What happens at the crossroads between Chemical Engineering and Mathematics? Prof. Gregory...
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What happens at the crossroads between Chemical Engineering
and Mathematics?Prof. Gregory S.Yablonsky
Parks College
Saint Louis University. USA
Dept. of Chemical Engineering,
Washington University in St. Louis, USA
“I gave my mind a thorough rest by plunging into a chemical analysis”
(Sherlock Holmes, “The Sign of Four”, Chapter 10)
Different points of view
David Hilbert(1862-1943), the greatest German mathematician : “Chemical stupidity”…
Auguste Comte (1798-1857), the French philosopher, founder of sociology:
“Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry…if mathematical analysis should ever hold a prominent place in chemistry--an aberration which happily almost impossible--it would occasion of rapid and widespread degeneration of that science”
What is Mathematics?What is Chemistry?
It is always difficult to answer simple questions.
One can say:
Mathematics is about special
symbolic reasoning or symbolic engineering
Chemistry is about transformation of substances
Or in another way:
Mathematics is Newton, Leibnitz, Hilbert,Hardy…
Chemistry is Lavoisier, Dalton, Avogadro, Mendeleev…
Chemical Engineering is Danckwerts, Damkoehler,
Aris, Amundson, Frank-Kamenetsky…
What is Chemical Engineering?
Chemical Engineering = Chemistry + Transport + Material Properties
Most of chemical processes (> 90%) occur with participation of special materials – catalysts, which composition is ill-defined.
Main topics:1) Chemistry2) Transport3) Catalyst properties
Mathematical Chemistry
•There are more than 6 millions of references on
“mathematical chemistry”(Internet)
•Journal of Mathematical Chemistry (since 1987)
•MaCKiE, “Mathematics in Chemical Kinetics and Chemical Engineering” (regular workshop
since 2002)
•MATCH, Communications in Mathematics and Computer Chemistry
The mathematical impact into chemistry is growing
•Models of quantum chemistry (DFT-modeling)
•Computational Fluid Dynamics
•Monte-Carlo modeling
•Statistical analysis
•FT (Fourier Transformation) based experiment
The most important chemical problems
Sustainability Problems = Energy via Chemistry,
e.g. development of the efficient C1 transformation system (CO2 sequestration, CO+H2, CO2+CH4), photocatalytic system of water splitting, hydrocarbon oxidation system, etc.
These problems have to be solved urgently.
Revealing chemical complexity
What is a chemical complexity?
There are many substances
which participate in many reactions.
Typically, chemical reactions
are performed over the catalysts.
Typically, chemical systems
are non-uniform and non-steady-state.
The chemical composition is changing in space and time.
Single Crystal
Single ComponentPolycrystalline
Well-defined Surface Structure
Increasing ComplexityIncreasing Pressure
Technical CatalystMulti-componentMulti-scalePolycrystallineHeterogeneous SurfaceDefectsChanges with Reaction
Structure-Activity Relationships“Materials-Pressure Gap”
?
However we still are very far from revealing chemical complexity,
from solving “chemical structure -activity”problem
Decoding Chemical Complexity: Questions
Questions before the decoding:
1.What we are going to decode?
2.What are experimental characteristics based on which we are going to decode the complexity?
3. In which terms we are going to decode?
Examples of chemical reactions:
Overall Reactions:
2 H2 + O2 2H2O
2SO2 +O2 2SO3
According to chemical thermodynamics,
Keq(T) = C2H2 0/ (C2
H2 C02)
Keq(T) = C2SO3 / (C2
SO2 CO2)
Detailed MechanismDetailed 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 = H20 + OH; 21) O + H2O2 = OH +H02; 22) H2 + O2 = H20 + O;
23) H2 + O2 + M = H202 + M; 24) OH +M = O + H + M; 25) HO2+OH=H2O+O2;
26) H2 + O +M = H2O +M; 27) O + H2O + M = H202 + M; 28) O + H2O2 = H20 + O2;
29) H2 + H2O2 = 2H2O; 30) H + HO2 + M = H2O2 +M
A matrix is the mathematical image of
complex chemical system
( a chemical graph as well)
1. “Atomic”(“molecular” matrix)
2. Stoichiometric matrix
3. Detailed mechanism matrixEnglish mathematician Arthur Cayley (1821-1895), one of the first founders of
linear algebra, applied its methods for enumerating isomers
Complexity1. Catalytic reaction is complex itself
E.g. mixed transition metal oxides used in the selective oxidation + support
2. Industrial catalysts are usually complex multicomponent solids
3. Catalyst composition changes in time under the influence of the reaction medium.
Multi step character of the reactionIncluding generation of different intermediates
A specifically prepared catalyst can exist in different catalyst states
that are functions of oxidation degree, water content, bulk structure, etc.
that have different kinetic properties (activity and selectivity)
Chemical Kinetics = Reaction Rate Analysis
Answers:
Our Holy Grail is the Detailed Mechanism
Our main experimental basis is
the Reaction Rate, R
(+data of some structural measurements)
Different goals of chemical kinetics:
1. To characterize chemical activity of reactive media and reactive materials, particularly catalysts; to assist catalyst design
2. To reveal the detailed mechanism
3. To be a basis of kinetic model for reactor design and recommendations on optimal regimes
Different types of chemical kinetics
1. Applied chemical kinetics
2. Detailed kinetics (Micro-kinetics)
3. Mathematical kinetics
Steady–state and non-steady-state measurements
(1) In most of previous studies, a focus was done on the steady-state experiments. Convectional transport was used as a ‘measuring stick’.(2) In our studies, a focus was done on non-steady-state experiments. Diffusional transport was used as a ‘measuring stick’.
Time Domain of Chemical Reactions
(Paul Weisz window)
Rates of reactions, moles product per cm3 of reactor volume per second
Petroleum geochemistry 5 x 10-14 - 5 x 10-13
Biochemical processes 5 x 10-9 - 5 x 10-8
Industrial catalysis 10-6 x 10-5
3 Methods to Kill Chemical Complexity
1. Describe it in detail (“kinetic screening”)
2. Panoramic description = Forget about it in a correct way (“thermodynamics”)
3. Recognize complexity via the fingerprints (analysis of informative domains, critical behavior etc.)
A statement:For revealing and describing chemical complexitythe following chemico-mathematical approaches
are extremely useful
•State-by-State Kinetic Screening of Active Complex Materials
•Description of Models of Complex Behavior assisted by Advanced Thermodynamic Approaches
•Analysis of Complexity Fingerprints
Chemico-Mathematical Idea #1 = “Chemical Calculus”State-by-State Transient Screening (e.g., Temporal
Analysis of Products, TAP)- John Gleaves, 1988
Three Requirements:1. Insignificant catalyst change during a single non-steady-state
experiment (e.g. one-pulse); 2. Control of reactant amount stored/released by catalyst in a series of
non-steady-state experiments (e.g. multi-pulse);3. Uniform chemical composition within the catalyst zone.
•To kinetically test a catalyst with a particular composition (particular state of the catalyst) that does not change significantly during this non-steady-state test.
•To perform such tests over a catalyst within a wide range of different composition prepared separately and scaled (the reduction/oxidation scale).
Kinetic Characterization
Preparing & Scaling
Simple State
of the Catalyst
Chemico-mathematical idea #2:
Comparison between characteristics related to transport-only model (standard transport curves) and characteristics related to transport-reaction model.
The goal is extracting the intrinsic chemical information.
Kinetic Model-Free Analysis
Reactor Model:Accumulation - Transport Term = Reaction Rate
Batch Reactor: Non SRdt
dCVg
CSTR: Convection SRCCVdt
dCVg )( 0
TAP: Diffusion SRx
CDV
t
CV rg
2
2
PFR: Convection SRx
Cv
t
CVg
Chemico-mathematical idea #3:
Propose a hypothesis about the reaction mechanism based on the kinetic fingerprints.
Typical Kinetic Dependence in Heterogeneous Catalysis(Langmuir Type Dependence)
Reaction
Rate
Concentration
Reaction
Rate
Concentration
Equilibrium
Irreversible Reaction Reversible Reaction
Critical Phenomena in Heterogeneous Catalytic Kinetics:
Multiplicity of Steady-States in Catalytic Oxidation(CO oxidation over Platinum)
Reaction
Rate
CO Concentration
A“Ignition”
B
C“Extinction”
D
Complexity1. Catalytic reaction is complex itself
E.g. mixed transition metal oxides used in the selective oxidation + support
2. Industrial catalysts are usually complex multicomponent solids
3. Catalyst composition changes in time under the influence of the reaction medium.
Multi step character of the reactionIncluding generation of different intermediates
A specifically prepared catalyst can exist in different catalyst states
that are functions of oxidation degree, water content, bulk structure, etc.
that have different kinetic properties (activity and selectivity)
Combinatorial catalysis(mostly steady-state procedure)
It is the most typical method of catalyst preparation.
1. Combination of catalyst compositions
2. Combination of regime parameters(temperature, pressure etc)
3. Testing under steady-state conditions using a battery of simple reactors (plug-flow reactors, PFR)
Our Key Idea of State-by-State Transient Screening
Three Requirements:1. Insignificant catalyst change during a single non-steady-state
experiment (e.g. one-pulse); 2. Control of reactant amount stored/released by catalyst in a series of
non-steady-state experiments (e.g. multi-pulse);3. Uniform chemical composition within the catalyst zone.
•To kinetically test a catalyst with a particular composition (particular state of the catalyst) that does not change significantly during this non-steady-state test.
•To perform such tests over a catalyst within a wide range of different composition prepared separately and scaled, (e.g. the reduction/oxidation scale).
Kinetic Characterization
Preparing & Scaling
Simple State
of the Catalyst
The main methodological and mathematical idea is to perform the integral analysis of data obtained using an insignificant perturbation:1) insignificant perturbation2) integral analysis
TC
Pulse
valve
Microreactor
Mass spectrometer
Catalyst
Vacuum (10-8 torr)
Reactant
mixture
Key Characteristics
Pulse intensity: 10-10 moles/pulse
Input pulse width: 5 x10-4 s
Outlet pressure: 10-8 torr
Observable: Exit flow (FA)
0.0 time (s) 0.5
Exit flow
(F
A)
Inert
Reactant
Product
TAP Pulse Response Experiment
Inert zone Catalyst zone
Thin-Zone (TZ) Idea
Dimensionless Axial CoordinateD
imen
sionless G
as Con
centration
0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.00.00
.25
.50
.75
1.00
1.25
1.50
1.75
2.00
0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.00.00
.25
.50
.75
1.00
1.25
1.50
1.75
2.00
Vacuum System
L/L
Thin-Zone (TZ) TAP Reactor
Thin-Zone Approach Matching Two Inert Zones Through
the Catalyst Zone:
CI(x,t)xLTZ
CII (x,t)xLTZ
CTZ(t)
For concentrations
DCI (x,t)
x xLTZ
DCII (x,t)
x xL TZ
Flux I (x, t)xLTZ
Flux II (x, t)xLTZ
LcatRTZ(CS,CTZ ,TZ )
For flows
TZ-model can be considered
as a diffusional CSTR:
X kappres, cat
diff
1 kappres, catdiff
Conversion
Apparent rate constant
Diffusional residence time in
the catalyst zone
Dim
ension
less C
oncen
tration
Inert zone Catalyst zone
TZTR vs CSTR
TAP:Diffusion
CSTR:Convection
SRVCVC 0
TZ-model can be considered
as a diffusional CSTR:
X kappres, cat
diff
1 kappres, catdiff
Conversion
Apparent rate constant
Diffusional residence time in
the catalyst zone
TZcatLx
II
Lx
I RLx
txCD
x
txCD
TZTZ
),(),(
Uniformity Is Achieved by Mixing
Uniformity Is Insured by Diffusion(finite gradient)
TZTR vs Differential PFR
TZTRDifferential PFR:
Conversion
x
Cx
C
Xg
g
g
g
CC
X
TAP: Diffusion SRx
CDV
t
CV rg
2
2
PFR: Convection SRx
Cv
t
CVg
The main idea is to combine two types of experiments:
Was firstly introduced in the paper:
Gleaves, J.T., Yablonskii, G.S., Phanawadee, Ph., Schuurman, Y.
“TAP-2: An Interrogative Kinetics Approach” Appl. Catal., A: General, 160 (1997) 55.
A state-defining experiment in which the catalyst composition and structure change insignificantly during a kinetic test
A state-altering experiment in which the catalyst composition is changed in a controlled manner
Interrogative Kinetics (IK) Approach
Inert zone Catalyst zone
State-defining Pulses
time
Insignificant change
0.0
State-defining Experiment
The State
The Small Amount of Gas in the Reactor
The Observed Responses Are
Essentially the Same in a Small train of Pulses
The Same Unsteady-State Kinetics
Observe Transient Responses
For the Reactant and Products
Unsteady-State Kinetics
The Gas Adsorbs, Reacts and Desorbs
There is Something about The Catalyst
That Stays the Same
State-Defining Kinetic Regime in a TAP Experiment
1. An insignificant change in the pulse responses within a train of pulses
Criteria of State Defining Experiment:
2. Independence of the shape of pulse response curves on pulse intensity.
Insignificant change
The same shape
Small number of pulses
Small number of pulses
State-defining Experiment
State-defining Experiment
Small number of pulses
Insignificant change
0.0
State-defining Experiment
State-Defining & State-Altering Experiment
Inert Reactant Product
Large number of pulses0.0
State-altering Experiment
TAP Multi-Pulse Experiment Combines
Single Particle Catalyst
• Platinum powder catalyst• Diameter ~400 µm• Packed in reactor middle
surrounded by inert quartz particles with diameters between ~250-300 µm
• Reaction: CO oxidation400 μm platinum particle
The main principle
We are not able to control the surface state
However we are able to control an amount of consumed reactants and released products
Knowing the total amount of consumed reactants, e.g. hydrocarbons, we introduce a catalyst scale
TAP Multipulse Data Reaction of Furan Oxidation over ‘Oxygen Treated’ VPO
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 2500 5000 7500 10000
Furan ConversionMA YieldCO2 YieldCO YieldAC YieldMA+CO2+CO+AC
Pulse Number
Fura
n C
onve
rsio
n/Pr
oduc
t Yie
ld
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8 1
Furan ConversionMA YieldCO2 YieldCO YieldAC YieldMA+CO+CO2+AC
Catalyst Alteration Degree
Fura
n C
onve
rsio
n/Pr
oduc
t Yie
ld
The total amount of transformed furan is
1.4 x 1018 molecules per 1 g of VPO catalyst
differs for different reactants
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
MACO2ACMA+ACMA+AC+CO2
Catalyst Alteration Degree
Por
tion
of
Fur
an M
olec
ules
Con
vert
ed
into
Par
ticl
ular
Pro
duct
s
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
O from MA
O from CO2
O from AC
MA+AC
MA+CO2+AC
Catalyst Alteration Degree
Am
ount
of O
xygn
Ato
ms
Rem
oved
pe
r F
uran
Mol
ecul
e C4H4O + (3MA + 2AC + 9CO2+ 5CO)Ocat MAC4H2O3+ACC3H4O +[CO2+(1/4) AC]4CO2 +CO4CO +[(1/2) MA + CO + CO2)] 2H2O
The total amount of active oxygen consumed in the reaction was approximately the same for all four reactant molecules: 7.7 x 1018 atoms per 1 g VPO
the same for different reactants
TAP Multi-pulse Characterization of Furan Oxidation over Oxygen-treated VPO
Apparent Kinetic Constants for Furan Oxidation as a Function of Oxidation State
0
200
400
600
800
1000
1200
1400
1600
00.20.40.60.81
FuranMACO2AC
Catalyst Oxidation Degree
App
aren
t K
inet
ic C
onst
ant,
r0
, (1/
s)
0
200
400
600
800
1000
1200
1400
1600
00.20.40.60.81
FuranMACO2AC
Catalyst Oxidation DegreeN
on-S
tead
y-St
ate
TO
F,
App
aren
t C
onst
ant /
Oxi
dati
on D
egre
e, (
1/s)
Non-steady-state TOF defined as the apparent constant divided by the oxidation degree, for furan and products (MA, CO2 and AC) versus
the catalyst oxidation degree.
Apparent “Intermediate-Gas” Constant and Time Delay
0
100
200
300
400
500
00.20.40.60.81
FuranMACO2AC
Catalyst Oxidation Degree
App
aren
t "I
nter
med
iate
-Gas
" C
onst
ant,
|r1|,
(abs
. val
ue)
0.0
0.5
1.0
1.5
2.0
2.5
00.20.40.60.81
FuranMACO2AC
Catalyst Oxidation Degree
App
aren
t T
ime
Del
ay, |
r 2/r
1|, (
s)
at least four intermediates can be involved
Detailed Mechanism of Furan Oxidation Over VPO
At least three independent routesAt least four specific intermediates
1) O2 + 2Z 2ZO;
2) Fr + ZO X; 3) Fr + ZO Y; 4) Fr + ZO U; 5) X MA+ Z + H2O;
6) YAC + Z + CO2 + H2O;
7) UZ + CO2+ H2O;
8) ZO + L LO + Z; 9) CO2 + Z1 Z1CO2.
where X, Y, U, Z1CO2 are different
surface intermediates, ZO and LO – surface and lattice oxygen respectively, Z and Z1 are different
catalyst active sites.Stoichiometric coefficients of surface substances will be specified in the course of reaction. Steps 2-4 are supposed to differ kinetically.
State-by-State Transient Screening DiagramMulti-Pulse Thin-Zone TAP Experiment
State-Altering ExperimentA long train of pulses
State-Defining ExperimentChecking state-defining regime for
one-pulse TAP experiment
Distinguishing MechanismsDependence of the coefficients on catalyst state substances
Structure-Activity RelationshipsConsiderations regarding the structure/activity of the active sites
Moment-based analysis of all pulse-response curves
Integral State CharacteristicsNumber of consumed/released gas
substances
Introduction of the Catalyst Scale Catalyst state substances
Kinetic Characteristics of Catalyst StatesBasic kinetic coefficients
Mechanism Assumptions(routes, intermediates, etc.)
Relationships between the coefficients
Chemical Kinetics.Textbook Knowledge (1)
The main law of chemical kinetics is
the Mass-Action Law
The first-order reaction: A B R = kCa
The second-order-reaction 2A B R = k (C )2a
or A+B C R= k Ca Cb
The third-order reaction
3A B; R= k (Ca)3;
2A + B C ; R=k (C )2a Cb
Chemical KineticsTextbook Knowledge (2)
All steps of complex chemical reactions
are reversible,
e.g. A B
Keq (T) =(k+/k-) = Ca / Cb
Chemical Kinetics
Textbook Knowledge (3) 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 = H20 + OH; 21) O + H2O2 = OH +H02; 22) H2 + O2 = H20 + O;
23) H2 + O2 + M = H202 + M; 24) OH +M = O + H + M; 25) HO2+OH=H2O+O2;
26) H2 + O +M = H2O +M; 27) O + H2O + M = H202 + M; 28) O + H2O2 = H20 + O2;
29) H2 + H2O2 = 2H2O; 30) H + HO2 + M = H2O2 +M
What about the General Law of Chemical Kinetics.
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 fuzzyPhysico-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
How to kill complexity, or “Pseudo-steady-state trick”
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)
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)
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)/(K-CH2CCO2)
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)
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;
K jc jipi
i
j
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
Mathematical basis
Our basis is algebraic geometry,
which provides the ideas of variable elimination1. Aizenberg L.A., and Juzhakov,A.P. “Integral representations and residues in multi-dimensional complex
analysis”, Nauka, Novosibirsk, 1979
2. Tsikh, A.K., Multidimensional residues and their applications, Trans. Math. Monographs, AMS, Providence, R.I., 1992
3. Gelfand,I.M., Kapranov, M.M., Zelevinsky, A.V., Discriminants, Resultants, and Multidimensional Determinants, Birkhauser, Boston, 1994
4. Emiris, I.Z., Mourrain,B. Matrices in elimination theory, Journal of Symbolic Computation, 1999, v.28, 3-43
5. 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, steady-state 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.
The Kinetic Polynomial
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
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
K jL iCimi , jL
i
jL
RL ... K j1 iCi
mi , j1
i
j1
R
B0 K f (C) K f (C) 0
Simplification of the polynomial: Four-term rate equation
• It is a “thermodynamic branch” of the kinetic polynomial
),(),(
))()(( 1
ckNck
cfKcfkR eq
Apparent “Kinetic Resistance”= Driving Force/Steady-State
Reaction Rate
KRapp = [ f +(c)- f -(c) /K eq ]/ R ,
where [ f +(c)- f -(c) /K eq ] – “driving force”, or “potential term” ,
R – reaction rate ,
KR app –”kinetic resistance”
Reverse and forward water-gas shift reaction
H2 + CO = H2O + CO2
-6
-4
-2
0
2
4
6
8
0 1 2 3 4 5 6 7 8 9
Partial Pressure CO (kPa)
rate
of
reacti
on
(m
mo
l/kg
.s)
P1
P2
P3P4
P12
P11P10P9
P8
P7
P6
P5
Equilibrium partial pressure of CO
Steady-state rate dependences at different temperatures
• Water –gas shift reaction
-12
-7
-2
3
8
13
0 1 2 3 4 5 6 7 8 9
Partial pressure CO (kPa)
To
tal R
ate
(mm
olk
g-1
ca
ts-1
) = The calculated equilibrium partial pressure
221°C
244°C
261°C
271°C
280°C
291°C
210°C
Apparent “Kinetic Resistance”= Driving Force/Steady-State
Reaction Rate
KRapp = [ f +(c)- f -(c) /K eq ]/ R ,
where [ f +(c)- f -(c) /K eq ] – “driving force”, or “potential term” ,
R – reaction rate ,
KR app –”kinetic resistance”
Kin. Resist.vs Pco
0
5
10
15
20
25
30
0 1 2 3 4 5 6 7 8 9
Partial pressure CO (kPa)
KR
(m
mo
l/m3 )2 (m
mo
lkg
ca
t-1s-1
)-1
221C
244C
261C
271C
280C
291C
Ln(Kin.Res) vs (1/T)
0
0.5
1
1.5
2
2.5
3
3.5
4
0.00175 0.0018 0.00185 0.0019 0.00195 0.002 0.00205
1/T (K-1)
ln K
R
8 kPa CO
7 kPa CO
6 kPa CO
5 kPa CO
4 kPa CO
3 kPa CO
Conclusion: A New Strategy:
• (1) Calculate the kinetic resistance based on the reaction net-rate and its driving force;
• (2) Present this resistance as a function of concentrations and temperature on” both sides of the equilibrium”.
An advantage of this procedure is that the kinetic resistance is just is a linear polynomial regarding its parameters in difference from the non-linear LHHW-kinetic models
Kinetic Fingerprints
Such temporal or parametric patterns that help to reveal or to distinguish the detailed mechanism
E.g., the fingerprint of consecutive mechanism is a concentration peak on the “concentration - time” dependence
Critical Phenomena in Heterogeneous Catalytic Kinetics:
Multiplicity of Steady-States in Catalytic Oxidation(CO oxidation over Platinum)
Reaction
Rate
CO Concentration
A“Ignition”
B
C“Extinction”
D
Critical Simplification:RC=k2
+CCO
RA=k2-
Critical SimplificationAnalyzing 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.
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
Relationships among scientistsDiscussion is a necessary part of the scientific process
•“Dog-eat-dog”
•Fighting
•Quarrel
•Argumentation
•Discussion
•Reconciliation
•Collaboration
•Mutual Understanding
•HARMONY
Collaboration between Sciences
•The most interesting events are occurred on the frontiers
•Scientists are collaborating, not Sciences
Ghent - St. Louis chemical-mathematical crossroads
Results1.Transfer matrix2.Y-procedure3.Coincidences.
Dramatis Personae
• Denis Constales, UGent
• Guy Marin, UGent
• Roger Van Keer, UGent
• Gregory S. Yablonsky, St.-Louis + UGent dr h.c.
1. Transfer Matrix for solving RD eqs.
Advantage: we can calculate the exit flow for
any configuration of reaction zones.
In TAP case:
2. Y-procedure
• Mathematically, it is a combination of the reverse Laplace Transformation method with the Fast Fourier Transformation Method for extracting Reaction Rate with no Assumption about the Detailed Mechanism (“Kinetic model”-free method)
Thin Zone TAP experiments
Inert zone Catalyst zone
(t)C(t)C(t)C
R(t)dx
(t)dC
dx
(t)dC
TZIII
III
Spatial uniformity and well defined transport in the inert zones allow “kinetically model-free” analysis via:
• Primary kinetic coefficients (r0, r1, r2)
• Y-Procedure - reconstruction of C(t) and R(t) (Constales, Yablonsky)
Y-Procedure
l
Direct Problem Inverse Problem
(t)C(t)C(t)C
R(t)dx
(t)dC
dx
(t)dC
TZIII
III
• Y-Procedure extracts gas concentration and reaction rate on the catalyst without any assumptions about the reaction kinetics
G.S.Yablonsky, D.Constales et al. (2007)
Y-Procedure
l
Direct Problem Inverse Problem
(t)C(t)C(t)C
R(t)dx
(t)dC
dx
(t)dC
TZIII
III
• Y-Procedure extracts gas concentration and reaction rate on the catalyst without any assumptions about the reaction kinetics
• Once the reaction rate is known, the surface coverage can be estimated as
t
AA dRS )(
G.S.Yablonsky, D.Constales et al. (2007)
Irreversible adsorption
AZZAk
AAZtotZA CSSkR )( ,
Addressing measurement of total number of active sites, SZ,tot :
• Usually SZ,tot is measured by simple titration
• Is titrated number of active sites different from total number of WORKING active sites?• Measurement of SZ,tot from intrinsic kinetics
0th M
omen
t
Pulse Number
State Defining experiment
Catalyst state remains unchanged (or relatively unchanged) during the pulse.
F
t
tt t
CR S
Exit flux
Concentration Reaction Rate Surface Coverage
State Altering experiment
Catalyst state changes significantly during the pulse.
F
t
tt t
CR S
Exit flux
Concentration Reaction Rate Surface Coverage
State Altering experiment
AAZtotZA CSSkR )( ,
C
R
R vs. C R/C vs. S
S
R/C
AZtotZA
A kSkSC
R ,
k
SZ,tot
Multipulse State Defining vs. State Altering
Sequence of state defining pulses(gradual change of the catalyst)
kap and S for each pulse
S
R/C,kap k
SZ,tot
Multipulse State Defining vs. State Altering
Sequence of state defining pulses(gradual change of the catalyst)
kap and S for each
pulse
S
R/C,kap k
SZ,tot
Sequence of state altering pulsesC
R
R vs. C
Thin Zone TAP experiments for transient catalyst characterization
with application to silica-supported gold nano-particles
Part II
Evgeniy Redekop, Gregory S. Yablonsky, Xiaolin Zheng, John T. Gleaves, Denis Constales, Gabriel M. VeithCREL , February 12th, 2010
Outline
• Summary of Part I (Y-Procedure theory)
• Application to the real data
- catalysis on gold
- CO adsorption
• Conclusions
Dimensionless variables
ALNp
cC
/
catALNp
sS
)1/(
rNpD
ALLR cat
2
D
L
Np
fF
2
2L
Dt
L
lx Length
Time
Flux
Concentration
Surface uptake
Reaction rate
Summary of Part I
Exit flow data
Y
C(t) R(t) S(t)
F(t)Thin-Zone TAP
Transient kinetics on the catalyst
“kinetically model-free”
10-8 torr
Summary of Part I
Exit flow data
Y
C(t) R(t) S(t)
F(t)Thin-Zone TAP
Transient kinetics on the catalyst
Model reaction mechanisms
1st order (Constales et al.)
Irreversible adsorption
Reversible adsorption
Data interpretation
“kinetically model-free”
?
0
40
80
120
160
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Cou
nts
nm
Avg. = 3.20 nm = 1.45 nm# particles = 845
Catalyst: n-Au/SiO2 (11 wt.%)
Gabriel M. Veith
Scanning Transmission Electron Microscopy
(STEM)
CO oxidation on the catalystIntroduction
CO + O2 → CO2
The overall reaction:
Probable mechanism:
O2 → O*2
CO ↔ CO*
CO* + O*2 → CO2
CO oxidation on the catalystIntroduction
CO + O2 → CO2
The overall reaction:
Probable mechanism:
O2 → O*2
CO ↔ CO*
CO* + O*2 → CO2
Oxygen is NOT activated on the surface under TAP conditions
CO adsorption on catalyst
t
F
Exit flux
t
t
R
R
Thin-Zone reaction rate
Yσ = 0
σ = 4, 6• Acceptable filtering at σ = 6
• Loss of peak rate value
Reaction rate:
C
R
Rate vs. Concentration
C
R
Rate vs. Concentration
From experiment From modeling
ZCOdCOZaCO SkCSkR
CO adsorption on catalyst
Catalyst development
Exit flow data
Y
C(t) R(t) S(t)
F(t)Thin-Zone TAP
Transient kinetics on the catalyst
Model reaction mechanisms
1st order (Constales et al.)
Irreversible adsorption
Reversible adsorption
Data interpretation(qualitative and
quantitative)
“kinetically model-free”
Mechanistic Understanding
Catalyst modification
The new phenomenological representation of
the transformation rate
The ‘Rate-Reactivity’Model (RRM)Rgi = ∑Rj (CM, , CMOx, Cad, Cint.r, NS, S, T) Cgj + Roj (CM, CMOx, Cad, Cint.r, NS, S, T) Ri, Roj are catalyst reactivities.Catalyst reactivities are functions of intermediate concentrations. The last ones can be estimated as (Integral uptake of
reactants – Integral release of products)
Coincidences (cont’d)
• New problem is posed: what do we know about the points of intersection, the maximum point of CB(t), and their ordering?
• Example: k1=k2
• we call it Euler point.
Coincidences (cont’d)
• Nonlinear problem, even for a linear system.
• Many analytical results can be obtained.
• Of 612 possible arrangements, only six can actually occur.
• We introduce separation points for domains
• A(cme), G(olden), E(uler), L(ambert),O(sculation), T(riad) points.
• Each point has special ordering or behavior.
Coincidences (cont’d)
• Inspecting the peculiarities of the experimental data, we may immediately infer the domain of the parameters.
• Intersections, extrema and their ordering are an important source of as yet unexploited information.
Different scenarios of interaction
1. Conceptual Transfer
2. “Spark”
3. Joint Activity
4. “Something”
Ideal scenario
•A“creative pair” , people who are able to share interests and values
•Optimal time of “knowledge circulation”
• A clear link to possible realization
Different scenarios of interaction
Inspiration
The great American mathematician J.J. Silvester wrote after becoming acquainted with the records odf Prof. Frankland’s lectures for students chemists:
“I am greatly impressed by the harmony of homology (rather than analogy) that exists between chemical and algebraical theories. When I look through the pages of “records”, I feel like Alladin walking in the garden where each tree is decorated by emeralds, or like Kaspar Hauser first liberated from a dark camera and looking into the glittering star sky. What unspeakable riches of so far undiscovered algebraic content is included in the results achieved by the patient and long-
term work of our colleagues -chemists even ignorant of these riches”.
Different scenarios of interaction
Transfer of concepts. Fick’s Law (1)Adolph Fick, “On liquid diffusion”,Ueber Diffusion, Poggendorff’s Annalen der Physik and Chemie, 94(1855)59-86, see also, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of
Science , Vol. X (1855) 30-39 “It was quite natural to suppose that this law for the diffusion of salt in its solvent must be identical with that, according to which the diffusion of heat in a conducting body takes place; upon this law Forier founded his celebrated thery of heat, and it is the same which Ohm applied with such extraordinary success, to the diffusion of
electricity in a conductor”…
Different scenarios of interaction
Transfer of concepts. Fick’s Law (2)
“…According to this law, the transfer of salt and water occurring in a unit of time, between two elements of space filled with differently concentrated solutions of the same salt, must be,
caeteris paribus, directly proportional to the difference of concentrations, and inversly proportional to the distance elements from one another”
Different scenarios of interaction
Transfer of concepts. Fick’s Law (3)
“The experimental proof just alluded to, consists in the investigation of cases in which the diffusion-current become stationary, in which a so-called dynamic equilibrium has been produced, i.e. when the diffusion-current no longer alters the concentration in the spaces through which it passes, or it other words, in each moment expels from each space-unit as much salt as enters that unit in the same time. In this case the analytical condition is therefore dy/dt=0.”…
Different scenarios of interaction
Transfer of concepts. Fick’s Law (4)“Such cases can be always, if by any means the concentration n two strata be maintained constant. This is most easily attained by cementing the lower end of the vessel filled with the solution, and in which the diffusion-current takes place, into the reservoir of salt, so that the section at the lower end is always end is always maintained in a state of perfect saturation by immediate contact with solid salt; the whole being then sunk in a relatively infinitely large reservoir of pure water, the section at the upper end, which passes into pure water, the section at the upper end, which passes into pure water, always maintains a concentration =0. Now, for a cylindrical vessel, the condition dy/dt =0 becomes by virtue of equation (2), 0 = d2y/dx2 (3)”…
Different scenarios of interaction
Transfer of concepts. Fick’s Law (5)
“The integral of this equation y=ax + b contains the following proposition: - “ If in a cylindrical vessel, dynamic equilibrium shall be produces, the differences of concentration between of any two pairs of strata must be proportional to the distances of the strata in the two pairs,” or in other words the decrease of concentration must diminish from below upwards as the ordinates of a straight line. Experiment fully confirms this proposition”.
“I gave my mind a thorough rest by plunging into a chemical analysis”
(Sherlock Holmes, “The Sign of Four”, Chapter 10)
Read in its context, it is clear that this phrase does not imply any deprecation of chemistry:
“Well, I gave my a thorough rest by plunging it into a chemical analysis. One of our greatest statesmen has said that a change of work is the best rest. So it is. When I had succeeded in dissolving the hydrocarbon which I was at work at, I came back to our problem of the Sholtos [etc.]”