Thermodynamics in the Limit of Irreversible Reactions

Transcription

1 Thermodynamcs n the Lmt of Irreversble Reactons A. N. Gorban Department of Mathematcs, Unversty of Lecester, Lecester, LE1 7RH, UK E. M. Mrkes Insttute of Space and Informaton Technologes, Sberan Federal Unversty, Krasnoyarsk, Russa G. S. Yablonsky Parks College, Department of Chemstry, Sant Lous Unversty, Sant Lous, MO 63103, USA Abstract For many real physco-chemcal complex systems detaled mechansm ncludes both reversble and rreversble reactons. Such systems are typcal n homogeneous combuston and heterogeneous catalytc oxdaton. Most complex enzyme reactons nclude rreversble steps. The classcal thermodynamcs has no lmt for rreversble reactons whereas the knetc equatons may have such a lmt. We represent the systems wth rreversble reactons as the lmts of the fully reversble systems when some of the equlbrum concentratons tend to zero. The structure of the lmt reacton system crucally depends on the relatve speeds of ths tendency to zero. We study the dynamcs of the lmt system and descrbe ts lmt behavor as t. The extended prncple of detaled balance provdes the physcal background of ths analyss. If the reversble systems obey the prncple of detaled balance then the lmt system wth some rreversble reactons must satsfy two condtons: () the reversble part satsfes the prncple of detaled balance and () the convex hull of the stochometrc vectors of the rreversble reactons does not ntersect the lnear span of the stochometrc vectors of the reversble reactons. These condtons mply the exstence of the global Lyapunov functonals and alow an algebrac descrpton of the lmt behavor. The thermodynamc theory of the rreversble lmt of reversble reactons s llustrated by the analyss of hydrogen combuston. Keywords: entropy, free energy, reacton network, detaled balance, rreversblty PACS: a, Qt, w, Hc 1. Introducton 1.1. The problem: non-exstence of thermodynamc functons n the lmt of rreversble reactons We consder a homogeneous chemcal system wth n components A, the concentraton of A s c 0, the amount of A n the system s N 0, V s the volume, N = Vc, T s the temperature. The n dmensonal vectors c = (c ) and N = (N ) belong to the closed postve orthant R n + n R n. (R n +. (The closed postve orthant s the set of all vectors x R n such that x 0 for all.) The classcal thermodynamcs has no lmt for rreversble reactons whereas the knetc equatons have. For example, let us consder a smple cycle k 1 k 2 k 3 A 1 A 2 A 3 A 1 k 1 k 2 k 3 wth the equlbrum concentratons c eq = (c eq and the detaled balance condtons: k c eq = k c eq +1 1, ceq 2, ceq 3 ) under the standard cyclc conventon, here, A 3+1 = A 1 and c 3+1 = c 1. The perfect free energy has the form F = RTVc ln c 1 + const. c eq Emal address: ag153@le.ac.uk (A. N. Gorban) Let the equlbrum concentraton c eq 1 0 for the Preprnt submtted to Elsever August 5, 2012

2 fxed values of c eq 2,3 > 0. Ths means that k 1 k 1 = ceq 1 c eq 0 and k 3 = ceq 1 k 2 3 c eq 3 0. Let us take the fxed values of the rate constants k 1, k ±2 and k 3. Then the lmt knetc system exsts and has the form: k 1 k 2 A 1 A2 A 3 A 1. k 2 k 3 It s a routne task to wrte a frst order knetc equaton for ths scheme. At the same tme, the free energy functon F has no lmt: t tends to for any postve vector of concentratons because the term c 1 ln(c 1 /c eq 1 ) ncreases to. The free energy cannot be normalzed by addng a constant term because the varaton of the term c 1 ln(c 1 /c eq 1 ) on an nterval [0, c] wth fxed c also ncreases to, t vares from c eq /e (for the mnmzer, c 1 = c eq 1 /e) to a large number c(ln c ln ceq 1 ) (for c 1 = c). The logarthmc sngularty s rather soft and does not cause a real physcal problem because even for c eq 1 /c 1 = the correspondng large term n the free energy wll be just 23RT per mole. Nevertheless, the absence of the lmt causes some mathematcal questons. For example, for perfect systems wth detaled balance under sochorc sothermal condtons the densty, f = F/(RTV) = 1 c (ln(c /c eq ) 1), (1) s a Lyapunov functon for a system of chemcal knetcs (here, c s the concentraton of the th component and c eq s ts equlbrum concentraton for a selected value of the lnear conservaton laws, the so-called reference equlbrum ). Ths functon s used for analyss of stablty, exstence and unqueness of chemcal equlbrum snce the work of Zeldovch (1938, reprnted n 1996 [26]). Detaled analyss of the connectons between detaled balance and the free energy functon was provded n [19]. Perhaps, the frst detaled proof that f s a Lyapunov functon for chemcal knetcs of perfect systems wth detaled balance was publshed n 1975 [22]. For the rreversble systems whch are obtaned as lmts of the systems wth detaled balance we should expect the preservaton of stablty of the equlbrum. More over, one can expect exstence of the Lyapunov functons whch are as unversal as the thermodynamc functons are. The unversalty means that these functons depend on the lst of components and on the equlbrum concentratons but do not depend on the reacton rate constants drectly. 2 The thermodynamc potental of a component A cannot be defned n the rreversble lmts when the equlbrum concentraton of A tends to 0. Nevertheless, n ths paper, we construct the unversal Lyapunov functons for systems wth some rreversble reactons. Instead of detaled balance we use the weaker assumpton that these systems can be obtaned from the systems wth detaled balance when some constants tend to zero The extended form of detaled balance condtons for systems wth rreversble reactons Let us consder a reacton mechansm n the form of the system of stochometrc equatons α r A β r j A j (r = 1,..., m), (2) j where α r 0, β r j 0 are the stochometrc coeffcents. The reverse reactons wth postve rate constants are ncluded n the lst (2) separately (f they exst). The stochometrc vector γ r of the elementary reacton s γ r = (γ r ), γ r = β r α r. We always assume that there exsts a strctly postve conservaton law, a vector b = (b ), b > 0 and b γ r = 0 for all r. Ths may be the conservaton of mass or of total number of atoms, for example. Accordng to the generalzed mass acton law, the reacton rate for an elementary reacton (2) s (compare to Eqs. (4), (7), and (14) n [14] and Eq. (4.10) n [7]) w r = k r n =1 a α r, (3) where a 0 s the actvty of A, a = exp µ µ 0. (4) RT Here, µ s the chemcal potental and µ 0 s the standard chemcal potental of the component A. Ths law has a long hstory (see [6, 24, 13, 7]). It was nvented n order to meet the thermodynamc restrctons on knetcs. For ths purposes, accordng to the prncple of detaled balance, the rate of the reverse reacton s defned by the same formula and ts rate constant should be found from the detaled balance condton at a gven equlbrum. It s worth mentonng that the free energy has no lmt when some of the reacton equlbrum constants tend to zero. For example, for the deal gas the chemcal potental s µ (c, T) = RT ln c + µ 0 (T). In the rreversble lmt some µ 0. On the contrary, the actvtes reman fnte (for the deal gases a = c ) and the

3 approach based on the generalzed mass acton law and the detaled balance equatons w + r = w r can be appled to fnd the rreversble lmt. The lst (2) ncludes reactons wth the reacton rate constants k r > 0. For each r we defne k r + = k r, w + r = w r, kr s the reacton rate constant for the reverse reacton f t s on the lst (2) and 0 f t s not, w r s the reacton rate for the reverse reacton f t s on the lst (2) and 0 f t s not. For a reversble reacton, K r = k r + /kr The prncple of detaled balance for the generalzed mass acton law s: For gven values k r there exsts a postve equlbrum a eq > 0 wth detaled balance, w + r = w r. Recently, t s found the extended form of the detaled balance condtons for the systems wth some rreversble reactons [12]. Ths extended prncple of detaled balance s vald for all systems whch obey the generalzed mass acton law and are the lmts of the systems wth detaled balance when some of the reacton rate constants tend to zero. It conssts of two parts: The algebrac condton: The prncple of detaled balance s vald for the reversble part. (Ths means that for the set of all reversble reactons there exsts a postve equlbrum where all the elementary reactons are equlbrated by ther reverse reactons.) The structural condton: The convex hull of the stochometrc vectors of the rreversble reactons has empty ntersecton wth the lnear span of the stochometrc vectors of the reversble reactons. (Physcally, ths means that the rreversble reactons cannot be ncluded n orented cyclc pathways.) Let us recall the formal conventon: the lnear span of empty set s {0}, the convex hull of empty set s empty The structure of the paper In Sec. 2 we study the systems wth detaled balance, ther multscale lmts and the lmt systems whch satsfy the extended prncple of detaled balance. The classcal Wegscheder denttes for the reacton rate constants are presented. Ther lmts when some of the equlbra tend to zero gve the extended prncple of detaled balance. We use the generalzed mass acton law for the reacton rates. For the analyss of equlbra for the general systems, the formulas wth actvtes are the same as for the deal systems and t s convenent to work wth actvtes unless we need to study dynamcs. The dynamcal varables are amounts and concentratons. In a specal 3 subsecton 2.3 we dscuss the relatons between concentraton and actvtes, formulate the man assumptons and present formulas for the dsspaton rate. We ntroduce attractors of the systems wth some rreversble reactons and study them n Sec. 3. It ncludes the central results of the paper. We fully characterze the faces of the postve orthant that nclude ω-lmt sets. On such a face, dynamcs s completely degenerated (zero rates) or t s drven by a smaller reversble system that obeys classcal thermodynamcs. Hydrogen combuston s the most studed and very mportant gas reacton. It has the modest complexty: n the usual models there are 6-8 components and elementary reversble reactons. Under varous condtons some of these reactons are practcally rreversble. We use ths system as a benchmark n Sec. 4 and gve an example of the correct separaton of the reactons nto reversble and rreversble part. The lmt behavor of ths system n tme s descrbed. In Concluson we brefly dscuss the results wth focus on the unsolved problems. 2. Multscale lmt of a system wth detaled balance 2.1. Two classcal approaches to the detaled balance condton There are two tradtonal approach to the descrpton of the reversble systems wth detaled balance. Frst, we can start from the ndependent rate constants of the elementary reactons and consder the solvablty of the detaled balance equatons as the addtonal condton on the admssble values of the rate constants. Here we have m constants (m should be an even number, m = 2l) and some equatons whch descrbe connectons between these constants. Ths approach was ntroduced by Wegscheder n 1901 [23] and developed further by many authors [20, 4]. Secondly, we can select a drect reacton n each par of mutually reverse elementary reactons. If a postve equlbrum s known then we can fnd the reacton rate constants for the reverse reacton from the constants for drect reacton and the detaled balance equatons. Therefore, the drect reacton rate constants and a set of the equlbrum actvtes form the complete descrpton of the reacton. Here we have l + n ndependent constants, l = m/2 rate constants of drect reactons and n (t s the number of components) equlbrum actvtes. For these l + n constants, the prncple of detaled balance produces no restrctons. Ths second approach s

4 popular n appled chemcal thermodynamcs and knetcs [17, 10, 25] because t s convenent to work wth the ndependent parameters from scratch. The Wegscheder condtons appear as the necessary and suffcent condtons of solvablty of the detaled balance equatons. (See, for example, the textbook [24]). Let us jon the drect and reverse elementary reactons and wrte α r A β r j A j (r = 1,..., l). (5) j The stochometrc matrx s Γ = (γ r ), γ r = β r α r (gan mnus loss). The stochometrc vector γ r s the rth row of Γ wth coordnates γ r = β r α r. Both sdes of the detaled balance equatons, w + r = w r, are postve for postve actvtes. The solvablty of ths system for postve actvtes means the solvablty of the followng system of lnear equatons: γ r x = ln k r + ln kr = ln K r (r = 1,... l) (6) (x = ln a eq ). Of course, we assume that f k r + > 0 then kr > 0 (reversblty) and the equlbrum constant K r > 0 s defned for all reactons from (5). Proposton 1. The necessary and suffcent condtons for exstence of the postve equlbrum a eq > 0 wth detaled balance s: For any soluton λ = (λ r ) of the system l λγ = 0.e. λ r γ r = 0 for all (7) r=1 the Wegscheder dentty holds: l (k r + ) λ r = r=1 l (kr ) λ r. (8) It s suffcent to use n (8) any bass of solutons of the system (7): λ {λ 1,, λ q } Multscale degeneraton of equlbra Let us take a system wth detaled balance and send some of the equlbrum actvtes to zero: a eq 0 when I for some set of ndexes I. Immedately we fnd a surprse: ths assumpton s not suffcent to fnd a lmtng rreversble mechansm. It s necessary to take nto account the rates of the convergency to zero of dfferent a eq. Indeed, let us study a very smple example, r=1 k 1 k 2 A 1 A 2 A 3 k 1 k 2 4 when a eq 1, aeq 2 0. If a eq 1, aeq 2 0, a eq 1 /aeq 2 = const > 0 and a eq 3 = k 1 const > 0 then the lmt system should be A 1 A 2 k 1 A 3 and we can keep k 1, 1,2 = const whereas k 2 0. If a eq 1, aeq 2 0, a eq 1 /aeq 2 0 then the lmt system should be A 1 A 2 A 3 and we can keep k 1,2 = const > 0 whereas k 1, 2 0. If a eq 1, aeq 2 0, aeq 2 /aeq 1 0 then n the lmt survves only one reacton A 2 A 3 (f we assume that all the reacton rate constants are bounded). We study asymptotcs a eq = const ε δ, ε 0 for varous values of non-negatve exponents δ 0 ( = 1,..., n). At equlbrum, each reacton rate n the generalzed mass acton law s proportonal to a power of ε: w eq+ r = k + r const ε α r δ, w eq r = k r const ε β r δ. Accordng to the prncple of detaled balance, w eq+ and w eq r k r + kr r = = const ε (γ r,δ), (9) where δ s the vector of exponents, δ = (δ ). There are three groups of reactons wth respect to the gven vector δ: 1. (γ r, δ) = 0; 2. (γ r, δ) < 0; 3. (γ r, δ) > 0. In the frst group ((γ r, δ) = 0) the rato k r + /kr remans constant and we can take k r ± = const > 0. In the second group ((γ r, δ) < 0) the rato kr /k r + 0 and we should take kr 0 whereas k r + may reman constant and postve. In the thrd group ((γ r, δ) > 0), the stuaton s nverse: k r + /kr 0 and we can take kr = const > 0, whereas k r + 0. These three groups depend on δ but ths dependence s pecewse constant. For every γ r, three sets of δ are defned: () hyperplane (γ r, δ) = 0, () hemspace (γ r, δ) < 0 and hemspace (γ r, δ) > 0. The space of vectors δ s splt n the subsets defned by the values of functons sgn(γ r, δ) (±1 or 0). We consder bounded systems, hence the negatve values of δ should be forbdden. At least one equlbrum actvty should not vansh. Therefore, δ j = 0 for some j. Below we assume that δ 0 and δ j = 0 for a non-empty set of ndces J 0. Moreover, the atom balance n equlbrum should be postve. Here, ths means that for the set of equlbrum concentratons c eq ( J 0 ) the correspondng values of all atomc concentratons are strctly postve and separated from zero. Let the vector of exponents, δ = (δ ) be gven and the three groups of reactons are found. For the reactons of

5 the thrd group (wth (γ r, δ) > 0) the drect reacton vanshes n the lmt ε 0. It s convenent to transpose the stochometrc equatons for these reactons and swap the drect reactons wth reverse ones. Let us perform ths transposton. After that, α r changes over β r, γ transforms nto γ, and the nequalty (γ r, δ) > 0 transforms nto (γ r, δ) < 0. Let us summarze. We use the gven vector of exponents δ and produce a system wth some rreversble reactons from a system of reversble reactons and detaled balance equlbrum a eq by the followng rules: 1. f δ > 0 then we assgn a eq = 0 and f δ = 0 then a eq does not change; 2. f (γ r, δ) = 0 then k r ± do not change; 3. f (γ r, δ) < 0 then we assgn kr = 0 and k r + does not change; 4. f (γ r, δ) > 0 then we assgn k r + = 0 and kr does not change. (In the last case, we transpose the stochometrc equaton and swap the drect reacton wth reverse one, for convenence, γ r changes to -γ r and kr becomes 0. Therefore, ths case transforms nto case 3.) Ths s a lmt system caused by the multscale degeneraton of equlbrum. The multscale character of the lmt a eq = const ε δ 0 (for some ) s mportant because for dfferent values of δ reactons may have dfferent domnant drectons and the set of rreversble reactons n the lmt may change. The general form of the knetc equatons for the homogeneous systems s dn dt = V w r γ r, (10) where N s the amount of A, N s the vector wth components N and V s the volume. Let us consder a lmt system for the degeneraton of equlbrum wth the vector of exponents δ. For ths system (γ r, δ) 0 for all r and, n partcular, (γ r, δ) < 0 for all rreversble reactons and (γ r, δ) = 0 for all reversble reactons. Proposton 2. A lnear functonal G δ (N) = (δ, N) decreases along the solutons of knetc equatons (10) for ths lmt system: dg δ (N)/dt 0 and dg δ (N)dt = 0 f and only f all the reacton rates for the rreversble reactons are zero. Proof. Indeed, dg δ (N) dt r = V w r (γ r, δ) 0, (11) r 5 because for reversble reactons (γ r, δ) = 0, and for rreversble reactons w r = w + r 0 and (γ r, δ) < 0. All the terms n ths sum are non-negatve, hence t may be zero f and only f each summand s zero. Ths Lyapunov functon may be used n a proof that the rates of all rreversble reactons n the system tend to 0 wth tme. Indeed, f they do not tend to zero then on a soluton of (10) G δ (N(t)) when t and N(t) s unbounded. Equaton (11) and Proposton (2) gve us the possblty to prove the extended prncple of detaled balance n the followng form. Let us consder a reacton mechansm that ncludes reversble and rreversble reactons. Assume that the reacton rates satsfy the generalzed mass acton law (3) and the set of reacton rate constants s gven. Let us ask the queston: Is t possble to obtan ths reacton mechansm and reacton rate constants as a lmt n the multscale degeneraton of equlbrum from a fully reversble system wth the classcal detaled balance. The answer to ths queston gves the followng theorem about the extended prncple of detaled balance. Theorem 1. A system can be obtaned as a lmt n the multscale degeneraton of equlbrum from a reversble system wth detaled balance f and only f () the reacton rate constants of the reversble part of the reacton mechansm satsfy the classcal prncple of detaled balance and () the convex hull of the stochometrc vectors of the rreversble reactons does not ntersect the lnear span of the stochometrc vectors of reversble reactons. Proof. Let the gven system be a lmt of a reversble system wth detaled balance n the multscale degeneraton of equlbrum wth the exponent vector δ. Then for the reversble reactons (γ r, δ) = 0 and for the rreversble reactons (γ r, δ) < 0. For every vector x from the convex hull of the stochometrc vector of the rreversble reactons (x, δ) < 0 and for any vector y from the lnear span of the stochometrc vectors of the reversble reactons (y, δ) = 0. Therefore, these sets do not ntersect. The reacton rate constants for the reversble reactons satsfy the classcal prncple of detaled balance because they do not change n the equlbrum degeneraton and keep ths property of the orgnal fully reverse system wth detaled balance. Conversely, let a system satsfy the extended prncple of detaled balance: () the reacton rate constants of the reversble part of the reacton mechansm satsfy the classcal prncple of detaled balance and () the convex hull of the stochometrc vectors of the rreversble

6 reactons does not ntersect the lnear span of the stochometrc vectors of reversble reactons. Due to the classcal theorems of the convex geometry, there exsts a lnear functonal that separates ths convex set from the lnear subspace. (Strong separaton of closed and compact convex sets.) Ths separatng functonal can be represented n the form (x, θ) for some vector θ. For the reversble reactons (γ r, θ) = 0 and for the rreversble reactons (γ r, θ) < 0. It s possble to fnd vector δ wth ths separaton property and non-negatve coordnates. Indeed, accordng to the basc assumptons, there exsts a lnear conservaton law wth strongly postve coordnates. Ths s a vector b (b > 0) wth the property: (γ r, b) = 0 for all reactons. For any λ, the vector θ + λb has the same separaton property as the vector θ has. We can select such λ that δ = θ + λb 0 and δ = θ + λb = 0 for some. Let us take ths lnear combnaton δ as a vector of exponents. Let us create a fully reversble system from the ntal partally rreversble one. We do not change the reversble reactons and ther rate constants. Because the reversble reactons satsfy the classcal prncple of detaled balance, there exsts a strongly postve vector of equlbrum actvtes a > 0 for the reversble reactons. Let us take one such a vector. (A smple remark s needed here: for the components A j that do not partcpate n the reversble reactons we have to select arbtrary postve values a j > 0.) For each rreversble reacton wth the stochometrc vector γ r and reacton rate constant k r = k r + > 0 we add a reverse reacton wth the reacton rate constant kr = k r + (a ) γ r. For ths fully reversble system the actvtes a > 0 provde the pont of detaled balance. In the multscale degeneraton process, the equlbrum actvtes depend on ε 0 as a eq = a εδ. For the reactons wth (γ r, δ) = 0 the reacton rate constants do not depend on ε and for the reactons wth (γ r, δ) < 0 the rate constant kr tends to zero as ε (γr,δ) and k r + does not change. We return to the ntal system of reactons n the lmt ε 0. Ths s a partcular form of the extended prncple of detaled balance. For more dscusson see [12] Actvtes, concentratons and affntes To combne the lnear Lyapunov functons G δ (N) = (δ, N) (11) wth the classcal thermodynamc potental and study the knetc equatons n the closed form we 6 have to specfy the relatons between actvtes and concentratons. We accept the assumpton: a = c g (c, T), where g (c, T) > 0 s the actvty coeffcent. It s a contnuously dfferentable functon of c, T n the whole dapason of ther values. In a bounded regon of concentratons and temperature we can always assume that g > g 0 > 0 for some constant g 0. Ths assumpton s vald for the non-deal gases and for lqud solutons. It holds also for the surface gas n knetcs of heterogeneous catalyss [24] and does not hold for the sold reagents (see for example, analyss of carbon actvty n the methane reformng [12]). The system of unts should be commented. Tradtonally, a s assumed to be dmensonless and for perfect systems a = c /c, where c s an arbtrary standard concentraton. To avod ntroducton of unnecessary quanttes, we always assume that n the selected system of unts, c 1. If the thermodynamc potentals exst then due to the thermodynamc defnton of actvty (4), ths hypothess s equvalent to the logarthmc sngularty of the chemcal potentals, µ = RT ln c +... where... stands for a contnuous functon of c, T (all the concentratons and the temperature). In ths case, the free energy has the form F(N, T, V) = RT N (ln c 1 + f 0 (c, T)), (12) where the functons f 0 (c, T) are contnuously dfferentable for all possble values of arguments. Functons f 0 n the rght hand sde of the representaton (12) cannot be restored unambguously from the free energy functon F(N, T, V) but for a small admxture A t s possble to ntroduce the partal pressure p whch satsfes the law p = RTc + o(c ). Ths s due to the terms N ln c n F. Indeed, P = F(N, T, V)/ V = RTc + o(c ) + P c =0. Connectons between the equaton of state, free energy and knetcs are dscussed n more detal n [7, 8]. There are several smple algebrac corollares of the assumed connecton between actvtes and concentratons. Let us consder an elementary reacton α A β A wth α, β 0. Then, accordng to the generalzed mass acton law, for any vector of concentratons c (c 0) 1. If, for some, c = 0 then γ w(c) 0; 2. If, for some, c = 0 and γ < 0 then α > 0 and w(c) = 0. Smlarly, for a reversble reacton α A β A 1. If, for some, c = 0 and γ > 0 then β > 0 and w (c) = 0;

7 2. If, for some, c = 0 and γ < 0 then α > 0 and w + (c) = 0. These statements as well as Proposton 3 and Corollary 1 below are the consequences of the generalzed mass acton law (3) and the connecton between actvtes and concentratons wthout any assumptons about extended prncple of detaled balance. Each set of ndexes J = { 1,..., j } defnes a face of the postve polyhedron, F J = {c c 0 for all and c = 0 for J}. By defnton, the relatve nteror of F J, r(f J ), conssts of ponts wth c = 0 for J and c > 0 for J. Proposton 3. Let for a pont c r(f J ) and an ndex J γ r w r (c) = 0. Then ths dentty holds for all c F J. r Proof. For convenence, let us wrte all the drect and reverse reactons separately and represent the reacton mechansm n the form (2). All the terms n the sum r γ r w r (c) are non-negatve, because c = 0. Therefore, f the sum s zero then all the terms are zero. The reacton rate w r (3) wth non-zero rate constant takes zero value f and only f α r j > 0 and a j = 0 for some j. The equalty a = 0 s equvalent to c = 0. Therefore, w r (c) = 0 for a pont c r(f J ) f and only f there exsts j J such that α r j > 0. If α r j > 0 for an ndex j J then w r (c) = 0 for all c F J because c j = 0 n F J. We call a face F J of the postve orthant R n + nvarant wth respect to a set S of elementary reactons f r S γ r j w r (c) = 0 for all c F J and every j J. Let us consder the reacton mechansm n the form (2) where all the drect and reverse reactons partcpate separately. Corollary 1. The followng statements are equvalent: 1. r S γ r w r (c) = 0 for a pont c r(f J ) and all ndexes J; 2. The face F J s nvarant wth respect to the set of reactons S ; 3. The face F J s nvarant wth respect to every elementary reacton from S ; 4. For every r S ether γ r j = 0 for all j J or α r j > 0 for some j J. We am to perform the analyss of the asymptotc behavor of the knetc equatons n the multscale degeneraton of equlbrum descrbed n Sec For ths 7 purpose, we have to answer the queston: how the relatons between actvtes a and concentratons c depend on the degeneraton parameter ε 0? We do no try to fnd the maxmally general approprate answer to ths queston. For the known applcatons, the answer s: the relatons between a and c do not depend on ε 0. In partcular, t s trvally true for the deal systems. The smple generalzaton, a = c g (c, T, ε), where g (c, T, ε) > g 0 > 0 are contnuous functons, s not a generalzaton at all, because we can use for ε 0 the lmt case that does not depend on ε, g (c, T) = g (c, T, 0). Ths ndependence from ε mples that the reversble part of the reacton mechansm has the thermodynamc Lyapunov functons lke free energy. If we just delete the rreversble part then the classcal thermodynamcs s applcable and the thermodynamc potentals do not depend on ε. For the generalzed mass acton law, the tme dervatve of the relevant thermodynamc potentals have very nce general form. Let, under gven condton, the functon Φ(N,...) be gven, where by... s used for the quanttes that do not change n tme under these condtons. It s the thermodynamcs potental f Φ(N,...)/ N = µ. For example, t s the free Helmholtz energy F for V, T = const and the free Gbbs energy G for P, V = const. Let us calculate the tme dervatve of Φ(N,...) due to knetc equaton (10). The reacton rates are gven by the generalzed mass acton law (3) wth defnton of actvtes through chemcal potental (4). We assume that the prncple of detaled balance holds (t should hold for the reversble part of the reacton mechansm accordng to the extended detaled balance condtons). More precsely, there exsts an equlbrum wth detaled balance for any temperature T, a eq (T): for all r, w + r (a eq ) = w r (a eq ) = w eq r (T). It s convenent to represent the reacton rates usng these equlbrum fluxes w eq r (T): w + r = w eq α r (µ µ eq ) r exp RT, w r = w eq β r (µ µ eq ) r exp RT. where µ eq = µ (a eq, T). These formulas gve mmedately the followng representaton of the dsspaton rate dφ Φ(N,...) dn dn = = µ dt N dt dt (13) = VRT (ln w + r ln w r )(w + r w r ) 0. r

8 The nequalty holds because ln s a monotone functon and, hence, the expressons ln w + r ln w r and w + r w r have always the same sgn. Formulas of ths knd for dsspaton are well known snce the famous Boltzmann H-theorem (1873 [2], see also [13]). The entropy ncrease n solated systems has the smlar form: ds dt Let us notce that = VR (ln w + r ln w r )(w + r w r ) 0. ln w + r ln w r = 1 RT r µ (α r β r ) = (γ r, µ) RT. The quantty (γ r, µ) s one of the central noton of physcal chemstry, affnty [5]. It s postve f the drect reacton prevals over reverse one and negatve n the opposte case. It measures the energetc advantage of the drect reacton over the reverse one (free energy per mole). The actvty dvded by RT shows how large s ths energetc advantage comparng to the thermal energy. We call t the normalzed affnty and use a specal notaton for ths quantty: A r = (γ r, µ) RT Let us apply an elementary dentty exp a exp b = (exp a + exp b) tanh a b 2 to the reacton rate, w r = w + r w r : w r = (w + r + w r ) tanh A r 2. (14) Ths representaton of the reacton rates gves mmedately for the dsspaton rate: dφ = VRT (w + r + w r )A r tanh A r dt 2 0. (15) r In ths formula, the knetc nformaton s collected n the postve factors, the sums of reacton rates (w + r + w r ), and the purely thermodynamcal multplers A r tanh(a r /2) are also postve. For small A r, the expresson A r tanh(a r /2) behaves lke A 2 r /2 and for large A r t behaves lke the absolute value, A r. So, we have two Lyapunov functons for two fragments of the reacton mechansm. For the reversble part, ths s just a classcal thermodynamc potental. For the rreversble part, ths s a lnear functonal G δ (N) = (δ, N). More precsely, the rreversble reactons decrease ths functonal, whereas for the reversble reactons t s the conservaton law. Therefore, t decreases monotoncally n tme for the whole system Attractors 3.1. Dynamcal systems and lmt ponts The knetc equatons (10) do not gve a complete representaton of dynamcs. The rght hand sde ncludes the volume V and the reacton rates w r whch are functons (3) of the concentratons c and temperature T, whereas n the left hand sde there s Ṅ. To close ths system, we need to express V, c and T through N and quanttes whch do not change n tme. Ths closure depends on condtons. The smplest expressons appear for sochorc sothermal condtons: V, T = const, c = N/V. For other classcal condtons (U, V = const, or P, T = const, or H, P = const) we have to use the equatons of state. There may be more sophstcated closures whch nclude models or external regulators of the pressure and temperature, for example. Proposton 2 s vald for all possble closures. It s only mportant that the external flux of the chemcal components s absent. Further on, we assume that the condtons are selected, the closure s done, the rght hand sde of the resultng system s contnuously dfferentable and there exsts the postve bounded soluton for ntal data n R n + and V, T reman bounded and separated from zero. The nature of ths closure s not crucal. For some mportant partcular closures the proofs of exstence of postve and bounded solutons are well known (see, for example, [22]). Strctly speakng, such a system s not a dynamcal system n R n + but a semdynamcal one: the solutons may lose postvty and leave R n + for negatve values of tme. The theory of the lmt behavor of the sem-dynamcal systems was developed for applcatons to knetc systems [9]. We am to descrbe the lmt behavor of the systems as t. Under the extended detaled balance condton the lmt behavor s rather smple and the system wll approach steady states but to prove ths behavor we need the more general noton of the ω-lmt ponts. By the defnton, the ω-lmt ponts of a dynamcal system are the lmt ponts of the motons when tme t. We consder a knetc system n R n +. In partcular, for each soluton of the knetc equatons N(t) the set of the correspondng ω-lmt ponts s closed, connected and conssts of the whole trajectores ([9], Proposton 1.5). Ths means that the moton whch starts from an ω-lmt pont remans n R n + for all tme moments, both postve and negatve. Proposton 4. Let N(t) be a postve soluton of the knetc equaton and x be an ω-lmt pont of ths soluton and x = 0. then at ths pont ẋ x = 0.

9 Proof. Let x(t) be a soluton of the knetc equatons wth the ntal state x(0) = x. All the ponts x(t) ( < t < ) belong to R n +. Indeed, there exsts such a sequence t j that N(t j ) x. For any τ (, ), N(t j + τ) x(τ). For suffcently large j, t j + τ > 0 and the value N(t j + τ) R n +. Therefore, x(τ) R n + ( < τ < ) and for any τ the pont x(τ) s an ω-lmt pont of the soluton N(t). Let x = 0 and ẋ x = v 0. If v > 0 then for small τ and τ < 0 the value of x becomes negatve, x (τ) < 0. It s mpossble because postvty. Smlarly, If v < 0 then for small τ > 0 the value of x becomes negatve, x (τ) < 0. It s also mpossble because postvty. Therefore, ẋ x = 0. We use Proposton 4 n the followng combnaton wth Proposton 3. Let us wrte the reacton mechansm n the form (2). Corollary 2. If an ω-lmt pont belongs to the relatve nteror rf J of the face F J R n + then the face F J s nvarant wth respect to the reacton mechansm and for every elementary reacton ether γ r j = 0 for all j J or α r j > 0 for some j J. Proof. If an ω-lmt pont belongs to rf J then at ths pont all ċ j = 0 for j J due to Proposton 4. Therefore, we can apply Corollary Steady states of rreversble reactons Under extended detaled balance condtons, all the reacton rates of the rreversble reactons are zero at every lmt pont of the knetc equatons (10), due to Proposton 2. In ths secton, we gve a smple combnatoral descrpton of steady states for the set of rreversble reactons. Ths descrpton s based on Proposton 2 and, therefore, uses the extended detaled balance condtons. We contnue to study multscale degeneraton of a detaled balance equlbrum. The vector of exponents δ = (δ ) s gven, δ 0 for all and δ = 0 for some. There are two sets of reacton. For one of them, (γ r, δ) = 0 and n the lmt both k r ± > 0. In the second set, (γ r, δ) < 0 and n the lmt we assgn kr = 0 and k r + s the same as n the ntal system (before the equlbrum degeneraton). If t s necessary, we transpose the stochometrc equatons and swap the drect reactons wth reverse ones. For convenence, let us change the notatons. Let γ be the stochometrc vectors of reversble reactons wth (γ r, δ) = 0 (r = 1,..., h), and ν l be the stochometrc vectors for the reactons from the second set, (ν l, δ) < 0 (l = 1,..., s). For the reacton rates and constants for the frst set we keep the same notatons: 9 w r, w ± r, k r ±. For the second set, we use for the reacton rate constants q l = q + l and for the reacton rates v l = v + l. (They are also calculated accordng to the generalzed mass acton law (3).) The nput and output stochometrc coeffcents reman α r and β r for the frst set and for the second set we use the notatons α ν l and βν l. Let the rates of all the rreversble reacton be equal to zero. Ths does not mean that all the concentratons a wth δ > 0 acheve zero. A bmolecular reacton A + B C gves us a smple example: w = ka A a B and w = 0 f ether a A = 0 or a B = 0. On the plane wth coordnates a A, a B and wth the postvty condton, a A, a B 0, the set of zeros of w s a unon of two sem-axes, {a A = 0, a B 0} and {a A 0, a B = 0}. In more general stuaton, the set n the actvty space, where all the rreversble reactons have zero rates, has a smlar structure: t s the unon of some faces of the postve orthant. Let us descrbe the set of the steady states of the rreversble reactons. Due to Proposton 2, f l v l ν l = 0 then all v l = 0. Let us descrbe the set of zeros of all v l n the the postve orthant of actvtes. For every l = 1,..., s the set of zeros of v l n R n + s gven by the condtons: at least for one α ν l 0 and a = 0. It s convenent to represent ths condton as a dsjuncton. Let J l = { α ν l 0}. Then the set of zeros of v l an a postve orthant of actvtes s presented by the formula J l (a = 0). The set of zeros of all v l s represented by the followng conjuncton form ( Jl (a = 0) ). (16) s l=1 To transform t nto the unons of subspaces we have to move to a dsjuncton form and make some cancelatons. Frst of all, we represent ths formula as a dsjuncton of conjunctons: ( Jl (a = 0) ) s l=1 = 1 J 1,..., s J s ( (a1 = 0)... (a s = 0) ). (17) For a cortege of ndexes { 1,..., s } the correspondent set of ther values may be smaller because some values l may concde. Let ths set of values be S {1,..., s }. We can delete from (17) a conjuncton (a 1 = 0)... (a s = 0) f there exsts a cortege { 1,..., s} ( l J l) wth smaller set of values, S {1,..., s } S { 1,..., s}. Let us check the corteges n some order and delete a conjuncton from (17) f there reman a term wth smaller (or the same) set of ndex values n the formula. We can also substtute n (17) the corteges by ther sets of values. The resultng mnmzed formula may become shorter. Each elementary conjuncton represents a coordnate subspace and after cancelatons each ths subspace does not belong to a

10 unon of other subspaces. The fnal form of formula (17) s j ( S j (a = 0)), (18) where S j are sets of ndexes, S j {1,..., n} and for every two dfferent S j, S p none of them ncludes another, S j S p. The elementary conjuncton S j (a = 0) descrbes a subspace. The steady states of the rreversble part of the reacton mechansm are gven by the ntersecton of the unon of the coordnate subspaces (18) wth R n +. For applcatons of ths formula, t s mportant that the equaltes a = 0, c = 0 and N = 0 are equvalent and the postve orthants of the actvtes a, concentratons c or amounts N represent the same sets of physcal states. Ths s also true for the faces of these orthants: F J for the actvtes, concentratons or amounts correspond to the same sets of states. (The same state may corresponds to the dfferent ponts of these cones, but the totaltes of the states are the same.) 3.3. Sets of steady states of rreversble reactons nvarant wth respect to reversble reactons In ths Sec. we study the possble lmt behavor of systems whch satsfy the extended detaled balance condtons and nclude some rreversble reactons. All the ω-lmt ponts of such systems are steady states of the rreversble reactons due to Proposton 2 but not all these steady states may be the ω-lmt ponts of the system. A smple formal example gves us the couple of reacton: A B, A + B C. Here, we have one reversble and one rreversble reacton. The condtons of the extended detaled balance hold (trvally): the lnear span of the stochometrc vector of the reversble reacton, ( 1, 1, 0), does not nclude the stochometrc vector of the rreversble reacton, ( 1, 1, 1). For the descrpton of the multscale degeneraton of equlbrum, we can take the exponents δ A = 1, δ B = 1, δ C = 0. The steady states of the rreversble reacton are gven n R n + by the dsjuncton, (c A = 0) (c B = 0) but only the ponts (c A = c B = 0) may be the lmt ponts when t. Indeed, f c A = 0 and c B > 0 then dc A /dt = k1 c B > 0. Due to Proposton 4 ths s not an ω-lmt pont. Smlarly, the ponts wth c A > 0 and c B = 0 are not the ω-lmt ponts. Let us combne Propostons 2, 4 and Corollary 2 n the followng statement. Theorem 2. Let the knetc system satsfy the extended detaled balance condtons and nclude some rreversble reactons. Then an ω-lmt pont x rf J exsts f and only f F J conssts of steady states of the 10 rreversble reactons and s nvarant wth respect to all reversble reactons. Proof. If an ω-lmt pont x rf J exsts then t s a steady state for all rreversble reactons (due to Propostons 2). Therefore, the face F J conssts of steady-states of the rreversble reactons (Proposton 4) and s nvarant wth respect to all reversble reactons (Proposton 4 and Corollary 2). To prove the reverse statement, let us assume that F J conssts of steady states of the rreversble reactons and s nvarant wth respect to all reversble reactons. The reversble reactons whch do not act on c j for j J defne a sem-dynamcal system on F J. The postve conservaton law b defnes an postvely nvarant polyhedron n F J. Dynamcs n such a compact set always has ω-lmt ponts. Let us fnd the faces F J that contan the ω-lmt ponts n ther relatve nteror rf J. Accordng to Theorem 2, these faces should consst of the steady states of the rreversble reactons and should be nvarant wth respect to all reversble reactons. Let us look for the maxmal faces wth ths property. For ths purpose, we always mnmze the dsjunctve forms by cancelatons. We do not lst the faces that contan the ω-lmt ponts n ther relatve nteror and are the proper subsets of other faces wth ths property. All the ω-lmt ponts belong to the unon of these maxmal faces. Let us start from the mnmzed dsjunctve form (18). Equaton (18) represents the set of the steady states of the rreversble part of the reacton mechansm by a unon of the coordnate subspaces S j (c = 0) n ntersecton wth R n +. It s the unon of the faces, j F S j. If a face F J conssts of the steady states of the rreversble reactons then J S j for some j. The followng formula (19) s true on a face F J f t contans ω-lmt ponts n the relatve nteror rf J (Theorem 2): (c = 0) [( r,γr >0 j,αr j >0 (c j = 0) ) ( r,γr <0 j,βr j >0 (c j = 0) )]. (19) Here, c = 0 n F J may be read as J. Followng the prevous secton, we use here the notatons γ r, β r and β r for the reversble reactons and reserve ν l, α ν l and β ν l for the rreversble reactons. The set of γ r n ths formula s the set of the stochometrc vectors of the reversble reactons. The requred faces F J may be constructed n an teratve procedure. Frst of all, let us ntroduce an operaton that transforms a set of ndexes S {1, 2,..., n} n a famly of sets, S(S ) = {S 1,..., S l }. Let us take formula

11 (19) and fnd the set where t s vald for all S. Ths set s descrbed by the followng formula: S [ (c = 0) ( r,γr >0 j,αr j >0 (c j = 0) ) ( r,γr <0 j,βr j >0 (c j = 0) )]. (20) Let us produce a dsjunctve form of ths formula and mnmze t by cancelatons as t s descrbed n Sec The result s j=1,...,k ( S j (c = 0) ). (21) Because of cancelatons, the sets S j do not nclude one another. They gve the result, S(S ) = {S 1,..., S l }. Each S j S(S ) s a superset of S, S S. Let us extend the operaton S on the sets of sets S = {S 1,..., S p } wth the property: S S j for j. Let us apply S to all S and take the unon of the results: S 0 (S) = S(S ). Some sets from ths famly may nclude other sets from t. Let us organze cancelatons: f S, S S 0 (S) and S S then retan the smallest set, S, and delete the largest one. We do the cancelatons untl t s possble. Let us call the fnal result S(S). It does not depend on the order of these operatons. Let us start from any famly S and terate the operaton S. Then, after fnte number of teratons, the sequence S d (S) stablzes: S d (S) = S d+1 (S) =... because for any set S all sets from S(S ) nclude S. The problems of propostonal logc that arse n ths and the prevous secton seem very smlar to elementary logcal puzzles [3]. In the soluton we just use the logcal dstrbuton laws (dstrbuton of conjuncton over dsjuncton and dstrbuton of dsjuncton over conjuncton), commutatvty of dsjuncton and conjuncton, and elementary cancelaton rules lke (A A) A, (A A) A, [A (A B)] A, and [A (A B)] A. Now, we are n poston to descrbe the constructon of all F J that have the ω-lmt ponts on ther relatve nteror and are the maxmal faces wth ths property. 1. Let us follow Sec. 3.2 and construct the mnmzed dsjunctve form (18) for the descrpton of the steady states of the rreversble reactons. 2. Let us calculate the famles of sets S d ({S j }) for the famly of sets {S j } from (18) and d = 1, 2,..., untl stablzaton. 3. Let S d ({S j }) = S d+1 ({S j }) = {J 1, J 2,... J p }. Then the famly of the faces F J ( = 1, 2,..., p) gves the answer: the ω-lmt ponts are stuated n rf J and for each there are ω-lmt ponts n rf J Smple examples In ths Sec., we present two smple and formal examples of the calculatons descrbed n the prevous sectons. 1. A 1 +A 2 A 3 +A 4, γ = ( 1, 1, 1, 1, 0); A 1 +A 2 A 5, ν = ( 1, 1, 0, 0, 1). The extended prncple of detaled balance holds: the convex hull of the stochometrc vectors of the rreversble reactons conssts of one vector γ 2 and t s lnearly ndependent of γ 1. The nput vector α for the rreversble reacton A 1 + A 2 A 5 s ( 1, 1, 0, 0, 0). The set J = J l from the conjuncton form (16) s defned by the non-zero coordnates of ths α ν : J = {1, 2}. The conjuncton form n ths smple case (one rreversble reacton) loses ts frst conjuncton operaton and s just (c 1 = 0) (c 2 = 0). It s, at the same tme, the mnmzed dsjuncton form (18) and does not requre addtonal transformatons. Ths formula descrbes the steady states of the rreversble reacton n the postve orthant R n +. For ths dsjuncton form, The famly of sets S = {S j } conssts of two sets, S 1 = {1} and S 2 = {2}. Let us calculate S(S 1,2 ). For both cases, = 1, 2 there are no reversble reactons wth γ r = 0. Therefore, one expresson n round parentheses vanshes n (20). For S = {1} ths formula gves and for S = {2} t gves (c 1 = 0) ((c 3 = 0) (c 4 = 0)) (c 2 = 0) ((c 3 = 0) (c 4 = 0)). The elementary transformatons gve the dsjunctve forms: [(c 1 = 0) ((c 3 = 0) (c 4 = 0))] [((c 1 = 0) (c 3 = 0)) ((c 1 = 0) (c 4 = 0))], [(c 2 = 0) ((c 3 = 0) (c 4 = 0))] [((c 2 = 0) (c 3 = 0)) ((c 2 = 0) (c 4 = 0))]. Therefore, S(S 1 ) = {{1, 3}, {1, 4}}, S(S 2 ) = {{2, 3}, {2, 4}} and S({S 1, S 2 }) = {{1, 3}, {1, 4}, {2, 3}, {2, 4}}. No cancelatons are needed. The teratons of S do not produce new sets from {{1, 3}, {1, 4}, {2, 3}, {2, 4}}. Indeed, f c 1 = c 3 = 0, or c 1 = c 4 = 0, or c 2 = c 3 = 0, or c 2 = c 4 = 0 then all the reacton rates are zero. More formally, for example for S({1, 3}) formula (20) gves [(c 1 = 0) ((c 3 = 0) (c 4 = 0))] [(c 3 = 0) ((c 1 = 0) (c 2 = 0))].

12 Ths formula s equvalent to (c 1 = 0) (c 3 = 0). Therefore, S({1, 3}) = {1, 3}. The same result s true for {1, 4}, {2, 3}, and {2, 4}. All the ω-lmt ponts (steady states) belong to the faces F {1,3} = {c, c 1 = c 3 = 0}, F {1,4} = {c, c 1 = c 4 = 0}, F {2,3} = {c, c 2 = c 3 = 0}, or F {2,4} = {c, c 2 = c 4 = 0}. The poston of the ω-lmt pont for a soluton N(t) depends on the ntal state. More specfcally, ths system of reactons has three ndependent lnear conservaton laws: b 1 = N 1 + N 2 + N 3 + N 4 + 2N 5, b 2 = N 1 N 2 and b 3 = N 3 N 4. For gven values of these b 1,2,3 vector N belongs to the 2D plane n R 5. The ntersecton of ths plane wth the selected faces depends on the sgns of b 2,3 : If b 2 < 0, b 3 < 0 then t ntersects F {1,3} only, at one pont N = (0, b 2, 0, b 3, b 1 + b 2 + b 3 ) (N 5 should be non-negatve, b 1 + b 2 + b 3 0). If b 2 = 0, b 3 < 0 then t ntersects both F {1,3} and F {2,3} at one pont N = (0, 0, 0, b 3, b 1 + b 3 ) (N 5 should be non-negatve, b 1 + b 3 0). If b 2 < 0, b 3 = 0 then t ntersects both F {1,3} and F {1,4} at one pont N = (0, b 2, 0, 0, b 1 + b 2 ) (N 5 should be non-negatve, b 1 + b 2 0). If b 2 > 0, b 3 < 0 then t ntersects F {2,3} only, at one pont N = (b 2, 0, 0, b 3, b 1 + b 2 + b 3 ) (N 5 should be non-negatve, b 1 + b 2 + b 3 0). If b 2 > 0, b 3 = 0 then t ntersects F {2,3} and F {2,4} at the pont N = (b 2, 0, 0, 0, b 1 + b 2 ) (N 5 s nonnegatve because b 1 + b 2 + b 3 0). If b 2 < 0, b 3 > 0 then t ntersects F {1,4} only, at one pont N = (0, b 2, b 3, 0, b 1 + b 2 + b 3 ) (N 5 should be non-negatve, b 1 + b 2 + b 3 0). Table 1: H 2 burnng mechansm [21] No Reacton Stochometrc vector 1 H 2 + O 2 2OH (-1,-1,2,0,0,0,0,0) 2 H 2 + OH H 2 O + H (-1,0,-1,1,1,0,0,0) 3 OH + O O 2 + H (0,1,-1,0,1,-1,0,0) 4 H 2 + O OH + H (-1,0,1,0,1,-1,0,0) 5 O 2 + H +M HO 2 +M (0,-1,0,0,-1,0,1,0) 6 OH + HO 2 O 2 + H 2 O (0,1,-1,1,0,0,-1,0) 7 H + HO 2 2OH (0,0,2,0,-1,0,-1,0) 8 O + HO 2 O 2 + OH (0,1,1,0,0,-1,-1,0) 9 2OH H 2 O + O (0,0,-2,1,0,1,0,0) 10 2H + M H 2 + M (1,0,0,0,-2,0,0,0) 11 2H + H 2 H 2 + H 2 (1,0,0,0,-2,0,0,0) 12 2H + H 2 O H 2 + H 2 O (1,0,0,0,-2,0,0,0) 13 OH + H + M H 2 O + M (0,0,-1,1,-1,0,0,0) 14 H + O + M OH + M (0,0,1,0,-1,-1,0,0) 15 2O + M O 2 + M (0,1,0,0,0,-2,0,0) 16 H + HO 2 H 2 + O 2 (1,1,0,0,-1,0,-1,0) 17 2HO 2 O 2 + H 2 O 2 (0,1,0,0,0,0,-2,1) 18 H 2 O 2 + M 2OH + M (0,0,2,0,0,0,0,-1) 19 H + H 2 O 2 H 2 + HO 2 (1,0,0,0,-1,0,1,-1) 20 OH + H 2 O 2 H 2 O + HO 2 (0,0,-1,1,0,0,1,-1) prncple of detaled balance holds. The steady-states of the rreversble reactons s gven by one equaton, c 5 = 0. Formula (20) gves for S({5}) just (c 5 = 0). The face F {5} ncludes ω-lmt ponts n rf {5}. Dynamcs on ths face s defned by the fully reversble reacton system and tends to the equlbrum of the reacton A 1 + A 2 A 3 + A 4 under the gven conservaton laws. On ths face, there exst the border equlbra, where c 1 = c 3 = 0, or c 1 = c 4 = 0, or c 2 = c 3 = 0, or c 2 = c 4 = 0 but they are not attractng the postve solutons. 4. Example: H 2 +O 2 system If b 2 = 0, b 3 > 0 then t ntersects F {1,4} and F {2,4} at one pont N = (0, 0, b 3, 0, b 1 + b 3 ) (N 5 s nonnegatve because b 1 + b 3 0). If b 2 > 0, b 3 > 0 then t ntersects F {2,4} only, at one pont N = (b 2, 0, b 3, 0, b 1 + b 2 + b 3 ) (N 5 s nonnegatve because b 1 + b 2 + b 3 0). As we can see, the system has exactly one ω-lmt pont for any admssble combnaton of the values of the conservaton laws. These ponts are the lsted ponts of ntersecton. For the second smple example, let us change the drecton of the rreversble reacton. 2. A 1 + A 2 A 3 + A 4, γ 1 = ( 1, 1, 1, 1, 0), A 5 A 1 + A 2, ν = (1, 1, 0, 0, 1). The extended 12 For the case study, we selected the H 2 +O 2 system. Ths s one of the man model systems of gas knetcs. The hydrogen burnng gves us an example of the medum complexty wth 8 components (A 1 =H 2, A 2 =O 2, A 3 =OH, A 4 =H 2 O, A 5 =H, A 6 =O, A 7 =HO 2, and A 8 =H 2 O 2 ) and 2 atomc balances (H and O). For the example, we selected the reacton mechansm from [21]. The lterature about hydrogen burnng mechansms s huge. For recent dscusson we refer to [16, 18]. A specal symbol M s used for the thrd body. It may be any molecule. The thrd body provdes the energy balance. Effcency of dfferent molecules n ths process s dfferent, therefore, the concentraton of the thrd body s a weghted sum of the concentratons of the components wth postve weghts. The thrd body does