Thermodynamic analysis of chemically reacting mixtures – comparison of first and second order models

HYPOTHESIS AND THEORY published: 01 Ma ch 2018 doi: 10.3389/ chem.2018.00035 F on ie s in Chemis y | www. on ie sin.o g 1Ma ch 2018 | Volume 6 | A icle 35 Edi ed by: Ramesh L. Ga das, Indian Ins i u e o Technology Mad as, India Re iewed by: E geni B. S a iko , Independen Resea che , Ge many Miguel Rubi, Uni e si a de Ba celona, Spain *Co espondence: Milosla Pekaˇ [email p o ec ed] Special y sec ion: This a icle was submi ed o Physical Chemis y and Chemical Physics, a sec ion o he jou nal F on ie s in Chemis y Recei ed: 02 Decembe 2017 Accep ed: 12 Feb ua y 2018 Published: 01 Ma ch 2018 Ci a ion: Pekaˇ M (2018) The modynamic Analysis o Chemically Reac ing Mix u es—Compa ison o Fi s and Second O de Models. F on . Chem. 6:35. doi: 10.3389/ chem.2018.00035 The modynamic Analysis o Chemically Reac ing Mix u es—Compa ison o Fi s and Second O de Models Milosla Pekaˇ * Ins i u e o Physical and Applied Chemis y and Ma e ials Resea ch Cen e, Facul y o Chemis y, B no Uni e si y o Technology, B no, Czechia Recen ly, a me hod based on non-equilib ium con inuum he modynamics which de i es he modynamically consis en eac ion a e models oge he wi h he modynamic cons ain s on hei pa ame e s was analyzed using a iangula eac ion scheme. The scheme was kine ically o he i s o de . He e, he analysis is u he de eloped o se e al i s and second o de schemes o gain a deepe insigh in o he he modynamic consis ency o a e equa ions and ela ionships be ween chemical he modynamic and kine ics. I is shown ha he he modynamic cons ain s on he so-called p ope a e coe icien a e usually simple sign es ic ions consis en wi h he supposed eac ion di ec ions. Cons ain s on he so-called coupling a e coe icien s a e mo e complex and weake . This means mo e eedom in kine ic coupling be ween eac ion s eps in a scheme, i.e., in he kine ic e ec s o o he eac ions on he a e o some eac ion in a eac ing sys em. When compa ed wi h adi ional mass-ac ion a e equa ions, he me hod allows a educ ion in he numbe o adi ional a e cons an s o be e alua ed om da a, i.e., a educ ion in he dimensionali y o he pa ame e es ima ion p oblem. This is due o iden i ying ela ionships be ween mass-ac ion a e cons an s ( ela ionships which also include he modynamic equilib ium cons an s) which ha e so a been unknown. Keywo ds: a ini y, en opic inequali y, independen eac ions, kine ics, a e cons an s, a e equa ions, he modynamics INTRODUCTION In es iga ing impac s o he modynamics on kine ics o chemical eac ions is an a ea o un lagging in e es and con inuous esea ch. Due o he inhe en non-equilib ium na u e o ongoing chemical eac ions, especially he non-equilib ium he modynamics b ings signi ican p og ess o he aim o pu he modynamics and kine ics in a common amewo k ( o example, Pagonaba aga e al., 1997; Bedeaux e al., 2010; Kannan and Rajagopal, 2011; Klika and G mela, 2013; Rubí e al., 2013; A a o and Mo o, 2014; Bo he and D eye , 2015; Ge and Qian, 2016; N’Guyen e al., 2016). Recen ly, a pape was published desc ibing he de ailed (non-equilib ium) he modynamic analysis o chemically eac ing mix u es wi h consequences o kine ic models (Pekaˇ , 2016). A mix u e o h ee isome s was used as an example o illus a e his app oach, i.e., a i s o de kine ic model was analyzed. An anonymous e iewe aised he ques ion o wha he e ec would be o including a second o de eac ion (a bimolecula eac ion, in his case, bu o he pu pose o his Pekaˇ The modynamic Analysis o Reac ing Sys ems wo k he dis inc ion be ween o de and molecula i y is no essen ial; only he e m o de will be used hence o h). Because he answe o his in e es ing and impo an ques ion is nei he sho no simple, p o iding i was pos poned un il his wo k. The abo emen ioned app oach (Pekaˇ , 2016) a emp s o ind some he modynamic es ic ions on chemical kine ics (Boyd, 1977). The majo i y o hese app oaches a e o an a pos e io i ype and use p inciples al eady es ablished in equilib ium he modynamics (Qian, 2007; Ede e and Gilles, 2008; Al- Kha eeb e al., 2009; Vlad and Ross, 2009; Fleming e al., 2010; He nández-Lemus e al., 2014; Jones e al., 2015). A pos e io i in his con ex means ha a eac ion scheme is designed i s and hen he modynamics is applied o co esponding kine ic ( a e) equa ions. The me hod o his (and he p eceding) pape is o an a p io i ype—i begins only wi h a lis o componen s o a eac ing mix u e. Then, he esul s o non-equilib ium he modynamics (Pekaˇ and Samohýl, 2014) a e combined wi h he ma hema ics o s oichiome y (Bowen, 1968) o ind independen eac ions and, subsequen ly, hei inal a e equa ions in he o m o a he modynamic polynomial. This polynomial is e y close o adi ional mass-ac ion a e equa ions. The a e equa ions a e hen analyzed o hei consis ency wi h a condi ion eme ging om he en opic inequali y ( he second law) gi en in mo e de ail below (see Equa ion 2). The me hod no only di ec ly de i es he modynamically consis en and su icien a e equa ions bu also gi es cons ain s on hei coe icien s ( ep esen ing, in ac , a e cons an s) equi ed o ensu e consis ency wi h he second law. A his s age o de elopmen he me hod is wo ked ou o a eac ing mix u e o linea luids (Pekaˇ and Samohýl, 2014), which, howe e , comp ises many sys ems o in e es o chemis s. The main s eps o he me hod a e b ie ly as ollows. Fi s , he numbe o independen eac ions is de e mined on he basis o he lis o componen s o a eac ing mix u e and hei a omic composi ions (Bowen, 1968; Pekaˇ and Samohýl, 2014; Pekaˇ , 2016). Non-equilib ium con inuum he modynamics p o es (Samohýl, 1982; Pekaˇ and Samohýl, 2014) ha eac ion a es (o independen eac ions) a e unc ions o concen a ions and empe a u e: J=J(T,c1,c2,..., cn)=J(T,c). This unc ion is app oxima ed by a polynomial in concen a ions: J= Z X β=1 kE νβ n Y α=1 cνβα α, n X α=1 νβα ≤M(1) He e, Jis he ec o whose componen s a e he a es o R independen eac ions, J=(J1,J2,..., JR), ciis he mola concen a ion o componen i, and nis he o al numbe o componen s. The ec o kE νβcon ains polynomial coe icien s dependen on empe a u e only, he ec o s E νβ=(νβ1,νβ2,..., νβn) con ain polynomial powe s and a e also used as subsc ip s o index a ious ec o s o polynomial coe icien s (kE νβ), and M is he polynomial deg ee. Fo he o al numbe o e ms Z, see Bowen (1968) and Pekaˇ and Samohýl (2014). The equilib ium condi ion is applied and he polynomial is modi ied o a simpli ied inal o m, called he he modynamic polynomial, which con ains also he modynamic equilib ium cons an s; no e e sed a e cons an s a e used. Al hough he second law was al eady applied in he applica ion o esul s o non-equilib ium he modynamics men ioned abo e, he e is s ill a condi ion esul ing om his law which is no used and is igno ed in o he wo ks. This condi ion e e s o eac ion a es exp essed as unc ions o (chemical) a ini ies (Ap) and eads (Pekaˇ and Samohýl, 2014): R X p=1 R X =1 (∂J /∂Ap)eqApA ≤0 (2) Equa ion (2) is a nega i e semide ini e quad a ic o m in a ini ies and he well-known condi ions o a quad a ic o m o be nega i e semide ini e can be applied. These condi ions gi e es ic ions on a e coe icien s ( a e cons an s) in he he modynamic polynomial; in o he wo ds, hey pu addi ional condi ions on he he modynamic consis ency o a e equa ions. Howe e , he ans o ma ion o he abo e unc ion o concen a ions, J(T,c), o he unc ion o a ini ies is a ma hema ical p ocedu e which equi es co ec and igo ous s eps. These s eps include exp essing chemical po en ials as unc ions o concen a ions, esul ing in he appea ance o bo h chemical and so-called cons i u i e a ini ies (Pekaˇ and Samohýl, 2014; Pekaˇ , 2016). The esul s a e gene ally J(T,c)=J(T,µ) =J(T,A,B), whe e he ec o µcon ains chemical po en ials, he ec o Acon ains chemical a ini ies, and he ec o B con ains cons i u i e a ini ies. Condi ion (2) is applied o he inal unc ion o ind es ic ions on a e pa ame e s (cons an s). OVERVIEW OF ANALYZED REACTING SYSTEMS The me hod desc ibed in p e ious wo ks (Pekaˇ and Samohýl, 2014; Pekaˇ , 2016) and ou lined in he in oduc ion is applied he e on ou di e en eac ing mix u es in o de o compa e i s ou comes o he esul ing i s and second o de kine ic models. Fi s , all ou eac ing sys ems a e o e iewed oge he wi h hei models o di e en o de s which esul om (1). Then, condi ion (2) is applied on all models. Al hough in his me hod he o de is a ma e o he deg ee o he app oxima ing polynomial in (1) and no a ma e o a p io i kine ic conside a ions, hese wo iews a e ela ed. To demons a e his is a u he aim o his wo k. Be o ehand, howe e , he ele an e minology should be cla i ied. S ic ly speaking, deg ee e e s o he polynomial in (1) and o de o adi ional a e equa ions. Howe e , hese wo e ms a e closely connec ed, as will be seen in wha ollows. Polynomial deg ees a e whole numbe s only and in a kine ic in e p e a ion o he he modynamic polynomial hey co espond o o de s—in his sense and acco ding o his in e p e a ion he wo concep s can be used and should be unde s ood in e changeably. The i s eac ing sys em is a simple mix u e o wo isome s, A and B. Chemically, he simples eac ion he e is assumed o be in he o m: A⇆B, (R1a) F on ie s in Chemis y | www. on ie sin.o g 2Ma ch 2018 | Volume 6 | A icle 35 Pekaˇ The modynamic Analysis o Reac ing Sys ems which, as w i en, in okes i s o de kine ics. I s second o de e sion can be exp essed in adi ional kine ics as 2A ⇆2B. (R1b) In his mix u e, only one independen eac ion is possible; he ec o Jis hus one-dimensional. Reac ion (R1a) is selec ed as he independen eac ion and he second-deg ee he modynamic polynomial can be w i en as ( o de i a ion see Supplemen a y Ma e ial): J=k10(cA−K−1cB) +k20(c2 A−K−1cAcB)+k02(c2 B−KcAcB) (3) whe e he ec o s ka e also one-dimensional and Kis he he modynamic equilib ium cons an o he independen eac ion. A he same ime, he i s e m on he igh hand side o (3) ep esen s he i s -deg ee he modynamic polynomial. The second analyzed sys em is composed o wo single componen s, A and B, and hei compound AB. He e, he combina ion eac ion A+B⇄AB (R2) occu s na u ally and is o he second o de ( o wa d di ec ion). Also he e, only one independen eac ion exis s— he mos na u al selec ion is jus he eac ion (R2). The i s -deg ee he modynamic polynomial anishes in his eac ing mix u e; consequen ly, only he second-deg ee (o possibly highe ) polynomial is easonable. I has he ollowing e y simple o m J=k110(cAcB−K−1cAB), (4) which esembles he adi ional mass-ac ion exp ession; K is he he modynamic equilib ium cons an o he selec ed independen eac ion (R2). No e ha , he e also, he ec o J is one-dimensional. The anishing i s -deg ee he modynamic polynomial is a na u al consequence o he ma hema ical non- exis ence o pai ed pu e i s -powe e ms in he polynomial. Howe e , his does no mean ha kine ically i s -o de e ms a e missing—c . he las e m on he igh hand side o (4). The hi d sys em is jus he mix u e o h ee isome s (A, B, and C) analyzed in he p e ious wo k, bu , he e, modeled wi h he second deg ee he modynamic polynomial: J=k100(cA−K−1 1cB)+k001(cC−K2cB)+k020(c2 B−K2 1c2 A) +k002(c2 C−K2 1K2 2c2 A)+k110(cAcB−K1c2 A)+k101(cAcC −K1K2c2 A)+k011(cBcC−K2 1K2c2 A) (5) He e, wo independen eac ions a e possible and a e selec ed (Pekaˇ , 2016) as A =B and B =C wi h he co esponding he modynamic equilib ium cons an s K1and K2, esp. The ec o s Jand ka e hus wo-dimensional, he o me con aining he a es o he wo independen eac ions: J=(J1,J2). The i s wo e ms on he igh hand side o (5) ep esen he i s -deg ee he modynamic polynomial which was analyzed in he p e ious wo k (Pekaˇ , 2016). T adi ionally, “ iangula ” eac ions, (R3), a e assumed o occu in his mix u e, one o hem being dependen . AB C 1 2 3 4 56 (R3) The las ( ou h) sys em is simila and is ha sugges ed by he e iewe o he p e ious wo k. He e, also, a compound o wo isome s is included and he mix u e hus con ains he ollowing componen s: A, B, AB, BA whe e (only) AB and BA a e isome s. Analogically o (R3), a iangula scheme (R4) is sugges ed he e in adi ional kine ics. The numbe o independen eac ions is s ill wo. The second- deg ee he modynamic polynomial is: J=k0001(cBA −K2cAB)+k1100(cAcB−K−1 1cAB) +k0002(c2 BA −K2 2c2 AB)+k0011(cABcBA −K2c2 AB) +k1001(cAcBA −K2cAcAB)+k0101(cBcBA −K2cBcAB)(6) The ec o s Jand ka e wo-dimensional, he i s e m on he igh hand side o (6) ep esen ing he only possible i s -deg ee e m. The wo independen eac ions a e selec ed as A +B=AB and AB =BA wi h co esponding he modynamic equilib ium cons an s K1and K2, esp. A + B AB BA 1 2 3 4 56 (R4) RESULTS AND DISCUSSION Now, he esul s o he he modynamic analysis o hese ou eac ing sys ems a e discussed. C ucial o his analysis is he applica ion o condi ion (2). Reac ing Sys em 1; A Mix u e o Two Isome s The i s -deg ee he modynamic polynomial con ains only one a e coe icien (also called he e “ a e cons an ”), namely k10 ( he bold symbol is no e ained due o he one-dimensionali y o k10 om (3), simila ly o o he one-dimensional quan i ies). The ans o med he modynamic polynomial co esponding o he unc ion J(T,A,B) is ( o de i a ion see Supplemen a y Ma e ial): J=k10 exp −µo A RT exp B RT exp A 2RT exp −A RT −1(7) F on ie s in Chemis y | www. on ie sin.o g 3Ma ch 2018 | Volume 6 | A icle 35 Pekaˇ The modynamic Analysis o Reac ing Sys ems whe e ◦deno es he s anda d s a e and Aand Ba e he wo one- dimensional a ini ies. Condi ion (2) esul s he e in he ollowing simple exp ession: ∂J ∂Aeq = −(1/RT)k10 exp −µo A RT exp Beq RT ≤0 (8) and om his i ollows ha k10 >0 (k10 =0 makes no sense in a eac ing mix u e), which is consis en wi h A being he eac an and wi h i s componen a e JAbeing equal o –J, c . also (3), whe e JA ep esen s eac an A’s o ma ion a e. Thus, he adi ional mass-ac ion kine ics exp essed in he o m o he i s -deg ee e sion o (3), including he sign o he a e cons an , is ully consis en wi h non-equilib ium he modynamics— pa icula ly, wi h en opic inequali y ( he second law). I should be no ed ha he p esen ed me hod does no inhe en ly include es ic ions on he non-nega i i y o concen a ions; hese should be added as addi ional cons ain s. In he case o he second-deg ee polynomial, he ans o ma ion o he unc ion J(T,A,B) is mo e complex han (7) and can be ound in Supplemen a y Ma e ial. He e, only he inal es ic ion esul ing om (2) is shown: k10 +k20cA,eq −k02K2cA,eq ≥0, (9) which should be ul illed o an a bi a y equilib ium concen a ion. This enables he ollowing heo em1 o be applied ( he heo em was no necessa y in he i s -deg ee models). Theo em. I he inequali y a+bx ≥0, (10) whe e a,b,xa e eal numbe s, is alid o any posi i e x, hen i is necessa y and su icien ha a≥0, b≥0. (11) P oo . The su iciency is ob ious. The necessi y is p o en by con adic ion. Fi s , (10) is no ul illed o any x>0 i a,ba e chosen as any o hese h ee combina ions: 1) a=0, b<0; 2) a< 0, b=0; 3) a<0, b<0. I a,ba e chosen as a>0, b<0 o a< 0, b>0 hen (10) is no ul illed o any (posi i e) x<–a/b>0. Fo all o he combina ions o aand b, i.e., hose sa is ying (11), a posi i e xno ul illing (10) does no exis . Q.E.D. No e ha he heo em can be easily modi ied o c+dx ≤0 ins ead o (10); in his case, c≤0, d≤0. The heo em e iden ly also enables he condi ion o he non-nega i i y o concen a ions o be included, albei in an indi ec way. The heo em hus gi es: k10 ≥0 (12) k20 −k02K2≥0 (13) The i s condi ion (12) is he same as ha ound abo e o he i s -deg ee polynomial. Thus, bo h polynomials gi e consis en 1The heo em and i s p oo a e p o ided by Ví Samohýl. condi ions o he i s o de a e cons an (k10). Re e ing o e minology in oduced in a p e ious wo k (Pekaˇ , 2016), he coe icien k10 is a p ope coe icien , while k20 and k02 a e examples o coupling coe icien s. Howe e , in his case, he coupling coe icien s a e o a somewha di e en ype han in Pekaˇ (2016), whe e hey coupled he selec ed independen eac ions in hei a e equa ions. He e, hey couple he a e o he independen eac ion wi h exp essions which could be iewed as a es o addi ional, dependen , eac ions, c . (3). Le us u he suppose ha k20 ≥0 (14) Then, condi ion (13) is ul illed (also) o k02 = −K−2k20, in which case k20 +k02K2=0. Consequen ly, eac ion a e (3) is ans o med in o he ollowing exp ession: J=k10(cA−K−1cB)+k20(c2 A−K−2c2 B) (15) This, wi h inequali y symbols only in (12) and (14), gi es he adi ional exp ession o JA(i.e., JA=–J) when (R1a) and (R1b) occu simul aneously. Thus, his adi ional exp ession is achie ed as a esul o necessa y condi ion (12) and a su icien e sion o condi ion (13), and, as such, his special case is hus compa ible wi h ou he modynamic equi emen s. Reac ing Sys em 2; A Mix u e o Th ee Componen s, One o Them Being Compound o he O he Two As s a ed abo e, he i s deg ee he modynamic polynomial anishes he e. The second deg ee polynomial in e ms o a ini ies is as ollows ( o de i a ion see Supplemen a y Ma e ial): J=k110 exp −µo A−µo B RT exp B1+B2 RT exp A 3RT exp −A RT −1 (16) Condi ion (2) is de ailed in Supplemen a y Ma e ial; i s main esul is ha k110 ≥0, which is ully consis en wi h he ac ha , o example, JA= −J( emembe he nega i e s oichiome ic coe icien s o eac an s), whe e JA ep esen s eac an A’s o ma ion a e and Jis he a e o ( he selec ed independen ) eac ion R2, c . (4). Because he ze o a e cons an is impossible o A eally eac ing wi h B, he posi i eness o he a e cons an , adi ionally supposed in mass-ac ion kine ics, is shown he e o be a condi ion o consis ency wi h he modynamics ( he second law). In his example, he he modynamic polynomial con ains only (one) p ope coe icien , c . (4). Reac ing Sys em 3; A Mix u e o Th ee Isome s Ou p e ious wo k (Pekaˇ , 2016) also compa ed adi ional, i s -o de , mass-ac ion a e equa ions wi h hose gi en by he he modynamic polynomial o he i s deg ee. The e o e, we s a he discussion o his example by simila compa ison in he case o he second o de (deg ee). The i s o de iangula F on ie s in Chemis y | www. on ie sin.o g 4Ma ch 2018 | Volume 6 | A icle 35 Pekaˇ The modynamic Analysis o Reac ing Sys ems scheme (R3) is added wi h i s second o de analog, i.e., scheme (R3) wi h all s oichiome ic coe icien s equal o wo. The mass- ac ion a e equa ions in a ba ch sys em a e as ollows: 11 =k1cA−k2cB, 12 =k7c2 A−k8c2 B 21 =k3cB−k4cC, 22 =k9c2 B−k10c2 C 31 =k5cC−k6cA, 32 =k11c2 C−k12c2 A(17) dcA/d = − 11 + 31 − 12 + 32 (18a) dcB/d = 11 − 21 + 12 − 22 (18b) dcC/d = 21 − 31 + 22 − 32 (18c) The ela ionship be ween he adi ional a es and he a es based on ou he modynamic app oach a e ( o de ails see Supplemen a y Ma e ial): J1= 11 − 31 + 12 − 32 (19a) J2= 21 − 31 + 22 − 32 (19b) which is mo e complex han wi hou he second o de (deg ee), c . Pekaˇ (2016), as expec ed. Howe e , he second o de a es a e ma hema ically included in he independen eac ion a es in he same way as hei i s o de coun e pa s. To be able o ind es ic ions on he adi ional a e cons an s, he he modynamic polynomial (5) should e ain only hose second deg ee e ms which co espond o he eac ions be ween iden ical isome s. In o he wo ds, we selec k110 =k101 =k011 = 0.Using he same p ocedu e as p e iously (Pekaˇ , 2016), we ind he same ela ionships be ween i s o de a e cons an s and i s deg ee polynomial coe icien s. In he case o he second o de (deg ee), he ollowing iden i ies a e ound: k8= −k1 020,k9=k2 020,k11 = −k1 002 (20) oge he wi h he addi ional es ic ions: k10 =k1 002 −k2 002 (21) k7+k12 = −k1 020K1−k1 002K2 1K2 2(22) (no e ha , o example, k1 020 e e s o he i s independen eac ion, whe eas k2 020 e e s o he second one). Thus, ou o six second o de adi ional a e cons an s, only ou a e independen and de e mine he alues o he emaining wo; o example: k7=(k8+k9)K1−k10K2 1K2 2(23) k12 = −k9K1+(k10 +k11)K2 1K2 2(24) This is simila o he i s o de scheme (R3), whe e wo i s o de a e cons an s we e also no independen (Pekaˇ , 2016). Fu he , i equilib ium is de ined by ij =0, hen om i2in (17) i is e y easy o de i e an addi ional, “de ailed balance” condi ion o he second o de a e cons an s: k7k9k11 =k8k10k12 (25) which, exac ly as in he case o he i s o de (Pekaˇ , 2016), dec eases he numbe o independen second o de a e cons an s o h ee. The ans o ma ion o (5) in o he unc ion o a ini ies is e y complex in his example, and can be ound in Supplemen a y Ma e ial as well as he esul s o applying condi ion (2) o he ull he modynamic polynomial, which a e a he complex and gene al. We he e o e es ic ou discussion o a simpli ied e sion which con ains only hose second o de e ms in he he modynamic polynomial which co espond o eac ions be ween iden ical isome s as abo e; o example, 2 A =2 B. In o he wo ds, we se k110 =k101 =k011 =0, again, and eac ions such as A +B=2 C a e no conside ed. Wi h his modi ica ion, condi ion (2) leads o es ic ions on wo (p ope ) i s o de a e cons an s which a e ully consis en wi h hose ob ained p e iously wi h he i s -o de polynomial (Pekaˇ , 2016): k1 100 ≥0, k2 001 ≤0. (26) Fu he , he ollowing explici es ic ions o h ee second o de cons an s can be de i ed om (2): k1 002 ≤0, k1 020 ≤0, k2 002 ≤0. (27) All h ee a e coupling cons an s bu he la e wo couple an independen eac ion wi h i s double. The e a e no simila simple es ic ions on he emaining h ee a e cons an s which a e o a coupling ype (coupling wi h a es o eac ions which do no belong o he selec ed independen eac ions). They should con o m o a mo e complex condi ion gi en in Supplemen a y Ma e ial. Reac ing Sys em 4; A Mix u e o Fou Componen s Capable o Isome iza ion and Combina ion Reac ions No e ha he i s e m on he igh hand side o (6) ep esen s he (only) i s o de con ibu ion and co esponds o he eac ion AB =BA (one o he wo selec ed independen eac ions). I only he i s deg ee he modynamic polynomial is conside ed, condi ion (2) gi es he ollowing es ic ions (see Supplemen a y Ma e ial o de ails): k1 0001 =0, k2 0001 ≤0. (28) This means ha J1=0, which is consis en wi h he ac ha he i s selec ed independen eac ion (A+B=AB) is o he second o de . Thus, ou analysis independen ly unde lines he ac ha he whole iangula scheme (R4) canno be modeled solely by a i s o de kine ic model. The second (inequali y) condi ion, which in eal eac ions is k2 0001 <0, is consis en wi h he ac ha JAB =–J2. All e ms in (6) o mally o m he ollowing eac ion scheme (Pekaˇ , 2009): 1. AB =BA, 2. A +B=AB, 3. 2 AB =2 BA, F on ie s in Chemis y | www. on ie sin.o g 5Ma ch 2018 | Volume 6 | A icle 35 Pekaˇ The modynamic Analysis o Reac ing Sys ems 4. 2 AB =AB +BA, 5. A +AB =A+BA, 6. B +AB =B+BA. Only he i s wo eac ions di ec ly e lec wha chemis s would expec o he iangula scheme (R4) ( he hi d eac ion in his iangle is no independen ). The e o e, simila ly as in he p eceding iangula example, we will es ic ou discussion o he simpli ied polynomial. This also enables compa ison wi h adi ional mass-ac ion kine ics. Thus, we pu k0002 =k0011 =k1001 =k0101 =0. The ec o o eac ion a es exp essed as a unc ion o a ini ies is hen: J=k0001 exp −µo BA RT exp B1+B2 RT exp A1−2A2 5RT exp A2 RT −1 +k1100 exp −µo A−µo B RT exp B1+B2 5RT exp A1−2A2 5RT exp −A1 RT −1 (29) Condi ion (2) hen gi es he ollowing simple and explici es ic ions (see Supplemen a y Ma e ial o de ails): k2 0001 ≤0;k1 1100 ≥0. (30) Bo h cons an s in (30) a e o he p ope ype. The condi ions in (30) a e consis en wi h he ac ha JAB =J1−J2. The i s condi ion in (30) also co esponds o he second condi ion in (28) ound o he i s o de model. The es ic ions on he emaining wo (coupling) a e cons an s a e no as simple, again. They should sa is y he ollowing condi ion (see Supplemen a y Ma e ial o de ails): 1 2k2 1100 −1 2k1 0001K1K22 ≤ −k2 0001k1 1100K1K2. (31) No e, ha due o (30), he igh hand side o (31) is posi i e, as equi ed. Also in his case we can de i e addi ional cons ain s on he a e cons an s o he adi ional mass-ac ion kine ic model o scheme (R4)—see Supplemen a y Ma e ial o de ails on his p ocedu e. The esul is ha he alues o a leas wo adi ional a e cons an s a e de e mined by he alues o he emaining ou cons an s ( oge he wi h he equilib ium cons an s). Fo ins ance: k2= −k5K2+(k1+k6)K−1 1,k3=(k4+k5)K2−k6K−1 1.(32) ( he numbe ing o he a e cons an s co esponds o he numbe s shown in scheme (R4)). Thus, he dimensionali y o pa ame e es ima ion p oblems is educed by wo. I can also be easily e i ied ha he adi ional mass-ac ion a e equa ions o scheme (R4) also lead o a ”de ailed balance” cons ain on hei a e cons an s, e y simila o ha o he “ i s o de ” iangula scheme (R3). This is ano he dimensionali y educ ion. The numbe o independen a e cons an s is he e o e h ee, he same as in he case o he i s o de scheme (R3) analyzed in ou p e ious wo k (Pekaˇ , 2016). CONCLUSIONS Second-o de kine ic models—o , mo e speci ically, second- deg ee he modynamic polynomials— u he demons a ed he powe o he p esen ed he modynamic analysis and also e ealed whe e simple, speci ic esul s ( es ic ions) canno be expec ed. Second-deg ee polynomial models led o mo e complex esul s a ising om he modynamic condi ion (2), bu in all cases, ega dless o he deg ee (o de ) and o e looking some o he simpli ica ions made abo e, simple sign es ic ions we e ound o p ope a e cons an s (coe icien s). In con as , es ic ions on coupling cons an s we e usually weake , gi ing no s ic es ic ions on hei sign, hus allowing hem mo e “ eedom.” This means mo e eedom in kine ic coupling be ween di e en eac ions o ming a eac ion scheme, i.e., in he kine ic e ec s o o he eac ions on he a e o some eac ion in a eac ing sys em; a he same ime, ze o coupling (ze o coupling a e cons an s) is no excluded by hese inal he modynamic es ic ions. The compa ison o he modynamic polynomials wi h adi ional mass-ac ion a e models e ealed cons ain s on he a e cons an s o he la e which could no be e ealed using he adi ional app oach. The he modynamic me hodology p esen ed in his wo k hus allows a educ ion in he numbe o kine ic pa ame e s o be e alua ed om da a, i.e., a educ ion in he dimensionali y o he pa ame e es ima ion p oblem. In his analysis, he uni e sali y and consis ency o he p esen ed me hod was hus u he unde lined. I no only allows a e equa ions o be de i ed on he basis o esul s o non- equilib ium he modynamics bu also enables he de i a ion o cons ain s on hei a e coe icien s which a e necessa y o ull consis ency wi h en opy inequali y ( he second law o he modynamics). To his end, he p ope ans o ma ion o a e equa ions o unc ions o chemical and cons i u i e a ini ies should be pe o med. AUTHOR CONTRIBUTIONS The au ho con i ms being he sole con ibu o o his wo k and app o ed i o publica ion. FUNDING This wo k was suppo ed by p ojec No. LO1211 “Ma e ials Resea ch Cen e a FCH BUT- Sus ainabili y and De elopmen ” unde he Na ional P og amme o Sus ainabili y I (Minis y o Educa ion, You h and Spo s). SUPPLEMENTARY MATERIAL The Supplemen a y Ma e ial o his a icle can be ound online a : h ps://www. on ie sin.o g/a icles/10.3389/ chem. 2018.00035/ ull#supplemen a y-ma e ial F on ie s in Chemis y | www. on ie sin.o g 6Ma ch 2018 | Volume 6 | A icle 35 Pekaˇ The modynamic Analysis o Reac ing Sys ems REFERENCES Al-Kha eeb, A. N., Powe s, J. M., Paolucci, S., Sommese, A. J., Dille , J. A., Hauens ein, J. D., and Menge s, J. D. (2009). One-dimensional slow in a ian mani olds o spa ially homogenous eac i e sys ems. J. Chem. Phys. 131:024118. doi: 10.1063/1.3171613 A a o, E., and Mo o, A. (2014). A he modynamic app oach o kine ics o eac ions. Z. Phys.Chem. 228, 793–815. doi: 10.1515/zpch-2014-0531 Bedeaux, D., Pagonaba aga, I., ,O izde Zá a e, J. M., Senge s, J. V., and Kjels up, S. (2010). Mesoscopic non-equilib ium he modynamics o non- iso he mal eac ion-di usion. Phys. Chem. Chem. Phys. 12, 12780–12793. doi: 10.1039/c0cp00289e Bo he, D., and D eye , W. (2015). Con inuum he modynamics o chemically eac ing luid mix u es. Ac a Mech. 226, 1757–1805. doi: 10.1007/s00707-014-1275-1 Bowen, R. M. (1968). On he s oichiome y o chemically eac ing sys ems. A ch. Ra ional Mech. Anal. 29, 114–124. doi: 10.1007/BF00281361 Boyd, R. K. (1977). Mac oscopic and mic oscopic es ic ions on chemical kine ics. Chem. Re . 77, 93–119. doi: 10.1021/c 60305a006 Ede e , M., and Gilles, M. E. D. (2008). The modynamic cons ain s in kine ic modeling: he modynamic-kine ic modeling in compa ison o o he app oaches. Eng. Li e Sci. 8, 467–476. doi: 10.1002/elsc.200800040 Fleming, R. M. T., Thiele, I., P o an, G., and Nasheue , H. P. (2010). In eg a ed s oichiome ic, he modynamic and kine ic modelling o s eady s a e me abolism. J. Theo . Biol. 264, 683–692. doi: 10.1016/j.j bi.2010.02.044 Ge, H., and Qian, H. (2016). Nonequilib ium he modynamic o malism o nonlinea chemical eac ion sys ems wi h Waage-Guldbe g’s law o mass ac ion. Chem. Phys. 472, 241–248. doi: 10.1016/j.chemphys.2016.03.026 He nández-Lemus, E., To a , H., and Mejía, C. (2014). Non-equilib ium he modynamics analysis o ansc ip ional egula ion kine ics. J. Non Equilib. The modyn. 39, 205–218. doi: 10.1515/jne -2014-0015 Jones, N. H., Cizmas, P. G. A., and Sla e y, J. C. (2015). C ea ing educed kine ics models ha sa is y he en opy inequali y. J. Eng. Gas Tu bines Powe 137:071504. doi: 10.1115/1.4029172 Kannan, K., and Rajagopal, K. R. (2011). A he modynamical amewo k o chemically eac ing sys ems. Z. Angew. Ma h. Phys. 62, 331–363. doi: 10.1007/s00033-010-0104-1 Klika, V., and G mela, M. (2013). Coupling be ween chemical kine ics and mechanics ha is bo h nonlinea and compa ible wi h he modynamics. Phys. Re . 87:012141. doi: 10.1103/PhysRe E.87.012141 N’Guyen, T. A., Lejeunes, S., Eyhe amendy, D., and Boukamel, A. (2016). A he modynamical amewo k o he he mo-chemo-mechanical couplings in so ma e ials a ini es ain. Mech. Ma e . 95, 158–171. doi: 10.1016/j.mechma .2016.01.008 Pagonaba aga, I., Pé ez-Mad id, A., and Rubí, J. M. (1997). Fluc ua ing hyd odynamics app oach o chemical eac ions. Physica A 237, 205–219. doi: 10.1016/S0378-4371(96)00377-9 Pekaˇ , M. (2009). The modynamic amewo k o design o eac ion a e equa ions and schemes. Collec . Czech. Chem. Commun. 74, 1375–1401. doi: 10.1135/cccc2009010 Pekaˇ , M. (2016). The modynamic analysis o chemically eac ing mix u es and hei kine ics – example o a mix u e o h ee isome s. ChemPhysChem 17, 3333–3341. doi: 10.1002/cphc.2016 00528 Pekaˇ , M., and Samohýl, I. (2014). The The modynamics o Linea Fluids and Fluid Mix u es. Cham: Sp inge . Qian, H. (2007). Phospho yla ion ene gy hypo hesis: open chemical sys ems and hei biological unc ions. Annu. Re . Phys. Chem. 58, 113–142. doi: 10.1146/annu e .physchem.58.032806.1 04550 Rubí, J. M., Bedeaux, D., Kjels up, S., and Pagonaba aga, I. (2013). Chemical cycle kine ics: emo ing he limi a ion o linea i y o a non- equilib ium he modynamic desc ip ion. In . J. The mophys. 34, 1214–1228. doi: 10.1007/s10765-013-1484-1 Samohýl, I. (1982). Ra ional The modynamics o Chemically Reac ing Mix u es. P ague: Academia (in Czech). Vlad, M. O., and Ross, J. (2009). The modynamically Based Cons ain s o Ra e Coe icien s o La ge Biochemical Ne wo ks. WIREs Sys . Biol. Med. 1, 348–358. doi: 10.1002/wsbm.50 Con lic o In e es S a emen : The au ho decla es ha he esea ch was conduc ed in he absence o any comme cial o inancial ela ionships ha could be cons ued as a po en ial con lic o in e es . Copy igh © 2018 Pekaˇ . This is an open-access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License (CC BY). The use, dis ibu ion o ep oduc ion in o he o ums is pe mi ed, p o ided he o iginal au ho (s) and he copy igh owne a e c edi ed and ha he o iginal publica ion in his jou nal is ci ed, in acco dance wi h accep ed academic p ac ice. No use, dis ibu ion o ep oduc ion is pe mi ed which does no comply wi h hese e ms. F on ie s in Chemis y | www. on ie sin.o g 7Ma ch 2018 | Volume 6 | A icle 35