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V alentin N. Parmon

THERMODYNAMIC FORM OF KINETIC EQUATIONS AND AN EXPERIENCE OF ITS USE FOR ANALYZING COMPLEX REACTION SCHEMES. V alentin N. Parmon. Boreskov Institute of Catalysis Novosibirsk State University Novosibirsk, Russia, 630090, parmon@catalysis.ru. May 30, 2012 Ghent, Belgium.

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V alentin N. Parmon

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  1. THERMODYNAMIC FORM OF KINETIC EQUATIONS AND AN EXPERIENCE OF ITS USE FOR ANALYZING COMPLEX REACTION SCHEMES Valentin N.Parmon Boreskov Institute of CatalysisNovosibirsk State UniversityNovosibirsk,Russia,630090, parmon@catalysis.ru May 30, 2012Ghent, Belgium

  2. Novosibirsk is the 3rd largest city in Russia (behind Moscow and St-Petersburg) Population >1,500,000Universities and academies 30 Great logistic center (Trans-Siberian railway, International airport)High-tech industriesThe highest density of science in Russia Russia St-Petersburg Siberia Moscow Siberian Branchof the Russian Academy of Sciences Novosibisrk Scientific Center – Akademgorodok • Population 130,000 • 35 academic research institutes of SB RAS with ca. 9,000 employees • 7 chemical research institutes • Novosibirsk State University Novosibirsk

  3. The Siberian school of mathematicians in chemistry (since the beginning of 60ths) M.G. Slin’ko(1914–2008) – initiator of the wide application of mathematical methods in catalysis G.S. Yablonsky – generalization of analysis of complex reaction schemes A.N.Gorban’ – coupling of kinetic analysis with thermodynamics V.I. Bykov – analysis of reaction schemes with singularitues M.Z. Lazman

  4. G.S. Yablonsky et al. (1970s-2000s) • Rigorous results based on assumed detailedreaction mechanisms withthe ideal mass-action-law dependences • (1) Linear Theory (1970s-1980s) • (2) Non-Linear Theory (1980s-2000s)

  5. Linear theory • ONE-ROUTE CATALYTIC REACTION with the linear mechanism General expression for the steady state reaction rate (Yablonsky, Bykov, 1976) where Cy is a “cyclic characteristics” Cy = K+f+(C) – K–f–(C) Cy corresponds to the overall reaction  presents a complexity of complex reaction • MULTI-ROUTE LINEAR MECHANISMS(Yevstignejev, Yablonsky, Bykov, 1979)

  6. Non-linear theory KINETIC POLYNOMIAL (Lazman, Yablonsky, 1980-2000s) It is considered as the most generalized form which includes Langmuir-Hinshelwood-Hougen-Watson equations and equations of enzyme kinetics as particular cases The kinetic polynomial of the one-route non-linear reaction scheme is BmRm +…+ B1R +B0Cy = 0 where R is the steady-state reaction rate, m are the integer numbers

  7. V.N. Parmon is a lecturer of Novosibirsk State University in chemical kinetics and both classical (equilibrium) and, since 1995, non-equilibrium thermodynamics

  8. Contents of the presentation: • Introducing the thermodynamic form of kinetic equations • Some interesting one consequences • A problem of the “bottle neck” (limiting step) and “rate controlling step” of a stepwise reaction • Few practical application

  9. A Thermodynamic Form of Kinetic Equations is an Inevitable Step for the Successive Unifying the Languages of Chemical Kinetics and Chemical Thermodynamics Chemical Kinetics: the main parameters are concentrations, c, of reactants A and rate constants, ki, Chemical Thermodynamics: the main parameters are chemical potentials, , of thermalized reactants A Note,however, that where  is an activity coefficient

  10. Introducing the Thermodynamic Form of Kinetic Equations For a substance A For an elementary reaction “ij” where is a “chemical potential” of reaction group i

  11. Properties of the Thermodynamic Form of Equations for an Elementary Reversible Reaction If ij is the chemical variable for reaction “ij” since ij = ji ! Indeed, for partial equilibrium of reaction ij or when thus ij = ji

  12. New proprietary definitions: – “truncated” rate constant of reaction ij which depends only on the properties of TS – thermodynamic “rush” of reaction group i Why “rush” ? TS

  13. Some Related Properties Inequality is equivalent tothe positive value of affinity Arij of reaction ij: at is equivalent to is equivalent to Direction of reaction ij coincides with the sign of Arij !

  14. Thermodynamic Criterium of Kinetic Irreversibility of a Reaction i j € So, the rate of reverse reaction is negligible in respect to (far from equilibrium) In this case double can be substituted by single (the vicinity to equilibrium) then

  15. Consequence 1 For a set of stationary consecutive reactions which occur from left to right • obligatory • the number of single arrows can not exceed the value of Here Ar  R – P is affinity of stoichiometric stepwise reaction R P

  16. Consequence 2 For any stoichiometric stepwise reaction “” which is linear in respect to its reaction intermediates Yi the rate is expressed in the same way as for elementary reaction: where R and P are initial and final reaction groups;  is an algebraic combination of ij and, in some cases, thermodynamic rushes of “external reactants” from either initial of final reactions groups Note:Stoichiometric stepwise reaction means steady state occurrence of the reaction in respect to its intermediates Yi This relation in catalysis is known as a Horiuti–Boreskov equation!

  17. Simple Example 1 Stepwise reaction occurs according to scheme In the steady state in respect to Y where

  18. Simple Example 2 Stepwise reaction occurs according to scheme In the steady state in respect to Y where

  19. Stepwise process occurs according to scheme Electrotechnical analog For an arbitrary set of monomolecular transformations of intermediates there is a total analogy with an electrotechnical equivalent scheme! is calculated in the same way as : is an algebraic combination of ij like R is that of Rij

  20. Main basis for “linear” non-equilibriumthermodynamics Flux Jiof a parameter ai where Xiis thermodynamic driving force for ai For a complex system where Lij are the Onsager’ coefficients of interrelation. A sequence: existence of the Raleigh-Onsager dissipation function According to the Prigogine theorem, P is the Lyapunov’ function which reaches a positively defined minimum at the stationary state of the system (when Jj = 0)

  21. Consequence 3: Existence of the Lyapunov’ functions  which are positively determined and minimazing at the steady state in respect to intermediates even far from equilibrium for any reaction schemes which are linear in respect to intermediates Example 1:Stepwise reaction occurs via the scheme where {Yi} means an arbitrary set of monomolecular transformations of Yi

  22. Stepwise process occurs according to scheme Electrotechnical analog Physical meaning of the Lyapunov’ function for an arbitrary set of monomolecular reactions far from equilibria Thus, the Lyapunov’ function corresponds to the power W of the dissipation of Ohmic heat in the electrical circuit

  23. Example 2:Stepwise reaction occurs via the scheme • Conclusions: • The Lyapunov function exists for any stepwise reactions which are linear in respect to intermediates • Steady state of above reaction is stable

  24. Consequence 4: According to the Prigogine theorem all systems near thermodynamic equilibrium have the stable steady state. All stepwise reactions linear in respect to intermediates have their Lyapunov’ functions and thus are also stable A contrary example:Stepwise reaction occurs via the nonlinear autocatalytic scheme in respect to Y: The Lyapunov function does not exist ! There are two steady states The steady state in respect to Y can be nonstable ! Thus,the necessary conditions for loosing the stability of the steady state of a kinetic scheme: (1) As least one elementary reaction has to be kinetically irreversible (2) This elementary reaction has to be non-linear in respect to the intermediates

  25. Necessary conditions for oscillation of the concentration of reaction intermediates: • far from equilibrium (at least one single in the reaction scheme) • at least two reaction intermediates • at least one step which is nonlinear in respect to intermediates

  26. Consequence 5: It is possible to write modified Onsager’ (the Horiuti-Boreskov-Onsager) equations of interrelation of parallel stepwise chemical reactions A simple example: Parallel step-wise reactions occur via mechanism At the steady state Thus, where • 11 = 12/(1 + 2 + 3) > 0 • 22 = 13/(1 + 2 + 3) > 0 • 12 = 21 = -23/(1 + 2 + 3)

  27. In a general case for parallel stepwise reactions Here ii > 0 Note: ij is not obviously symmetrical in respect to indexes i and j as it is the case for reprocisity coefficients Lij in the classic Onsager equations in the vicinity of equilibrium

  28. The problem of “rate controlling” (“rate determining”) step and “rate limiting” step (“bottle neck” of the stepwise reaction) An unambigous interpretation of the notion “rate determining” (rate controlling) step by IUPAC Rate controlling factor but In the thermodynamic representation – contains parameters of only the transient states – contains parameters of only the thermalized reactions groups and reactants Thus, – rate controlling factor of transient states – rate controlling factor of the reactant

  29. How to define correctly the “rate limiting” step (the “bottle neck”)?

  30. The “rate determining step” and “bottle neck” in a consequtive monomolecular reaction where So rate-determining step is the step with minimal i Note: in the steady state for i = 0,…, n+1 Thus For i = lim the value of is maximal ! It means that the “bottle neck” (limiting step) is the step with the maximum drop of !

  31. Application to catalytic reactions A simple example: occurs via catalytic Michaelis–Menten scheme where K and K1 are free catalytically active site and the catalytic intermediate The balance equation can be rewritten: Thus where and corresponds to at Here

  32. Finally, at the steady state in respect to K1 • At small extent of occupation of the active site with catalytic intermediates and does not depend on standard thermodynamic parameters of K1 • At large extent of occupation of the active site with catalytic intermediates depends on

  33. Note that for one can have the situation when the rate determining step does not coincide with the bottle neck ! Let: In this situation independently on whether So, the rate determining step is always step 2 But obligatory the “bottle neck” is the step with minimum i !

  34. An unexpected conclusion:there are situations when the rate-limiting step can not be the rate-controlling step!

  35. Conclusions • The thermodynamic form of kinetic equations allows a dramatic simplification of analysis of complex reaction schemes • Indeed, the main application of this approach is possibility to extend fruitful and systematic analysis of chemical reaction schemes for the area “far from equilibrium” • Among few principal problems which are resolving via this approach this is a mathematically correct definition of the “bottle neck” (the limiting step) of a stepwise reaction and “rate determing step” • Unexpectedly, for some particular cases (e.g. for catalytic reactions) these steps can not coinside

  36. Few examples of practical interest

  37. o 300 A An example of a practical application:Super low temperature of melting of active componentof operating metal catalysts due to their oversaturationwith carbon Electron microscopy “in situ” videotape of Fe–C fluidized particles migration over amorphous carbon supportat 650 °C time (sec): The melting temperature is 500 °C (!)lower than that of the Fe–C eutectics time (sec): O.P.Krivoruchko, V.I.Zaikovskij, K.I.Zamaraev, Dokl.Akad.Nauk, v.329, 744 (1992) (in Russian)

  38. Formation ofMetastableOversaturated Solutions of Carbon in Metals at Catalytic Graphitizationof Amorphous Carbon Camorph®Cgraphite DG»–12 kJ/mol ( >RT) met Melting Temperatures, oC equilibriumeutecticswith graphite steadystate puremetal metal solution of C in metal Camorph Cgraphite Fe 1539 1145 640 Co 1493 1320 600 mc(amorf) > mc(in metal) >mc(graphite) Ni 1453 1318 670 Result:steady-state concentration of xC in metal>>concentration of C in stable eutectics If the rate determining step is formation of graphite from the melt: xC (eq. with amorph. C) = xC (eq. with graphite)exp(–GR/RT)   4xC(eq. with graphite)  4xC (eq. eutectics) Hm and To are the melting heat and melting temperature for pure metal V.N.Parmon. Catalysis Letters, 42, 195 (1996), O.P.Krivoruchko, V.I.Zaikovskij (1995)

  39. Fe3C Fe2C Metastable Phase Equilibria for Fe–C Systems during Occurrence of the Catalytic Reaction T, K T Parameters, influencing the melting temperature: Graphite liquids 2000 1.Oversaturation: theSchröderequation Tx = TomH/{mH – RToln(1 – x)} Solution of Fe in C 1600 eutectics (T = 1420K, x = 0.173) 2.Metal particle size r: Schröder 1200 Solution of C in Fe steady state (920K) 800 3.Size r’ of a crystallization center (= size of the catalyst particle) 0 50 100 Content of C (mol. %)

  40. o A 500 o A 1000 Formation of Filamentous Carbon together with Hydrogen at the Moderate-Temperature Catalytic Pyrolysis of Methane and Low Hydrocarbons Ni-catalyst Ni/Cu-catalyst The weight of the catalyst can be increased by a factor of 400 due to formation of carbonaceous filamentous material The growth of the filament corresponds to diffusion of carbon through the active component with D>10–10 cm2/s L.B.Avdeeva, V.A.Likholobov, G.G.Kuvshinov, at al. (1994)

  41. Size effects in catalysis over metal nanoparticles Au/Al2O3 Pt/Al2O3 CО + O2 CO2 CН4 + 2 O2 CO2 + 2 Н2О 430 °C 400 °C TOF (s–1) 10–4 TOF (s–1) <d>, nm <d>, nm Conclusion:There may occur size effects in catalytic reactions, which are many time increase in the activity of metal catalysts upon a decrease of the active component particles in size to several nanometers I. Beck, V.I. Bukhtiyarov, I.Yu. Pakharukov, V.I. Zaikovsky, V.V. Kriventson, V.N. Parmon, Journal of Catalysis 268 (2009) 60-67

  42. Influence of the active component particle size on the catalytic activity (an energy correlation approach) S A BMechanism: A + K K1 (1) K1 B + K (2) ¥ The increment of chemical potential of a nanoparticle of radius r 2 V r r = ¥ Here  – surface excess energy, V – molar volume of the catalyst active phase, TS2  = ær TS1 æ < 1 is the Brønsted-Polyany correlation coefficient æ at low coverage with K1 Result: æ at large coverage with K1 V.N. Parmon, Doklady Physical Chemistry, vol. 413 (2007) 42-48

  43. Correlation of the measured TOF values for the complete CH4 oxidation over Pt/Al2O3with the Pt size 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,0 –1,0 140 120 –1,4 100 –1,8 80 log(TOF) (TOF ins–1) –2,2 60 Ea, kJ/mol 40 –2,6 lg (TOF) = 3,304 (1/d) – 2,981 20 0,0 0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,0 1/d, nm–1 at temperature 700 K and of the apparent activation energies Eawith the reciprocal to the active component size (diameter) d For both lines the correlation coefficient is the same: æ 0.75 I. Beck, V.I. Bukhtiyarov, I.Yu. Pakharukov, V.I. Zaikovsky, V.V. Kriventson, V.N. Parmon, Journal of Catalysis 268 (2009) 60-67

  44. Thermodynamic conjugation of parallel chemical transformations via a common catalytic intermediate {X} – catalytic intermediates m A mA C* {X} mX mB B mC C Reaction coordinate The Horiuti-Boreskov-Onsager coupling equations: kij – formal rate constants kB and KC – equilibrium constants Conclusion:To change selectivity of a catalytic process one has to generate some thermodynamic driving forces V.N. Parmon, Thermodynamics of Non-Equilibrium Processes, Elsevier, 2010

  45. Thermodynamic control of selectivity at the decomposition of methanol 2 Methanol 2 Methanol Gibbs energy 2CO + 4 H2 (II) {X} {X} Gibbs energy Methylformate + 2 H2 (I) Methylformate + 2 H2 (I) 2 CO + 4 H2 (II) {X} – catalytic intermediate Reaction coordinate Reaction coordinate Two independent stepwise reactions – two channels of decomposition: 2 CH3OH CH3OOCH + 2 H2 (I) CH3OH CO + 2 H2 (II) Conclusion:an increase in the partial pressure of CO has to result in improving the selectivity in respect to methylformate

  46. main products CH4 as a byproduct Usually: hydrocarbons An example of an important practical application: Development of principally new one-step catalytic processes of direct insertion of methane higher hydrocarbons Due to existence of Onsager’s interrelation, one can reverse the direction of the process of CH4formation Now: hydrocarbons + CH4 heavier hydrocarbons Examples: Process “Bicyclar” CH4 + C3,C4 alkanes  aromatics + 5 H Process “Biforming” CH4 + linear C5+  aromatics + 5 H2

  47. Putative one-stage processes for conversion of light paraffins CH4andC3–C4 (methane and propane-butanes) to aromatic compounds T*, K Reactions of light hydrocarbons 1. 6 CH4C6H6 + 9 H21630 2. 2 C3H8C6H6 + 5 C2H6760 3. 2 n-C4H10 p-C6H4(CH3)2+5H2800 4. C3H8 +n-C4H10C6H5CH3 + 5H2710 5.3 C2H6C6H6 + 5 H2930 6. CH4 + 2 C3H8C6H5CH3 + 6 H2880 7. CH4 + C3H8 + n-C4H10 p-C6H4(CH3) + 6 H21060 8. CH4 + C2H6 + C3H8C6H6 + 6 H2940 9. CH4 + 3C3H8C10H8 + 10 H2830 Observation: Aromatization of C2–C4 paraffins is accompanied by the methane co-production.

  48. Performance of the BICYCLAR process depending on the C1/C4 ratio CH4 + 2 C3H8C6H5CH3 + 6 H2 CH4 + 3C3H8C10H8 + 10 H2 1.5 1.0 Yield of aromatichydrocarbons, t/t C4 0.5 0 0 3 6 9 12 15 18 Molar ratioC1/C4 Catalyst Zn-ZSM-5, temperature 550 °C The coupled conversion of butane and methane allows the yield of aromatic hydrocarbons to be 2.5 times increased – up to 1.7 tonn per 1 tonn of involved C4 G.V. Echevsky, E.G. Kodenev, O.V. Kikhtyanin, V.N. Parmon, Appl. Catal. A: General 258 (2004) 159-171

  49. An example of a practical application: Natural selection in simple autocatalytic systems at diminishing the concentration of food R follows in one-directional progressive evolution of the system There are two steady states: At diminishing the concentration of food R, one proceeds a consecutive and irreversible (due to disappearance of seeds) “death” of all autocatalysts with the largervalues of Thus, a one-directional and progressive(toward diminishing the parameterRcri) natural selection takes place in the system. This is analogous to appearance of a prototype of biological memory An extremely important conclusion:existenceof a prototype of biological memoryin the absence of RNA or DNA ! V. Parmon, Doklady Phys. Chem., 377, 4 (2001) 510-515

  50. Thank you for your attention !

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