Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 3(ISC-2013), 74-78 (2014) Res. J. Recent. Sci. International Science Congress Association 74 Stability of Autocatalytic Reaction by Lyapunov Function Analysis Dethe Pragati, Burande Chandrakant, Burande Bharati and Megha Sawangikar4 Priyadarshini College of Engineering, Nagpur, INDIAVilasrao Deshmukh College of Engineering and Technology, Mouda, Nagpur, INDIA Priyadarshini Indira Gandhi College of Engineering, Nagpur, INDIA Datta Meghe College of Engineering, Technology and Research, Wardha, INDIAAvailable online at: www.isca.in, www.isca.me Received 26th September 2013, revised 14th January 2014, accepted 27th February 2014 AbstractThermodynamic stability of autocatalytic reaction by Lyapunov function analysis using the framework of CTTSIP has been investigated in this paper. The stability analysis of autocatalytic reactions is complex in mechanism. The feedback mechanism in autocatalytic reaction plays vital role in some biological and industrial processes. This autocatalytic feedback mechanism also leads to the oscillatory chemical reaction. Keywords: Irreversible thermodynamics, stability, autocatalytic reaction, Lyapunov analysis. Introduction Lyapunov’s direct method of stability of motion1, 2 also known as second method aptly deals the necessary aspects of stability determination of dynamic systems. It is found that Lyapunov’s theory of stability of motion inherits thermodynamic type of generality. The main ingredient of Lyapunov’s direct method of stability of motion is the identification of Lyapunov function. The sign of the function and its time derivative then tell us whether the dynamic system is stable or unstable. The gist this method is described below. Let the given differential equations in the perturbation space be ( ) ()()12 ,,,.......,1,2,......, n Xtxxxin ddd== (1) where 0 iii xxx d =- is the coordinate in perturbation space, i x is the perturbation coordinate and 0 i x is the coordinate of unperturbed motion (real trajectory). On real trajectory, coordinate i x d vanishes, that is 0. iiixxx =-º (2) Similarly, on real trajectory ( ) ,0,0,........,00. Xt = (3) Let ( ) 12,,,........, n Vtxxx dddbe a differentiable Lyapunov function such that, ( ) 12 ,,,........,0. Vtxxxddd  (4) Note that V is the function of only time, and coordinate, i x d . The function, V has a strict minimum at the origin (on real trajectory), that is ( ) ,0,0,........,00. Vt = (5) The total time derivative of Lyapunov function, V is then written as iii i dVVV VX dttx  ¶¶==+  ¶¶  (6) According to Lyapunov’s direct method of stability of motion, the system is stable and asymptotically stable if ()12 ,,,........,0,0 VtxxxVdddb �£- (7) and ()12 ,,,........,0,0 VtxxxVdddb ³&#x-505;.18; (8) where b is positive definite. The system under investigation is unstable if ()12 ,,,........,0,0 VtxxxVdddb �³� (9) and ()12 ,,,........,0,0. VtxxxVdddb £- (10) The main feature of Lyapunov’s direct method of stability of motion is that it directly assigns the stability of motion without actual finding the solution of the function. Thus, because of its simplicity, this theory finds wider applications in the fields of science and technology. Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 3(ISC-2013), 74-78 (2014) Res. J. Recent. Sci. International Science Congress Association 75 Representative reaction In this paper we discussed the thermodynamic stability of autocatalytic reactions using Lyapunov’s direct method of stability of motion. We consider the selected autocatalytic reaction proceeding at finite rates, at constant and . The representative reaction3, 4 of this category is given below. 2 kk AXX -+   (11) where and are the chemical species, 1 k is the rate constants of forward reactions and 1 k - is that of reverse reaction. Thermodynamic space For applying Lyapunov’s direct method of stability of motion to a real process one first needs to choose an appropriate thermodynamic space. For chemically reacting spatially uniform closed systems, the traditional Gibbs relation gives an appropriate expression for rate of entropy production5, 6. The traditional Gibbs relation for a spatially uniform closed system undergoing chemical conversions at finite rates read as, 11 k kk dn dsdUpdV dtTdtTdtTdt =+- (12) where is the entropy, is the temperature, is the pressure, is the internal energy, is the volume, k m is the chemical potential per mole of the component and ' k n s are the mole numbers. Notice that the first and second terms on the right-hand-side of Equation (12) are generated due to the thermal and mechanical interactions of the system with its surrounding and the last term originates due to the occurrence of a chemical reaction proceeding at finite rate. Further if the irreversibility is only due to a single chemical reaction occurring at a finite rate, then from Dalton’s law, we have , kk dnd nx (13) where x is the extent of advancement of the chemical reaction and ' k n s are the stoichiometric coefficients and by convention are taken positive for the products and negative for the reactants. Further, the standard expression for chemical affinity,  , is . kk k mn =   (14) On substitution of Equation(13) and (14) in Equation(12) gives 1 dsdUpdVAd dtTdtTdtTdt x =++ (15) Equation (15) is the De Donderian equation9,10. Thus, in the absence of irreversibility in thermal and mechanical interactions, the rate of entropy production, S S , due to a single chemical reaction reads as 0. Tdt S=�  (16) Equation(16) conforms the positive definite rate of entropy production for any non-equilibrium processes as per the second law of thermodynamics. Using the expression for rate of entropy production given in Equation(16), we now discuss the stability of autocatalytic chemical reaction at constant and in different physical situations. In stability analysis our first job is to identify Lyapunov function, S L . In this case we generate operative S L in following way: ( ) SS dd L TdtTdtTdt dx xdx dd=S==+ (17) Perturbation before chemical equilibrium As per the chemical kinetics11, 12, the rate equation for the chemical reaction, given by Equation(11), in terms of extent of advancement of chemical reaction, reads as [ ] [ ] [][]()[]11 0, dAdXkAkXXdtdtdt -===-� (18) where [ ] A and [ ] X are the mole numbers of reaction species and respectively, x is the extent of chemical reaction. Notice that in Equation(11) the sum of concentration of reactant species and remains constant in reaction mixture, that is [ ] [ ] constant AX+= (19) On perturbation in mole numbers of and , Equation (19) gives, [ ] [ ] 0, AXdd += (20) where [ ] A and [ ] X are sufficiently small. Equation (20) in terms of extent of reaction express as, [ ] [ ] AX dddx -== (21) Therefore, in perturbation space the relevant expression of rate equations is obtained from Equation(18), that is [ ] ( ) [ ] ( ) () [][]()[][][]()[]1111dAdX kAkXXkAkXX dtdtdtdddxddd---===-+- (22) From Equation(21), the rate expression, Equation(22) in perturbation space is modified as Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 3(ISC-2013), 74-78 (2014) Res. J. Recent. Sci. International Science Congress Association 76 ( ) ()()()1111111kAkXkkXkAkkXdtdx dxdxdx ---=--+=-+ \n (23) The expression for the chemical affinity on unperurbed trajectory are given by chemical thermodynamics and in this case that reads as [ ] [] [ ] [] 11 lnln0, AkARTRTXkX =+=�  (24) where q  is the chemical affinity of standard state,  is the chemical affinity of corresponding state and is the gas constant. In perturbation space the expression for chemical affinity is given by [ ] [] [ ] [] [] [] 11 0 AXRTRTAXAXdddx =-=-+  (25) From Equation(17), the required expression for Lyapunov function, S L is obtained as [] [] [][]()[] []()[]()1111111LRkAkXXkAkkXAXT dx -\r=-+-+-+ (26) Notice that in the perturbation space, S L is the function of perturbation coordinate, dx and time, , that is ( ) SSLLt dx (27) The total time derivative of S L from Equation(27), reads as, () ( ) , SSSdLLL dttdt dx dx¶¶=+¶¶ (28) where S Lt ¶¶ is the local time derivative and ( ) SL dx ¶¶ is the gradient of Lyapunov function, S L . From Equation(26), we easily obtain the local time derivative and gradient of S L .The local time derivative, 0 Lt ¶¶= , because the ( ) SSLLt dx is function of dx . However, the gradient of S L is obtained as, () [] [] [][]()[] []()[]()1111111RkAkXXkAkkXAXTdx- \r   =-+-+-+      (29) Thus from Equation(28) and (29), the total derivative of Lyapunov function, S L now becomes [] [] [][]()[] []()[]()()111111111122dLRkAkXXkAkkXkAkkXdtAXT dx --- \r=-+-+-+-+   \n  (30) On substituting [] [] [][]()[][]1111 0positivedefinite kAkXXPAX+-=£ (31) and [ ] ( ) [ ] ( ) [ ] 111 2either positive or negative, kAkkXQ-+= (32) Equation(26) and (30), respectively recast as, LRPQ dx  =-+  \n  (33) and . dLRPQQdtT dx  =-+\n  (34) Now we identify the stability of the given reaction as per the fabrics of Lyapunov’s direct method of stability of motion. i. If 0andthen0,0. dLQRPQLTdt ³³££  In this case both, 0,0 SSLdLdt ££ , have same sign and hence the process is unstable. ii. If 0andthen0,0. dLQRPQLTdt ³£³³  Again this situation leads to instability to the process as both, andSS LdLdt , have same sign. iii. If 0then0,0 dLQLdt ££³ In this condition the given process is stable because andSS LdLdt have opposite signs. In first two cases Lyapunov’s stability of motion is not guaranteed while in third case stability is ensured provided [ ] ( ) [ ] ( ) 111 20 kAkkXQ -+=£ (35) Therefore, from Equation(35), the physical conditions for stability is [] [] 1. k AX £+ (36) For [ ] [ ] 11 ,3 kkXA»³, that is the stability the given process, ensured if the concentration of is at least one third of the concentration of . For [ ] [ ] 11 ,, kkXA -³  in this case the stability is guaranteed if and only the concentration of is greater that the concentration of . Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 3(ISC-2013), 74-78 (2014) Res. J. Recent. Sci. International Science Congress Association 77 Stability when the concentration of is higher than The concentration of is kept constant and higher than by continuous supplying it from outside. In this case small fluctuation in concentration of is insignificant as compare to . Then, only perturbation in is significant. In this situation the rate equation in perturbation space is obtained as, ( ) [][]11 2. kAkXdtdx dx  =-\n (37) Similarly, the expression for chemical affinity in perturbation space is obtained as, [] 0. RTddx =- (38) From Equation(37) and (38), the required expression for Lyapunov function, S L is obtained as, [][]()[] [][]()1111 2. LRkAkXXkAkX dx -\r=--+-  (39) Thus, the total time derivative of S L from Equation(28) and (39) is obtained as, [][]() [][]()[][]()111111 22. dLRkAkXkAkXkAkXdtT dx --- \r=--+--  (40) On substituting [ ] [ ] ( ) [ ] 11 0positivedefinite kAkXM-=£ and (41) [ ] [ ] ( ) [ ] 11 2either positive or negative, kAkXN-= (42) in Equation(39) and (40), we obtain LRMN dx  =-+  \n  and (43) dLRMNNdtT dx  =-+\n  . (44) Now, we consider Equation(43) and (44) for the stability analysis in different physical conditions. i. If 0,0,0 dLNLdt ³££ . Then process is unstable because both, 0,0 SSLdLdt ££ have same sign. ii. If 0then0,0 dLNLdt ££³ . In this case the process is stable as per the fabrics of Lyapunov’s direct method of stability of motion. Thus the stability of motion is guaranteed for all values of and if one satisfy the condition, [ ] [ ] ( ) 11 20 kAkXN -=£ . Therefore, the physical conditions for stability is [][]()[] []1120 kAkXAx -£. (45) For [ ] [ ] 11 ,2 kkXA»³, that is the stability ensured if the concentration of is at least half the concentration of . For [ ] [ ] 11 ,, kkXA -³  in this case the stability is guaranteed if and only the concentration of is greater that the concentration of . Conclusion In this paper we have explored the tools of Lyapunov’s direct method of stability of motion (direct method) to investigate the stability of autocatalytic reactions. We have discussed almost all possibilities of perturbations of reactants away from chemical equilibrium. The domain of stability and instability is clearly revealed in our exercise. Autocatalytic reactions in equilibrium state show stability. It is found that away from equilibrium, the stability is guaranteed under certain constrains. The constraints are mostly the relative concentration of reactants and the ratio of forward rate to reverse rate at the time of perturbation. Reference 1.N.G. Chetayev, The Stability of Motion, Transl. M. Nadler, Pergamon, Oxford, (1961)2.J. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method with Applications, Academic Press, New York, (1961)3.J.D. Murray, Mathematical Biology, 19, Springer-Verlag, Berlin (1990)4.S.K. Scott, Chemical Chaos, Clarendon Press, Oxford, (1990)5.H.B. Callen, Thermodynamics, Wiley, New York, (1960)6.R. Hasse, Thermodynamic of Irreversible Processes, Addison-Wisley, Reading, MA, (1969)7.S.R. De Groot and P. Mazur, Nonequilibrium Thermodynamics, North-Holand, Amsterdam, (1962) Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 3(ISC-2013), 74-78 (2014) Res. J. Recent. Sci. International Science Congress Association 78 8.P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience, A division of John Wiley and Sons, Ltd. New York, (1971)9.D. Kondepudy and I. Prigogine, ‘Modern Thermodynamics”, John Wiley and Sons, New York, (1998)10.I. Prigogine and R. Defay, “Chemical Thermodynamics”, trans. D. H. Everett, Longmans Green, London, (1954)11.A.A. Frost and R.G. Pearson, Kinetics and Mechanism, A Study of Homogeneous Chemical Reactions, Second edition, Wiley Eastern Private Limited, New Delhi, (1961)12.M. Eigen and L. Demaeyer in Investgation of Rates and Mechanisms of Reactions, Vol. VIII-Part-II of Technique of Organic Chemistry, Eds. S. L. Fries, E. S. Lewis and A. Weissberger and Ser. Ed. A. Weissberger, Wiley, New York, (1963)