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**Prof. Gregory S.Yablonsky**

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

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“I gave my mind a thorough rest by plunging into a chemical analysis” (Sherlock Holmes, “The Sign of Four”, Chapter 10)

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**Different points of view**

David Hilbert( ), the greatest German mathematician : “Chemical stupidity”… Auguste Comte ( ), 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”

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**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…

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**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

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**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

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**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

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**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.

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**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.

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**Structure-Activity Relationships “Materials-Pressure Gap”**

Well-defined Surface Structure Single Crystal Single Component Polycrystalline Increasing Complexity Increasing Pressure One of the goals in our lab is to understand the relationship between catalyst structure and its activity. One way for us to understand this relationship is to bridge the pressure and materials gap. Under process conditions, it is often difficult to extract intrinsic properties of a catalyst. Intrinsic properties are properties that are directly related to the catalyst composition. Therefore, in order to extract these intrinsic properties, scientists must use surface science techniques under vacuum conditions to study single crystals. By performing single particle experiments using a single component polycrystalline material, we are developing an initial methodology to bridge this pressure and materials gap. Using the TAP reactor, we can define single particles as either single crystals or an industrial catalyst and perform identical experiments under the same conditions on both materials and directly compare the two results. ? Multi-component Multi-scale Polycrystalline Heterogeneous Surface Defects Changes with Reaction Technical Catalyst

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However we still are very far from revealing chemical complexity, from solving “chemical structure -activity”problem

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**Decoding Chemical Complexity: Questions**

Questions before the decoding: What we are going to decode? What are experimental characteristics based on which we are going to decode the complexity? 3. In which terms we are going to decode?

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**Examples of chemical reactions:**

Overall Reactions: 2 H2 + O2 2H2O 2SO2 +O2 2SO3 According to chemical thermodynamics, Keq(T) = C2H2 0/ (C2H2 C02) Keq(T) = C2SO3 / (C2SO2 CO2)

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**An example:Hydrogen Oxidation 2H2 +O2 = 2H2O**

Detailed Mechanism 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

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**A matrix is the mathematical image of complex chemical system **

( a chemical graph as well) “Atomic”(“molecular” matrix) Stoichiometric matrix Detailed mechanism matrix English mathematician Arthur Cayley ( ), one of the first founders of linear algebra, applied its methods for enumerating isomers

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**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.

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**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)

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**Different goals of chemical kinetics:**

To characterize chemical activity of reactive media and reactive materials, particularly catalysts; to assist catalyst design To reveal the detailed mechanism To be a basis of kinetic model for reactor design and recommendations on optimal regimes

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**Different types of chemical kinetics**

Applied chemical kinetics Detailed kinetics (Micro-kinetics) Mathematical kinetics

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**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

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**Time Domain of Chemical Reactions (Paul Weisz window)**

Rates of reactions, moles product per cm3 of reactor volume per second Petroleum geochemistry 5 x x 10-13 Biochemical processes x x 10-8 Industrial catalysis x 10-5

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**3 Methods to Kill Chemical Complexity**

Describe it in detail (“kinetic screening”) Panoramic description = Forget about it in a correct way (“thermodynamics”) Recognize complexity via the fingerprints (analysis of informative domains, critical behavior etc.)

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**State-by-State Kinetic Screening of Active Complex Materials **

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

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**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: Insignificant catalyst change during a single non-steady-state experiment (e.g. one-pulse); Control of reactant amount stored/released by catalyst in a series of non-steady-state experiments (e.g. multi-pulse); Uniform chemical composition within the catalyst zone. Kinetic Characterization Preparing & Scaling Simple State of the Catalyst

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**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.

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**Accumulation = Transport term + Reaction Term**

Mass - Balance Accumulation = Transport term + Reaction Term

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**Kinetic Model-Free Analysis**

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

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**Chemico-mathematical idea #3:**

Propose a hypothesis about the reaction mechanism based on the kinetic fingerprints.

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**Typical Kinetic Dependence in Heterogeneous Catalysis (Langmuir Type Dependence)**

Irreversible Reaction Reversible Reaction Reaction Rate Reaction Rate Equilibrium Concentration Concentration

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**Critical Phenomena in Heterogeneous Catalytic Kinetics:**

Multiplicity of Steady-States in Catalytic Oxidation (CO oxidation over Platinum) C “Extinction” Reaction Rate B A “Ignition” D CO Concentration

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**KINETIC CHARACTERIZATION OF ACTIVE MATERIALS, **

PARTICULARLY OF CATALYSTS

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**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.

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**Combinatorial catalysis (mostly steady-state procedure)**

It is the most typical method of catalyst preparation. Combination of catalyst compositions Combination of regime parameters(temperature, pressure etc) Testing under steady-state conditions using a battery of simple reactors (plug-flow reactors, PFR)

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**Kinetic Characterization**

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: Insignificant catalyst change during a single non-steady-state experiment (e.g. one-pulse); Control of reactant amount stored/released by catalyst in a series of non-steady-state experiments (e.g. multi-pulse); Uniform chemical composition within the catalyst zone. Kinetic Characterization Preparing & Scaling Simple State of the Catalyst

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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

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**TAP Pulse Response Experiment**

0.0 time (s) 0.5 Exit flow (FA) Inert Reactant Product Pulse valve Reactant mixture Microreactor Catalyst Key Characteristics Pulse intensity: moles/pulse Input pulse width: 5 x10-4 s Outlet pressure: 10-8 torr Observable: Exit flow (FA) Mass spectrometer Vacuum (10-8 torr)

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**“Don’t stop questioning !”**

(A. Einstein)

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**Dimensionless Gas Concentration Dimensionless Axial Coordinate**

Thin-Zone (TZ) Idea Catalyst zone Inert zone 0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 0.00 .25 .50 .75 1.00 1.25 1.50 1.75 2.00 0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 0.00 .25 .50 .75 1.00 1.25 1.50 1.75 2.00 Dimensionless Gas Concentration L/L Vacuum System Dimensionless Axial Coordinate

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**Thin-Zone (TZ) TAP Reactor**

Inert zone Catalyst zone Thin-Zone Approach Matching Two Inert Zones Through the Catalyst Zone: Dimensionless Concentration For concentrations For flows TZ-model can be considered as a diffusional CSTR: Conversion Apparent rate constant Diffusional residence time in the catalyst zone

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**Uniformity Is Achieved by Mixing Uniformity Is Insured by Diffusion**

TZTR vs CSTR CSTR: TAP: Convection Diffusion TZ-model can be considered as a diffusional CSTR: Conversion Apparent rate constant Diffusional residence time in the catalyst zone Uniformity Is Achieved by Mixing Uniformity Is Insured by Diffusion (finite gradient)

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**TZTR vs Differential PFR**

Convection TAP: Diffusion Conversion Differential PFR: TZTR

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**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

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**Unsteady-State Kinetics The Same Unsteady-State Kinetics**

Inert zone Catalyst zone State-defining Pulses time Insignificant change 0.0 State-defining Experiment The State State-Defining Kinetic Regime in a TAP Experiment Observe Transient Responses For the Reactant and Products Unsteady-State Kinetics The Gas Adsorbs, Reacts and Desorbs 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 There is Something about The Catalyst That Stays the Same

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**Criteria of State Defining Experiment:**

Insignificant change The same shape Small number of pulses State-defining Experiment An insignificant change in the pulse responses within a train of pulses Independence of the shape of pulse response curves on pulse intensity.

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**TAP Multi-Pulse Experiment Combines**

State-Defining & State-Altering Experiment Inert Reactant Product Small number of pulses Insignificant change 0.0 State-defining Experiment Large number of pulses 0.0 State-altering Experiment

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**Thin-zone and Single Particle Reactor Configurations**

Previously, the most recent reactor configuration has been the thin zone configuration in which the catalyst zone is made very thin in comparison to the whole length of the reactor. By changing to the single particle configuration, we are able to create a more uniform temperature and composition profile in the catalyst zone. These pictures are actually drawn to scale showing the size of the active particle surrounded by a sea of inactive, inert particles. Single-particle

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**Single Particle Catalyst**

Platinum powder catalyst Diameter ~400 µm Packed in reactor middle surrounded by inert quartz particles with diameters between ~ µm Reaction: CO oxidation 400 μm platinum particle The catalyst used for all of the experiments I will talk about today is a platinum powder catalyst. The size of the platinum particle was up to 400 micrometers in diameters. This picture gives you a sense of the small size of the particle in relation to a pencil point. Here I have some SEM images of the platinum particle showing that the particle is a polycrystalline material. For all my experiments, the platinum particle was packed in the middle of the reactor surrounded by inert quartz particles with diameters between 250 to 300 micrometers and the reaction used was always CO oxidation.

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

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**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

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**The total amount of transformed furan is **

TAP Multipulse Data Reaction of Furan Oxidation over ‘Oxygen Treated’ VPO The total amount of transformed furan is 1.4 x 1018 molecules per 1 g of VPO catalyst differs for different reactants

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**the same for different reactants**

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

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**Apparent Kinetic Constants for Furan Oxidation as a Function of Oxidation State**

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.

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**Apparent “Intermediate-Gas” Constant and Time Delay**

at least four intermediates can be involved

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**Detailed Mechanism of Furan Oxidation Over VPO**

At least three independent routes At least four specific intermediates O2 + 2Z 2ZO; Fr + ZO X; Fr + ZO Y; Fr + ZO U; X MA+ Z + H2O; YAC + Z + CO2 + H2O; UZ + CO2+ H2O; ZO + L LO + Z; 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.

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**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 Introduction of the Catalyst Scale Catalyst state substances Kinetic Characteristics of Catalyst States Basic kinetic coefficients 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

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

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**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

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**Chemical Kinetics Textbook Knowledge (2)**

All steps of complex chemical reactions are reversible, e.g. A B Keq (T) =(k+/k-) = Ca / Cb

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**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

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**What about the General Law of Chemical Kinetics. What is a LAW?**

Dependence Correlation MODEL Equation LAW !!!

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**This definition is too fuzzy**

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

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**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)

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**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

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**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)

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**Catalytic mechanism. Two-step mechanism: Temkin-Boudart**

Z + H2O ZO + H2; ZO + CO CO2 + Z 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)

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**Conversion of methane CH4 + Z ZCH2 +H2 ; ZCH2 +H2O ZCHOH +H2 ;**

ZCHOH ZCO +H2 ; ZCO Z + CO Overall reaction : CH4 + H2O CO + 3 H2 R = (K+CCH4CH2O - K-CCOCH23) / ; R= R+ - R- (R+ / R- ) = (K+CCH4CH2O) / ( K-CCOCH23)

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**R = Cy / , Cy = K+ f+(C) - K- f- (C) ,**

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;

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**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

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**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 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

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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.

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**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

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**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

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**Simplification of the polynomial: Four-term rate equation**

It is a “thermodynamic branch” of the kinetic polynomial

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**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”

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**Reverse and forward water-gas shift reaction H2 + CO = H2O + CO2**

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**Steady-state rate dependences at different temperatures**

Water –gas shift reaction

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**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”

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Kin. Resist.vs Pco

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

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**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

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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

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**Critical Phenomena in Heterogeneous Catalytic Kinetics:**

Multiplicity of Steady-States in Catalytic Oxidation (CO oxidation over Platinum) C “Extinction” Reaction Rate Critical Simplification: RC=k2+CCO RA=k2- B A “Ignition” D CO Concentration

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**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.

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**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

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**SCIENTISTS ARE JUST PEOPLE**

A SIMPLE IDEA SCIENTISTS ARE JUST PEOPLE

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

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

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**Collaboration between Sciences**

The most interesting events are occurred on the frontiers Scientists are collaborating, not Sciences

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**Ghent - St. Louis chemical-mathematical crossroads**

Results Transfer matrix Y-procedure Coincidences.

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**Dramatis Personae Denis Constales, UGent Guy Marin, UGent**

Roger Van Keer, UGent Gregory S. Yablonsky, St.-Louis + UGent dr h.c.

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**1. Transfer Matrix for solving RD eqs.**

Advantage: we can calculate the exit flow for any configuration of reaction zones. In TAP case:

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**Transfer Matrix (cont’d)**

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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)

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**Thin Zone TAP experiments**

Inert zone Catalyst zone 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)

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

Y-Procedure Direct Problem Inverse Problem 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) 94

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

Y-Procedure Direct Problem Inverse Problem 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 G.S.Yablonsky, D.Constales et al. (2007) 95

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**First order irreversible reaction**

D.Constales, G.S.Yablonsky, et al. (2007) 96

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**Irreversible adsorption**

Addressing measurement of total number of active sites, SZ,tot : Usually SZ,tot is measured by simple titration 0th Moment Pulse Number Is titrated number of active sites different from total number of WORKING active sites? Measurement of SZ,tot from intrinsic kinetics

98
**State Defining experiment**

Catalyst state remains unchanged (or relatively unchanged) during the pulse. Exit flux F t Concentration Reaction Rate Surface Coverage R S C t t t

99
**State Defining experiment**

R vs. C kap R C 99

100
**State Altering experiment**

Catalyst state changes significantly during the pulse. Exit flux F t Concentration Reaction Rate Surface Coverage R S C t t t

101
**State Altering experiment**

R vs. C R C 101

102
**State Altering experiment**

R vs. C R/C vs. S R R/C k C S SZ,tot 102

103
**Multipulse State Defining vs. State Altering**

kap and S for each pulse R/C, kap k Sequence of state defining pulses (gradual change of the catalyst) S SZ,tot

104
**Multipulse State Defining vs. State Altering**

kap and S for each pulse R/C, kap k Sequence of state defining pulses (gradual change of the catalyst) S SZ,tot R vs. C R Sequence of state altering pulses C

105
**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 12th, 2010 105

106
**Outline Summary of Part I (Y-Procedure theory)**

Application to the real data catalysis on gold CO adsorption Conclusions 106

107
**Dimensionless variables**

Length Concentration Time Surface uptake Flux Reaction rate

108
**“kinetically model-free”**

Summary of Part I Exit flow data F(t) Thin-Zone TAP 10-8 torr Y Transient kinetics on the catalyst C(t) R(t) S(t) “kinetically model-free”

109
**“kinetically model-free”**

Summary of Part I Exit flow data F(t) Thin-Zone TAP Y Transient kinetics on the catalyst C(t) R(t) S(t) “kinetically model-free” Model reaction mechanisms Data interpretation 1st order (Constales et al.) Irreversible adsorption ? Reversible adsorption

110
**Scanning Transmission Electron Microscopy (STEM)**

Catalyst: n-Au/SiO2 (11 wt.%) Scanning Transmission Electron Microscopy (STEM) Gabriel M. Veith 110

111
**CO oxidation on the catalyst Introduction**

The overall reaction: CO + O2 → CO2 Probable mechanism: O2 → O*2 CO ↔ CO* CO* + O*2 → CO2 111

112
**CO oxidation on the catalyst Introduction**

The overall reaction: CO + O2 → CO2 Probable mechanism: O2 → O*2 CO ↔ CO* CO* + O*2 → CO2 Oxygen is NOT activated on the surface under TAP conditions 112

113
**CO adsorption on catalyst**

Exit flux F t

114
**CO adsorption on catalyst**

Exit flux Thin-Zone reaction rate Y F R σ = 0 t t Reaction rate: Acceptable filtering at σ = 6 R σ = 4, 6 Loss of peak rate value t

115
**CO adsorption on catalyst**

Rate vs. Concentration Rate vs. Concentration R R C C From experiment From modeling

116
**CO adsorption on catalyst**

From experiment From modeling

117
**CO adsorption on catalyst**

From experiment From modeling

118
**“kinetically model-free”**

Catalyst development Exit flow data F(t) Thin-Zone TAP Y Transient kinetics on the catalyst C(t) R(t) S(t) “kinetically model-free” Model reaction mechanisms Data interpretation (qualitative and quantitative) Catalyst modification 1st order (Constales et al.) Irreversible adsorption Reversible adsorption Mechanistic Understanding

119
**CO adsorption on catalyst**

K ΔHads ≈ kJ/mol log(K) 1/T

120
**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)

121
3. Coincidences Surprising properties of the simple kinetic models; in particular, A->B->C.

122
**Coincidences (cont’d)**

Solutions (known before)

123
**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.

124
**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.

125
**Coincidences (cont’d)**

Acme, k2=k1/2

126
**Coincidences (cont’d)**

Golden, k2=k1/ø

127
**Coincidences (cont’d)**

Lambert, k2=1.1739… k1

128
**Coincidences (cont’d)**

Osculation, k2=2k1

129
**Coincidences (cont’d)**

Triad, k2=3k1

130
**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.

131
**Different scenarios of interaction**

Conceptual Transfer “Spark” Joint Activity “Something”

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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

133
**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”.

134
**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”…

135
**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”

136
**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.”…

137
**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)”…

138
**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”.

139
“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.]”

140
The Tower of Babel

141
Leaning Tower of Pisa

142
Gulliver’s Watch

143
Gulliver’s Travels

144
Elementary, Watson…. Ideal Science

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