WHAT CAN KINETICS LEARN FROM NONSTATIONARY

THERMODYNAMICS

Miloslav Pekař

Faculty of ChemistryInstitute of Physical and Applied Chemistry

Brno University of TechnologyBrno, Czech Republic

INTRODUCTION

• Evolution of experimental methods in chemical kinetics.

• Fine details of chemical processes and their mechanism can be obtained.

• Evolution of thermodynamics up to several general theories not limited by space or time homogeneity.

• Kinetics and thermodynamics – two different and independent, though complementary approaches to description of chemically reacting systems.

INTRODUCTION

• Most often thermodynamic-equilibrium restrictions on chemical processes and rate equations are discussed.

• Rational thermodynamics + Samohýl new approach to chemical kinetics.

• Direct derivation of rate equations with new insights on the relationships between mechanism and kinetics.

• Overview of this method and its potential relevance in kinetic research and data treatment.

ESSENCE – function

Rational thermodynamics proof:

for fluids with linear transport properties

reaction rate is function of only temperature and densities

or, alternatively, concentrations:

J = J(T,c)

ESSENCE – approximationProved general function is approximated by a

polynomial of degree M:

McnZ n

01 0

,ν

kJ

),,,( 21 n ν

J = (J1, J2…, Jp)

p…number of independent reactions

n…number of components

),,,( 21 pkkk

ννννk

rate constants vector:

Z…number of polynomial terms

ESSENCE – consistency

Fundamental thermodynamic requirement:

equilibrium concentrations found fromkinetic equilibrium condition, J = 0,

must accord with the values of equilibrium constants

restrictions on approximating polynomial follow

ESSENCE – procedure

1. Independent reactions are selected.

2. Approximating polynomial of selected degree is constructed.

3. Equilibrium constants are used to express some concentrations as functions of remaining ones.

4. Modified polynomial should be zero for arbitrary equilibrium values of remaining concentrations.

5. Simplified polynomial results.

EXAMPLES – adsorption “Atoms”: A, S“Components”: A, S, ASOne independent reaction; selected: A + S = AS

• second-degree polynomial standard mass-action law

• third-degree polynomial gives following rate equation:

)()()()( ASS12

SA120ASSA2AS002S

2AASA101AS

1SA110 ccKcckccKcckcKccckcKcckJ

JA = – J

EXAMPLES – adsorption

interpretation of individual terms as reactions

A + AS = 2 A + S

2 AS = A + S + AS

A + 2 S = S + AS

)()()()( ASS12

SA120ASSA2AS002S

2AASA101AS

1SA110 ccKcckccKcckcKccckcKcckJ

EXAMPLES – dissociation adsorption

for components A2, S, AS, A2S

two independent reactions are possible

selected

A2 + 2 S = 2 AS

A2 + S = A2S

second degree polynomial – two terms

A2 + S = A2S

A2S + S = 2 AS

EXAMPLES – dissociation adsorption

third degree polynomial – nine terms

• both independent adsorptions included

• desorption by impact: A2 + A2S = 2 A2 + S

• desorption by surface mobility: AS + A2S = A2 + S + AS

• surface rearrangement: S + 2 A2S = 2 AS + A2S

EXAMPLES – isomerisation

“Atoms”: A, S

“Components”: A, S, AS, RS, R lead to three independent reactions, e.g.:

A + S = AS

AS = RS

RS = R + S

EXAMPLES – isomerisation

second degree polynomial may contain up to 9 terms, e.g.

A + S = ASA = R

A + AS = A + RSA + R = 2 R

AS + S = RS + SR + S = RS

AS + RS = 2 RSRS + R = AS + R

2 A = 2 R

EXAMPLES – isomerisation

the term corresponding to AS + S = RS + S reads

)()()( RS1

2AS01100RS1

2ASS01100SRS1

2SAS01100 cKccKccccKcc kkk

and can be interpreted as surface reaction with concentration dependent rate “constant”

EXAMPLES - CO oxidation

CO + ½ O2 = CO2

Atoms: C, O, S

Components: CO, O2, CO2, S, OS, COS

3 independent reactions are possible, e.g.:

O2 + 2 S = 2 OS

CO + S = COS

OS + COS = 2 S + CO2

EXAMPLES - CO oxidation

second degree polynomial approximation:

)(

)()(

OSCO1

31

1COSO010001

SCO1

31

2OSCO100010COS1

2SCO100100

22

2

ccKKcc

ccKKcccKcc

k

kkJ

three reactions “appear”

CO + S = COS

CO + OS = CO2 + S

O2 + COS = CO2 + OS

O2 + 2 S = 2 OS

CO + S = COS

OS + COS = 2 S + CO2

EXAMPLES - CO oxidation

)(

)()(

OSCO1

31

1COSO010001

SCO1

31

2OSCO100010COS1

2SCO100100

22

2

ccKKcc

ccKKcccKcc

k

kkJ

appearing in rate equation

CO + S = COS

CO + OS = CO2 + S

O2 + COS = CO2 + OS

selected

O2 + 2 S = 2 OS

CO + S = COS

OS + COS = 2 S + CO2

reactions “appearing” are combinations of reactions “selected”, but the terms in rate equation could not be combined in this way

EXAMPLES - CO oxidation

rates of formation:

)(

)()(

OSCO1

31

1COSO010001

SCO1

31

2OSCO100010COS1

2SCO100100

22

2

ccKKcc

ccKKcccKcc

k

kkJ

3CO

1O

2CO 22 JJJJJJ

EXAMPLES - CO oxidation

dissociative oxygen adsorption requires third degree polynomial:

• resulting rate equations are rather complicated and may contain up to twenty terms,

• many steps describe various displacements by attack of gaseous species onto adsorbed ones or various trimolecular reactions of low probability,

• rate constants of all these steps set to zero:

)()(

)()()(2OS

11

2SO010200

2SCO

13COSOS000011

OSCO1

31

1COSO010001SCO1

31

2OSCO100010COS1

2SCO100100

22

222

cKccccKcc

ccKKccccKKcccKcc

kk

kkkJ

EXAMPLES - CO oxidation

five reactions can be identified:

CO + S = COS

CO + OS = CO2 + S

O2 + COS = CO2 + OS

OS + COS = 2 S + CO2

O2 + 2 S = 2 OS

)()(

)()()(2OS

11

2SO010200

2SCO

13COSOS000011

OSCO1

31

1COSO010001SCO1

31

2OSCO100010COS1

2SCO100100

22

222

cKccccKcc

ccKKccccKKcccKcc

kk

kkkJ

O2 + 2 S = 2 OS

CO + S = COS

OS + COS = 2 S + CO2

3CO

1O

2CO 22 JJJJJJ

EXAMPLES - CO oxidation

adding also non-dissociatively adsorbed oxygen (O2S)

to the component list

four independent reactions are possible, e.g.:

O2 + 2 S = 2 OS

CO + S = COS

OS + COS = 2 S + CO2

O2 + S = O2S

EXAMPLES - CO oxidation

in second degree polynomial approximation following reactions are found:

CO + S = COS

O2 + COS = CO2 + OS

O2 + S = O2S

S + O2S = 2 OS O2 + 2 S = 2 OS

CO + S = COS

OS + COS = 2 S + CO2

O2 + S = O2S

EXAMPLES - CO oxidation

• For the oxidation itself only the first two steps are necessary.

• The last two can be viewed, in their reversed direction, as liberation of active sites occupied by oxygen

• For oxidation of one CO molecule only “a half” of oxygen molecule is needed.

• In the forward directions, the last two steps block the active sites.

CO + S = COS

O2 + COS = CO2 + OS

O2 + S = O2S

S + O2S = 2 OS

PRINCIPAL FEATURES

• operates with independent reactions only, but resulting rate equations “contain” also other reactions, relevant for the kinetic description,

• kinetic equilibrium criteria are fulfilled with more general equation than usual Guldberg-Waage with no need for “kinetic” equilibrium constant,

• reaction mechanism for given set of species directly appears,

PRINCIPAL FEATURES

• effects of other reactions or inert species on the rate of particular reaction are possible and naturally included,

• rates of all dependent reactions can be unambiguously expressed from the rates of selected independent reactions,

• independent reactions with “single simple” intermediate, i.e. intermediate which is sole product in one reaction (step) and sole reactant in another step, are not supported,

PRINCIPAL FEATURES

• reaction orders are only whole numbers,

• importance of “additional” terms in the rate equation should be well assessed when the equilibrium constants of selected independent reactions are known,

• no need to determine the backward rate constant from kinetic experiments,

• not all rate constants must be positive (second law of thermodynamics).

)( ijkk

APPLICATION SUGGESTIONS

(…see the following flowchart)

Detection of “all” components in reacting mixture

atoms, components

number of independent reactions

Proposal of independent reactions stoichiometric matrix— chemical intuition— algebraic method (e.g. Hooyman’s)

Selection of degree of approximating polynomial (usual 2 or 3)+

rational thermodynamics method

rate equations for independent reactions

Rates of components’ reactions

“hidden” mechanism with kinetically significant reactions

Experimental tests and verificationsome

LIMITATIONS

• function form is valid strictly only for the fluids with linear transport properties,

• polynomial approximation is purely formal, although it can be given (classical) kinetic interpretation,

• more complex rate equations with higher number of constants to be determined than usual,

• “all components” should be known,

• (non-unique rate equation).

Acknowledgements

Ivan Samohýl(Institute of Chemical Technology, Prague)

Milan Roupec(Brno University of Technology, Brno)