves acknowledge that their theories are tenable in the situations close to equilibrium [4–6,8–14] and hence no attempt has been made herein to compare our results with those ones if could be arrived at for the presently investigated cases using any one of the earlier theories. In contrast, it gets lucidly revealed from the above discussion that in principle there is no restriction at all for the application of the presently proposed thermodynamic Lyapunov function along with its manipulation in systematic steps on the lines of Lyapunov’s direct method of stability of motion in terms of whether the system happens to be close to or far away from equilibrium or whether the process falls in the linear or nonlinear regime (from the thermodynamic or from the constitutive theory point of views). The main demand is to first identify the correct expression of the entropy source strength for the process to be investigated for thermodynamic stability, which indeed implies the identification of the appropriate thermodynamic space, which in certain cases could be a formidable task [24].

Acknowledgements One of us (A.A.B.) is thankful to CSIR (New Delhi) and DAE-BRNS (Mumbai) for their partial financial support to this work. We gratefully acknowledge the critical and constructive suggestions of the anonymous referees. References [1] LaSalle J, Lefschetz S. Stability by Lyapunov’s direct method with applications. New York: Academic Press; 1961; Chetayev NG. The stability of motion. Pergamon Press; 1961 [Nadler M, Trans.]; Elsgolts L. Differential equations and the calculus of variations. Moscow: Mir Publications; 1970 [Yankuvsky G, Trans.]. [2] Malkin IG. Theory of stability of motion. ACE-tr-3352 physics and mathematics. Washington: US Atomic Energy Commission; 1952. [3] Malkin IG. Stability and dynamic systems. Translation Ser. 1 vol. 5. Providence (RI): American Mathematical Society; 1962. [4] Callen HB. Thermodynamics. New York: Wiley; 1960. [5] De Groot SR, Mazur P. Nonequilibrium thermodynamics. Amsterdam: North-Holland; 1962.

C.S. Burande, A.A. Bhalekar / Energy 30 (2005) 897–913 [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

[17]

[18] [19] [20] [21]

[22] [23] [24]

913

Haase R. Thermodynamics of irreversible processes. Reading (MA): Addison-Wesley; 1969. Prigogine I, Defay R. Chemical thermodynamics. London: Longmans Green; 1954 [Everett DH, Trans.]. Schlo¨gl F. On stability of steady states. Z Phys 1971;243:303–13. Lavenda BH. Thermodynamics of irreversible processes. London: The Macmillan Press; 1978. Glansdorff P, Prigogine I. Thermodynamic theory of structure, stability and fluctuations. London: Wiley; 1971. Jou D, Casas-Va´zquez J, Lebon G. Extended irreversible thermodynamics. Berlin: Springer-Verlag; 1996. Silhavy M. The mechanics and thermodynamics of continuous media. Berlin: Springer-Verlag; 1997. Stratonovich RL. Nonlinear nonequilibrium thermodynamics. II. Advanced theory. Berlin: Springer-Verlag; 1994. Beck C, Schlo¨gl F. Thermodynamics of chaotic systems: an introduction. Cambridge: Cambridge University Press; 1993. Bhalekar AA. On a comprehensive thermodynamic theory of stability of irreversible processes. A brief introduction. Far East J Appl Math 2001;5(2):199–210. Bhalekar AA. Comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP). I. The details of a new theory based on Lyapunov’s direct method of stability of motion and the second law of thermodynamics. Far East J Appl Math 2001;5(3):381–96. Bhalekar AA. Comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP). II. A study of thermodynamic stability of equilibrium and nonequilibrium stationary states. Far East J Appl Math 2001;5(3):397–416. Bhalekar AA. On the irreversible thermodynamic framework for closed systems consisting of chemically reactive components. Asian J Chem 2000;12:433–44. Muschik W. Formulations of the second law—recent developments. J Phys Chem Solids 1988;49:709–20. Laidler KJ. Chemical kinetics. New York: McGraw-Hill; 1961. Burande CS, Bhalekar AA. Thermodynamic stability of some elementary chemical reactions investigated within the framework of comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP). J Indian Chem Soc 2003;80:583–96. Keizer J, Fox RF. Qualms regarding the range of validity of the Glansdorff-Prigogine criterion for Stability of non-equilibrium states. Proc Natl Acad Sci USA 1974;71(1):192–6. Landsberg PT. The fourth law of thermodynamics. Nature 1972;238:229–31. Casas-Va´zquez J, Jou D. Temperature in non-equilibrium states: a review of open problems and current proposals. Rep Prog Phys 2003;66:1937–2023.