Download THE GENERAL LAW OF CHEMICAL KINETICS, DOES IT EXIST?

Boreskov-Horiuti Problem and General
Law of Chemical Kinetics
Gregory S. Yablonsky
Saint Louis University and
Washington University in St. Louis,
USA
What is a LAW?
Dependence
Correlation
MODEL
Equation
LAW !!!
Some definitions
“A physical law is a scientific generalization based on
empirical observations” (Encyclopedia)
This definition is too fuzzy
Physico-chemical law is a mathematical
construction(functional dependence)with the following
properties:
1) It describes experimental data in some domain
2) This domain is wide enough
3) It is supported by some basic considerations.
4) It contains not so many unknown parameters
5) It is quite elegant
Our goal:
We will discuss the general law of complex chemical
reaction.A special attention will be paid to the
analysis of the Horiuti - Boreskov’s problem:
Famous Horiuti’s question was “How can a kinetic
equation be found for a back reaction?” (Juro
Horiuti,1939)
Boreskov tried to find answer the same question
analyzing “SO2 oxidation - SO3 decomposition”
processes.
General Kinetic Expression
for the Complex Reaction
R =R+ - R- ,
where (R+ / R- ) = Keq (f+(C) / f- (C)),
A similar equation was tested for some oxidation
reactions, especially for SO2 oxidation and for
ammonia synthesis (Boreskov, 1950-1960ies).
(The backward reaction rate was estimated using
isotope exchange data, especially data of the
homomolecular isotope exchange in the ammonia
synthesis )
It was found that this equation did not fit
experimental data.
So, what to do?
Other ideas, other assumptions
SO2 oxidation:
1) 2 K + O2  2 KO;
1
2) KO + SO2  K + SO3 2
Overall reaction 2 SO2 + O2  2 SO3
Numbers on the right side (1,2) are Horiuti’s numbers ()
Assumptions:
1) Non-linearity of some step (the first step is nonlinear).
2) One step is limiting, other steps are under fast equilibrium
conditions.
New equation
Boreskov’s molecularity was introduced
in his hystorical paper by 1945)
M
R (C) Keq f (C) 
  
 ; M  (1/  lim . );

R (C)  f (C) 


M
  


f (c)

R  R (C)1
K f  (c) 
 
  eq
Molecularity
SO2 oxidation
1) 2 K + O2  2 KO;
1
2) KO + SO2  K + SO3 2
Overall reaction 2 SO2 + O2  2 SO3
In SO2 oxidation, molecularity is equal to 1/2.
Therefore, the limiting step is the second one.
Rigorous Status of Boreskov’s
Equation
The rigorous status of Boreskov’s equation
was found based on algebraic ideas (concept
of kinetic polynomial)
See Marin, Yablonsky, “Kinetics of Chemical
Reactions: Decoding Complexity”, 2011, J.
Wiley
Chemical Kinetics. Textbook Knowledge
Detailed mechanism is a set of elementary reactions
which law is assumed , e. g. the mass-action-law
An example:Hydrogen Oxidation 2H2 +O2 = 2H2O
1)H2 + O2 = 2 OH ; 2) OH + H2= H2O + H ; 3) H + O2 = OH + O;
4) O + H2 = OH + H ; 5)O + H20 = 2OH; 6) 2H + M = H2 + M ; 7) 2O + M = O2 + M;
8)H + OH + M = H2O + M; 9) 2 OH + M = H2O2 + M; 10) OH + O + M = HO2 + M;
11) H + O2 + M = HO2 + M; 12) HO2 + H2 = H2O2 + H;13)HO2 +H2 = H2O +OH;
14) HO2 + H2O = H2O2 + OH; 15) 2HO2 = H2O2 + O2; 16) H + HO2 = 2 0H;
17) H + HO2 = H2O + O; 18) H + HO2 = H2 + O2; 19) O + HO2 = OH +H;
20) H + H2O2 = H2O + OH; 21) O + H2O2 = OH +HO2; 22) H2 + O2 = H2O + O;
23) H2 + O2 + M = H2O2 + M; 24) OH +M = O + H + M; 25) HO2+OH=H2O+O2;
26) H2 + O +M = H2O +M; 27) O + H2O + M = H2O2 + M; 28)O + H2O2 = H20 + O2;
29) H2 + H2O2 = 2H2O; 30) H + HO2 + M = H2O2 +M
Simple reactions
Simple reactions are rather exceptions than rule
e.g., first-order reactions of monomolecular
decomposition
C2H5Br  C2H4 + HBr
or
CH3-N2- CH3  C2H6 + N2;
the second-order reaction
2 NOI  2 NO + I2
A Question:
Imagine that hydrogen oxidation reaction is performed
under steady-state or pseudo-steady-state conditions.
It means that concentrations of intermediates are governed
by gas concentrations.
Do we know, how to present the reaction rate as
a function of main reactant and product concentrations
(hydrogen, oxygen, water)?
The answer is: NO !
We do not know the law of chemical kinetics for the
complex reaction, we even do not know its form.
Nice Quotation
Daniels:
“Despite Eyring and Arrhenius, chemical kinetics is
all-in-all confusion. But through all the confusion of
complications some promising perspective can be seen.
Numerous consecutive, competing and reverse reactions by
themselves are simple mono- or bimolecular reactions that
obey simple laws. Hence we are fighting not so much with
primary steps as with the problem of their mutual
coordination to interpret the observed facts and to make
practical predictions”.
Hidden History of Chemical Kinetics,I
Gul’dberg and Waage , Norway, 1862-1867
Mass-Action-Law( M.A.L.)
Equilibrium formulation
“ In chemistry like in mechanics the most natural
methods will be to determine forces in their
equilibrium states”.
Kpq  Kp'q', where p, q,p'q' are the " action masses"
Initially, Guldberg and Waage used an expression
Kp q  K(p') (q')
Hidden History of Chemical Kinetics, II
Gul’dberg and Waage, 1879
Dynamic Formulation of the Mass-Action-Law
(M.A.L.)
R = K pqr
Hidden History of Chemical Kinetics, III
Van’t Hoff, Netherlands, the first winner of
the Nobel award (1901) on chemistry
1884, “Essays on chemical kinetics”
Idea of normal transformations
“The process of chemical transformations is characterized
solely by the number of molecules whose interaction provides this
transformation” (AB; 2A B; A+B  C; 2A+B  C)
Strong discussion with Gul’dberg and Waage: “As a
theoretical foundation I have accepted not the concept of
mass action ( I had to leave this concept in the course of my
experiment)”. Van’t Hoff tried to eliminate mechanics from chemistry.
Hidden History of
Chemical Kinetics, IV
Van’t Hoff believed that he found
the chemical (not mechanical)
LAW OF CHEMICAL KINETICS
However, his normal transformation dependences
did not fit many real experimental data,
e.g. hydrogen oxidation data
Hidden History of Chemical Kinetics, V
Idea of the complex mechanism:
“Reaction is not a single act drama” (Schoenbein)
Intermediates (X) and Pseudo-Steady-State-Hypothesis
According to the P.S.S.H.,
Rate of intermediate generation = Rate of
intermediate consumption
Ri.gen (X, C) = Ri.cons(X, C)
Then, X = F(C)
and Reaction Rate R(X, C)=R (C, F(C))=R(C)
The term “chemical mechanism” has an
obvious “mechanical origin”
In 1879, a vivid interpretation of complex systems as mechanical systems was
given by Maxwell. “In an ordinary chime every bell has a rope that is drawn
through a hole in the floor into the bell-ringer room. But let us imagine that every
rope instead of putting into motion one bell participates in the motion of many parts
of the mechanism and that the motion of every bell is determined not only by the
motions of its own rope but the motions of several ropes; then let us assume that all
this mechanism is hidden and absolutely unknown for the people standing near the
ropes and capable of seeing only the holes ceiling above them”.
Hidden History of Chemical Kinetics, VI
In the XX century, the history of
chemical kinetics is a series of
attempts to hide the failure of
Van’t Hoff’s paradigm
P.S.S.H, or Bodenstein’s Principle
A paradox of PSSH.
Reflecting complexity, we are introducing new
unobserved substances (intermediates).
At the same time, we are eliminating intermediates
searching for simplicity.
“The first who applied this theory was S. Chapman and
half the year later Bodenstein referred to it in the paper
devoted to the hydrogen reaction with clorine. His efforts
to confirm his view point were so energetic that this theory
is quite naturally associated with his name” (Christiansen)
P.S.S.H. has been applied in many areas of
chemical kinetics:
Reactions in gaseous phase
Heterogeneous catalytic reactions
Enzyme reactions
Etc.
Chain Reaction
Fragment of the mechanism:
1) H + Cl2  HCl + Cl
2) Cl + H2 HCl +H
Overall reaction: H2 + Cl2  2HCl
R=(k 1k2CH2CCl2- k-1k-2C2HCl) / ,
where  = k 1CH2 +k2CCl2 + k-1CHCl +k-2 CHCl
Thermodynamic validity
The equation
R=(k 1k2CH2CCl2 - k-1k-2C2HCl) /  ,
where  = k 1CH2 +k2CCl2 + k-1CHCl +k-2 CHCl
is valid from the thermodynamic point of view.
Under equilibrium conditions, R=0,and
(C2HCl / CH2 C Cl2) = (k1k2 /k-1k-2)= K eq(T)
Forward and backward
rates of complex reaction
In the case H2 +Cl2 = 2 HCl reaction,
Rate can be represented as
a difference of two terms :
R =R+ - R- ,
where (R+ / R- ) = Keq (f+(C) / f- (C)),
and f+(C) and f- (C) correspond to the overall
reaction, forward and backward respectively.
Heterogeneous Catalytic Reaction
Langmuir- Hinshelwood equation
K  ci
ni
R
i
1  K jiCimi
j
i
Langmuir ( Langmuir-Hinshelwood ) equation is
related to the irreversible case: the“back term” is
absent.
For simple molecules oxidation (H2, CO)
over noble catalysts (Pt, Pd), Langmuir made an
assumption on the fast adsorption - desorption
equilibrium.
Hougen-Watson Equations.
Typically, there is no assumptions
about the detailed mechanism
R
K  CC6H12  K  CC6H10 CH 2
(K1C C6H10  K 2C  H 2  K3C C6H12 )m
 R  R
Enzyme cycle (King, Altman,1956)
First application of the graph theory
1) S + E  X1; 2) X1  X2;
3)X2  X3 ;…
n-1) Xn-2  Xn-1 ; n) Xn-1  P+ E
E
X1
X2
Xn
Xn-1
…
Overall reaction is : S  P
X = f (Cs, Cp )
According to the P.S.S.H,
R=[ (k1Cs)(k2)…(kn) - (k-1) (k-2)… (k-n)Cp] / ( )=
R= (K+CS - K-Cp) /( ), where  =  ( Ki(Ci)P)
R = R+ - R-
Catalytic mechanism. Two-step
mechanism: Temkin-Boudart
•
•
•
•
1) Z + H2O  ZO + H2; 1
2) ZO + CO  CO2 + Z 1
The overall reaction is CO + H2O= CO2 +H2,
R=[(k1Cco)(k2CH2O)- (k1CH2)(k2CCO2)] / ,
where  = k1CH2O +k2CCO+ (k-1CH2)(k-2CCO2)],
R = R+ - R- ; (R+/ R- ) = (K+CcoCH2O)/(KCH2CCO2)
Conversion of methane
1) CH4 + Z ZCH2 +H2 ;
2) ZCH2 +H2O  ZCHOH +H2 ;
3) ZCHOH  ZCO +H2 ;
4) ZCO  Z + CO
Overall reaction : CH4 + H2O  CO + 3 H2
R = (K+CCH4CH2O - K-CCOCH23) /  ;
R= R+ - R(R+ / R- ) = (K+CCH4CH2O) / ( K-CCOCH23)
Linear mechanisms
What is general about these different reactions:
H2+Cl2 = 2 HCl, enzyme reaction, two-step catalytic
reaction, catalytic conversion of methane?
1) All these reactions have a cycle (circuit) in the
detailed mechanism.
2) In any reaction, only one molecule of the
intermediate participates.
Such mechanisms are called linear ones. The
corresponding PSSH-equations for intermediated can
be solved easily.
One-route catalytic reaction with the linear mechanism.
General expression (Yablonsky, Bykov, 1976)
R = Cy / ,
where Cy is a “cyclic characteristics”,
Cy = K+ f+(C) - K- f- (C) ,
Cy corresponds to the overall reaction;
 presents complexity of complex reaction;
pi
   K j c ji
j
i
Main Properties of the General
Expression (I)
1. Cyclic Characteristic, Cy (numerator) does not
depend on the details of the mechanism
2. All the information about the mechanism
complexity is reflected by the denominator, ,
which is the complexity term.
3. Another form of the rate equation is:
( ) R = Cy, or ( ) R - Cy = 0
Main Properties
of the General Expression(II)
4. R =R+ - R- ,
where (R+ / R- ) = Keq (f+(C) / f- (C)),
5. These equations can be derived very easily
(with ‘a rate of pen’) using the methods of the
graph theory.
Life is Tough
(that’s its stuff)
A similar equation was tested for some oxidation
reactions, especially for SO2 oxidation and for
ammonia synthesis (Boreskov, 1950-1960ies).
(The backward reaction rate was estimated using
isotope exchange data, especially data of the
homomolecular isotope exchange in the ammonia
synthesis )
It was found that this equation did not fit
experimental data.
So, what to do?
Other ideas, other assumptions
SO2 oxidation:
1) 2 K + O2  2 KO;
1
2) KO + SO2  K + SO3 2
Overall reaction 2 SO2 + O2  2 SO3
Numbers on the right side (1,2) are Horiuti’s numbers, 
Assumptions:
1) Non-linearity of some step (the first step is nonlinear).
2) One step is limiting, other steps are under fast equilibrium
conditions.
New equation
(Boreskov’s molecularity)
M
R (C) Keq f (C) 
  
 ; M  (1/  lim . );

R (C)  f (C) 


M
  


f (c)

R  R (C)1
K f  (c) 
 
  eq
Molecularity
SO2 oxidation
1) 2 K + O2  2 KO;
1
2) KO + SO2  K + SO3 2
Overall reaction 2 SO2 + O2  2 SO3
In SO2 oxidation, molecularity is equal to 1/2.
Therefore, the limiting step is the second one.
Life is even more tough than we expected
In the kinetics of heterogeneous catalysis, in the late
1960ies and 1970ies, various critical phenomena
under isothermic conditions were observed. A
typical example of these phenomena is the
multiplicity of steady states which was observed in
many oxidation reactions, particularly CO oxidation,
SO2 oxidation etc. This means that the different
values of reaction rate can correspond to one or the
same composition of reaction mixture
Critical Phenomena in Heterogeneous
Catalytic Kinetics:
Multiplicity of Steady-States in Catalytic Oxidation
(CO oxidation over Platinum)
Reaction Rate
C
“Extinction”
B
A
“Ignition”
D
CO Concentration
Typical catalytic system
with critical phenomena
Adsorbed mechanism (Langmuir’s mechanism)
1) 2 K + O2  2 KO
2) K + B  KB
3) KO + KB  2 K + BO
Overall reaction 2B+O2  KB
Particular cases of this mechanisms are mechanisms of
CO oxidation over noble metals of VIII group and
SO2 oxidation over Pt.
Kinetic model of the adsorbed mechanism
1) 2 K + O2  2 KO
2) K + SO2  KSO2
3) KO + KSO2  2 K + SO3
Steady state (or pseudo-steady-state) kinetic model is
KO : 2k1CO2(CK )2 - 2 k-1 (CKO )2 - k3 (CKO ) (CKSO2 )+
k-3 CSO3 (CK )2 = 0 ;
KSO2: k2CSO2CK - k-2 (CKSO2) - k3 (CKO ) (CKSO2 )+
+k-3CSO3 (CK )2 = 0 ;
CK + CKO +CKSO2 =1
The same question:
What is the general form of kinetic description
taking into account the thermodynamic validity
and, ideally,with no assumptions about step
limiting, fast equilibrium etc.?
The obvious difficulty
In general, i.e. non-linear, case, an
explicit presentation
for steady (or pseudo-steady-state)
reaction rate can be obtained only for
special non-linear kinetic models.
Generally, it is impossible.
The obvious requirement!
Any representation should be valid from the
thermodynamic point of view, i. e. under equlibrium
conditions (R=0), we should have
Keq = (f+(C) / f- (C))
Thermodynamics does not care about our
mathematical difficulties
Our strategy
Our strategy was to perform an analysis of the
concrete non-linear kinetic models, especially
kinetic model corresponding to adsorbed
mechanism of CO oxidation, and to develop the
general theory based on the ideas of algebraic
geometry.
Mathematical basis
Our basis is algebraic geometry,
which provides the ideas of variable elimination
1.
2.
3.
4.
5.
Aizenberg L.A., and Juzhakov,A.P. “Integral representations and residues in
multi-dimensional complex analysis”, Nauka, Novosibirsk, 1979
Tsikh, A.K., Multidimensional residues and their applications, Trans. Math.
Monographs, AMS, Providence, R.I., 1992
Gelfand,I.M., Kapranov, M.M., Zelevinsky, A.V., Discriminants, Resultants, and
Multidimensional Determinants, Birkhauser, Boston, 1994
Emiris, I.Z., Mourrain,B. Matrices in elimination theory, Journal of Symbolic
Computation, 1999, v.28, 3-43
Macaulay, F.S. Algebraic theory of modular systems, Cambridge, 1916
Our main result
In the case of mass-action-law model, it is always
possible to reduce our polynomial algebraic
system to a polynomial of only variable, steadystate reaction rate.
For this purpose, an analytic technique of variable
elimination is used. Computer technique of
elimination is used as well.
Mathematically, the obtained polynomial is a system
resultant. We term it a kinetic polynomial.
First publications
The idea of kinetic polynomial was firstly
emphasized in the paper
M.Z. Lazman, G.S. Yablonskii, and V.I. Bykov, Sov.
J. Chem. Phys.,2(1985)404
The mathematical apparatus was described in the
paper
M.Z. Lazman, and G.S. Yablonski, Kinetic Polynomial: a
New Concept of Chemical Kinetics”, in “Patterns and
Dynamics in Reactive Media”, IMA Proceedings,1989;
Springer-Verlag, Berlin-N.Y.,1991
The Kinetic Polynomial (I)
The kinetic model which corresponds to the single-route
mechanism has the form:
rs(z1,…,zn) - R = 0, s=1,…,n
1-  zj =0; j = 1,…,n,
where R is the rate of complex reaction (rate of reaction
route), rs is the rate of the reaction step
rs(z1,…,zn) = bszj s - b-s zjs;  is the Horiuti’s number
X is the concentration of the intermediate;
bs ,b-s are the reaction weights, i.e. reaction rates at the unitary
concentrations of intermediates
Kinetic polynomial (II)
In analysis of non-linear kinetic model, the concept
of the limiting subsystem was introduced.
The kth limiting subsystem is such system in which
all steps are considered to be in an equilibrium
except the kth step. Then, the set of k limiting
subsystems is analyzed.
The Kinetic Polynomial (III)
For the linear mechanism, the kinetic polynomial has a
traditional form: R = (K+ f+(C) - K- f- (C))/ ( ) ,
or ( ) R = Cy, or ( ) R - Cy = 0,
where Cy is the cyclic characteristic;  is the
“Langmuir term” reflecting complexity
For the typical non-linear mechanism the kinetic
polynomial is represented as follows:
BmRm+…+ B1R +BoCy=0 ,
where m are the integer numbers
The Kinetic Polynomial(IV)
Coefficients B have the same “Langmuir’ form as in
the denominator of the traditional kinetic
equation,i.e. they are concentration polynomials
as well. Therefore, the kinetic polynomial can
be written as follows




mi , jL
L
mi , j1
 K j L iCi
R  ...  K j1 iCi
R 
j L i

 j1 i







 B0 K f (C)  K f (C)  0
The Kinetic Polynomial (V)
The coefficients of kinetic
polynomial are obtained using
solutions of k limiting subsystems
Properties
of the Kinetic Polynomial
1) It is a non-linear polynomial, an
implicit function of the reaction rate;
2) It is thermodynamically correct.
At R=0 (equilibrium),
K+f+(C) - K-f- (C) = 0
Properties
of kinetic polynomial(II)
3)Kinetic polynomial is the generalized form which
includes many typical cases: Langmuir-Hinshelwood
equation, Hougen-Watson equation, equations of enzyme
kinetics etc.
4)It can be used for the rigorous analysis of special cases:
limiting, kinetic behavior in the domain which is close to
the equilibrium.
5)Kinetic polynomial may have several physical solutions.
Therefore it can be used for for description of critical
phenomena, especially multiplicity of steady-states.
Uniqueness of the
“thermodynamic branch”
Generally, the kinetic polynomial may have some
physical solutions and, correspondingly, some physical
branches.
However, only one branch (“thermodynamic branch”)
passes through the equilibrium point.
Otherwise, the system will have more than one
equilibrium. It is impossible.
Analysis of two limiting cases:
1)LIMITATING STEPS;
2)NEIGHBORHOOD OF
EQUILIBRIUM
In both cases, results are obtained on the
basis of solutions of limiting subsystems.
Limiting Steps




1/ lim
R / R  [ f (C)/ f (C)]

RR R

Neighborhood of equilibrium
K  f  (C)  K  f  (C)
R
,
Aeq
where :
2  n
pk
'( k )



/
p
N
 


ji
k
k
A  K f (C) k  [(b j ) /(b j )]

b

k1 k i1 jk
n
n
R+(C) / R- (C ) =[ K+f+(C) / K-f-(C) ]
R(C)=R+(C) - R- (C )
eq
Boreskov-Horiuti Problem: Conclusions
Generally, the rate of complex reaction
cannot be presented as a difference of the forward and back
reaction rates. Therefore, in a general case the representation
R (C) = R+(C) - R- (C) is not valid
Such representation is valid only for the thermodynamic branch.
1. R+(C) / R- (C) = K+f+(C) / K-f-(C)
This representation is valid for the non-linear mechanism in the
domain “close to equilibrium”and for the linear mechanism in any
domain
2. R+(C) / R- (C) =[ K+f+(C) / K-f-(C) ](1/lim)
This representation is valid for the non-linear mechanism in the
domain of limitation of some steps
Bm
4
R +…+
B1R +BoC=0 ,
R= k3 CKO CKCO - k-3CCO2 CK2 = 0
Conditions:
PO2>PCO; k3 > ki , k-i
Multiplicity of steady states
Low branch (1), Middle branch (2),
Upper Branch(3)
R1  k2 ( H  (H 1))2 ;R2  k2 ( H  (H  1))2 ;
2
R3  (k2Cco ) /2k1CO 2 ;
(k 2Cco  k2 )
H
8k2k1CO 2
2
Branches and rates (I)
The first (low) and third (upper) branches are stable,
the second (middle) branch is unstable.
The second and middle rate expressions
can not be presented
in the form of differences of the forward and back
reaction rates
R =R + - R - ,
where (R + / R- ) = Keq (f+(C) / f- (C)),
Branches and rates (II)
The first rate expression (low branch) can be
presented as a difference of values which can be
termed as forward and back terms, e.g. (R+)1/2 and
(R-)1/2
However, their ratio is not KeqC2CO2/C2COCO2
Reaction Rate Dependence on CO Concentration
0.7
0.6
Non-Physical
Branch
Reaction Rate
0.5
0.4
0.3
Physical Branch
0.2
Physical Branches
0.1
0
-0.1
Non-Physical Branches
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
CO Concentration
1.6
1.8
2
Critical Simplification
Analyzing kinetic polynomial,
critical simplification was found
At the extinction point Rext. = k+2Cco
At the ignition point Rign = k-2
Therefore, the interesting relationship is fulfilled
Rext / Rign = k+2Cco /k-2 = Keq Cco
It can be termed as a “Pseudo-equilibrium constant of
hysteresis”
Therefore, we have the similar equation for (R + / R- ) in
terms of bifurcation points.
Critical Phenomena in Heterogeneous
Catalytic Kinetics:
Multiplicity of Steady-States in Catalytic Oxidation
(CO oxidation over Platinum)
Reaction Rate
C
“Extinction”
Critical Simplification:
RC=k2+CCO
RA=k2-
B
A
“Ignition”
D
CO Concentration
Experimental evidence
It was found theoretically that at the point of ignition the
reaction rate is equal to the constant of CO desorption.
It was found experimentally, that the temperature
dependence of reaction rate at this point equals to the the
activation energy of the desorption process.
(Wei, H.J., and Norton, P.R, J. Chem. Phys.,89(1988)1170;
Ehsasi, M., Block, J.H., in Proceedings of the International
Conference on Unsteady-State Processes in Catalysis, ed.by
Yu.Sh. Matros, VSP-VIII, Netherlands, 1990, 47
Evolution of kinetic behavior of the complex
reversible reaction with the increase of parameter
(concentration)
Typical scenario: from the “close to equilibrium”domain to the “far from equilibrium”- domain
• “Close to Equilibrium ”- Domain. Kinetic behavior in
accordance with the overall reaction
•Domain of Limitation by the controlled parameter
• Domain(s) of Limitation by the other concentrations (or
intermediates)
•Domain of Irreversibility (“Far from Equilibrium”)
•Domain(s) of Critical Phenomena (“Far from
Equilibrium”), e.g. multiplicity of steady states
References (I)
1.
2.
3.
4.
Lazman, M.Z., Yablonsky, G.S., and Bykov V.I. , Sov. J. Phys. Chem.,
2(1985)404
Lazman, M.Z., and Yablonskii G.S., Kinetic Polynomial: a New Concept
of Chemical Kinetics, in Patterns and Dynamics in Reactive Media, The
IMA Volumes in Mathematics and its Applications, Springer-Verlag,
New-York, 1991, 117-150
Yablonsky, G.S., and Lazman, Non-Linear Steady-State Kinetics of
Complex Catalytic Reactions: Theory and Applications, Proceedings of
International Symposium “Dynamics of Surface and Reaction Kinetics in
Heterogeneous Catalysis”, Antwerp, Belgium, 1997, In Studies in Surface
Science and Catalysis, 109, 371-378
Yablonsky, G.S., Mareels, M.Y., and Lazman, M.Z. The Principle of
Critical Simplification in Chemical Kinetics, Chemical Engineering
Science, 58 (2003) 4833-4842
References (II)
5. Yablonskii,G.S., Lazman M.Z, and Bykov V.I., Stoichiometric Number,
Molecularity and Multiplicity, Reaction Kinetics and Catalysis Letters, 20
(1982)73-77
6. Aizenberg, L.A, Bykov V.I., Kytmanov A.M., and Yablonsky, G.S., Search for
all steady-states of chemical kinetic equations with the modified method of
elimination, Chemical Engineering Science, 38 (1983)1555-1568
7. Yablonskii G.S, Bykov V.I., Gorban A.N., and Elokhin V.I., Kinetic Models of
Catalytic Reactions//Comprehensive Chemical Kinetics, V.32, Elsevier,
Amsterdam, N.-Y, 1991
8. Yablonskii, G.S. and Lazman, M.Z., New Correlations to Analyze Isothermal
Critical Phenomena in Heterogeneous Catalytic Reactions, Reactions Kinetics and
Catalysis Letters, 59(1996)145-150
8. Bykov, V.I., Kymanov A.M. and Lazman M.Z., Elimination methods on
Computational Computer Algebra, Kluwer Academic Publishers, Dordrecht, 1998