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What happens at the crossroads between Chemical Engineering and Mathematics?

Prof. Gregory S.Yablonsky

Parks College Saint Louis University. USA [email protected]

Dept. of Chemical Engineering, Washington University in St. Louis, USA [email protected]

“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

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) Chemistry 2) Transport 3) 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.

Structure-Activity Relationships “Materials-Pressure Gap”

Well-defined Surface Structure Single Crystal Increasing Complexity Increasing Pressure Multi-component Multi-scale Polycrystalline Heterogeneous Surface Defects Changes with Reaction Single Component Polycrystalline

Technical Catalyst

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 H 2 2SO 2 + O 2 +O 2  2H 2 O  2SO 3 According to chemical thermodynamics, K eq (T) = C 2 H2 0 / (C 2 H2 C 02 ) K eq (T) = C 2 SO3 / (C 2 SO2 C O2 )

Detailed Mechanism

Detailed mechanism is a set of elementary reactions which law is assumed , e. g. the mass-action-law An example:

Hydrogen Oxidation 2H 2 +O 2 = 2H 2 O

1) H 2 + O 2 = 2 OH ; 2) OH + H 2 = H 2 O + H ; 3) H + O 2 = OH + O; 4) O + H 2 = OH + H ; 5) O + H 2 0 = 2OH; 6) 2H + M = H 2 + M ; 7) 2O + M = O 2 + M; 8) H + OH + M = H 2 O + M; 9) 2 OH + M = H 2 O 2 + M; 10) OH + O + M = HO 2 + M; 11) H + O 2 + M = HO 2 + M; 12) HO 2 + H 2 = H 2 O 2 + H;13) HO 2 +H 2 = H 2 O +OH; 14) HO 2 + H 2 O = H 2 O 2 + OH; 15) 2HO 2 = H 2 O 2 + O 2 ; 16) H + HO 2 = 2 0H; 17) H + HO 2 = H 2 O + O; 18) H + HO 2 = H 2 + O 2 ; 19) O + HO 2 = OH +H; 20) H + H 2 O 2 = H 2 0 + OH; 21) O + H 2 O 2 = OH +H0 2 ; 22) H 2 + O 2 = H 2 0 + O; 23) H 2 + O 2 + M = H 2 0 2 + M; 24) OH +M = O + H + M; 25) HO 2 +OH=H 2 O+O 2 ; 26) H 2 + O +M = H 2 O +M; 27) O + H 2 O + M = H 2 0 2 + M; 28) O + H 2 O 2 = H 2 0 + O 2 ; 29) H 2 + H 2 O 2 = 2H 2 O; 30) H + HO 2 + M = H 2 O 2 +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 matrix English mathematician Arthur Cayley (1821-1895), one of the first founders of linear algebra, applied its methods for enumerating isomers

Complexity

1. Catalytic reaction is complex itself

Multi step character of the reaction Including generation of different intermediates

2. Industrial catalysts are usually complex multicomponent solids

E.g. mixed transition metal oxides used in the selective oxidation + support 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)

3. Catalyst composition changes in time under the influence of the reaction medium.

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 cm 3 of reactor volume per second Petroleum geochemistry 5 x 10 -14 - 5 x 10 -13 Biochemical processes 5 x 10 -9 Industrial catalysis 10 -6 - 5 x 10 -8 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 complexity the 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 •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).

Three Requirements : 1.

Insignificant catalyst change experiment (e.g. one-pulse); during a single non-steady-state

Kinetic Characterization

Control of reactant amount stored/released by catalyst in a series of non-steady-state experiments (e.g. multi-pulse);

Preparing & Scaling

Uniform chemical composition within the catalyst zone.

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.

Mass - Balance

Accumulation = Transport term + Reaction Term

Kinetic Model-Free Analysis

Reactor Model:

Accumulation - Transport Term = Reaction Rate Batch Reactor: CSTR: PFR: TAP: Non Convection Convection Diffusion

dC V S dt dC V dt



g C V t C v x R



V t



C D x R

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) Irreversible Reaction Reversible Reaction

Equilibrium Concentration Concentration

Critical Phenomena in Heterogeneous Catalytic Kinetics:

Multiplicity of Steady-States in Catalytic Oxidation (CO oxidation over Platinum)

C “Extinction” B A “Ignition” D

CO Concentration

I. KINETIC CHARACTERIZATION OF ACTIVE MATERIALS, PARTICULARLY OF CATALYSTS

Complexity

1. Catalytic reaction is complex itself

Multi step character of the reaction Including generation of different intermediates

2. Industrial catalysts are usually complex multicomponent solids

that have different kinetic properties (activity and selectivity)

3. Catalyst composition changes in time under the influence of the reaction medium.

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 •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).

Three Requirements : 1.

Insignificant catalyst change experiment (e.g. one-pulse); during a single non-steady-state

Kinetic Characterization

Control of reactant amount stored/released by catalyst in a series of non-steady-state experiments (e.g. multi-pulse);

Preparing & Scaling

Uniform chemical composition within the catalyst zone.

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 perturbation 2) integral analysis

TAP Pulse Response Experiment Pulse valve Microreactor

Vacuum (10 -8 torr) Inert 0.0

time (s) 0.5

Reactant Reactant mixture Catalyst Mass spectrometer Key Characteristics Pulse intensity: 10 -10 moles/pulse Input pulse width: 5 x10 -4 s Outlet pressure: 10 -8 torr Observable: Exit flow (F A )

“Don’t stop questioning !”

(A. Einstein)

Thin-Zone (TZ) Idea Catalyst zone Inert zone 

L/L

0.00

0.0

Dimensionless Axial Coordinate Vacuum System

    Inert zone Thin-Zone (TZ) TAP Reactor Catalyst zone Thin-Zone Approach Matching Two Inert Zones Through the Catalyst Zone: For concentrations

C x

 )

x TZ

For flows 







)

 

TZ-model can be considered as a diffusional CSTR:

Conversion X

 1



diff res

 ,

ca t diff res Diffusional residence time in the catalyst zone Apparent rate constant



TZTR vs CSTR

CSTR: Convection TAP: Diffusion 0

I VC x

 TZ-model can be considered as a diffusional CSTR:

Conversion X

  

diff res



diff res Diffusional residence time in the catalyst zone Apparent rate constant

Uniformity Is Achieved by Mixing Uniformity Is Insured by Diffusion (finite gradient)

PFR: TAP:

TZTR vs Differential PFR

Convection Diffusion

g C V t C v x R



V t



C D x R Differential PFR:



C g

Conversion

TZTR X C g



C g

Interrogative Kinetics (IK) Approach

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. The main idea is to combine two types of experiments: 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

State-Defining Kinetic Regime in a TAP Experiment Catalyst zone Inert zone

Unsteady-State Kinetics

Observe Transient Responses For the Reactant and Products

The State

The Gas Adsorbs, Reacts and Desorbs

State-defining Pulses The Small Amount of Gas in the Reactor 0.0

State-defining Experiment Insignificant change

The Same Unsteady-State Kinetics

The Observed Responses Are Essentially the Same in a Small train of Pulses

There is Something about The Catalyst That Stays the Same

time

Criteria of State Defining Experiment:

State-defining Experiment 1.

An insignificant change in the pulse responses within a train of pulses Insignificant change 2. Independence of the shape of pulse response curves on pulse intensity.

Small number of pulses State-defining Experiment The same shape Small number of pulses

Inert TAP Multi-Pulse Experiment Combines State-Defining & State-Altering Experiment Reactant Product State-defining Experiment Insignificant change 0.0

Small number of pulses State-altering Experiment 0.0

Large number of pulses

Thin-zone and Single Particle Reactor Configurations

Thin-zone Single-particle

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 oxidation 400 μm platinum particle

“Needle in a Haystack” “Pt Needle in a Quartz Haystack”

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

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

0 TAP Multipulse Data Reaction of Furan Oxidation over ‘Oxygen Treated’ VPO 2500 5000

Pulse Number

Furan Conversion MA Yield CO2 Yield CO Yield AC Yield MA+CO2+CO+AC 7500 1.0

0.8

0.6

0.4

0.2

0.0

10000 -0.2

0 Furan Conversion MA Yield CO2 Yield CO Yield AC Yield MA+CO+CO2+AC 0.2

0.4

0.6

Catalyst Alteration Degree

0.8

1 The total amount of transformed furan is 1.4 x 10 18 molecules per 1 g of VPO catalyst

differs for different reactants

TAP Multi-pulse Characterization of Furan Oxidation over Oxygen-treated VPO 1 0.8

0.6

0.4

0.2

MA CO2 AC MA+AC MA+AC+CO2 3 2 1 0 0 6 5 4 O from MA O from CO2 O from AC MA+AC MA+CO2+AC 0 0 0.2

0.4

0.6

Catalyst Alteration Degree

0.8

1 0.2

0.4

0.6

Catalyst Alteration Degree

0.8

C 4 H 4 O + (3  MA + 2  AC + 9  CO2 + 5  CO )O cat   MA C 4 H 2 O 3 +  AC C 3 H 4 O +[  CO2 +(1/4)  AC ]4CO 2 +  CO 4CO +[(1/2)  MA +  CO +  CO2 )] 2H 2 O The total amount of active oxygen consumed in the reaction was approximately the same for all four reactant molecules: 7.7 x 10 18 atoms per 1 g VPO

the same for different reactants

Apparent Kinetic Constants for Furan Oxidation as a Function of Oxidation State 1600 1400 1200 1000 800 600 400 200 0 1 0.8

0.6

0.4

Catalyst Oxidation Degree

0.2

Furan MA CO2 AC

0 1600 1400 1200 1000

Furan MA CO2 AC

800 600 400 200 0 1 0.8

0.6

0.4

Catalyst Oxidation Degree

0.2

Non-steady-state TOF defined as the apparent constant divided by the oxidation degree, for furan and products (MA, CO 2 and AC) versus the catalyst oxidation degree.

500 400 300 200 100 0 1 Apparent “Intermediate-Gas” Constant and Time Delay 0.8

0.6

0.4

Catalyst Oxidation Degree

0.2

Furan MA CO2 AC

2.5

2.0

1.5

1.0

0.5

0 0.0

Furan MA CO2 AC

0.8

0.6

0.4

Catalyst Oxidation Degree

0.2

0 at least four intermediates can be involved

Detailed Mechanism of Furan Oxidation Over VPO

At least three independent routes At least four specific intermediates 1) O 2 + 2Z  2) Fr + ZO  3) Fr + ZO  4) Fr + ZO  2ZO; X; Y; U; 5) X  MA+ Z + H 2 O; 6) Y  AC + Z + CO 2 + H 2 O; 7) U  Z + CO 2 + H 2 O; 8) ZO + L  LO + Z; 9) CO 2 + Z 1  Z 1 CO 2 . where X, Y, U, Z 1 CO 2 are different surface intermediates, ZO and LO – surface and lattice oxygen respectively, Z and Z 1 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 Diagram

Multi-Pulse Thin-Zone TAP Experiment State-Altering Experiment

A long train of pulses

State-Defining Experiment

Checking state-defining regime for one-pulse TAP experiment Moment-based analysis of all pulse-response curves

Integral State Characteristics

Number of consumed/released gas substances

Kinetic Characteristics of Catalyst States

Basic kinetic coefficients

Introduction of the Catalyst Scale

Catalyst state substances

Mechanism Assumptions

(routes, intermediates, etc.) Relationships between the coefficients

Distinguishing Mechanisms

Dependence of the coefficients on catalyst state substances

Structure-Activity Relationships

Considerations regarding the structure/activity of the active sites

Complexity, General Kinetic Law and Thermodynamic Validity: Algebraic Analysis in Chemical Kinetics

Chemical Kinetics.

Textbook Knowledge (1)

The main law of chemical kinetics is the Mass-Action Law The first-order reaction: A  B R = kC a The second-order-reaction 2A  or A+B  C R= k C a C b B R = k (C ) 2 a The third-order reaction 3A  B; R= k (C a ) 3 ; 2A + B  C ; R=k (C ) 2 a C b

Chemical Kinetics

Textbook Knowledge (2)

All steps of complex chemical reactions are reversible, e.g. A  B K eq (T) =(k + /k ) = C a / C b

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 2H 2 +O 2 = 2H 2 O

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 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

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 R i.gen

(X, C) = R i.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 + Cl 2  HCl + Cl 2) Cl + H 2  HCl +H Overall reaction: H 2 + Cl 2  2HCl R=(k 1 k 2 C H2 C Cl2 - k -1 k -2 C 2 HCl ) /  , where  = k 1 C H2 +k 2 C Cl2 + k -1 C HCl +k -2 C HCl

Thermodynamic validity

The equation R=(k 1 k 2 C H2 C Cl2 - k -1 k -2 C 2 HCl ) /  where  = k 1 C H2 +k 2 C Cl2 + k -1 C HCl +k -2 , C HCl is valid from the thermodynamic point of view.

Under equilibrium conditions, R=0,and (C 2 HCl / C H2 C Cl2 ) = (k 1 k 2 /k -1 k -2 )= K eq (T)

Catalytic mechanism. Two-step mechanism: Temkin-Boudart

1) Z + H 2 O  2) ZO + CO  ZO + H 2 ; 1 CO 2 + Z 1 The overall reaction is CO + H 2 O= CO 2 +H 2 , R=[(k 1 C co )(k 2 C H2O )- (k 1 C H2 )(k 2 C CO2 )] /  , where  = k 1 C H2O +k 2 C CO + (k -1 C H2 )(k -2 C CO2 )], R = R + - R ; (R + / R ) = (K + C co C H2O )/(K C H2 C CO2 )

Conversion of methane

1) CH 4 + Z  ZCH 2 2) ZCH 2 +H 2 O  +H 2 ; ZCHOH +H 2 3) ZCHOH  ZCO +H 2 ; 4) ZCO  Z + CO ; Overall reaction : CH 4 R = (K + C CH4 C H2O + H 2 O  - K C CO C H2 3 ) /  CO + 3 H 2 ; R= R + - R (R + / R ) = (K + C CH4 C H2O ) / ( K C CO C H2 3 )



One-route catalytic reaction with the linear mechanism. General expression (Yablonsky, Bykov, 1976)

R = C

/



,

where

C

y is a “cyclic characteristics”,

C

K + f + (C) - K f (C) ,

C

y corresponds to the overall reaction;  presents complexity of complex reaction;  

p ji j i

Kinetic model of the adsorbed mechanism

1) 2 K + O 2 2) K + SO 2   3) KO + KSO 2 2 KO KSO 2  2 K + SO 3

Steady state

(or

pseudo-steady-state

) kinetic model is KO : 2k 1 C O2 (C K ) 2 k -3 C SO3 (C K ) 2 = 0 ; - 2 k -1 (C KO ) 2 - k 3 (C KO ) (C KSO2 )+ KSO 2 : k 2 C SO2 C K - k -2 (C KSO2 ) - k 3 (C KO ) (C KSO2 )+ +k -3 C SO3 (C K ) 2 = 0 ; C K + C KO +C KSO2 =1

Mathematical basis

Our basis is algebraic geometry

which provides the ideas of variable elimination

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 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, 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 = C y , or (  ) R - C y = 0, where C y is the cyclic characteristic;  “Langmuir term” reflecting complexity is the For the typical non-linear mechanism the kinetic polynomial is represented as follows:

B m R m +…+ B 1 R +B o C y

=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   

i K L C



   0     

)  



K C

   )  0

Simplification of the polynomial: Four-term rate equation

• It is a “thermodynamic branch” of the kinetic polynomial

( ( 

f eq

, ) )) , )

Apparent “Kinetic Resistance”= Driving Force/Steady-State Reaction Rate

KR app

= [ 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 H 2 + CO = H 2 O + CO 2 8 6 4 2 0 0 -2 P1 P2 P3 P4 P5 1 P6 2

Equilibrium partial pressure of CO

P7 3 4 5 6 7 8 9

Partial Pressure CO (kPa)

P8 -4 P9 P12 P10 P11 -6

Steady-state rate dependences at different temperatures • Water –gas shift reaction

291°C 280°C

271°C

8 3 -2 0 -7 1 -12 = The calculated equilibrium partial pressure 2 3 4 5 6

Partial pressure CO (kPa)

7 8

210°C 221°C

244°C 261°C

Apparent “Kinetic Resistance”= Driving Force/Steady-State Reaction Rate

KR app

= [ 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

30 25 20 15 10 5 0 0 221C 244C 261C 271C 280C 291C 1 2 3 4 5

Partial pressure CO (kPa)

6 7 8 9

Ln(Kin.Res) vs (1/T)

4 3.5

3 2.5

2 1.5

1 0.5

0 0.00175

0.0018

0.00185

0.0019

1/T (K -1 )

0.00195

0.002

0.00205

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)

C “Extinction” B

Critical Simplification: R C =k 2 + C CO R A =k 2 -

A “Ignition” D

CO Concentration

Critical Simplification

Analyzing kinetic polynomial,

critical simplification

was found At the extinction point R ext.

= k + 2 C co At the ignition point R ign = k -2 Therefore, the interesting relationship is fulfilled R ext / R ign = k + 2 C co /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

(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

A SIMPLE IDEA

SCIENTISTS ARE JUST PEOPLE

Relationships among scientists Discussion is a necessary part of the scientific process

•“Dog-eat-dog” •Fighting •Quarrel •Argumentation •Discussion •Reconciliation •Collaboration •Mutual Understanding •

HARMONY

Collaboration between Sciences

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Transfer matrix 2.

Y-procedure 3.

Coincidences.

Results

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:

Transfer Matrix (cont’d)

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 

I dx



(t) dx R(t)

 

(t)

Spatial uniformity and well defined transport in the inert zones allow “kinetically model-free” analysis via: • Primary kinetic coefficients (r 0 , r 1 , r 2 ) • Y-Procedure - reconstruction of C(t) and R(t) (Constales, Yablonsky)

Y-Procedure

Direct Problem 

Inverse Problem 

I (t) dx



II (t) dx R(t) I



II C



TZ C (t)

• 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

Direct Problem 

Inverse Problem 

I (t) dx



II (t) dx R(t) I



II C



TZ C (t)

• 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

S A





R A

(  )  G.S.Yablonsky, D.Constales et al. (2007)

First order irreversible reaction

A k



B R A



kC A

D.Constales, G.S.Yablonsky, et al. (2007)

Irreversible adsorption



Z k R A

( )

AZ A

Addressing measurement of total number of active sites,

S Z,tot

• Usually

S Z,tot

is measured by simple titration : Pulse Number • Is titrated number of active sites different from • total number of WORKING Measurement of

S Z,tot

active sites?

from intrinsic kinetics

C State Defining experiment

Catalyst state remains unchanged (or relatively unchanged) during the pulse.

Exit flux F Concentration Reaction Rate t Surface Coverage R S t t t

R State Defining experiment

R A k S

R vs. C

AZ A k ap

C State Altering experiment

Catalyst state changes significantly during the pulse.

Exit flux F Concentration Reaction Rate t Surface Coverage R S t t t

R State Altering experiment

R A k S

R vs. C

AZ A

R State Altering experiment

R A k S

R vs. C

AZ A R A C A kS

tot kS AZ

R/C vs. S R/C

C S

S Z,tot

Multipulse State Defining vs. State Altering

k ap

and S for each pulse

R/C, k ap

Sequence of state defining pulses (gradual change of the catalyst)

S Z,tot

Multipulse State Defining vs. State Altering

k ap

and S for each pulse

R/C, k ap

Sequence of state defining pulses (gradual change of the catalyst)

R vs. C S

S Z,tot

Sequence of state altering pulses

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. Veith CREL , February 12 th , 2010

Outline

• Summary of Part I (Y-Procedure theory) • Application to the real data catalysis on gold CO adsorption • Conclusions

Dimensionless variables

Length Time



l L t

 



2 Flux



f Np



Concentration



c Np

/ 

Surface uptake



/( 1 

 )

AL cat

Reaction rate

 

L cat A r NpD

Summary of Part I Exit flow data F(t) Thin-Zone TAP

10 -8 torr “kinetically model-free”

Y Transient kinetics on the catalyst C(t) R(t) S(t)

Summary of Part I Exit flow data F(t) Thin-Zone TAP

“kinetically model-free”

Y Transient kinetics on the catalyst C(t) R(t) S(t) Data interpretation ?

Model reaction mechanisms 1 st order (Constales et al.) Irreversible adsorption Reversible adsorption

Catalyst: n-Au/SiO 2 (11 wt.%)

Scanning Transmission Electron Microscopy (STEM) Gabriel M. Veith 160 120 Avg. = 3.20 nm  = 1.45 nm # particles = 845 80 40 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 nm

CO oxidation on the catalyst Introduction

The overall reaction: CO + O 2 → CO 2 Probable mechanism: O 2 → O * 2 CO ↔ CO * CO * + O * 2 → CO 2

CO oxidation on the catalyst Introduction

The overall reaction: CO + O 2 → CO 2 Probable mechanism: O 2 → O * 2 CO ↔ CO * CO * + O * 2 → CO 2 Oxygen is NOT activated on the surface under TAP conditions

F CO adsorption on catalyst Exit flux t

CO adsorption on catalyst Exit flux Thin-Zone reaction rate F Y R t

Reaction rate: • Acceptable filtering at σ = 6 • Loss of peak rate value

R t t σ = 0 σ = 4, 6

CO adsorption on catalyst Rate vs. Concentration Rate vs. Concentration R R C From experiment

 

S CO Z

C From modeling

S ZCO

CO adsorption on catalyst R S From experiment C From modeling

CO adsorption on catalyst C S R S From experiment C S C From modeling

Catalyst development Thin-Zone TAP Exit flow data F(t) Y Transient kinetics on the catalyst C(t) R(t) S(t) Catalyst modification Data interpretation (qualitative and quantitative)

“ kinetically model-free”

Model reaction mechanisms 1 st order (Constales et al.) Irreversible adsorption Reversible adsorption Mechanistic Understanding

CO adsorption on catalyst

K log(K) ΔH ads ≈ - 24 kJ/mol 1/T

The new phenomenological representation of the transformation rate

The ‘Rate-Reactivity’Model (RRM) Rgi = ∑Rj (C M , , C MOx , Cad, Cint.r, N S , S, T) Cgj + Roj (C M , C MOx , Cad, Cint.r, N S , 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)

3. Coincidences

• Surprising properties of the simple kinetic models; in particular, A->B->C.

Coincidences (cont’d)

• Solutions (known before)

Coincidences (cont’d)

• New problem is posed: what do we know about the points of intersection, the maximum point of C B (t), and their ordering?

• Example:

k 1 =k 2

• 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)

• Acme,

k 2 =k 1 /2

Coincidences (cont’d)

• Golden,

k 2 =k 1 /ø

Coincidences (cont’d)

• Lambert,

k 2 =1.1739… k 1

Coincidences (cont’d)

• Osculation,

k 2 =2k 1

Coincidences (cont’d)

• Triad,

k 2 =3k 1

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

(3)”… becomes by virtue of equation (2), 0 =

d 2 y/dx 2

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.]”