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)
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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)
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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
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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
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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.
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