What happens at the crossroads between Chemical Engineering and Mathematics? Prof. Gregory S.Yablonsky Parks College Saint Louis University. USA [email protected].

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 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.

Mass - Balance

Accumulation =

Transport term + Reaction Term

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

I.

KINETIC CHARACTERIZATION OF ACTIVE MATERIALS,

PARTICULARLY OF CATALYSTS

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

“Don’t stop questioning !”

(A. Einstein)

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


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


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.


The same shape

Small number of pulses






0.0


State-Defining & State-Altering Experiment

Inert Reactant Product

Large number of pulses0.0

State-altering Experiment

TAP Multi-Pulse Experiment Combines

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 oxidation400 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

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


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


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


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


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

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 = 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

A SIMPLE IDEA

SCIENTISTS ARE JUST PEOPLE

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:

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


(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)

First order irreversible reaction

BAk

AA kCR

D.Constales, G.S.Yablonsky, 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 Defining experiment

AAZtotZA CSSkR )( ,

C

R

R vs. C

kap

State Altering experiment

Catalyst state changes significantly during the pulse.

F

t

tt t

CR S

Exit flux

Concentration Reaction Rate Surface Coverage


AAZtotZA CSSkR )( ,

C

R

R vs. C


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


Model reaction mechanisms

1st order (Constales et al.)


Reversible adsorption

Data interpretation


?

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

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



From experiment

From modeling

R

SC


From experiment

From modeling

R

SC

S

C

S

C

Catalyst development

Exit flow data

Y

C(t) R(t) S(t)

F(t)Thin-Zone TAP


Model reaction mechanisms

1st order (Constales et al.)


Reversible adsorption

Data interpretation(qualitative and

quantitative)


Mechanistic Understanding

Catalyst modification


K

log(K)

1/T

ΔHads ≈ - 24 kJ/mol

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)

3. Coincidences

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

Coincidences (cont’d)

• Solutions (known before)


• 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.


• 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.


• Acme, k2=k1/2


• Golden, k2=k1/ø


• Lambert, k2=1.1739… k1


• Osculation, k2=2k1


• Triad, k2=3k1


• 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


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”.


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”…


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”



“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.”…


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)”…



“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.]”

The Tower of Babel

Leaning Tower of Pisa

Gulliver’s Watch

Gulliver’s Travels

Elementary, Watson….

Ideal Science

What happens at the crossroads between Chemical Engineering and Mathematics? Prof. Gregory S.Yablonsky Parks College Saint Louis University. USA [email protected].

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