Kinetic modeling of scission and recombination...

Jonas Van Belleghem

compoundsreactions of hydrocarbons and of O- and S-containingKinetic modeling of scission and recombination

Academiejaar 2011-2012Faculteit Ingenieurswetenschappen en ArchitectuurVoorzitter: prof. dr. ir. Guy MarinVakgroep Chemische Proceskunde en Technische Chemie

Master in de ingenieurswetenschappen: chemische technologieMasterproef ingediend tot het behalen van de academische graad van

Begeleiders: Aäron Vandeputte, prof. dr. ir. Guy MarinPromotor: prof. dr. Marie-Françoise Reyniers

Kinetic modeling of scission and recombination

reactions of hydrocarbons and of O- and S- containing

compounds

Jonas Van Belleghem

Promotor: prof. dr. M.-F. Reyniers

dr. ir Aäron Vandeputte

Thesis submitted to obtain the degree of

Master of science in Chemical Engineering

Vakgroep Chemische Proceskunde en Technische Chemie

Voorzitter: prof. Dr. Ir. G.B. Marin

Faculteit Ingenieurswetenschappen en Architectuur

Universiteit Gent

Academiejaar 2011-2012

Summary

This work focuses on the determination of accurate rate coefficients for recombination reactions

using theoretical methods. Canonical variational transition state theory was selected for this

study as it offers an ideal trade-off between accuracy and computational efforts. In total, 34 rate

coefficients have been calculated and used to construct a GA model. Both calculated and GA

predicted rate coefficients are in good agreement with experimental data. A new network

generated with RMG, based on this GA model, succeeds to predict experimentally observed

yields of ethane and n-butane steam cracking.

Keywords

scission reactions; recombination reactions; multireference ab initio methods; CASSCF;

CASMP2; W1bd; canonical transition state theory; group additivity method; reaction networks;

ethane steam cracking; n-butane steam cracking

FACULTY OF ENGINEERING AND ARCHITECTURE

Department of Chemical Engineering & Technical Chemistry Laboratory for Chemical Technology

Director: Prof. Dr. Ir. Guy B. Marin

Laboratory for Chemical Technology • Krijgslaan 281 S5, B-9000 Gent • www.lct.ugent.be

Secretariat : T +32 (0)9 264 45 16 • F +32 (0)9 264 49 99 • GSM: +32 (0)475 83 91 11 • [email protected]

Laboratory for Chemical Technology

Declaration concerning the accessibility of the master thesis

The undersigned, Jonas Van Belleghem ......................................................................

graduate at the UGent in the academic year 2011-2012 and author of the master

thesis with title: Kinetic modeling of scission and recombination reactions of

hydrocarbons and of O- and S- containing compounds .................................................

......................................................................................................................................

hereby declares:

1. he opted for the possibility checked below concerning the consultation of his

master thesis:

X the thesis can always be put at the disposal of any applicant

the thesis can only be disposed of with an explicit, written approval of

the author

the thesis can be put at the disposal of any applicant after a waiting

period of ………… year(s).

the thesis can never be disposed of by any applicant

2. that any user will always be obligated to correctly and fully quote the source

Gent,

(Signature)

Acknowledgment

Vooreerst zou ik graag de promotor, Prof. Dr. Marie-Françoise Reyniers, van dit proefschrift

willen bedanken voor het vertrouwen dat ik de modellering van recombinatiereacties tot een

goed einde zou brengen.

Ook een woord van dank gaat uit naar prof. dr. Ir. G. B. Marin om mij als directeur van het

LCT de mogelijk te geven deze thesis uit te voeren.

Daarnaast bedank ik Dr. Ir. Aäron Vandeputte voor mij te helpen met Gaussian berekeningen

te doen convergeren, het verstaan van theorieën zoals VRC-FTST en het grondig nalezen van

mijn thesis. Zonder hem zou ik nooit zo ver geraakt zijn. Ik wil hierbij even zijn ongelooflijke

bereidheid en onbaatzuchtige inzit om altijd iedereen te willen helpen in de verf zetten.

Ik bedank ook mijn medemasterstudenten zonder wie de tijd op de 918 een pak saaier zou zijn

geweest. In het bijzonder bedank ik Maxime Van den Bossche, Cederik Vancouillie en Steven

Vandermeersch met wie ik veel tijd heb doorgebracht op het tweede verdiep van de 918.

Ik bedank ook mij ouders voor mij in eerste instantie de mogelijkheid te geven om te studeren

en daarnaast voor de steun gedurende de voorbije 5 jaar.

Als laatste bedank ik al mijn andere vrienden om mij af en toe eens het thesiswerk volledig te

doen vergeten.

Kinetic modeling of scission and recombination reactions of

hydrocarbons and of O- and S- containing compounds Jonas Van Belleghem

Promotors: prof. dr. M.-F. Reyniers and dr. ir. Aäron Vandeputte

Abstract This work focuses on the determination of

accurate rate coefficients for recombination reactions

using theoretical methods. Canonical variational transition

state theory was selected for this study as it offers an ideal

trade-off between accuracy and computational efforts. In

total, 34 rate coefficients have been calculated and used to

construct a GA model. Both calculated and GA predicted

rate coefficients are in good agreement with experimental

data. A new network generated with RMG, based on this

GA model, succeeds to predict experimentally observed

yields of ethane and n-butane steam cracking.

I. INTRODUCTION

Optimization of reactors used in large scale chemical

processes is based on models which combine a description

of physical and chemical phenomena.1 The physical

phenomena are accounted for by an adequate reactor model

that accounts for the conservation laws and physical

transport phenomena.1 In these conservation laws, the rates

of production of the chemical compounds emerge which are

described by kinetic models which must include the required

level of detail of the chemistry involved.

For processes based on gas phase radical chemistry, e.g.

the steam cracking of hydrocarbons, the reactive nature of

the intermediates results in huge reaction networks.1

Providing such networks with thermodynamic and kinetic

data is one of the major challenges in developing kinetic

models.

For gas phase chemistry, it is possible to determine such

data on an ab initio basis. However, for reaction networks

describing gas phase radical chemistry it is impossible to

determine all the thermodynamic and kinetic data based on

first principles. Therefore, engineering approximations were

developed to extrapolate the ab initio data from small species

to large species. The group additivity (GA) model for

Arrhenius parameters developed at the LCT2-6

has proven its

accuracy for this purpose.

In this work, one of the main reaction families occurring in

gas phase radical chemistry is modeled, i.e.

scission/recombination reactions. The ab initio determined

rate coefficients are cast into a GA model. The validity of

the first principle method and the GA model is verified by

simulating pilot experiments of steam cracking of ethane and

n-butane with a new reaction network generated with RMG.

II. METHODOLOGY

Modeling recombination reactions. Recombination

reactions differ from other commonly encountered gas phase

reactions due to the absence of a clear barrier in the potential

energy surface. This necessitates a variational application of

transition state theory (VTST)7. In literature several

implementations of TST are described of which three have

been considered: (i) the Gorin model8, (ii) a canonical

Jonas Van Belleghem, Student at Ghent University (UGent), Ghent,

Belgium. E-mail: [email protected]

variational TST (CVTST)9 and (iii) variable reaction

coordinate for a flexible transition state theory (VRC-FTST) 7. The models are applied to 5 recombination reactions:

C•H3+H

•, C

•2H5+H

•,CH3O

•+H

•,C

•H3+C

•H3 and C

•H3+

•OH to

select one of the considered models based on a trade-off

between accuracy and computational effort.

The Gorin model used is the simplest of the three and is

described by an analytic formula. The implementation of

CVTST is based on the work of Vandeputte et al 9. However

small modifications were made to improve its accuracy: (i)

the CBS-QB3 level of theory is replaced with the more

accurate W1bd composite method, (ii) CASMP2

calculations are performed on the CASSCF geometries to

capture more electron correlation and (iii) vibrational modes

are calculated for each point along the reaction coordinate.

The third method, VRC-FTST, is the generally accepted

theoretical framework to calculate quantitatively the rate

coefficient of recombination reactions of two radical

fragments. A software package, VARIFLEX, is available for

application of this theory.

GA model. In the group additivity method, a group is

defined as a polyvalent atom surrounded with all its ligands.

In the GA model for the Arrhenius parameters, the Arrhenius

parameters are found as the sum of the Arrhenius parameters

of a reference reaction and the sum of contributions

accounting for differences in the ligands on the polyvalent

atoms that form a bond during the recombination reaction3:

( ) ( ) ∑ ( )

(1)

( ( )) ( ( ) ) ∑ ( ) ( )

(2)

with Ea the activation energy, ΔGAV0 the group additivity

value of a kinetic parameter, relative to the value of the

reference reaction and Ã the single-event pre-exponential

factor obtained after dividing the pre-exponential A by the

numbers of single-events, defined as:

∏

∏

(3)

with nopt the number of optical isomers, σ the product of

internal and external symmetry number. In first instance the

GA modeling of Arrhenius parameters was applied to

experimental available rate coefficients for recombination

reactions to check if recombination reactions can be modeled

with a GA model.

Reaction network generation and reactor modeling. A new

network to simulate steam cracking of ethane and n-butane

was generated with RMG 3.0. The 1D continuity equations

are integrated with CHEMKIN

Figure 1: Parity plots of ab initio predicted yields simulated with the new network generated with RMG. Left: the intermediate and major

products of ethane steam cracking. Right: intermediate and major products of n-butane steam cracking. Axes are in mass fractions

III. RESULTS

The GA model was applied to 18 experimental

recombination rate coefficients between carbon centered

radicals. 8 reactions were used as a training set to determine

ΔGAV0’s to predict the rate coefficients of the other

reactions. It was found that: (i) ΔGAV0’s for recombination

reactions are temperature dependent and (ii) for

recombination of bulky fragments, an additional contribution

needs to be included in the expressions (1) and (2) that

accounts for gauche interactions.

The average ρ (=kmax/kmin) value based on the calculated

data with the Gorin model, CVTST and VARIFLEX for the

5 recombination reactions amounts to 3.2, 2.4 and 2.4,

respectively. CVTST yields results that are comparable to

FTST at a fraction of the required computational cost. Based

on this result, it was opted to use the CVTST method to

calculated rate coefficients for 34 recombination reactions

from which the ∆GAV°’s can be derived.

19 recombination reactions of hydrogen with carbon

centered radicals and 15 recombination reactions of methyl

with carbon centered radicals have been studied. The

agreement with experimental data is good, i.e. generally

within a factor 3. The 4 additional recombination reactions

involve ring structures and allow to assess the influence of

ring strain effects. It is illustrated that for 5- and 6-membered

rings, ring strain effects do not have an influence on the rate

coefficients for recombination reactions.

34 ’s and 34 ( )

were determined from the

calculated Arrhenius parameters. It was found that the

ΔGAV0’s derived for H

• recombinations can be used for

recombinations involving methyl and vice versa. Based on

this, rate rules for recombination reactions involving oxygen

and sulfur centered radicals are presented.

A new reaction network was generated with RMG 3.0

based on the ΔGAV0’s developed in this work. This reaction

network was used to simulate the steam cracking of ethane

and n-butane. The results are presented on Figure 1 as parity

plots for the intermediate, i.e. up to 5 wt%, and major

product yields. The reaction network is capable to predict the

experimental yields, with exception of the ethane yield of

steam cracking of n-butane.

IV. CONCLUSIONS

The applied CVTST allows to obtain accurate rate

coefficients for recombination reactions at only a fraction of

the computational cost required for more advanced methods.

In total, 34 rate coefficients for recombination reactions have

been calculated. A group additivity method was constructed

to model this reaction family based on the ab initio

determined rate coefficients. The GA model allows to

construct a reaction network that can predict the

experimental yields the steam cracking of ethane and n-

butane.

V. REFERENCES

1. Sabbe, M.K., et al., AIChE J., 2011. 57(2): p. 482-496.

2.Sabbe, M.K., et al., Journal of Physical Chemistry A,

2008. 112(47): p. 12235-12251.

3.Sabbe, M.K., et al., ChemPhysChem, 2008. 9(1): p. 124-

140.


210.


210.

6.Sabbe, M.K., et al., Journal of Physical Chemistry A,

2005. 109(33): p. 7466-7480.

7.Fernandez-Ramos, A., et al., Chemical Reviews, 2006.

106(11): p. 4518-4584.

8.Sumathi, R. and W.H. Green, Theoretical Chemistry

Accounts, 2002. 108(4): p. 187-213.

9.Vandeputte, A.G., M.F. Reyniers, and G.B. Marin, Journal

of Physical Chemistry A, 2010. 114(39): p. 10531-10549.

Kinetisch modeleren van scissie- en recombinatiereacties van

koolwaterstoffen en O- en S- houdende verbindingen. Jonas Van Belleghem

Promotor: prof. dr. M.-F. Reyniers en dr. ir. Aäron Vandeputte

Abstract Het hoofddoel van dit werk is de ab initio

bepaling van accurate snelheidscoëfficiënten. Canonische

variationele transitietoestandstheorie is geselecteerd voor dit

doel. Deze theorie leidt tot accurate schattingen gepaard

gaand met relatief beperkte computationele vereisten. In

totaal zijn 34 snelheidscoëfficiënten bepaald geworden en

gebruikt voor het opstellen van een groepadditief model.

Zowel berekende als GA voorspelde snelheidscoëfficiënten

zijn in goede overeenkomst met experimentele data. Dit GA

model werd gebruikt om een nieuw reactienetwerk te

generen met RMG. Dit netwerk slaagt erin de experimentele

productopbrengsten voor het stoomkraken van ethaan en n-

butaan te voorspellen.

I. INLEIDING

De optimalisatie van reactoren die worden gebruikt in

grootschalige chemisch productieprocessen, is gebaseerd op

modellen die de optredende fysische en chemische

verschijnselen beschrijven.1 Hierbij wordt een zogenaamd

reactormodel gebruikt dat zijn oorsprong vindt in de

behoudswetten. Deze behoudswetten beschrijven het behoud

van energie en materie en hierin komen de productiesnelheden

van al de verschillende chemische verbindingen voor. Deze

worden beschreven door middel van een kinetisch model die

de onderliggende chemie beschrijft.

Voor processen gebaseerd op radicalaire gasfasechemie,

zoals bijvoorbeeld het stoomkraken van koolwaterstoffen,

leidt de reactieve aard van het intermediair tot grote

reactienetwerken.1 Eén van de grootste uitdagingen in

kinetische modelering is zulke grote reactienetwerken

voorzien van de nodige thermodynamische en kinetische data.

Voor gasfasechemie is het mogelijk om deze data te

verkrijgen op basis van ab inito berekeningen. Op zich is het

natuurlijk onmogelijk om de duizenden reacties

computationeel te berekenen. Daarom werden benaderingen

ontwikkeld die toelaten om op basis van data verworven voor

kleine moleculen de thermodynamica en kinetica te

voorspellen van grotere moleculen. Het groepsadditief model

voor Arrhenius parameters dat ontwikkeld is geworden aan het

LCT, heeft zijn accuratesse voor dit doel bewezen.2-6

In dit werk wordt een van de belangrijkste reactiefamilies

die optreden in gasfasechemie gemodelleerd, meer bepaald

scissie/recombinatiereacties. Uit de ab initio bepaalde

snelheidscoëfficiënten wordt een GA-model geabstraheerd. De

geldigheid van de ab initio methode en het GA-model wordt

geverifieerd door pilotexperimenten voor het stoomkraken van

ethaan- en n-butaan te simuleren met een nieuw

reactienetwerk dat werd gegenereerd met RMG.

II. METHODOLOGY

Modellering van recombinatiereacties.

Recombinatiereacties verschillen van andere veel

voorkomende gasfasereacties door het feit dat er geen

Jonas Van Belleghem, Student aan de University van Gent (UGent), Gent,

België. E-mail: [email protected]

duidelijke barrière aanwezig is in het potentieel

energieoppervlak. Dit maakt een variationele benadering van

de transitietoestandstheorie noodzakelijk.7 In de literatuur zijn

verschillende implementaties van TST beschreven waarvan er

hier 3 zijn beschouwd geworden: (i) het Gorin model8, (ii) een

canonische variationele transitietoestandstheorie (CVTST)9 en

(iii) flexibele transitietoestandstheorie met een variabele

reactiecoördinaat (VRC-FTST).7 De modellen zijn gebruikt

om 5 recombinatiereacties, C•H3+H

•, C

•2H5+H

•, CH3O

•+H

•,

C•H3+C

•H3 en C

•H3+

•OH, te berekenen om zo een selectie te

kunnen maken tussen de modellen op basis van accuratesse en

computationele vereisten.

Het Gorin model is het simpelste van de drie en kan worden

vertaald in één enkele analytische formule. De implementatie

van CVTST is gebaseerd of een werk van Vandeputte et al.9 ,

hoewel kleine aanpassingen werden aangebracht: (i) de CBS-

QB3 methode is vervangen door de veel accuratere W1bd

methode, (ii) CASMP2 berekeningen worden doorgevoerd op

de CASSCF geoptimaliseerde geometrieën om zo meer

elektron correlatie te vatten en (iii) de vibrationele modes

worden voor elk punt langsheen het reactiecoördinaat

berekend. De derde methode, VRC-FTST, is de algemeen

aanvaardde methode om kwantitatief de snelheidscoëfficiënt

te bepalen van twee recombinerende radicalen. Een software

pakket, VARIFLEX, is beschikbaar om deze theorie te kunnen

toepassen.

GA-model. In de groepadditiviteitsmethode wordt een groep

gedefinieerd als zijnde een polyvalent atoom dat wordt

omgeven door al zijn liganden. In het GA-model voor

Arrheniusparameters worden Arrheniusparameters gevonden

als de som van Arrheniusparameters van een referentiereactie

en bijdragen die moeten corrigeren voor verschillen in de

liganden die op het polyvalent atoom aanwezig zijn3:

( ) ( ) ∑ ( )

(1)

( ( )) ( ( ) ) ∑ ( ) ( )

(2)

met Ea de activeringsenergie, ΔGAV0 de groepadditieve

waarde van een kinetische parameter, relatief t.o.v. van de

referentie reactie en Ã de single-event preëxponentiële factor

verkregen na delen van de preëxponentiële factor, A, door het

aantal enkelvoudige gebeurtenissen, gedefinieerd als:

∏

∏

(3)

met nopt het aantal optische isomeren en σ het product van

interne en externe symmetriegetallen. In eerste instantie werd

de GA modelering van Arrheniusparameters toegepast op

experimenteel beschikbare snelheidscoëfficiënten voor

recombinatiereacties. Hierbij werd nagegaan of het mogelijk is

om recombinatiereacties groepsadditief te modelleren.

Reactienetwerkgeneratie en reactormodelering. Een nieuw

netwerk is gegeneerd geworden met RMG 3.0 dat kan

gebruikt worden om het stoomkraken van ethaan- en n-butaan

te simuleren. Het softwarepakket CHEMKIN werd gebruikt

voor de simulaties.

Figuur 1: Pariteitsdiagrammen van de ab initio voorspelde productopbrengsten gesimuleerd met het nieuw RMG-reactienetwerk. Links: de

belangrijkste producten bij het stoomkraken van ethaan. Rechts: de belangrijkste producten van stoomkraken van n-butaan.

III. RESULTATEN

Aan de hand van 18 experimentele snelheidscoëfficiënten

voor recombinaties tussen C• radicalen werd gecontroleerd

indien een GA model bruikbaar is voor deze reactiefamilie. 8

reacties werden gebruikt als trainingset om ΔGAV0’s te

bepalen die daarna konden worden gebruikt om de andere

snelheidscoëfficiënten te schatten. Er werd gevonden dat: (i)

ΔGAV0’s van recombinatiereacties variëren als functie van de

temperatuur en (ii) dat voor recombinatiereacties van

volumineuze fragmenten een extra bijdrage in rekening moet

worden gebracht voor gauche-interacties in vergelijking (1) en

(2).

De gemiddelde ρ (=kmax/kmin) waarde van de berekende data

met het Gorin model, CVTST en VARIFLEX voor een testset

die 5 recombinatiereacties bevat, bedraagt respectievelijk 3.2,

2.4 en 2.4. De resultaten van CVTST zijn vergelijkbaar met de

resultaten verkregen met VARIFLEX en dit aan een fractie

van de vereiste computationele kost. Gebaseerd op dit

resultaat werd CVTST geselecteerd om 34

recombinatiereacties te berekenen op basis waarvan ∆GAV°’s

kunnen worden berekend.

19 recombinatiereacties van waterstof met C• radicalen en

15 recombinatiereacties van methyl met C• radicalen werden

bestudeerd. De overeenkomst met experimentele data is goed,

i.e. in het algemeen binnen een factor 3. De 4 extra

recombinatiereacties met H•, zijn reacties waarin

ringstructuren voorkomen. Deze laten toe om na te gaan wat

de invloed is van ringspanning op de snelheidscoëfficiënten.

In het algemeen blijkt voor 5- en 6-ringen dat ringspanningen

geen invloed hebben op de snelheidscoëfficiënten.

34 ’s en 34 ( )

’s zijn bepaald geworden op

basis van de berekende Arrheniusparameters. Er werd

gevonden dat de ΔGAV0’s bepaald voor H

•

recombinatiereacties kunnen worden gebruikt voor

recombinaties van methyl en omgekeerd. Gebaseerd op deze

vaststelling werden snelheidscoëfficiënten bepaald voor

recombinatiereacties van radicalen met zuurstof en zwavel.

Een nieuw reactienetwerk werd gegenereerd met RMG 3.0

dat gebruikt maakt van het GA model, ontwikkeld in dit werk.

Dit reactienetwerk is gebruikt geweest om experimentele data

voor het stoomkraken van ethaan en n-butaan te simuleren. De

resultaten zijn voorgesteld op Figuur 1 als pariteitdiagrammen

voor de belangrijkste producten. Het reactienetwerk is in staat

om de experimentele productopbrengsten te reproduceren met

uitzondering van de ethaanopbrengst bij n-

butaanstoomkraken.

IV. BESLUIT

Het CVTST model laat toe om accurate

snelheidscoëfficiënten voor recombinatiereacties te bepalen en

dit aan een fractie van de computationele kost vereist voor

meer geavanceerde methodes. In totaal werden 34

snelheidscoëfficiënten berekend. Een GA-model is opgesteld

om deze reactiefamilie te modeleren gebaseerd op de ab initio

bepaalde snelheidscoëfficiënten. Het GA-model laat toe om

een reactienetwerk te genereren met RMG dat de

experimenteel waargenomen productopbrengsten voor het

stoomkraken van ethaan- en n-butaan voorspelt.

V. REFERENTIES

1. Sabbe, M.K., et al., AIChE J., 2011. 57(2): p. 482-496.

2.Sabbe, M.K., et al., Journal of Physical Chemistry A, 2008.

112(47): p. 12235-12251.


140.


210.


210.

6.Sabbe, M.K., et al., Journal of Physical Chemistry A, 2005.

109(33): p. 7466-7480.

7.Fernandez-Ramos, A., et al., Chemical Reviews, 2006.

106(11): p. 4518-4584.

8.Sumathi, R. and W.H. Green, Theoretical Chemistry

Accounts, 2002. 108(4): p. 187-213.

9.Vandeputte, A.G., M.F. Reyniers, and G.B. Marin, Journal

of Physical Chemistry A, 2010. 114(39): p. 10531-10549.

Table of Content i

Table of Content List of Figures ........................................................................................................................... iv

List of Tables ........................................................................................................................... viii

List of Symbols ......................................................................................................................... xi

Chapter 1: Introduction .......................................................................................................... 1

1.1. Objectives .................................................................................................................... 2

1.2. Structure of the work ................................................................................................... 3

Chapter 2: Literature Review ................................................................................................. 5

2.1. Wave function based electronic property calculation methods ................................... 5

2.1.1 Fundamental Concepts ......................................................................................... 5

2.1.2 The Hartree-Fock (HF) self consistent field method ........................................... 7

2.1.3 Electron Correlation ............................................................................................. 9

2.1.3.1 Non-Dynamical correlations: Multiconfiguration Self-Consistent Field

Theory (MCSCF) ........................................................................................................... 9

2.1.3.2 Full Configuration Interaction (Full CI) ......................................................... 10

2.1.3.3 Dynamical Correlation ................................................................................... 10

2.1.3.4 Parameterized methods ................................................................................... 13

2.1.4 Computational methods used in this master thesis ............................................. 13

2.2. Some elements of statistical mechanics ..................................................................... 15

2.3. Transition state theory (TST) .................................................................................... 19

2.3.1 Conventional transition state theory (CTST) ..................................................... 19

2.3.2 Variational transition state theory ...................................................................... 21

2.3.2.1 General considerations when modeling reactions without a pronounced

potential energy barrier ................................................................................................ 21

2.3.2.2 The Gorin model ............................................................................................. 23

2.3.2.3 Canonical variational transition state theory (CVTST) .................................. 24

Table of Content ii

2.3.2.4 Variable reaction coordinate for a flexible transition state theory (VRC-FTST)

........................................................................................................................ 24

2.4. Group additivity ......................................................................................................... 28

2.4.1 implementation ................................................................................................... 28

2.4.2 A group additive scheme for the Arrhenius parameters ..................................... 29

2.4.3 Group additivity values based on experimental rate equations .......................... 39

Chapter 3: Method selection to study recombination reactions ........................................... 47

3.1. Implementation of the canonical variational transition state theory .......................... 47

3.1.1 Previous implementation of CVTST .................................................................. 47

3.1.2 Implementation of CVTST used in this master thesis ........................................ 49

3.2. Comparison of transition state theories ..................................................................... 53

3.2.1 Recombination of hydrogen with methyl ........................................................... 53

3.2.2 Recombination of hydrogen with ethyl .............................................................. 55

3.2.3 Recombination of hydrogen with methoxy radical ............................................ 57

3.2.4 Recombination of two methyl radicals ............................................................... 58

3.2.5 Recombination of hydroxyl and methyl radical ................................................. 60

3.3. Selection of an accurate yet cost-effective TST to model recombination reactions . 62

Chapter 4: Recombination reactions involving hydrocarbons ............................................. 65

4.1. Determination of the groups present in the steam cracking network ........................ 65

4.2. Recombination reactions of hydrogen and carbon centered radicals ........................ 68

4.2.1 Alkanes ............................................................................................................... 68

4.2.2 Alkenes ............................................................................................................... 69

4.2.2.1 Scission of a vinylic C–H bond ...................................................................... 69

4.2.2.2 Scission of an allylic C–H bond ..................................................................... 73

4.2.3 Alkynes ............................................................................................................... 75

4.2.4 Ring structures .................................................................................................... 75

4.3. Recombination reactions of carbon centered radicals ............................................... 81

iii

4.3.1 Alkanes ............................................................................................................... 81

4.3.2 Alkenes ............................................................................................................... 84

4.3.2.1 Scission of vinylic C-C bond .......................................................................... 84

4.3.2.2 Scission of allylic C-C bond ........................................................................... 85

4.3.3 Alkynes ............................................................................................................... 86

4.3.4 Ring structures .................................................................................................... 87

4.4. Conclusions ............................................................................................................... 91

Chapter 5: Determination of group additivity values ........................................................... 92

5.1. ΔGAV0’s for recombination reactions of hydrogen centered and carbon centered

radicals ................................................................................................................................. 93

5.2. ΔGAV0’s for recombination reactions of two carbon centered radicals .................... 95

5.3. Group additive modeling of recombination reactions involving oxygen compounds ...

................................................................................................................................... 99

5.4. Group additive modeling of recombination reactions involving sulfur compounds 102

Chapter 6: Modeling steam cracking of ethane and n-butane ............................................ 107

6.1. Reactor modeling ..................................................................................................... 107

6.2. Reaction networks ................................................................................................... 107

6.3. Steam cracking of ethane ......................................................................................... 109

6.4. Steam cracking of n-butane ..................................................................................... 113

6.5. Conclusions ............................................................................................................. 118

Chapter 7: Conclusion and future work ............................................................................. 119

7.1. Future work .............................................................................................................. 121

References .............................................................................................................................. 123

Appendix A: Reaction network .............................................................................................. 128

Appendix B: W1bd calculations ............................................................................................ 134

List of Figures iv

List of Figures

Figure 2–1: Schematic depiction of a potential energy surface with a chemical barrier. ........ 20

Figure 2–2: Rocking modes of two recombining methyl fragments. ....................................... 23

Figure 2–3: Representation of the procedure implemented in VARIFLEX. VARIFLEX

samples geometries of which the energy is calculated by an external software package, e.g.

Gaussian. .................................................................................................................................. 27

Figure 2–4: Transition state for a general recombination reaction of two radical carbon

fragments. The carbon atoms in the full line will form a bond during the course of the

reaction. The Xi and Yi atoms have the C1 or C2 atom as a ligand and, hence, also influence the

reaction. .................................................................................................................................... 33

Figure 2–5: The first reaction, recombination of methyl radicals, is the reference reaction for

the group additive modeling of radical recombination reactions. The second reaction is the

recombination of an ethyl with a methyl radical. ..................................................................... 36

Figure 2–6: The gauche interactions arising in the test reactions 1 – 4 and 6.......................... 43

Figure 2–7: Temperature dependence of . ............................................................ 45

Figure 2–8: Temperature dependence of

. .................................................................. 46

Figure 3–1: Summary of the algorithm used to calculated rate coefficients. ........................... 53

Figure 3–2: The bonding (left) and anti-bonding (right) orbitals of the σ bond that is broken

during the scission reaction. The two radical fragments are at an interfragmental distance of

300 pm from each other. .......................................................................................................... 54

Figure 3–3: Comparison of the recombination rate coefficient calculated using CVTST (full

line), VARIFLEX (dashed line) and the Gorin algorithm (dotted line) with experimental and

theoretical data (symbols). ....................................................................................................... 55

Figure 3–4: Results for the recombination reaction of a hydrogen and ethyl radical obtained

with CVTST (full line), VARIFLEX (dashed line) and the Gorin algorithm (dotted line).

Experimental and theoretical data is also indicated. ................................................................ 56

Figure 3–5: The results obtained with CVTST (full line), VARIFLEX (dashed line) and the

Gorin algorithm (dotted line) for the rate coefficient for recombination of a hydrogen and

methoxy radical. The experimental and theoretical data are presented with symbols. ............ 58

Figure 3–6: The orbitals of the active space. The active space includes the bonding (left) and

anti-bonding orbital (right) of the σ bond that is broken during the course of the reaction. .... 59

List of Figures v

Figure 3–7: The rate coefficient for the recombination of 2 methyl fragments as calculated

with CVTST (full line), and the Gorin model (dotted line) are presented together with

experimental and theoretical data. ............................................................................................ 60

Figure 3–8: Depiction of the results obtained with CVTST, VARIFLEX and the Gorin

algorithm. Experimental and theoretical data are also presented for comparison.................... 61

Figure 4–1: Rate coefficients for the recombination of a hydrogen radical and an iso-propyl

radical. The CVTST rate coefficient is indicated by the full line, the experimental and

theoretical data are indicated by the symbols. .......................................................................... 69

Figure 4–2: Depiction of the orbitals included in the active space calculations for the

recombination of a hydrogen and 1,2-propadiene-3-yl radical. Top: bonding and anti-bonding

orbitals of the resonance effect due to interference of the orbitals of the double bond with the

orbitals of the forming radical. Middle: bonding and anti-bonding orbitals of the double bond

orthogonal to the breaking bond. Bottom: the bonding and anti-bonding orbitals of the

breaking σ bond. ....................................................................................................................... 70

Figure 4–3: Depiction of the orbitals involved during the active space calculations for the

scission of 1,3-butadiene into a hydrogen and 1,3-butadiene-3-yl radical. Top and middle: the

orbitals of the conjugated π system. Bottom: bonding and anti-bonding orbital of the σ bond

that is broken during the reaction. ............................................................................................ 71

Figure 4–4: The CVTST rate coefficient for the recombination of a vinyl and hydrogen

radical compared with experimental data and data calculated by Harding et al. 13

................. 73

Figure 4–5: The orbitals involved in the multi-reference calculations for the scission of

propene into hydrogen and allyl. Top: the bonding and anti-bonding orbitals of the

conjugating π system. Bottom: the bonding and anti-bonding orbitals of the bond that is

broken during the scission reaction. ......................................................................................... 74

Figure 4–6: Orbitals present spanning the active space of the multi-reference calculations. Top

and middle: the orbitals that make up the conjugated π system. Bottom: bonding and anti-

bonding orbital of the breaking σ bond. ................................................................................... 76

Figure 4–7: The results of the CVTST calculations for the rate coefficient for the

recombination of a methyl with an ethyl radical are presented together with experimental and

theoretical data. ........................................................................................................................ 82

Figure 4–8: Representation of calculated, theoretical and experimental rate coefficients for the

recombination reaction of methyl with iso-propyl. .................................................................. 83

List of Figures vi

Figure 4–9: Depiction of the results for the CVTST rate coefficient for the recombination of a

methyl and tert-butyl radical together with experimental data and data calculated by

Klippenstein et al 11

. ................................................................................................................. 84

Figure 4–10: Comparison of the CVTST rate coefficient for the recombination of a methyl

and allyl radical with experimental data reported by Tsang 71

. ................................................ 86

Figure 5–1: The

’s of the recombination reactions involving a hydrogen and a carbon

centered radical as function of the

’s determined from the rate coefficients for

recombination of a methyl and carbon centered radical........................................................... 97

Figure 5–2: the ’s determined from the rate coefficients for recombination of a

hydrogen radical and carbon centered radical as function of the ’s obtained from

the rate coefficients for the recombination of methyl with a carbon centered radical. ............ 98

Figure 6–1: Parity plots for the two main components during ethane steam cracking. Red dots

are simulation results obtained with Reaction Network 1 of Sabbe et al. Orange dots are

simulation results obtained with Reaction Network 2. This is the network of Sabbe et al. in

which the rate coefficients for recombinations are substituted by estimates based on the GA

scheme developed in this work. Green dots are simulation results obtained with Reaction

Network 3, i.e. the network generated with RMG 3.0. .......................................................... 110

Figure 6–2: Parity plots for dihydrogen and methane. Red dots are simulation results obtained

with Reaction Network 1 of Sabbe et al. Orange dots are simulation results obtained with

Reaction Network 2. This is the network of Sabbe et al. in which the rate coefficients for

recombinations are substituted by estimates based on the GA scheme developed in this work.

Green dots are simulation results obtained with Reaction Network 3, i.e. the network

generated with RMG 3.0. ....................................................................................................... 111

Figure 6–3: Parity plots of products with minor yields. Red dots are simulation results

obtained with Reaction Network 1 of Sabbe et al. Orange dots are simulation results obtained

with Reaction Network 2. This is the network of Sabbe et al. in which the rate coefficients for


Green dots are simulation results obtained with Reaction Network 3, i.e. This is the network


Figure 6–4: Parity plots of the four main products. Red dots are simulation results obtained

with Reaction Network 1 of Sabbe et al. Orange dots are simulation results obtained with

Reaction Network 2. This is the network of Sabbe et al. in which the rate coefficients for


List of Figures vii



Figure 6–5: Parity plots of the products with yields between the 1 and 5 wt%. Red dots are

simulation results obtained with Reaction Network 1 of Sabbe et al. Orange dots are

simulation results obtained with Reaction Network 2. This is the network of Sabbe et al. in

which the rate coefficients for recombinations are substituted by estimates based on the GA

scheme developed in this work. Green dots are simulation results obtained with Reaction

Network 3, i.e. the network generated with RMG 3.0. .......................................................... 116

Figure 6–6:Parity plots of the minor products. Red dots are simulation results obtained with

Reaction Network 1 of Sabbe et al. Orange dots are simulation results obtained with Reaction

Network 2. This is the network of Sabbe et al. in which the rate coefficients for




List of Tables viii

List of Tables

Table 2–1: Carbon can have different bonding patterns. The first column lists the different

symbols that are used to distinguish between the different carbon atoms atoms that can be

encountered. The second column explains the meaning of the symbol used. .......................... 29

Table 2–2: Summary of the data abstracted from NIST. [A in m³ mol-1

s-1

and Ea in kJ mol-1

]

.................................................................................................................................................. 40

Table 2–3: ’s derived from the Arrhenius parameters of the reactions belonging to the

training set presented in Table 2. [Ã in m³ mol-1

s-1

and Ea in kJ mol-1

] .................................. 41

Table 2–4: The single-event pre-exponential factors and the activation energies for the

reactions of the test set based on the ‘s of Table 2–3 [Ã in m³ mol-1

s-1

and Ea in kJ

mol-1

] ........................................................................................................................................ 42

Table 2–5: average values for and

. [Ã in m³ mol-1

s-1

and Ea in kJ mol-1

]

.................................................................................................................................................. 44

Table 2–6: Improvements obtained by introducing the . [Ã in m³ mol-1

s-1

and Ea in kJ

mol-1

] ........................................................................................................................................ 44

Table 2–7: ’s for the low temperature range based on Arrhenius parameters of the

reactions belonging to the training set presented in Table 2–2 [Ã in m³ mol-1

s-1

and Ea in kJ

mol-1

] ........................................................................................................................................ 44

Table 2–8: The single-event pre-exponential factors and the activation energies for the

reactions of the test set based on the ‘s of Table 2–7 [Ã in m³ mol-1

s-1

and Ea in kJ

mol-1

] ........................................................................................................................................ 45

Table 3–1: ρ=kmax/kmin for the three studied TST’s. X: Computational too expensive. ........... 62

Table 3–2: Results of the 5 recombination reactions which were calculated to test the three

TST’s. ....................................................................................................................................... 64

Table 4–1: Required groups for the determination of the recombination reactions between a

hydrogen and carbon centered radical occuring in the steam cracking network...................... 65

Table 4–2: Required groups for the determination of the recombination reactions between

carbon centered radicals occuring in the steam cracking network. .......................................... 67

Table 4–3: Ratio of the scission rate for 5- or 6-membered rings to the scission rate of the

alkane or alkene analogue. ....................................................................................................... 77

List of Tables ix

Table 4–4: Comparison of reaction rates for recombination reactions involving 5- or 6-

membered ring radicals to reaction rates for recombination of the alkyl or alkenyl equivalent.

.................................................................................................................................................. 78

Table 4–5: Results of the CVTST calculations for the scission of alkylic, vinylic, allylic and

propargylic C–H bonds (second and third column). Comparison with most recent review

values (fifth and sixth column) or most recent experimental data (seventh, eighth, ninth and

tenth column) available from NIST.......................................................................................... 79

Table 4–6: CVTST results for the scission of alkylic, vinylic, allylic and propargylic C–C

bonds (second and third column). Comparison with most recent review values (fifth and sixth

column) or most recent experimental data (seventh, eighth, ninth and tenth column) available

from NIST. ............................................................................................................................... 88

Table 5–1:

’s and ’s of the rate coefficients for the recombination of a

hydrogen and carbon centered radical. ..................................................................................... 94

Table 5–2:

’s and ’s of the rate coefficients for the recombination of a

methyl and carbon centered radical. ......................................................................................... 95

Table 5–3: Arrhenius parameters for the recombination of a hydroxyl and carbon centered

radical based on the ’s determined for the recombination of hydrogen with carbon

centered radicals. .................................................................................................................... 100

Table 5–4: recombination rate coefficients for recombination of methoxy radical with a

carbon centered radical based on the ’s determined for the recombinations of a

hydrogen radical and a carbon centered radical. .................................................................... 101

Table 5–5: Comparison of rate coefficients obtained with the GA model with the experimental

data from NIST. ...................................................................................................................... 102

Table 5–6: Arrhenius parameters for the recombination of a sulfanyl radical and a carbon

centered radical based on the ’s presented in Most reactions have rate coefficients that

decrease with increasing temperature, leading to negative activation energies. It can be seen

that an adjacent methyl group generally increases the activation energy for recombination

with a few kJ mol-1

, with exception for the reaction H• + CH2=CHC

•HCH3. Similar activation

energies and pre-exponential factors are obtained for recombinations leading to vinylic C–H

bonds. The single-event pre-exponential factor for these recombination reactions range around

6 107 m

3 mol

-1 s

-1 and the activation energy amount to ±2 kJ mol

-1. ..................................... 102

Table 5–7: Arrhenius parameters for the recombination of a methylsulfanyl radical and a

carbon centered radical based on the ’s presented in Most reactions have rate

List of Tables x

coefficients that decrease with increasing temperature, leading to negative activation energies.

It can be seen that an adjacent methyl group generally increases the activation energy for

recombination with a few kJ mol-1


•HCH3.

Similar activation energies and pre-exponential factors are obtained for recombinations

leading to vinylic C–H bonds. The single-event pre-exponential factor for these

recombination reactions range around 6 107 m

3 mol

-1 s

-1 and the activation energy amount to

±2 kJ mol-1

. ............................................................................................................................. 104

Table 6–1: Experimental conditions during the ethane cracking experiments. The HC feed is

the hydrocarbon feed and is in g s-1, the steam dilution δ is in g g-1, CIT and COT stand for

coil inlet and outlet temperature and are in °C, the Max Temp is the maximum temperature

observed along the reaction tube and is in °C, CIP and COP stand for coil inlet and outlet

pressure and are in bar. ........................................................................................................... 109

Table 6–2: Experimental conditions during the steam cracking of n-butane. The HC feed is

the hydrocarbon feed and is in g s-1

, the steam dilution δ is in g g-1

, CIT and COT stand for

coil inlet and outlet temperature and are in °C, the Max Temp is the maximum temperature

observed along the reactor tube and is in °C, CIP and COP stand for coil inlet and outlet

pressure and are in bar. ........................................................................................................... 113

Introduction xi

List of Symbols

Roman Symbols

Pre-exponential factor m3 mol

-1 s

-1 or s

-1

Single-event pre-exponential factor m3 mol

-1 s

-1 or s

-1

Single-event pre-exponential factor of the reference reaction m3 mol

-1 s

-1 or s

-1

E Energy kJ mol-1

Activation barrier at 0 K (including the ZPVE) kJ mol-1

Electronic activation barrier (excluding the ZPVE) kJ mol-1

Activation energy kJ mol-1

Activation energy of reference reaction kJ mol

-1

G Gibbs energy kJ mol-1

Gibbs activation energy kJ mol-1

Difference of Group Additivity Value in the transition state

and reactant value

Group additivity value for one of the two Arrhenius

parameters, relative to the value of the reference reaction

Group additivity value of the activation energy, relative to the

activation energy of the reference reaction

Group additivity value of the single-event pre-exponential

factor, relative to the single-event pre-exponential factor of the

reference reaction

Degeneracy of energy state i -

Plank’s constant 6.62 x 10 -34

J.s

ħ Reduced Plank’s constant 1.05 x 10 -34

J.s

Hamiltonian operator J

Principle moment of inertia along the principle axis of inertia i kg m2

Reduced moment of inertia of an internal rotation kg m2

Ionization potential of fragment X

Reaction rate coefficient m3.mol

-1.s

-1 or s

-1

Single-event rate coefficient m3 mol

-1 s

-1 or s

-1

Introduction xii

Boltzmann constant 1.38 10

-23 J

molecule-1

K-1

Mass of electron 9.109 10-31

kg

Mass of fragment X

Molecularity of the reaction -

Number of single-events -

Number of optical isomers -

Pressure Pa

Molecular partition function -

Canonical partition function of a system of N indistinguishable

particles -

Distance between particle i and particle j m

Ideal gas constant 8.314 J mol-1

K-1

Set of coordinates of all the nuclei present in a molecular

system -

Reaction coordinate m

Activation entropy J mol-1

K-1

Symmetry-independent single-event activation entropy J mol-1

K-1

Temperature K

Operator used in coupled cluster theory

Volume of a system m3

Set of coordinates of particle i (including spatial and spin

coordinates) -

Atomic number of atom i -

Greek and other Symbols

‡ Transition state -

Steam dilution kg kg-1

Permittivity of free space 8.854 10

-12

C2 N

-1 m

-2

Frequency of internal mode cm-1

Introduction xiii

Product of internal rotational and external rotational symmetry

numbers -

Internal rotational symmetry number -

External rotational symmetry number -

Product of symmetry numbers of fragment X and Y -

Eulerian angles (θi, φi, χi) of fragment i -

Spherical polar coordinates of the line connecting the centers of mass

of two recombining fragments -

Acronyms

CAS+1+2 Multireference configuration interaction method with first and

second order excitations

CASPT2 Multireference method with perturbation theory corrections

CASSCF Complete active space self consistent field

CC Coupled cluster

CCSD Coupled cluster with single and double excitations

CCSD(T) Coupled cluster with single and double excitations with

pertubative treatment of the third excitations

CI Configuration Interaction

CIP Coil inlet pressure bar

CISD Configuration interaction with single and double excitation

CIT Coil inlet temperature °C

COP Coil outlet pressure bar

COT Coil outlet temperature °C

CTST Conventional transition state theory

Introduction xiv

CVTST Canonical variational transition state theory

FR Free rotor

FTST Flexible transition state theory

GA Group additive

GAV Group additivity value

Gn Gaussian composite methods

HF Hartree Fock theory

HO Harmonic oscillator

HR Hindered rotor

MCSCF Multiconfiguration self consistent field

MPn Møller-Pleset perturbation theory with corrections of order n

MRCI Multireference configurational interaction

NNI Non nearest neighbor interaction

PES Potential energy surface

RMG Reaction mechanism generator

SCF Self consistent field

TST Transition state theory

VRC-FTST Variable reaction coordinate for a flexible transition state theory

Wn Weizmann composite methods

W1BD Weizmann-1 composite methods with Breuckner doubles

ZPVE Zero point vibrational energy

Introduction 1

Chapter 1: Introduction

Optimization of reactors used in large scale chemical processes is based on models which

combine a description of physical and chemical phenomena.1 The physical phenomena are

accounted for by an adequate reactor model that accounts for the conservation laws and

physical transport phenomena.1 In these conservation laws, the rates of production of the

chemical compounds emerge which are described by a kinetic model.

For commonly encountered industrial reactors, reactor models are rather well established.1

However, the kinetic models mostly lack the required level of detail of the chemistry involved

which is essential if one wants to control a process on a molecular level. These almost

chemistry-free kinetic models lump the chemical compounds in groups based on global

properties such as boiling point.1

Detailed kinetic models are, however, very difficult to construct as there are frequently

several hundreds of species involved and mostly thousands of reactions are occurring between

these species.2, 3

This is certainly the case for chemical processes based on gas phase radical

chemistry. The reactive nature of the radical intermediates results in complex chemistry, i.e.,

in huge reaction networks.1

The importance of gas phase radical chemistry is evident as the steam cracking process, by

which the building blocks for the petrochemical industry are produced, is based on this type

of chemistry. For steam cracking of hydrocarbons, it is generally accepted that there are three

reaction families involved:1 (i) carbon-carbon and carbon-hydrogen bond scission and the

reverse radical-radical recombinations, (ii) hydrogen abstraction reactions which can be

intermolecular and intramolecular and (iii) radical addition to olefins and the reverse β

scission of radicals which can also be intermolecular or intramolecular.

Based on these reaction families, reaction networks can be constructed. As already stated,

these reaction networks contain several hundreds of species and several thousands of

reactions. It is evident that such reaction networks, when constructed by hand, can miss many

of the important reactions. Therefore, several research groups have developed computer tools

to automatically generate these reaction networks.4, 5

Introduction 2

Once these reaction networks are constructed, they need to be provided with the necessary

thermodynamic and kinetic data. Experimental determination of these data is very time

consuming and when obtained by fitting the predicted yields to experimental yields,

deficiencies in the network might be compensated by a bias on the rate coefficients. This can

lead to a reaction network that performs well within the limited range of experimental

conditions for which the fitting parameters are determined, however, predictions for

conditions outside this limited range can be off.

In this respect, the use of quantum chemistry to calculate the required data is attractive as it

avoids time consuming experimental work and the need to rely on assumed reaction

schemes.1 Furthermore, the data is intrinsic in nature. However, calculating the

thermodynamic and kinetic data for thousands of reactions on a first principle basis is

practically impossible, certainly for larger species. Therefore, in previous work, group

additivity models have been constructed that allow to predict the thermodynamics of the

species and the kinetics of two of the three important reaction families occurring in the steam

cracking process, i.e. radical additions and hydrogen abstractions based on accurate ab initio

calculations involving only small species.6-12

Up to now, rate coefficients for recombination

reactions were obtained from theoretical work performed by Klippenstein et al.13-15

or

obtained by application of the geometric mean rule.

1.1. Objectives

Theoretical and experimental data for bond scission/recombination reactions are limited and

often large discrepancies exist between reported rate coefficients. Many authors have

addressed the complexity to calculate rate coefficients for this reaction family, however only

few reactions are well documented. This work therefore aims at calculating reliable rate

coefficients for recombination reactions involving hydrocarbons and O- and S-containing

compounds on a feasible way. A group additivity method is constructed that allows to model

these recombination reactions and the applicability of the GA model is illustrated for the

simulation of pilot plant experiments, conducted at the Laboratory for Chemical Technology

(LCT).

Although only hydrocarbons are involved during steam cracking, oxygen and sulfur

containing compounds are also studied as they are becoming more and more important, e.g.

for the pyrolysis of biomass which typically contains oxygen compounds. Sulfur components

are also frequently added to the hydrocarbon feed for steam cracking as S has proven to

Introduction 3

reduce CO and coke formation during the process.16

Despite the growing interest towards O-

and S-containing compounds, the main focus in this work is on hydrocarbons.

1.2. Structure of the work

In Chapter 2, literature available on the topic has been discussed. In a first part different

computational quantum chemistry methods are discussed, focusing mainly on multi-reference

techniques and composite methods. This is followed with a discussion of some important

concepts of statistical physics, required to understand the calculation of rate coefficients.

Next, variational transition state theory is discussed and its use for modeling radical-radical

recombination reactions is explained. The chapter ends with a discussion of the group

additivity method that is previously developed to predict rate coefficients. Its use is illustrated

by applying the method to an extensive set of experimental rate coefficients for recombination

reactions obtained from the NIST Chemical Kinetics Database.17

Doing so, the validity of a

GA method for recombination reactions can be demonstrated

In Chapter 3, three transition state theories (TST’s) that have been developed to predict rate

coefficients for recombination reactions are reviewed. Based on a set containing 5 reactions,

the method yielding the optimal trade-off between accuracy and computational cost will be

selected for further use in this work. The three theories considered are: the Gorin model,

canonical variational transition state theory (CVTST) and variable reaction coordinate for a

flexible transition state theory (VRC-FTST). The Gorin model can be reduced to an analytic

formula and is the most simple method considered here. For CVTST various calculations

need to be performed, which were automized. The method is founded on the CVTST method

reported by Vandeputte et al.18

The VRC-FTST is implemented by other researchers in a

software package called VARIFLEX.19, 20

In Chapter 4, rate coefficients for an extensive set of bond scission/recombination reactions,

i.e. 34 reactions, are presented. The studied reactions suffice to construct a group additive

model which allows to make an estimate of the rate coefficients for all the recombination

reactions present in the previously developed steam cracking network.1

In the next chapter, Chapter 5, group additive values for the 34 calculated recombination

reactions are presented and discussed. Rate rules are presented for recombination reactions

involving O- and S-compounds.

Introduction 4

In Chapter 6, pilot plant experiments for ethane and n-butane steam cracking are simulated

with various reaction networks. One of the three networks is the network developed by Sabbe

et al.1 The other two reaction networks make use of the group additivity model presented in

this work to estimate the rate coefficients for recombination reactions. These 2 networks are:

(i) the reaction network constructed by Sabbe et al.1 in which the rate coefficients for

recombination reactions were modified and (ii) a reaction network obtained using an

automated reaction network generator, i.e. RMG 3.0, containing ab initio determined

thermodynamic and kinetic for the three most important reaction families.

The last chapter, Chapter 7, concludes the work and makes suggestions for possible future

work.

Literature Review 5

Chapter 2: Literature Review

This chapter starts with a discussion on wave function based electronic property calculation

methods. The strengths and flaws of each method will be highlighted and based on this

comparison a method will be selected for further use in this work. In a second part of this

chapter, some elements of statistical mechanics are highlighted as they form the bridge to the

next paragraph dealing with conventional and variational transition state theory. The chapter

ends with a discourse on the group additivity method developed by Sabbe et al.8-10, 21

and an

application of the method to experimentally obtained recombination rate coefficients acquired

from the NIST Chemical Kinetics Database17

.

2.1. Wave function based electronic property calculation

methods

As mentioned in the introduction of this master thesis, the aim of the presented work is to

calculate accurate rate coefficients for recombination/addition reactions using first principles.

A method is said to be from first principles if it starts from the laws of physics without

empirical corrections or fitted parameters. The postulates and theorems of quantum

mechanics, hence, form the rigorous foundation for the prediction of the observable chemical

properties.22

Going into the subtle details of quantum mechanics is far beyond the scope of

this literature review. Rather, it will give an overview of a few fundamental concepts of

quantum mechanics and use these concepts to give a general impression of the wave function

based electronic property calculation methods used in this master thesis.

2.1.1 Fundamental Concepts

The governing equation of quantum mechanics is the time-independent Schrödinger equation:

[2–1]

Hamiltonian operator

Ψ the wave function which is function of the set of coordinates of all the

particles of the system

E energy corresponding to the wave equation Ψ

Literature Review 6

Solving equation [2–1] results in a complete set of eigenvalues Ei and corresponding

eigenfunctions Ψi that are orthonormal. This feature of the eigenfunctions results in:

∫

∫

∫

[2–2]

Equation [2–2] offers a prescription for determining the energy of a quantummechanical

system: with a wave function in hand one simply constructs and solves the integral on the

right.22

On the other hand even if one obtains a wave function which is not a solution of [2–1],

the corresponding expectation value for the energy can still be calculated based on [2–2].

For the systems of interest, i.e. molecular systems, the Hamiltonian takes into account five

contributions to the total energy of a system: the kinetic energy of the electrons and nuclei, the

attraction of the electrons to the nuclei and the interelectronic and internuclear repulsions:22

∑

∑

∑∑

∑∑

∑∑

[2–3]

where i and j run over the electrons and k and l run over the nuclei.

ħ reduced Planck’s constant (1.055 10-34

Js)

mass of an electron

the elementary charge

atomic number of nuclei k

distance between particle i and particle j

Laplacian acting upon particle i

Solving equation [2–1] is extremely difficult due to the pairwise attraction and repulsion

terms, implying that no particle is moving independently of all of the others.22

To simplify the

problem, one generally assumes that the electrons will adiabatically follow the motion of the

nuclei, which allows to separate the motion of the electrons from the motion of the nuclei.

This assumption is often referred to as the Born-Oppenheimer approximation and leads to:23

[2–4]

spatial coordinates of all the nuclei of the molecular system

spatial and spin coordinates of all the electrons of the molecular system

Literature Review 7

wave function describing the motion of the nuclei

wave function describing the motion of the electrons, for a fixed position of

the nuclei

Esys the total energy of the system (electrons and nuclei) within the Born-

Oppenheimer approximation.

with:

[ ∑

∑∑

∑∑

∑∑

]

[2–5]

[ ∑

] [2–6]

The Born-Oppenheimer approximation, hence, divides the Hamiltonian of equation [2–3] into

an electronic part, equation [2–5], in which the positions of the nuclei act as parameters.

Equation [2–5] can be seen as a Schrödinger equation for the electrons and, hence, a set of

eigenvalues Eel,i and corresponding eigenfunctions, Ψel,i, will result upon solving this

equation. If, for every combination of parameters, , the lowest eigenvalue is retained, a so

called potential energy surface (PES) is constructed. The motion of the nuclei on this PES can

be described by equation [2–7], obtained after substituting equation [2–5] in equation [2–4]

and dividing the right and left hand side by :

[2–7]

2.1.2 The Hartree-Fock (HF) self consistent field method

Due to interelectronic repulsion, solving equation [2–5] is not straightforward. However, one

can significantly reduce the complexity of the problem by separating the movement of the

various electrons. This results in a Hamiltonian which is a sum of one particle Hamiltonians.

The interelectronic repulsion is then replaced by a mean field, which interacts with all

electrons. The wave function of such a sum of one particle Hamiltonians can be written as a

Slater determinant of one particle state wave functions:

Literature Review 8

√ |

| [2–8]

one particle state wave function i

For the Slater determinant to be a good approximation of the true wave function, ,

corresponding with the electronic Hamiltonian, an accurate treatment of the interelectronic

interactions is needed. Due to the antisymmetric character of the slater determinant, the HF

mean field includes a coulomb term and a correlation term.

The HF equations are derived based on the variational principle. For every constructed Slater

determinant, one can calculate the expectation value for the energy (equation [2–2]):

∫ [2–9]

In order to evaluate equation [2–9] an optimal set of one particle state wave functions is

required. This optimal set can be obtained by minimizing [2–9] with respect to the one

particle state wave functions and the boundary condition that the one particle state functions

are orthonormal.24

This variational principle in quantum mechanics guarantees that

minimization of [2–9] leads to an upper limit of the true ground state energy. The variational

principle then results in the Hartree-Fock equation which allows to determine the one particle

state wave functions:23

∑

∑ [∫

∫

]

[2–10]

In order to solve equation [2–10] the one particle state wave functions are generally projected

on an orthonormal set of basis functions. This leads to a matrix equation. However, solving

this matrix equation is not straightforward as solving the equation to the expansion

coefficients requires that the expansion coefficients are already known. This matrix equation

is, hence, solved iteratively: an initial set of values for the expansion coefficients is assumed

and new expansion coefficient are determined. This procedure is repeated until the difference

between the old and new expansion coefficients is below a certain threshold. This solution

method is referred to as self consistent.

Literature Review 9

The two major shortcomings in the Hartree Fock method are: (i) the one electron nature of the

Hartree Fock equation, used to determine the one particle state wave functions and (ii) other

than the exchange – second integral of [2–10] – , all electron correlation is ignored.22

2.1.3 Electron Correlation

To overcome the major shortcoming of the Hartree Fock method, ways have been developed

to include some of the correlated motion of the electrons. With a single determinant, one

cannot do better than the HF wave equation, so an obvious choice is to construct a wave

function as a linear combination of multiple determinants:22

[2–11]

There are two types of electron correlation: dynamical and non-dynamical correlation. The

dynamical electron correlation methods try to compensate for the correlated motion of the

electrons. These correlations tend to be made up from a sum of individually small

contributions from other determinants.22

As a consequence, the Hartree Fock wave function is

a leading term in [2–11] and c0 is much larger than any other coefficient.22

However, in some instances, one or more of these other determinants may have coefficients of

similar magnitude to that for the HF wave function.22

In case multiple degenerate orbitals are

available, one of them will be chosen during the HF calculation to be occupied. The SCF

cycle will optimize the shapes of all of the occupied orbitals and one will end up with a best

possible single-Slater-determinantal wave function based on the initial choice.22

However, an

equally good wave function was obtained if the original guess had chosen to populate one of

the other degenerate orbitals.22

Thus, it might be expected that each of these different

determinants contribute roughly equally to an expansion of the kind represented by [2–11].22

It is important to emphasize that the error here is not so much that the HF approximation

ignores the correlated motion of the electrons, but rather that the HF process is constructed in

a fashion that is intrinsically single-determinantal which is insufficiently flexible for some

systems.22

2.1.3.1 Non-Dynamical correlations: Multiconfiguration Self-Consistent Field

Theory (MCSCF)

MCSCF is an advanced computational method that allows to include non-dynamical electron

correlation when the ground state has more than one dominant determinant. An important

issue when using MCSCF calculations is the selection of the orbitals and electrons that should

Literature Review 10

be included in the MCSCF optimization procedure. These orbitals and electrons are called the

active orbitals and electrons and together they form the active space, denoted by (m,n), where

m represents the number of active electrons and n refers to the number of active orbitals.22

After the selection of the active space, the next question that needs consideration is how many

configurations should be included in the MCSCF procedure. The selection of the most

important configuration can be done on a rational basis. However, an alternative to picking

and choosing amongst configurations is simply to include all possible configurations in the

expansion.22

For a (4,4) active space this would lead to 20 singlet configurations. When all

possible arrangements of electrons are allowed to enter into the MCSCF expansion the

method is referred to as a complete active space self-consistent field (CASSCF) calculation.22

2.1.3.2 Full Configuration Interaction (Full CI)

A full CI is a CASSCF calculation for which the active space contains all the electrons and all

the orbitals. Within the choice of basis set, it is the best possible calculation that can be done,

because it considers the contribution of every possible configuration.22

If a full CI would be

performed with an infinite basis set, an exact solution for the – time-independent, Born-

Oppenheimer – Schrödinger equation would be obtained.23

2.1.3.3 Dynamical Correlation

In general, there are three different ways to include dynamical correlation: configuration

interaction (CI), perturbation methods and coupled cluster theory.

2.1.3.3.1 Configuration interaction

Performing a full CI on a large molecule with a large basis set is practically impossible.

However, one can choose to reduce the number of excitations allowed. To proceed, it is useful

to rewrite the full-CI wave function as a linear combination of excited configurations:22

∑∑

∑∑

[2–12]

where i and j are occupied MO’s in the HF reference wave function, r and s are virtual MOs

in the HF reference wave function, and the additional configurations appearing in the

summations are generated by exciting an electron from the occupied orbital(s) into the virtual

orbital(s).22

As only the single reference HF wave function is used, this method is referred to

as a single reference CI method.


If it is assumed that the error between the HF energy and the true electronic energy is due to

dynamical correlations, there is little need to reoptimize the MOs for every configuration

included in [2–12]. The problem is then reduced to determining all the coefficients in the

expansion [2–12]. This is done on a variational way and leads to the typical encountered

secular equation.22

In practice, the expansion of [2–12] is mostly truncated after the double excitations and the

resulting method is referred to as CISD. This method has one very appealing feature: it is

variational. On the other hand CISD also has a problem referred to as “size consistency”.22

This formalism can also be applied to systems that require a MCSCF wave function. The

method is then called multireference configuration interaction (MRCI). The idea is quite

similar to that for single-reference CI, except that instead of the HF wave function serving as

reference, a MCSCF wave function is used.22

2.1.3.3.2 Perturbation Theory

The starting point of perturbation theory is to replace a difficult to handle operator with an

operator for which the resulting eigenvalue problem can be solved by removing an unpleasant

portion of the initial operator. Using the exact eigenvalues and eigenfunctions of the

simplified operator, it is possible to estimate the eigenfunctions and eigenvalues of the more

complete operator.

With respect to the electronic Schrödinger equation, the troubling term is the coulombic

repulsion term between the electrons. Møller and Plesset25

proposed, as a more tractable

operator, the sum of the one-electron fock operators with corresponding eigenfunction and

eigenvalue of this operator the HF Slater determinant and the sum of the energies of the

occupied one particle state wave functions.22

Depending on the number of correction terms taken into account, the method is referred to as

MP2, MP3…. Taking only first order corrections (MP1) into account leads to the HF

electronic energy. A big advantage of the MPn correction is that they are size-consistent22

,

however they are no longer variational in nature which means that it is possible that the

correlation energy is overestimated.22

Perturbation theory can also be applied to systems that require a MCSCF wave function. The

obvious choice for the eigenfunction of the simplified operator is the MCSCF wave function.

However, it is much less obvious what should be chosen for the simplified operator itself.

22


One of the more popular choices is the so-called CASPT2N method of Roos and co-

workers.26

An appealing feature of multireference perturbation theory is that it can correct for

some deficiencies associated with an incomplete active space 22, 27

.22, 27

2.1.3.3.3 Coupled Cluster Theory (CC Theory)

The central idea of CC theory is that the full-CI wave function can be described by:22

[2–13]

with

∑

[2–14]

where n is the total number of electrons and the various operators generate all possible

determinants having i excitations form the reference.22

For example:

∑∑

[2–15]

in which the amplitudes t are determined by the constraint that [2–13] is met.

When operates on the ΨHF, the full CI will already result. The question is, hence, why

making the operator more complex by taking the exponential of it. This is done because the

operator will be truncated in order to limit the number of excitation. If, for example,

would be simplified to , then only double excitation would be incorporated in the new wave

function. It has already been mentioned that this leads to problems like size consistency.

However, if the exponential of is taken, the powers of are also generated and this

resolves the size consistency problem.22

The next step is the determination of the amplitudes t. This is done in the usual way by left

multiplying the Schrödinger equation with trial wave functions expressed as determinants of

the HF orbitals. This makes the CC method, however, no longer variational.22

The most commonly used implementation of CC theory are the CCSD and the CCSD(T)

methods.22

In the latter, the triple excitations are treated perturbatively.


2.1.3.4 Parameterized methods

Some of the post-HF methods are very powerful. For example, a full CI carried out with large

and flexible basis sets leads to highly accurate solutions of the Schrödinger equation.

However, a full CI with a large and flexible basis set cannot be applied to more than the

smallest fraction of chemically interesting systems because of their computational expense.22

As a result, theories emerged in which parameters are introduced to improve predictive

accuracy.22

There are roughly three important groups in the multilevel methods: the Gn methods, the CBS

methods and the Wn methods22

.

Within the Gn methods one tries to account for errors due to basis-set incompleteness and

correlation energy in an additive fashion. Generally speaking, the geometry of a molecule is

optimized on a certain post-HF method and this geometry is used for all other high level post-

HF calculations. The final electronic energy is computed as a sum of an electronic energy

resulting from a high level post-HF method and small contributions. These small

contributions are made up of differences between several high level post-HF methods with

several basis sets.22

The CBS methods try to compensate for incomplete basis-sets by extrapolating the result for

different levels of theory to the complete basis-set limit. The electronic energy is calculated

also in a composite way.22

The Wn methods are similar to the CBS methods in that extrapolation schemes are used to

estimate the infinite basis set limits.22

A key difference between the two is that the Wn models

set as a benchmark goal an accuracy of 1 kJ mol-1

on thermochemical quantities.22

This

resulted in the very accurate, but computational expensive, W1bd method which is

recommended when very accurate energies are requisite.28

2.1.4 Computational methods used in this master thesis

Calculating rate coefficients requires that some information of the PES is obtained (see 2.3).

For bond scission reactions it is clear that during the reaction the electrons of the breaking

bond initially occupy the bonding orbital of the breaking bond. However, at certain

interfragmental distances, at least two configurations become important,29

(i) a configuration

in which. the one electrons occupy the bonding orbital and (ii) a configuration having one

electron in both the bonding and antibonding orbital. Hence, to describe the energy profiles


during bond scission, multireference methods will have to be applied to guarantee the required

flexibility to describe the wave function. It this work CASMP2 calculations were chosen for

calculations along the reaction coordinate as these calculations are able to capture both

dynamical as non-dynamical electron correlation.

It is noted that also other methods have been used to study bond scission. For example, a

specific MRCI method, CAS+1+2, has been used by Klippenstein et al.30-32

Certainly for

smaller systems, this method is more accurate than the CASPT2 method. However, for larger

systems CAS+1+2 suffers from size-extensivity whereas CASPT2 is approximately size-

extensive.29

Furthermore, CASPT2 calculations are less sensitive to the choice of the active

space.22, 29

Next to this, CASPT2 calculations scale with N5 whereas CAS+1+2 scales with

N6.29


2.2. Some elements of statistical mechanics

The aim of statistical mechanics is to relate microscopic properties of particles, obtained from

quantum mechanics, to macroscopic thermodynamic properties such as enthalpy, entropy…..

To derive thermodynamic properties, statistical mechanics make use of the concept ensemble,

which is nothing more than a large number of copies of the system, each representing a

possible state. Imagine an ensemble of systems having a fixed volume V, containing N

identical and indistinguishable particles and in thermal contact with a heat reservoir at

temperature T. This is called a canonical ensemble or a NVT ensemble as from a macroscopic

point of view, this system is defined by the parameters T, V and N. For such a system,

statistical mechanics state that the thermodynamic properties can be calculated as:33

( )

[2–16]

( ( ) ( )

) [2–17]

with

Q the total canonical partition function of the system as defined in equation

[2-18]

R ideal gas constant [J mol1 K

-1]

E thermodynamic internal energy [J mol-1

]

S entropy [J K-1

mol-1

]

∑

[2–18]

The summation in equation [2–18] goes over all the possible states of the system and:

the energy of state n

the degeneracy of state n

Boltzmann constant [J molecule-1

K-1

]

To calculate Q(T,V,N), it is assumed that the particles which make up the system behave as a

classical ideal gas. In this case the total partition function can be found as:33

[2–19]

with


q is the molecular partition function defined as in equation [2–20]

∑

[2–20]

where the summation goes over all the possible states of the molecule.

These molecular energy levels can be calculated using computational chemistry. The Born-

Oppenheimer approximation (see 2.1.1), allows to split the electronic energy from the energy

related to the motion of the nuclei:

[2–21]

The energy levels related to the motion of the nuclei can be divided in contributions caused by

rotational, translational and rovibrational movement. If the coupling between the translational

motion of the molecule and the internal motion of the nuclei is neglected, equation [2–21] can

be written as:24

[2–22]

The third term in equation [2–22] relates to the rotational motion of the molecule and the

internal motion of the nuclei which are coupled in most cases.

With this split-up of the energy levels, the molecular partition function can be rewritten as:

[2–23]

the electronic partition function is calculated as:

[2–24]

For this partition function, only the ground state energy is taken into account as it can be

assumed that the population of excited states will be very low at moderate temperatures.

The translational partition function can be calculated as:34

[2–25]

The standard way to calculate the qext rot+rovib is to neglect the coupling between the rotational

motion of the molecule and the internal rovibrational motion of the nuclei:


[2–26]

The external rotational partition function is calculated as:34

√

(

)

√ [2–27]

with

external rotational symmetry number

principle moment of inertia i along the principle axis of inertia i [kg m2]

The internal partition function is calculated based on the approximation that the internal

motion of the nuclei can be approximated as harmonic oscillators (the HO approximation). In

this case the internal partition function is also called the vibrational partition function and is

calculated as34

:

∏

[2–28]

with

frequency corresponding to internal mode i [s-1

]

However, for molecules that are characterized with low frequencies for internal modes that

are not pure vibrational in nature, it has been found that the thermodynamic properties are not

reproduced well if these modes are modeled as harmonic oscillators.33

These low-frequency

modes are most of the time similar to internal rotations. More accurate partition functions for

these low-frequency modes, are obtained by treating these modes as hindered rotors.

In previous work, computational methods have been developed to take these considerations

into account7. In this previous work, the coupling between all the internal rotations is

neglected. For every internal rotation a relaxed scan is performed as function of the dihedral

angle. The resulting profile is interpolated and leads to a potential as a function of the dihedral

angle. The corresponding Schrödinger equation is solved and the partition function

corresponding to this internal rotation is calculated as:33

∑

[2–29]

internal rotational symmetry number


There is one special case of internal rotations that deserves some more attention: the free rotor

(FR). If the energy barrier is well below the value RT, the energy profile can be assumed flat.

The corresponding approximate partition function is:33

[2–30]

with

I Reduced moment of inertia for the internal rotation [kg m2]


2.3. Transition state theory (TST)

The aim of this master thesis is the a priori modeling of scission/recombination reactions for

hydrocarbons and O- and S-containing compounds in the gas phase. Scission/recombination

reactions are different from the other commonly encountered reaction families that occur

during gas phase kinetics, e.g. β-scissions, isomerization, eliminations et cetera, as a

pronounced potential energy barrier along the reaction coordinate is missing.35, 36

The

implications of this will be briefly pointed out, followed by a discussion of several transition

state theories that are developed during the past decades.

2.3.1 Conventional transition state theory (CTST)

Most theoretical rate coefficients reported in literature are obtained using CTST.36

The

assumptions that lead to a closed analytic expression for the rate coefficient are:37

i. Molecular systems that have surmounted the col in the direction of products cannot

turn back and form reactant molecules again.

ii. The energy distribution among the reactant molecules is in accordance with the

Boltzmann distribution. Furthermore, it is assumed that even when the whole system is

not at equilibrium, the concentration of those activated complexes that are becoming

products can also be calculated using equilibrium theory.

iii. It is possible to separate the motion of the system over the col from the other motions

associated with the activated complex.

iv. A chemical reaction can be satisfactorily treated in terms of classical motion over the

barrier, quantum effects being ignored.

Several derivations exist 35, 37

that lead to Equation [2–31] for the rate coefficient within the

framework of CTST:

∏ (

) [2–31]

q the molecular partition function per unit volume[m-3

]

electronic barrier (excluding ZPVE) [kJ mol-1

]

Boltzmann constant [J molecule-1

K-1

]

plank constant [J s]

rate coefficient as calculated by CTST [m3 mol

-1 s

-1 for a bimolecular

reaction or s-1

for a monomolecular reaction]


T Temperature [K]

universal gas constant [J mol-1

K-1

]

From Equation [2–31], one of the biggest advantages of CTST is evident: there is a limited

amount of information required to calculate the rate coefficient.36

It suffices to calculate the

ground state energy of the reactants and the transition state, the frequency of the normal

modes of the reactants and the transition state and the geometry of the reactants and the

transition state. This means that the amount of ab initio calculations is reduced to some

sampling points on the potential energy surface (PES) in order to find the transition state. This

can be done on a relatively low level of theory. Once the transition state is found, only a few

calculations need to be performed on a higher level of theory: one for each reactant and one

for the transition state.

However, there is one question that remains: “Where to locate the transition state?”. This can

be deduced from the first assumption stating that a critical surface has to be found so that

every trajectory passing through this surface started in the reactant valley and that these

reactive trajectories do not re-cross the surface.35

There is, thus, a surface that separates the

reactant valley from the product valley. For reactions with a pronounced saddle point in the

potential energy surface(see Figure 2–1) , i.e. a maximum for the reaction coordinate and a

minimum for all the other coordinates, it is clear that the best choice of placing this surface is

at the saddle point.

Figure 2–1: Schematic depiction of a potential energy surface with a chemical

barrier.

However, in the absence of a potential barrier, it is not clear as where to place the transition

state. This has led to the development of several new theoretical frameworks which make use

of the variational principal, as it has been recognized that the expression for the CTST is an

upper bound to the exact classical rate coefficient that would be obtained from classical

trajectory calculations.35

Minimizing the rate coefficient expression with respect to the

location of the transition state, hence, results in a more accurate estimation of the rate of the


reaction step. This, however, makes an application of so called variational TST more tedious

as compared to the CTST as a lot more information is required to perform the minimization

procedure.

2.3.2 Variational transition state theory

As stated, the absence of a potential barrier introduces certain complications in the application

of conventional transition state theory.35

A variational implementation of transition state

theory is essential as the potential energy profile does not provide any guidance where to

locate the transition state. Moreover, the location of the transition state will be very sensitive

to the temperature.36, 38

This means that, in order to obtain accurate rate coefficients over a

wide temperature interval, it is by no means sufficient to determine the transition state for one

temperature and use this transition state for other temperatures. There are hence a lot of subtle

details involved in the modeling of scission/recombination reactions which make it

surprisingly difficult to predict barrierless reaction rate coefficients quantitatively.36

In the following paragraphs, three transition state theories, i.e. the Gorin model, canonical

variational transition state theory (CVTST) and variable reaction coordinate for a flexible

transition state theory (VRC-FTST) are discussed. These three methods were selected as they

present three of the most commonly used methods of transition state theories that are used to

model barrierless reactions.36

However, before discussing these transition state theories, some

general aspects of modeling reactions without a pronounced barrier are discussed.

2.3.2.1 General considerations when modeling reactions without a pronounced

potential energy barrier

In order to describe the partition function of the transition state accurately, it is necessary to

identify the modes that need to be modeled and how they can be modeled correctly. For

reactions without a clear saddlepoint in the potential energy surface, the following modes

should be treated:35

i. the vibrations that are present in the fragments

ii. the internal torsional rotation of the fragments relative to each other

iii. the two dimensional rocking motions of the fragments perpendicular to the axis that

connects the two fragments(see Figure 2–2) .

Due to the absence of a barrier in the potential energy surface for a scission reaction, the

transition state is fairly product-like. This makes the determinations of the first kind of modes,


i.e. the vibrations, easier than for reactions with a clear barrier for reaction as the vibrations of

the fragments can, to a good approximation, be taken equal to the vibrations of the separated

fragments.35, 38

The lack of a saddlepoint also simplifies the determination of the second modes. As the

distance between the fragments is quite large, the internal torsional modes can be treated quite

accurately as free rotors.35

The third type of modes, i.e. the rocking modes (see Figure 2–2), that need to be modeled,

however, introduce a lot of complications. This is due to two factors.35

The fragments will

interfere with each other as they rotate about the axes that are perpendicular to the axis that

connects the two fragments. This introduces steric effects that reduce the freedom of the

fragments considerably compared with the fragments at very large distances. The other effect

is due to the overlap of the orbitals in which the electrons of the single bond that is broken

during the course of the reaction are positioned. During the rocking modes, the overlap

between these orbitals will clearly reduce and, as a consequence, the potential energy will

increase.


Figure 2–2: Rocking modes of two recombining methyl fragments.

2.3.2.2 The Gorin model

For the recombination of two radical fragments one can use as a first approximation the Gorin

algorithm.36

Gorin derived an analytic formula for the high pressure limit rate coefficient for

the recombination reaction A + B → AB:36

(

) (

)

[2–32]

polarizability of fragment X [m³]

ionization potential of fragment X [J mol-1

]

symmetry number of the two free radicals

mass of fragment X [kg]

In this model, the rocking modes are modeled as unhindered until a hard sphere interaction

occurs.36

The interacting potential is assumed to be spherical symmetric.36

The model is

correct in a sense that it includes angular momentum conservation as the transition state is

placed at the centrifugal barrier. However, the steric effects might not be modeled correctly

due to the hard-sphere model. Also the potential energy effects due to less overlap of the

orbitals in which the electrons are positioned during the rocking modes are neglected.


2.3.2.3 Canonical variational transition state theory (CVTST)

Numerous implementations of canonical variational transition state theory exist.36, 39

All these

are based on seeking the best transition state to describe the dynamics at a given

temperature.35

This is done by minimizing the rate expression of CTST at a given temperature

T (see equation [2–31]), as a functions of the reaction coordinate. For scission reactions, the

reaction coordinate can be chosen as the distance between the two reacting fragments:

[2–33]

the rate coefficient as calculated by CVTST [m3 mol

-1 s

-1 for a bimolecular

reaction s-1

for a monomolecular reaction]

s reaction coordinate

A more theoretically correct approach is to vary the location of the TS as function of the

energy of the reactants and angular momentum. The shift of the TS as function of temperature

is due to a changing equilibrium distribution of these two properties as function of

temperature.35

However, this is a complex procedure and is not standard practice at present.

As mentioned in previous section, the biggest difficulty in modeling scission reactions is to

construct a model that allows to calculate the contribution of the rocking modes of the two

fragments. Various methods to accurately treat these rocking modes have been documented.

In this master thesis, the name CVTST refers to an implementation of CVTST that was

previously used at the LCT.18

The method was slightly modified in order to calculate rate

coefficients on a more routinely basis. The details are explained elsewhere (3.1.2).

2.3.2.4 Variable reaction coordinate for a flexible transition state theory (VRC-

FTST)

This is the generally accepted theoretical framework to calculate quantitatively the rate

coefficient of recombination reactions of two radical fragments. The theory is still under

development as the application of the theory has revealed more and more subtleties involved

in the precise modeling of recombination reactions. Some of these will be briefly highlighted.

The explanation of the theory starts with discussing the FTST part first. Later on, the VRC

part is introduced as an improvement to FTST.

Important in FTST is the dividing of all the modes into so called conserved modes and

transitional modes.38

The conserved modes are the modes that correspond to the vibrational

modes which are also present in the separated fragments. The transitional modes correspond


to the relative and overall rational modes.38

This corresponds with the discussion in section

2.3.2.1

The separation of modes allows to evaluate the transition state partition function as the

product of the conserved and transitional mode partition function:38

[2–34]

The conserved modes are treated quantum mechanically using the HO approach. This is

allowed as the corresponding frequencies are high. Furthermore, it is assumed that these

modes do not vary as a function of the reaction coordinate which is, in FTST, measured as the

distance between the centers of mass of the recombining fragments.38

The promising part of the theory is the accurateness with which it treats the transitional

modes, i.e. the internal torsional modes, the rocking modes and the coupling between these

modes and external rotational modes. For a canonical ensemble, the transitional part of the

transition state partition function can be written as a function of the reaction coordinate:20, 38

(

)∭

[2–35]

Euler angles (θi, φi, χi) of fragment i

sperical polar angles of the line connecting the centers of mass of the

fragments

Equation [2–35] shows that the transitional modes are treated classically which is allowed as

the corresponding frequencies are low.38

In FTST equation [2–35] is minimized with respect

to the reaction coordinate in order to obtain the rate coefficient.

The difficulty in calculating equation [2–35] is to find an analytic function to accurately

describe the interaction potential V as function of the transitional modes. For radical-radial

recombinations, this potential energy surface generally must span the region from 200 to 400

pm in the incipient bond distance and cover all orientations of the two fragments.38

For this

type of reactions, ab initio multi-reference calculations are needed as discussed in (2.1.4)

As stated, in FTST the reaction coordinate is defined as the separation between the centers of

mass of the 2 reacting fragments.38

It is found that a more accurate reaction coordinate is the

distance between the atoms or between the orbitals involved in the incipient bond.38, 40

As

such FTST can overestimate the rate of the reaction by a factor 2.38


This has led to the development of VRC-FTST,40, 41

i.e. variable reaction coordinate for a

flexible transition state theory. In this theory a more general reaction coordinate is considered:

the reaction coordinate is defined by a fixed distance between the so called pivot points,

which can, in general, be located randomly on the two fragments. The location of these pivot-

points is defined by vectors that point from the center of mass of the recombining fragments

to the pivot points.

This, however, introduces several complications. First, minimization of the rate coefficient in

VRC-FTST now means that this has to be done with respect to seven variables: one variable is

the distance between the pivot points, the other six variables come from the vectors that are

used to locate the pivot points.38

The second complication comes from the fact that the reaction coordinate is no longer

separable form the remaining orientational coordinates of the transitional modes.38

As a result,

the integral expression of the transition state (see equation [2–40]) is no longer valid as it was

based on the separation of the reaction coordinate from all the other coordinates used to

describe the transitional modes. This results in more complicated integral expressions for the

partition function of the transition state which will now no longer depend on a reaction

coordinate but on a definition of a dividing surface.38

Fortunately, as several studies of VRC-FTST are performed,38

some general statements have

been formulated which simplify, to a certain extent, the whole procedure. For example, for the

location of the pivot points, it has been found that the vectors that define the location of the

pivot points are pointing from the atom involved in the incipient bond to the center of its

radical.38

This reduces the amount of parameters which need to be taken into account for the

minimization procedure from seven to three, namely: the distance between the pivot points

and the magnitude of the pivot point position vectors.

Less fortunately, it has also become clear that many reactions have multiple sites where the

two reacting fragments can bind together.38, 42

These multiple binding sites are also termed

channels. To be theoretically correct, these channels have to be treated simultaneously38, 42

.38,

42 In this procedure, the overall transition state dividing surface is expressed in terms of a

composite of individual surfaces with one surface for each of the different binding sites.38, 42

The global rate expression is then found by minimization of the VRC-FTST parameters for

each of the individual surfaces 38, 42

.38, 42


VRC-FTST has been implement in a software package called VARIFLEX. This software

package integrates equation [2–35], or a more difficult version of it, by sampling geometries

of which the energy is calculated by an external ab initio software package, e.g. Gaussian.43

Figure 2–3 illustrates the procedure.

Figure 2–3: Representation of the procedure implemented in VARIFLEX.

VARIFLEX samples geometries of which the energy is calculated by an external

software package, e.g. Gaussian.


2.4. Group additivity

The simulation of industrially relevant, large-scale processes needs to account for the physical

transport phenomena on one hand and for the chemical reactions on the other hand.1 An

adequate description of the chemistry involved is required in order to guarantee the

applicability of the model to a wide range of operating conditions. The kinetic model must

hence be able to grasp all of the underlying chemistry.

For radical gas phase reactions, such a kinetic model results in a vast and complex reaction

network involving hundreds of species and thousands of reactions taking place among these

species.1 A big issue is to provide these reaction networks with a proper set of thermodynamic

and kinetic data.

Although computational chemistry has reached a high level of accuracy, in particular

concerning thermodynamic data, it is not feasible to calculate the required data for each

species and each elementary reaction step, involved in the network.

Additive structure-property relations are therefore developed able to link the requisite

thermochemical and kinetic data to structural subunits, regardless of the position within the

molecule. A simple example of these structure-property relations can be found when one

calculates the molecular weight of a molecule as the sum of the atomic masses of the atoms

that constitute the molecule. This is called atom additivity and is, in the hierarchy of additive

methods, the lowest level of approximation.44

The next three levels are, in order of attainable

accuracy: bond additivity, group additivity and component additivity.44

Bond additivity can

provide only a crude estimate of the required data. Group additivity results in a much higher

level of accurateness by accounting for the ligands present on the atoms in the molecule. The

highest level, component additivity, includes the ligands of the ligands. However,

improvement is limited and at the expense of a huge increase in required parameters.

2.4.1 implementation

In the group additivity method, a group is defined as a polyvalent atom with all its ligands and

is represented by:7

[2–36]

With X the central atom surrounded by i ligands A, j ligands B, k ligands C and l ligands D.


Several atoms, e.g. carbon, can have different bonding patterns. The distinction between these

different kinds of carbon atoms is made by adding an additional sub- or superscript; An

overview of the sub- and superscripts used in this work is given in Table 2–1.6, 7

Table 2–1: Carbon can have different bonding patterns. The first column lists the

different symbols that are used to distinguish between the different carbon atoms

atoms that can be encountered. The second column explains the meaning of the

symbol used.

symbol meaning

C single bonded carbon atom

Cd double bonded carbon atom

Ct triple bonded carbon atom

Cb carbon in a benzene ring

Ca carbon in allene

C•

radical carbon atom

In the group additivity method, a molecular property is obtained by breaking down a molecule

into the constituting groups and by adding the group addivity values (=GAV’s) of all the

groups together.44

As an example, n-butane can be partitioned into the following groups: C–

(C)(H)3, C–(C)2(H)2, C–(C)2(H)2 and C–(C)(H)3. A molecular property , e.g., the enthalpy of

formation, can now be found as:

[2–37]

This example points out one of the shortcomings of this group additive scheme, i.e., a group

only accounts for local effects as it is defined as a local quantity. This means that eclipse-

butane or gauche-butane has the same enthalpy of formation as the more stable trans-butane.

To account for these non-nearest-neighbor interactions (=NNI), NNI corrections are added to

the group additive scheme.6

2.4.2 A group additive scheme for the Arrhenius parameters

Methods to predict the Arrhenius parameters, i.e., the pre-exponential factor, A, and the

activation energy, Ea, have been developed since the beginning of the 20th

century. A simple

and widespread known method is the Evans-Polanyi relation which correlates the Ea to the

reaction enthalpy, , for a set of homologous reactions.

45 This simple method, however,

does not give very accurate results.


This Evans-Polanyi relation has been extended by Truong et al. and resulted in the reaction

class transition state theory.2 In this method, the reaction rate coefficient is calculated, based

on:

[2–38]

k rate coefficient of the target reaction

k0 the rate coefficient of a reference reaction

fκ transmission coefficient

fσ symmetry coefficient

fQ partition function coefficient

fV potential energy coefficient

For the potential energy coefficient, fV, an Evans-Polanyi relation or a barrier height grouping

method is used. In the latter method, an average barrier is used within a reaction class.2 For

this method to give accurate results, the definition of a reaction class needs to be very narrow

and is, hence, not practical for application to reaction networks that contain hundreds of

species and thousands of elementary reactions as it would imply the definition of a lot of

reaction classes.2

The number of reaction classes is limited when methods are introduced that explicitly account

for structural effects on the reactivity. To do so, group additivity offers a powerful tool as it

can be done in a very consistent way. In the next paragraphs, the group additivity method,

previously developed at the LCT,8-10, 21, 46, 47

is explained and extended to radical

recombination reactions.

The method starts from the thermodynamic formulation of the transition state:8

(

)

(

) [2–39]

kB Boltzmann’s constant[J molecule-1

K-1

]

T temperature [K]

h Planck’s constant [J s]

R ideal gas constant [J mol-1

K-1

]

p pressure [Pa]

the change in number of moles at the formation of the transition state (-1 for

a bimolecular reaction) [-]


enthalpy of activation [J mol-1

]

entropy of activation [J mol-1

K-1

]

The entropy of activation contains two contributions: one term is related to the symmetrical

and optical contributions to the entropy and the other term is the symmetry-independent

single-event activation entropy:8

[

∏

∏

] [2–40]

the symmetry-independent single-event activation entropy [J mol-1

K-1

]

nopt the number of optical isomers

σ the product of the internal and external symmetry number

ne the number of single events

Substituting [2–40] in [2–39] leads to:8

(

)

(

) [2–41]

the single-event rate coefficient [m3 mol

-1 s

-1 or s

-1 ]

The Arrhenius expression for the rate coefficient is:

(

) [2–42]

Ea Activation energy [kJ mol-1

]

A pre-exponential factor [m3 mol

-1 s

-1 or s

-1 ]

The Ea can, hence, be found as:

[2–43]

Substituting [2–41] in [2–43] yields for the activation energy:8

( ) [2–44]

[2–44] relates the activation energy solely to the enthalpy of activation and the molecularity of

the reaction.

Introduction of [2–44] for the activation energy in [2–41], results in an expression for the

single-event pre-exponential factor, :8


(

)

( ) (

) [2–45]

[2–45] relates the single-event pre-exponential factor solely to the molecularity of the reaction

and the entropy of activation.

The Arrhenius parameters are a function of: the molecularity of the reaction, two

thermodynamic property functions of the transition state, i.e., the activation enthalpy and the

activation entropy, and the temperature and pressure. For a given reaction family, the

molecularity of a reaction, belonging to the reaction family, will not change and, hence, all the

reaction specific information is included in the two thermodynamic property functions.

Important to note is that the thermochemistry of the transition state can be poured into the

same group additive scheme that is used for stable species. This means that the enthalpy and

entropy of formation at 1 bar and a temperature T can be found by adding all the GAV’s and

NNI corrections, corresponding to the different Benson groups occurring in the transition

state.8

However, from a kinetic viewpoint, the interesting properties are the enthalpy and entropy of

activation which are the difference between the enthalpy of formation and entropy of the

transition state and the corresponding value for the reactant(s).8 This leads to an expression for

the enthalpy and entropy of activation as a function of differences in GAV’s and NNI

corrections. However, a transition state typically contains breaking/forming bonds. Additional

Benson-type groups hence need to be introduced, centered on those atoms that have their

bonding pattern changed during reaction. Before continuing, special attention is therefore paid

to the reactive moiety (see Figure 2–4).8


Figure 2–4: Transition state for a general recombination reaction of two radical

carbon fragments. The carbon atoms in the full line will form a bond during the

course of the reaction. The Xi and Yi atoms have the C1 or C2 atom as a ligand and,

hence, also influence the reaction.

Figure 2–4 represents the transition state for a general recombination reaction of two radical

carbon fragments. The reactive moiety comprises all the groups that change during the

reaction. Such a group is also referred to as a transition state specific group.46, 47

These groups

can be categorized into two classes:

The first class consists of the transition state specific groups that are centered on the

atoms C1 and C2, i.e. the two carbon atoms indicated by the full line on Figure 2–4. The

atoms change their bonding pattern during the course of the reaction. This class, hence,

comprises two groups which are denoted by:

These groups are called the primary groups.

The second category of transition state specific groups contains the groups having C1 or

C2 as a ligand. These groups are denoted Xi and Yi in Figure 2–4 and are indicated by the

dotted line. This category includes 6 groups:


These groups are called the secondary groups.

The groups that do not belong to one of these two categories are assumed to have the same

GAV’s in the transition state and in the reactant(s) and, hence do not contribute to the

activation enthalpy or activation entropy.8 This leads to equation [2–46] and equation [2–47]

as expression for the enthalpy and entropy of activation. For simplicity of notation, the

indication of the ligands is discarded.

∑ ( )

∑

∑

∑

[2–46]

∑ ( )

∑

∑

∑

[2–47]

the difference in GAV’s between transition state and reactant(s)

the difference in NNI corrections when going from reactant(s) to

transition state

The are also called the tertiary contributions.46, 47

From expressions [2–46] and [2–47], the corresponding expressions for the Arrehenius

parameters can be found as:8

∑ ( )

∑

∑

∑

( )

[2–48]

( )

(

) (

)

(( )

∑ (

) ∑

∑

∑

)

[2–49]

The ∆GAV’s can hence be introduced in order to calculate the Arrhenius parameters.

However, these GAV’s are temperature dependent which necessitates the use of cp values as


in the method of Sumathi et al.48

In previous work, this has been circumvented by the

introduction of a reference reaction.8 This results in the following expressions for the

Arrhenius parameters:8

∑

( )

∑

∑

∑

[2–50]

( ) ( ) ∑ (

)

∑

∑

∑

[2–51]

with

[2–52]

( ) [2–53]

The temperature dependence of the ’s and ’s for recombination reactions will

be further investigated in section 2.4.3 of this work.

For some reaction families, e.g. addition reactions, it is already shown that the most important

contribution to the activation energy and single-event pre-exponential factor comes from the

primary groups and that the secondary and tertiary contributions are as good as negligible.46

If this is also the case for radical recombination reactions, the expressions for the activation

energy and single-event pre-exponential factor can, hence, be truncated after the contribution

of the primary groups:

∑

( )

[2–54]

( ) ( ) ∑ (

)

[2–55]

In the transition state of recombination reactions, the bond length of the breaking/forming C-

C bonds typically range between 250 and 400 pm. At this large interfragmental distance NNI


contributions are expected to be relatively small. As also no NNI contributions are present in

the separated fragment, the ∆NNI°’s should be small. However, for the reverse reaction, i.e.

bond scission, ∆NNI°’s may become important.

In order to illustrate the physical background of the ’s and in order to discuss some

practical aspects of the method, the use of the ’s is demonstrated for the recombination

of an ethyl and a methyl radical. The following discussion is based on the example provided

in Sabbe2 but applied to recombination reactions instead of hydrogen abstraction reactions.

The reference reaction for radical recombination reactions is the recombination of two methyl

radicals. Figure 2–5 illustrates both transition states. The following discussion is limited to

enthalpies and activation energies. An extension to entropies and the single-event pre-

exponential factor is, more or less, straightforward.

Figure 2–5: The first reaction, recombination of methyl radicals, is the reference

reaction for the group additive modeling of radical recombination reactions. The

second reaction is the recombination of an ethyl with a methyl radical.

The activation enthalpies of the reference reaction and the recombination of an ethyl radical

with a methyl radical are respectively(see [2–46]):

(

) (

) [2–56]

(

)

(

)

(

)

[2–57]

The corresponding activation energies are(see [2–48]):


(

) (

)

( ) [2–58]

(

)

(

)

(

) ( )

[2–59]

When these two equations are subtracted from each other and the contribution of the third

term in equation [2–57] is neglected, which is a contribution of a secondary group, the

difference between the activation energy of the reference reaction and the activation energy

for the recombination of an ethyl radical with a methyl radical is found as:

(

)

(

)

(

)

(

)

[2–60]

The difference in activation energies can be attributed to:2

different ligands on the C1 atom in the transition state

different ligands on the C2 atom in the transition state

different ligands on the C1 atom in the reactants

different ligands on the C2 atom in the reactants

In equation [2–60] the second and fourth term are zero, as it comprises two times a methyl

radical. The first and the third term together form the

which hence

includes the structural differences that are present on the carbon atom C1 in the transition state

and the reactants.

Neglecting the contributions from the secondary groups implies that this single

can be used to model all possible recombination reactions between methyl and

primary alkyl radicals. This can be seen when the same derivation is made for the

recombination of a methyl and a n-propyl radical. Under the condition that contributions of

secondary groups are negligible, equation [2-63] would again be obtained. In this way, the

most straightforward way to find the ’s is to vary only one substituent at a time.2


However, in practice, the contributions of the secondary groups are not entirely negligible.

This means that if one determines the ’s by varying one substituent at a time, the

secondary contributions are included in this . For example, if the

is determined with reference to the recombination of an ethyl radical with a methyl

radical, this

includes the contributions of the secondary group present

in the ethyl radical. When this value is used for the other primary alkyl radicals, it is, hence,

assumed that the secondary contributions of the primary alkyl radicals are the same compared

to the secondary contribution of the ethyl radical.

An important issue when describing the kinetics in a reaction network is: are the rate

coefficients thermodynamic consistent? For example, the standard reaction enthalpy should be

equal to the difference between the activation enthalpy of the forward and reverse reaction

(assuming the molecularity of the forward and reverse reaction is the same). For example, the

reaction enthalpy for the recombination of an ethyl radical with a methyl radical, without

neglecting the secondary contributions, amounts to:

[2–61]

[(

) (

)

(

)]

[(

)

(

) (

)]

[2–62]

[2–63]

This example shows that thermodynamic consistency is incorporated in the GA model as it is

possible to regroup the GAV’s of equation [2–63] into the GAV’s needed for the enthalpy of

formation of the reactants and the products:

(

)

[2–64]

[2–65]

However, the secondary contributions are normally neglected in order to limit the amount of

ΔGAV0’s that should be determined. As already mentioned, ignoring the secondary


contributions for recombination reactions might be acceptable but ignoring them for bond

scission reactions not. Nevertheless, neglecting secondary contributions in describing the

recombination reactions does not lead to thermodynamic inconsistency. If the secondary

contributions are negligible for the recombination reaction, then [2–62] becomes:

[(

) (

)]

[(

)

(

) (

)]

[2–66]

[2–67]

Assuming that the secondary contributions are negligible for the recombination reactions

means that:

[2–68]

On substituting [2–68] into [2–67], [2–64] is obtained, which basically means that

thermodynamic consistency is still guaranteed.

2.4.3 Group additivity values based on experimental rate equations

The benchmark for ab initio calculated rate equations is, and should always be, experimentally

obtained rate equations. An extended set of rate coefficients for recombination reactions are

available on the NIST Chemical Kinetics Database. To avoid deriving ∆GAV°’s from

calculated data and afterwards concluding that a GA model does not work for this reaction

family, it was initially tried to study validity of a GA model for recombination reactions based

on experimental data.

The reactions for which data is available on NIST can be found in Table 2–2. The reactions

are divided into two sets: i.e. a training set and a test set. Generally the latest review

expression is used. However, in rare cases, i.e. when that review value is only valid at a

certain temperature, the next most recent experimental or review expression is used which can

then be used over a broad temperature interval.

The activation energies and the pre-exponential factors, corresponding to the rate expressions

abstracted from the NIST chemical database, are also listed in Table 2–2. These values are

determined both at low (300 – 400 K) and higher (900 – 1100 K) temperatures. The latter


temperature interval was investigated because of its importance for the steam cracking of

hydrocarbons.

Table 2–2: Summary of the data abstracted from NIST. [A in m³ mol-1

s-1

and Ea in

kJ mol-1

]

Reaction 300 K – 400 K 900 K - 1100 K

log(A) Ea log(A) Ea

Training set

1

7.26 -1,8 6.74 -7,8

2 6.99 -1,4 6.97 -4,1

3

7.12 -1,9 6.86 -4,7

4

7.22 -2,5 7.21 -2,5

5

7.07 -1,5 6.91 -3,2

6

7.15 0,2 7.14 0,2

7

7.09 0,9 7.08 0,9

8

7.25 0,9 7.24 0,9

Test set

1

7.03 -1,0 6.86 -2,9

2

6.62 -2,1 6.26 -6,2

3

6.5 -1,6 6.11 -5,8

4

6.36 -3,1 5.84 -9,1


5

7.03 -1,6 6.86 -3,4

6

5.64 -4,3 4.94 -12,4

7

6.64 -2,7 6.29 -6,8

8

7.02 -1,1 7.01 -1,1

9

7.16 0,5 - -

10

7.18 -1,3 - -

The ’s valid from 900 – 1100 K were derived from the reactions of the training set and

are represented in Table 2–3.The number of single-events was accounted for to get the single-

event pre-exponential factor.

Table 2–3: ’s derived from the Arrhenius parameters of the reactions

belonging to the training set presented in Table 2. [Ã in m³ mol-1

s-1

and Ea in kJ

mol-1

]

training reaction log(Ã ) E a group ΔGAV0

log(Ã) ΔGAV0

Ea

1 6.14 -7.8 ref. reaction

2 6.37 -4.1 CTS

-(C)(H)2 0.23 3.6

3 6.26 -4.7 CTS

-(C)2(H) 0.13 3.0

4 6.61 -2.5 CTS

-(C)3 0.47 5.3

5 6.31 -3.2 CTS

-(Cd)(H)2 0.17 4.6

6 6.84 0.2 CTS

b 0.70 7.9

7 6.47 0.9 CTS

-(Cb)(H)2 0.34 8.7

8 6.64 0.9 CTS

-(Cb)(C)(H) 0.50 8.7

The single-event pre-exponential factors and the activation energies for the reactions of the

test set are calculated using the ’s of Table 2–3.The results are presented in Table 2–4.

In the fourth column of this table, the ratio of the GA rate coefficient at 1000 K to the

experimental rate coefficient is presented. In the fifth and sixth column, the deviations on the

Arrhenius parameters are presented.


Table 2–4: The single-event pre-exponential factors and the activation energies for

the reactions of the test set based on the ‘s of Table 2–3 [Ã in m³ mol-1

s-1

and Ea in kJ mol-1

]

test reaction log(Ã )ΔGAV° E a,ΔGAV°

k ΔGAV°(1000K)/k(

1000K)

log(Ã) ΔGAV° -

log(Ã)E a,ΔGAV° -E a

1 6.50 -1.1 1.39 0.24 1.8

2 6.84 1.1 7.21 1.18 7.3

3 6.39 -1.7 4.65 0.88 4.1

4 6.74 0.5 9.91 1.50 9.6

5 6.43 -0.2 1.01 0.18 3.3

6 7.08 2.8 89.20 2.74 15.1

7 6.78 2.1 4.32 1.10 8.8

8 6.47 1.4 0.87 0.07 2.5

9 - - - - -

10 - - - - -

Large deviations (more than a factor 5) between GA modeled and experimental rate

coefficients are obtained for test reactions 2, 4, and 6. This can be resolved by also accounting

for NNI corrections or more specifically by accounting for gauche interactions. These gauche

interactions arise in test reactions 1 – 4 and 6. On Figure 2–6, the counting scheme for gauche

interactions is illustrated together with the number of gauche interactions that occur in the

transition state of test reactions 1 - 4 and 6.


Figure 2–6: The gauche interactions arising in the test reactions 1 – 4 and 6.

The average and

for 1 gauche interaction is calculated by taking the

average of 1 gauche interaction present in test reaction 1 - 4 and 6 obtained after dividing the

difference between the Arrhenius parameters based on ’s and the experimental

Arrhenius parameters by the number of gauche interactions. The average values are listed in

Table 2–5 and the improvements obtained by introducing these ’s are represented in

Table 2–6.


Table 2–5: average values for

and

. [Ã in m³ mol-1

s-1

and Ea in

kJ mol-1

]

Test reaction

( log(Ã) ΔGAV° -

log(Ã))/number of

gauche interaction

(E a,ΔGAV° -E a )/number

of gauche interaction

1 0.24 1.79

2 0.59 3.67

3 0.44 2.04

4 0.37 2.40

6 0.46 2.52

average 0.42 2.49

Table 2–6: Improvements obtained by introducing the . [Ã in m³ mol-1

s-1

and Ea in kJ mol-1

]

test reaction log(Ã) ΔGAV° E a,ΔGAV°

k ΔGAV° (1000K)/k(

1000K)

log(Ã) ΔGAV° -

log(Ã)E a,ΔGAV° -E a

1 6.08 -3.58 0.71 -0.18 -0.70

2 6.00 -3.84 1.65 0.34 2.37

3 5.55 -6.67 1.22 0.04 -0.89

4 5.06 -9.41 0.68 -0.18 -0.33

6 4.56 -12.15 1.62 0.22 0.23

By introducing a for gauche interactions, the ratio of the GA modeled rate coefficients

to the experimental rate coefficients of the test reactions is now within a factor 2.

The results for the lower temperature area are represented on Table 2–7 and Table 2–8.

Table 2–7: ’s for the low temperature range based on Arrhenius

parameters of the reactions belonging to the training set presented in Table 2–2 [Ã

in m³ mol-1

s-1

and Ea in kJ mol-1

]

training reaction log(Ã) Ea group ΔGAV0

log(Ã) ΔGAV0

Ea

1 6.64 -1.8 ref. reaction

2 6.37 -1.4 CTS

-(C)(H)2 -0.27 0.4

3 6.51 -1.9 CTS

-(C)2(H) -0.13 -0.1

4 6.61 -2.5 CTS

-(C)3 -0.03 -0.7

5 6.45 -1.5 CTS

-(Cd)(H)2 -0.19 0.4

6 6.84 0.2 CTS

b 0.20 2.0

7 6.47 0.9 CTS

-(Cb)(H)2 -0.17 2.7

8 6.64 0.9 CTS

-(Cb)(C)(H) 0.00 2.7


Table 2–8: The single-event pre-exponential factors and the activation energies for

the reactions of the test set based on the ‘s of Table 2–7 [Ã in m³ mol-1

s-1

and Ea in kJ mol-1

]

test reaction log(Ã) ΔGAV° E a ,ΔGAV°

k ΔGAV°(300K)/k(

300K)

log(Ã) ΔGAV° -

log(Ã)E a,ΔGAV°-E a

1 6.24 -1.56 0.83 -0.18 -0.55

2 6.34 -3.55 3.79 0.33 -1.40

3 6.38 -2.07 3.74 0.49 -0.47

4 6.48 -2.62 4.43 0.74 0.53

5 6.32 -1.59 0.82 -0.09 -0.04

6 6.58 -3.17 22.59 1.55 1.13

7 6.42 -2.14 1.98 0.39 0.56

8 6.26 -1.12 0.73 -0.14 -0.01

9 7.04 2.20 0.39 -0.11 1.73

10 6.31 3.65 0.08 -0.26 4.94

The results listed in Table 2–8 are without NNI corrections. It seems to be more difficult to

assign a value to for the gauche interactions in this lower temperature range as the

Arrhenius parameters based on the ’s now sometimes overestimate and sometimes

underestimate the Arrhenius parameters of the test set. This could be due to the fact that some

of the experimental data is not very reliable in the lower temperature area.

Figure 2–7 and Figure 2–8 represent the temperature dependence of ’s and

’s.


.

-0.400

-0.200

0.000

0.200

0.400

0.600

0.800

200 400 600 800 1000

ΔG

AV

°log(Ã

)

Temperature [K]

CTS-(C)(H)2

CTS-(C)2(H)

CTS-(C)3

CTS-(Cd)(H)2

CTSb

CTS-(Cb)(H)2

CTS-(Cb)(C)(H)



.

These two figures show a strong temperature dependence for the ’s and the

’s. Although some data may not be very reliable for the lower temperature area, the

strong temperature dependence of the ’s can be attributed to the fact that the position

of the transition state is also strongly temperature dependent.

This study, hence, illustrates that a group additivity model for Arrhenius parameters of

recombination reactions might work. NNI corrections will be required in order to describe

steric effects which seem to have a big influence on the rate coefficients. Furthermore, as the

position of the transition state varies as function of temperature, the derived ∆GAV°’s are

expected to show more temperature dependence than those obtained for other reaction

families.

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

200 400 600 800 1000

ΔG

AV

° Ea

Temperature [K]

CTS-(C)(H)2

CTS-(C)2(H)

CTS-(C)3

CTS-(Cd)(H)2

CTSb

CTS-(Cb)(H)2

CTS-(Cb)(C)(H)

Method selection to study recombination reactions 47

Chapter 3: Method selection to study

recombination reactions

In this chapter, the three transition state theories that were previously discussed are compared

to each other based on calculations performed on a test set of reactions: CH3•+

•H, C2H5

•+

•H,

CH3•+

•CH3, CH3

•+

•OH and CH3O

•+

•H. For all these reactions, both experimental rate

coefficients and rate coefficients calculated by Klippenstein and coworkers making use of the

VARIFLEX program, are available. In this way, this test set should provide enough

information as to which transition state theory should be used in order to determine rate

coefficients of recombination reactions with sufficient accuracy, at an affordable

computational cost.

Reactions involving resonance stabilized radicals are not considered in this chapter, although

experimental and FTST-VRC data is available for this type of reactions. The reason for this, is

that resonance stabilized molecules contain at least three carbon atoms which would increase

the computational time considerably.

Before the results on the five reactions are discussed, the implementation of canonical

variational transition state theory is discussed.

3.1. Implementation of the canonical variational transition

state theory

3.1.1 Previous implementation of CVTST

The calculation of the high pressure limit rate coefficient for homolytic bond scission

reactions is based on a minimization procedure of the CTST rate coefficient expression [2–

31]. The minimization is performed with respect to the interatomic distance of the breaking

bond which is, hence, considered as the reaction coordinate s:18

[3–1]

Equation [3–1] expresses that the high pressure limit CVTST rate coefficient at a certain

temperature is found by minimization of the CTST rate coefficient expression at that


temperature as function of the reaction coordinate s. From equation [3–1] it is also evident

that, at first instance, the scission reaction is calculated as only the partition function of one

reactant molecule is considered.

The features of the potential energy surface along the reaction coordinate are obtained by

performing CASSCF calculations at several points along the reaction coordinate. These

energies are rescaled according to equation [3–2].18

Rescaling is necessary as the CASSCF

bond dissociation energies are generally less accurate than those calculated with high level

composite methods, regardless of the basis set and active space employed.49, 50

( )

( )

( )( ( ))

[3–2]

seq is the equilibrium distance between both fragments in the reactant

The potential energy surface is obtained as a functions of the reaction coordinate by cubic

spline interpolation. ΔE(0 K)(s) is then acquired by adding the ZPVE to the calculated energy

surface.18

It should be noted that the ZPVE also changes along the reaction coordinate.

The expression for the CTST partition function of the transition state in equation [3–2] can be

written as the product of six contributions:18

[3–3]

number of optical isomers in the transition state

electronic partition function

translational partition function

external rotational partition function

vibrational partition function of the conserved modes

vibrational partition function of the transitional modes

The authors tried to account for spin crossing by including an electronic degeneracy in the

electronic partition function, which varies from 1 to 3.18

The transitional partition function

depends solely on the mass of the compound and does not vary as a function of the reaction

coordinate. The external rotational partition function varies during reaction as the principal

moments of inertia change. This has been taken into account by a polynomial fit to the


product of the three moments of inertia obtained from the CASSCF/6-311G(2d,d,p) optimized

geometries along the reaction coordinate.18

The modes are also split into a transitional part, , and a conserved part,

. The transitional modes are treated as described by Grabowy and Mayer:

51 the

internal mode corresponding with the interfragmental stretching represents the motion along

the reaction coordinate, the lowest frequency mode, which corresponds with a torsion mode,

is treated as free rotor, while the other four modes are treated as harmonic oscillators. The

frequencies of these four modes are expressed as a function of the reaction coordinate:

[3–4]

the frequency at s [s-1

]

The parameter β is obtained by fitting of equation [3–4] to the CASSCF frequencies

calculated at the initial bond length and at an interfragmental distance of 300 or 350 pm

depending on the studied bond scission. The part of the partition function corresponding to the

conserved modes is treated as in VRC-FTST, this is by using the frequencies of the free

fragments.

3.1.2 Implementation of CVTST used in this master thesis

The starting point of the implementation used in this master thesis is also equation [3–1]. The

minimization is implemented in a Maple worksheet. There are two main steps: first the

various contributions to and

are derived, in the second step, the CVTST

rate coefficient is obtained by minimization of expression [3–1] as function of the reaction

coordinate s, which is the distance between the two atoms involved in the breaking bond.

kCVTST is calculated at 300, 400, …, 1000 K.

In order to obtain

, 21 Gaussian43

calculations are performed. 18 of these 21

calculations consist of 16 calculations along the reaction coordinate: starting at 250 pm up to

400 pm, in steps of 10 pm. These calculations are performed on the CASSCF/6-311G(2d,d,p)

level of theory with an appropriate selection of the complete active space. None of the internal

coordinates are kept fixed during the optimization of these 16 points with exception of the

interfragmental distance. The remaining two of the 18 GAUSSIAN calculations are

CASSCF/6-311G(2d,d,p) calculations on the stable molecule and the two fragments at a


distance of 1000 pm, both with the same active space used for the calculation of the other 16

points. To include dynamical electrons correlations, CASMP2 calculation with the same basis

set are performed on the CASSCF geometries in order to obtain more accurate energies.

The three remaining Gaussian calculations are W1bd calculations on the stable molecule and

on the two radical fragments. These high level of theory calculations are performed in order to

do the necessary rescaling according to:

(

)

[3–5]

with

s the distance along the reaction coordinate, varying here from 250 pm to 400

pm

E the electronic energy, not ZPVE corrected [kJ mol-1

]

The point at 1000 pm is calculated for a proper rescaling as is observed that at 400 pm not all

the chemical relevant interactions between the two fragments will have disappeared. A cubic

spline interpolation is used through the rescaled energies to obtain an energy profile as

function of the reaction coordinate, s.

The canonical partition function of the transition state is written as the product of four

contributions instead of six, as has been done previously:18

[3–6]

On comparing equation [3–6] with equation [3–3] for the expression of the canonical

transition state partition function, it is evident that no distinction is made between the

transitional and conserved modes. Instead of using the same frequencies for the conserved

modes at each point along the reaction coordinate, frequency calculations were performed for

every point along the reaction coordinate. The internal mode corresponding with the

interfragmental stretching mode was kicked out while the lowest frequency mode

corresponding with the torsion mode is treated as a free rotor as the barrier for internal

rotation is typically lower than 1 kJ mol-1

. All the other modes, transitional or conserved, are

treated quantummechanically within the harmonic oscillator approximation using equation


[2–28]. The frequencies that are needed to calculate the rovibrational partition functions of the

16 points along the reaction coordinate are obtained from CASSCF/6-311G(2d,d,p)

calculations. The frequencies are scaled with a factor of 0.92. This is the scaling factor that is

commonly used for Hartree-Fock calculations52

and agrees well with a scaling factor of 0.915

reported by Brouwer et al.53

For each temperature considered, the total rovibrational partition

function is calculated at each point along the reaction coordinate. A cubic spline interpolation

is used to calculate the rovibrational partition function at intermediate distances.

This has clearly simplified the previous implementation as it is often hard to distinguish the

modes in the reactant that correspond to the transitional modes at interfragmental distances of

300 or 350 pm. One way of doing this is working backwards, i.e. start at the larger

interfragmental distances and check where the frequencies corresponding to the transtitional

modes shift to in the spectrum at shorter distances. Until 250 pm, the transitional frequencies

are mostly found at the lower frequency-end of the spectrum, though, they are not just always

the lowest frequencies. However, when looking at equation [3–4], one most find the

frequencies corresponding to the transitional modes in the reactant molecule. This seems to be

very difficult and requires a lot of points along the reaction coordinate for interfragmental

distances between the equilibrium bond length and the 250 pm if one wants to make a correct

guess of the frequencies of the transitional modes in the reactants. In the absence of these

additional points, these frequencies are very difficult to be found in the reactant molecule and

makes the determination of these frequencies rather arbitrarily which has its repercussions on

reproducibility. It should also be noted that it is very well possible that the transitional

frequencies are not purely present in the stable molecule as it is possible that in the reactant

molecule these modes are mixed up with other frequencies. This is why the procedure

involving equation [3–4] has been discarded and replaced by considering all the vibrations at

each point. It is noted that this is by no means in contradiction with the normally applied split

up of the modes into conserved and transitional modes. In fact, it should even result in a more

accurate treatment of the conserved modes as the little variation of the conserved modes is

now also taken into account.

From equation [3–6], the electronic partition function which accounted for electronic

degeneracy is discarded as it is expected that most of the rate coefficients that will be

calculated in this master thesis will not suffer from intersystem crossing as only light elements

are considered.35


For the translational partition function and external rotational partition function, the same

considerations as have been discussed in 3.1.1 apply, with the exception that instead of using

a polynomial interpolation, a cubic spline interpolation is used through the product of the

three moments of inertia as a function of the reaction coordinate.

Now, the minimization procedure can be carried out in Maple at each temperature as all the

contributions to the CTST rate expression that show reaction coordinate dependence or

contribute to the partition function of the transition state have been considered. The results of

the Maple worksheet are the distance at which the transition state occurs according to the

CVTST calculations and the corresponding value of the product of and

.

These results are used in an excel sheet to calculate the rate coefficient. For the calculation of

scripts have been previously developed. These scripts account for hindered

rotor corrections.

The recombination rate coefficient is calculated based on thermodynamic consistency. For

the calculation of the required thermodynamic property functions, i.e. enthalpy and entropy,

hindered rotor corrections are taken into account. The complete procedure is summarized on

Figure 3–1.


Figure 3–1: Summary of the algorithm used to calculated rate coefficients.

3.2. Comparison of transition state theories

3.2.1 Recombination of hydrogen with methyl

The most simple recombination reaction involving a hydrogen centered and a carbon centered

radical is the recombination of the hydrogen radical with methyl. As discussed in 3.1.2, the

first step is to perform 18 Gaussian calculations along the reaction path. For this

recombination reaction the active space consists of 2 electrons and 2 orbitals which is the

minimum required for a bond scission.15

Figure 3–2 depicts the CAS orbitals at an

interfragmental distance of 300 pm.


Figure 3–2: The bonding (left) and anti-bonding (right) orbitals of the σ bond that

is broken during the scission reaction. The two radical fragments are at an

interfragmental distance of 300 pm from each other.

The pre-exponential factor and activation energy for the scission reaction can be found in

Table 3–2. A comparison between experimental and the CVTST data can also be found in this

table. The experimental data used for this comparison includes the most recent review values

and the two most recent experimental rate coefficients at the studied temperatures.17

Averaged

over 300 and 1000 K, this ratio amounts to 5, which means that the calculated rate coefficient

overestimates the experimental data.

The results for rate coefficient for the reverse reaction are presented on Figure 3–3, where

they are compared with the rate coefficients reported in the most recent review article,54

the

three most recent experimental studies55-57

and the rate coefficient as calculated by Harding et

al.15

When available, error bars are indicated. The rate coefficient of the recombination

reaction was also calculated using the software package VARIFLEX and the Gorin algorithm.

The results of these calculations are also presented on Figure 3–3. The VARIFLEX

calculations are done without pivot point optimization and only one binding site in

considered.


Figure 3–3: Comparison of the recombination rate coefficient calculated using

CVTST (full line), VARIFLEX (dashed line) and the Gorin algorithm (dotted line)

with experimental and theoretical data (symbols).

From Figure 3–3, it follows that the CVTST rate coefficient overestimates the experimental

and theoretical data. The temperature dependence is more pronounced than compared with the

other data. The ratio of the calculated rate coefficient to the other data is on average a factor

3.3 at 300 K and a factor 1.7 at higher temperatures. The VARIFLEX rate coefficient only

overestimates the data of Harding et al.15

with 10% over the entire temperature range and is in

good agreement with most of the reported experimental data. The Gorin algorithm results in a

rate coefficient that underestimates the experimental and theoretical data. The kGorin/kreview

varies from 0.17 to 0.21.

3.2.2 Recombination of hydrogen with ethyl

The second reaction that has been calculated with CVTST is the recombination of a hydrogen

radical and an ethyl radical. As for C•H3 + H

•, the minimum active space is used, i.e. two

electrons and two orbitals. The pre-exponential factor and activation energy for this reaction

can be found in Table 3–2, together with a comparison with experimental data from the NIST

Chemical Kinetics Database.17

The calculated rate coefficient underestimates the experimental

data as the ratio of the calculated rate coefficient to the experimental rate coefficient is lower

than one.

0.00E+00

1.00E+08

2.00E+08

3.00E+08

4.00E+08

5.00E+08

6.00E+08

7.00E+08

8.00E+08

300 500 700 900

k [m

3m

ol

-1s-1

]

T [K]

CVTST

VARIFLEX

Gorin

1994BAU/COB847-1033

1991FOR3612-3620

1989TSA71-86

1989PIL267-291

2005HAR/GEO4646-4656


Rate coefficients for the reverse reaction are presented on Figure 3–4 together with the rate

coefficients calculated with VARIFLEX and the Gorin algorithm. The recombination rate

coefficient as calculated by Harding et al.15

is also indicated. Furthermore, there are also two

experimental values for the recombination reaction available from NIST.17

Figure 3–4: Results for the recombination reaction of a hydrogen and ethyl radical

obtained with CVTST (full line), VARIFLEX (dashed line) and the Gorin

algorithm (dotted line). Experimental and theoretical data is also indicated.

The CVTST recombination rate coefficient predicts the experimental value of Sillesen et al.58

just within the error bars. The ratio of this rate coefficient to the rate coefficient as calculated

by Harding et al.15

varies from 0.66 at 300 K to 0.63 at 1000 K. The temperature dependence

predicted by CVTST hence agrees well with the temperature dependence of the rate

coefficient calculated by Harding et al.15

The rate coefficient as calculated with VARIFLEX

corresponds very well with the rate coefficient calculated by Harding et al.15

at the low

temperature. At the higher temperature, the VARIFLEX rate coefficient overestimates the

data of Harding et al.15

by 10%. The Gorin rate coefficient overestimates all the rate

coefficients: compared to the data reported by Harding et al., the rate coefficient is a factor 4

higher over the entire temperature range.

0.00E+00

1.00E+08

2.00E+08

3.00E+08

4.00E+08

5.00E+08

6.00E+08

7.00E+08

8.00E+08

300 500 700 900

k [m

3m

ol

-1s-1

]

T [K]

1993SIL/RAT171-177

1970KUR/PET2776

2005HAR/GEO4646-4656

CVTST

VARIFLEX

Gorin


3.2.3 Recombination of hydrogen with methoxy radical

The scission reaction of methanol into a hydrogen radical and a methoxy radical was

calculated to see whether it is possible or not for the implementation of CVTST used in this

master thesis to predict scission reactions in which oxygen atoms are involved. The

calculations along the reaction path are carried out using an active space containing two

electrons and two orbitals, although an active space of four electrons and three orbitals is also

reported in literature.59

However, no considerable differences in energy were observed when

this larger active space was used. Furthermore, geometry convergence issues appeared using

the larger active space as frequencies calculated with this larger active space were negative.

The kinetic parameters of the scission rate coefficient are listed in Table 3–2. No experimental

data is available for this scission reaction.

The rate coefficient for the reverse reaction, i.e. recombination of a hydrogen and methoxy

radical, was calculated based on the CVTST calculations, the Gorin model and VARIFLEX.

The results are presented on Figure 3–5 together with the rate coefficient as obtained by

Jasper et al. 59

and one experimental rate coefficient.60


Figure 3–5: The results obtained with CVTST (full line), VARIFLEX (dashed line)

and the Gorin algorithm (dotted line) for the rate coefficient for recombination of

a hydrogen and methoxy radical. The experimental and theoretical data are

presented with symbols.

The rate coefficient for the recombination reaction calculated with CVTST corresponds well

with the experimental data. However, the CVTST rate coefficient underestimates the rate

coefficient as calculated by Jasper et al.:59

the ratio of the CVTST rate coefficient to the data

of Jasper et al. varies from 0.06 at 300 K to 0.15 at 1000 K. For this recombination reaction it

was not possible to reproduce the data of Jasper et al. with the software package VARIFLEX.

The ratio of the former to the latter is, on average, 1.7 for the entire temperature range. The

Gorin model overestimated considerably all the other data. .

3.2.4 Recombination of two methyl radicals

The scission of an ethane molecule into two methyl fragments constitutes the most simple

scission reaction within the family of carbon-centered radical scission/recombination

reactions. The active space for the calculation along the reaction path was constructed using 2

electrons and 2 orbitals. The orbitals that are present in the CAS are presented in Figure 3–6

for an interfragmental distance of 300 pm.

0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

300 500 700 900

k [m

3m

ol

-1s-1

]

T [K]

2007JAS/KLI3932-3950

Dobe et al.

CVTST

VARIFLEX

Gorin


Figure 3–6: The orbitals of the active space. The active space includes the bonding

(left) and anti-bonding orbital (right) of the σ bond that is broken during the

course of the reaction.

The results for the scission reaction are listed in Table 3–2. The agreement with the review

data and the most recent experimental data in the high temperature area is good: the ratio of

the CVTST calculated rate coefficient to the review and experimental rate coefficients is

almost equal to one. In the low temperature area, this ratio stays within a factor 3. The

calculated date lies within the reported uncertainty interval on the review rate coefficient,

except at 300 K.

The results obtained with CVTST and the Gorin model for the recombination reaction are

presented on Figure 3–7. The CVTST rate coefficient overestimates the experimental and

theoretical rate coefficient calculated by Klippenstein et al.13

over the entire temperature area.

The average ratio of the CVTST rate coefficient to the experimental and theoretical rate

coefficient equals 5.5 in the low temperature area, but reduces to a factor 1.7 at higher

temperatures. The Gorin model underestimates the reported rate coefficients. The deviation

from the rate coefficient obtained by Klippenstein et al. is 0.24 at low temperatures and

amounts to 0.47 at higher temperatures. The temperature dependence is hence less

pronounced than the temperature dependence of the CVTST rate coefficient for the

recombination reaction.


Figure 3–7: The rate coefficient for the recombination of 2 methyl fragments as

calculated with CVTST (full line), and the Gorin model (dotted line) are presented

together with experimental and theoretical data.

3.2.5 Recombination of hydroxyl and methyl radical

The rate coefficient for the scission of the carbon oxygen bond in methanol is calculated based

on CASSCF calculations containing 4 electrons and 3 orbitals in the active space, this is in

accordance with data found in literature59

. It was tried to perform the CASSCF calculations

with an active space of two electrons and two orbitals, however, convergence difficulties

arisen making it impossible to find enough converged points along the reaction coordinate.

The kinetic parameters for the scission reaction are summarized in Table 3–2. The agreement

with experimental data is good, certainly when it is noted that from 400 K onwards, the ratio

of the calculated rate coefficient to experimental rate coefficients is smaller than 2, which is

also the reported uncertainty on the experimental rate coefficient.

Rate coefficients for the recombination of a methyl radical with a hydroxyl radical obtained

with CVTST, VARIFLEX and the Gorin model are presented on Figure 3–8. Experimental

data and theoretical data as calculated by Jasper. et al.59

are also indicated.

0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

300 500 700 900

k

[m3

mo

l -1

s-1]

T [K]

1995ROB/PIL13452-13460

2003WAN/HOU11414-11426

1996DU/HES974-983

1991WAL/GRO107-114

2006KLI/GEO1133-1147

CVTST

Gorin


Figure 3–8: Depiction of the results obtained with CVTST, VARIFLEX and the

Gorin algorithm. Experimental and theoretical data are also presented for

comparison.

From Figure 3–8 it follows that the three models succeed to predict the rate coefficient

intermediate to the experimental data. The CVTST and VARIFLEX rate coefficients for

recombination show a too strong temperature dependence in the lower temperature area. The

rate coefficient obtained with the Gorin model shows a temperature dependency which is in

better agreement with the experimental and theoretical data.

0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

250 450 650 850 1050

k [m

3m

ol

-1s-1

]

T [K]

1994BAU/COB847-1033

1994HUM/OSE721-731

1993FAG/LUN226-234

1992OSE/STO597-604

1992OSE/STO5359-5363

1987DEA/WES207

2007JAS/KLI3932-3950

CVTST

VARIFLEX

Gorin


3.3. Selection of an accurate yet cost-effective TST to

model recombination reactions

In Table 3–1, the ρ (=kmax/kmin)12

value of the CVTST, VARIFLEX and Gorin model are

presented.

Table 3–1: ρ=kmax/kmin for the three studied TST’s. X: Computational too

expensive.

Reaction CVTST VARIFLEX Gorin

CH4 CH3H +

2.9 1.5 4.7

CH2H +

1.8 2.5 10.5

OH +H O

2.5 3.2 8.5

CH3CH3 +

2.7 X 2.8

OH +CH3 OH

2.0 2.4 2.1

<ρ> 2.4 2.4 5.7

Based on the average ρ value, it is decided to use the CVTST model to calculate rate

coefficients for scission/recombination reactions. The CVTST scission rate coefficients

reproduce the experimental data rather well, certainly when the higher temperatures are

considered. For recombinations, the CVTST model predicts the experimental data and rate

coefficients as reported by Klippenstein et al. within a factor 2 at higher temperatures.

The Gorin model always predicts the same temperature dependence whereas CVTST or

VARIFLEX include more flexibility to account for an increasing or decreasing rate

coefficient. The basic assumption that the radicals in the transition state have the same

vibrational and rotational degrees of freedom is a huge approximation and does not allow to

capture the steric interaction between the fragments.61

This has a big influence on the rate

coefficients.

The rate coefficients obtained from the calculations performed with VARIFLEX in general

agree with experimental and Klippenstein data. However, the temperature dependence can

also be too pronounced. Furthermore, the computational resources required for the

VARIFLEX calculations are, compared with CVTST calculations, much higher. Next to this,

as the VARIFLEX procedure is automated, it is not possible to check if the Gaussian


calculations made a right guess of the orbitals used in the active space calculations. This can

introduce considerable errors in the calculated energies. Besides, the VARIFLEX calculations

are carried out in the absence of pivot point optimization or consideration of multiple binding

sites. This would further increase the computational resources considerably and it is not clear

how this can be done in an efficient way. Moreover, the VARIFLEX software package of the

LCT is outdated. During the past years several modifications have been made to the

VARIFLEX software package,62

however, these updates are not publicly available.


Table 3–2: Results of the 5 recombination reactions which were calculated to test the three TST’s.

Reaction A [s-1

] Ea[kJ/mol]

kcalc/knist

Review experimental

300 K 1000 K 300 K 1000 K 300 K 1000 K

CH4 CH3H +

1986TSA/HAM1087 1989STE/SMI923-945 1985DEA4600-4608

2.97E+16 436.92 2.79 6.21 5.02 5.39 7.70 4.21

CH2H +

1989STE/LAR25-31 1985DEA4600-4608

2.75E+16 422.36 - - 0.24 0.28 0.02 0.54

OH +H O

6.11E+14 444.03 - - - - - -

OH +CH3 OH

1994BAU/COB847-1033 2005OEH/DAV1119-1127 1989STE/LAR25-31

1.01E+17 370.46 2.88 0.87 1.79 1 0.92 18.86 2.22

CH3CH3 +

1987TSA471

1.98E+16 382.03 2.57E+00 1.32E+00 - - - -

1 valid at 700 K

Recombination reactions involving hydrocarbons 65

Chapter 4: Recombination reactions

involving hydrocarbons

In this chapter, the rate coefficients which have been calculated with canonical variational

transition state theory are presented. However, before discussing these calculations, the

various recombination reactions occurring in the steam cracking network of ethane are briefly

discussed. A second part of this chapter presents rate coefficients for scission reactions and

highlights the agreement with other values (calculated or experimental) that could be found in

the literature.

4.1. Determination of the groups present in the steam

cracking network

One of the objectives of this master thesis is to implement the ab initio determined rate

coefficients in available reaction networks. This allows to validate the calculated rate

coefficients: if the network succeeds to reproduce the experimental product yields more

accurately it can be concluded that the new rate coefficients are reliable. The first reaction

network which is aimed at, is the ab initio network that was recently constructed to simulate

ethane steam cracking.1

In appendix A the 90 recombination reactions occurring in the ethane steam cracking network

are listed. Table 4–1 lists all the groups that are required to model the occurring

recombination reactions between hydrogen radicals and carbon centered radicals in the first

column. The second column shows the most simple reaction to which these groups apply.

Table 4–1: Required groups for the determination of the recombination reactions

between a hydrogen and carbon centered radical occuring in the steam cracking

network.

Alkanes


Alkenes

Alkynes

5-rings

6-rings


The required groups to model all recombination reactions between two carbon centered

radicals are presented in the first column of Table 4–2. As in Table 4–1, the second column

depicts the most simple reaction that corresponds with the considered group.

Table 4–2: Required groups for the determination of the recombination reactions

between carbon centered radicals occuring in the steam cracking network.

Alkanes

Alkenes

5-rings

6-rings

Table 4–1 and Table 4–2 illustrate that steam cracking networks need to account for a wide

range of recombination reactions between alkylic, allylic, propargylic and cyclic radicals.

However, the amount of groups required to model all of them is quite limited, i.e. 26. It can be

noted that all primary groups appearing in Table 4–2 can also be retrieved in Table 4–1 for

recombination with H•.

Reaction paths towards the formation of benzene, include components containing 5- and 6-

membered rings. For hydrogen abstraction from 5- and 6-membered ring structures it is

generally observed that ring effects are small.63

However, for recombination reactions it is


possible that additional ring strain alters the dynamics of the reaction making it impossible to

obtain accurate rate coefficients for recombination reactions involving 5- or 6- membered ring

radicals based on the group additivity values determined for their non cyclic analogues.

4.2. Recombination reactions of hydrogen and carbon

centered radicals

In this paragraph, the recombination reactions listed in Table 4–1 between a hydrogen radical

and a radical centered on a carbon atom are discussed. Where possible, comparison with

experimental data or rate coefficients as reported by Klippenstein et al. is made.

4.2.1 Alkanes

Rate coefficients for scission of a C–H bond in methane and ethane were studied in a previous

chapter. However, in order to aid the discussion in this paragraph, the calculated Arrhenius

parameters for both reactions are presented in Table 4−5.

Two additional scissions of C–H bonds of alkanes have been calculated. These two reactions

are C–H scissions in propane and isobutane resulting in hydrogen and an iso-propyl and tert-

butyl radical, respectively. As hyperconjugation does not have a significant influence on the

PES, the rate coefficients were obtained using a minimum active space containing 2 electrons

distributed over 2 orbitals.

From Table 4−5, it follows that each additional methyl group lowers the activation energy for

bond scission with approximately 12 kJ mol-1

. The activation energies for scission of a C–H

bond resulting in a methyl, primary, secondary and tertiary C radical amount to respectively

437, 422, 410 and 403 kJ mol-1

.

For the scission of propane, experimental data could be obtained from the NIST Chemical

Kinetics database. The ratio between calculated and experimental data at 300 and 1000 K can

be found in the sixth and seventh column of Table 4−5. The CVTST rate coefficient for

CH3CH2CH3 → CH3CH•CH3 + H

• deviates from the experimental data by two orders of

magnitude at 300 K. However at higher temperatures, the calculated and experimental data

agree within a factor 4. Rate coefficients for the reverse reaction are presented in Figure 4–1,

together with experimental and calculated data. Large discrepancies between the experimental

values are observed: Munk et al.64

report rate coefficients of 1.5 108 m³ mol

-1 s

-1, while

Warnatz65

reports values that are 6 times lower. Our calculated data are in good agreement


with the latter ones. Harding et al.15

report values that are almost a factor 5 larger than our

values.

Figure 4–1: Rate coefficients for the recombination of a hydrogen radical and an

iso-propyl radical. The CVTST rate coefficient is indicated by the full line, the

experimental and theoretical data are indicated by the symbols.

No experimental data could be retrieved for the reaction H• + tert-butyl ↔ isobutane.

However, for the recombination reaction the CVTST rate coefficients correspond within a

factor 2 at 300 K and 2.5 at 1000 K with the data reported by Harding et al.15

4.2.2 Alkenes

4.2.2.1 Scission of a vinylic C–H bond

In this work rate coefficients are calculated for scission of a vinylic C–H bond in ethene,

propene, allene and butadiene. For scission of ethene and propene the active space included

two electrons in two orbitals.

For allene and 1,3-butadiene larger active spaces were used that allow to account for the

expected π conjugation. For allene, the active space included 6 electrons and 6 orbitals,

although 4 electrons and 4 orbitals are sufficient to describe the resonance effect.

Nevertheless, this larger active space was used to facilitate the Gaussian calculations. By


adding an additional pair of electrons and orbitals it is guaranteed that the active space

includes the correct orbitals to account for the conjugating π system. An active space

containing 4 electrons and 4 orbitals often converges to a state including the π and π* orbital

orthogonal to the σ bond that is broken during the reaction. A depiction of the active space is

provided on Figure 4–2.

Figure 4–2: Depiction of the orbitals included in the active space calculations for

the recombination of a hydrogen and 1,2-propadiene-3-yl radical. Top: bonding

and anti-bonding orbitals of the resonance effect due to interference of the orbitals

of the double bond with the orbitals of the forming radical. Middle: bonding and

anti-bonding orbitals of the double bond orthogonal to the breaking bond.

Bottom: the bonding and anti-bonding orbitals of the breaking σ bond.

For 1,3-butadiene, calculations were performed with an active space containing 6 electrons

and 6 orbitals, instead of four electrons and four orbitals as would be expected from the 1,3-

butadiene-2-yl radical resonance structures. An active space containing 6 electrons and 6

orbitals is required to accurately describe the stabilizing interaction of conjugated dienes.

According to Hückel theory,24

the two degenerate energy levels of the π bonds interfere and,


as a consequence, the total energy of the molecule is lowered. Using an active space of four

electrons and four orbitals cannot grasp this interaction and will lead to a lower bond

dissociation energy . A depiction of the orbitals along the reaction coordinate is presented on

Figure 4–3 for an interatomic distance of 300 pm.

T

Figure 4–3: Depiction of the orbitals involved during the active space calculations

for the scission of 1,3-butadiene into a hydrogen and 1,3-butadiene-3-yl radical.

Top and middle: the orbitals of the conjugated π system. Bottom: bonding and

anti-bonding orbital of the σ bond that is broken during the reaction.

The resulting Arrhenius parameters can be found in Table 4−5. As vinylic radicals are

relatively unstable, higher activation energies are obtained than for C–H scission in saturated

components. For example, the activation energy for C2H4 → C•2H3 + H is 30 kJ mol

-1 higher

than obtained for scission of a C–H bond in methane. Comparing the activation energy for

scission in ethene and propene, once more illustrates that an adjacent methyl ligand lowers the

activation energy for C–H bond scission with approximately 12 kJ mol-1

. Due to the π-

conjugation in the forming 1,2-propadien-1-yl and 1,3-butadien-2-yl radical, the activation

energies for these two reactions are respectively 85 kJ mol-1

and 40 kJ mol-1

lower.


The higher barrier for scission in butadiene can be explained by the loss of overlap of the two

conjugated π systems during reaction. A rotational scan around the single bond in 1,3-

butadiene illustrates that more than 30 kJ mol-1

is required to break the conjugation and rotate

one of the π systems out of plane with the other one.

Experimental data was retrieved for the scission reactions C2H4 → C•2H3 + H

• and

CH2=CHCH3 → CH2=C•CH3 + H

•. The ratio kCVTST/kexp for the scission reactions can be

found in the sixth and seventh column of Table 4−5 at 300 and 1000 K, respectively. For C–H

scission in ethene the calculated rate coefficient deviates a factor 5 at 300 K, while at higher

temperatures the agreement improves to a factor 2. For propene, large discrepancies between

the calculated and experimental data are observed, i.e. a factor 107 at 300 K. However, as the

rate coefficient for the C–H scission in ethylene agrees well with the experimental data and as

the activation energy for C–H scission in propene is 12 kJ mol-1

lower, which is in agreement

with the results obtained for alkanes, it is expected that the experimental data is faulty. In

Figure 4–4 experimental and theoretical rate coefficients are presented for the recombination

reaction C•2H3 + H

• → C2H4. The CVTST rate coefficients underestimate the other data by a

factor 2. However, it is observed that for this reaction the temperature dependence predicted

by FTST-VRC15

and CVTST are in excellent agreement.


Figure 4–4: The CVTST rate coefficient for the recombination of a vinyl and

hydrogen radical compared with experimental data and data calculated by

Harding et al. 15

4.2.2.2 Scission of an allylic C–H bond

The scission of an allylic C-H bond is studied in propene, 1-butene and 1,4-pentadiene. The

active space for the former two included 4 electrons distributed over 4 orbitals. The CAS

orbitals for propene → allyl + H• are shown in Figure 4–5. For the scission of 1,4-pentadiene,

the active space contained 6 electrons and 6 orbitals. The rate coefficient for scission of an

allylic hydrogen atom in 1-butene, is multiplied with a factor two to account for the

stereogenic center in the transition state. This is due to the fact that an ab initio calculated rate

coefficient is calculated for a R- or S-configuration in the transition state which cannot be

distinguished during an experiment. To take this effect into account, the calculated rate

coefficient has to be multiplied by 2.66


Figure 4–5: The orbitals involved in the multi-reference calculations for the

scission of propene into hydrogen and allyl. Top: the bonding and anti-bonding

orbitals of the conjugating π system. Bottom: the bonding and anti-bonding

orbitals of the bond that is broken during the scission reaction.

Arrhenius parameters for all three reactions can be found in Table 4−5. The resonance effect

has a pronounced influence on the activation energy: compared to the scission of methane into

a methyl and hydrogen radical, the activation energy for propene → allyl + H• is lowered with

73 kJ mol-1

. The additional methyl group in 1-butyne lowers the activation energy with an

additional 17 kJ mol-1

. This is of the same magnitude as observed for C–H bond scission in

alkanes and vinylic C–H bonds. The resonance in the transition state for scission of a diallylic

C–H bond in 1,4-pentadiene lowers the activation energy with an additional 53 kJ mol-1

compared to the activation energy observed for the scission of propene into a hydrogen and

allyl radical.

For the scission reaction of propene experimental data are available on the NIST Chemical

Database. The ratio kCVTST/kexp for these two reactions can be found in Table 4−5. The ratio

reported in the fourth column is 0.5. This is within the indicated uncertainty on the

experimental data, which amounts to a factor 3. The agreement with other experimental data

is much worse. However, as the similarity with the review rate coefficient is within the


uncertainty range and as the ratio of the CVTST recombination rate coefficient to the rate

coefficient as calculated by Harding et al.14

varies from a factor 3 at 300 K to a factor 0.8 at

1000 K, it can be concluded that the rate coefficients reported by Barbe et al.67

and Naroznik

et al.68

are not accurate. For scission in 1-butene, the ratio of the CVTST rate coefficient to the

experimental rate coefficient varies from a factor 0.3 at 300 K to a factor 0.39 at 1000 K.

4.2.3 Alkynes

Two reactions in which an propargylic C–H bond is broken have been calculated: the scission

of propyne and 1-butyne into hydrogen and a 1-propyn-3-yl or 1-butyn-3-yl radicals,

respectively. The active space contained 6 electrons spanned over 6 orbitals in order to help

Gaussian choosing the right orbitals to construct the complete active space , as explained in

section 4.2.2.1.

The pre-exponential factor and activation energy for both reactions are reported in Table 4−5.

The activation energy for the scission of propyne is 18 kJ mol-1

higher than the activation

energy for the scission of an allylic C–H bond in propene. This can be understood by looking

at the two resonance structures that can be drawn for a 1-propyn-3-yl radical. One of the

resonance structure has a radical centered on a sp carbon center which is less stable than a

radical centered on a sp2 carbon center. The additional methyl ligand lowers the activation

energy of the C–H bond scission in 1-butyne with 13 kJ mol-1

. This results agrees with the

previously noted change in activation energy due to an adjacent methyl group in alkanes and

alkenes.

No experimental data is available for scission nor recombination reactions involving

propargylic C–H bonds. The ratio of the CVTST recombination rate coefficient to the rate

coefficient reported by Hardine et al. 14

for the recombination of a 1-butyn-3-yl radical with

hydrogen varies from 5.5 at 300 K to 2.2 at 1000 K.

4.2.4 Ring structures

In total 6 rate coefficients have been calculated for recombination reactions of ring structure

radicals and hydrogen radicals. 4 of them involve 5-membered ring structures: cyclopenten-3-

yl, 1-cyclopenten-4-yl, 2,4-cyclopentadien-1-yl and 1,4-cyclopentadien-1-yl. The two

remaining reactions are recombination reactions of hydrogen with 6-membered ring

structures, i.e. cyclohexen-3-yl and 2,5-cyclohexadien-1-yl. The rate coefficients for the

scission and recombination reaction will be compared with the alkane or alkene analogue,


having similar adjacent groups. This allows to study the influence of ring strain/steric effect

induced by the ring structure on the Arrhenius parameters for scission and recombination

reactions. The active space calculations contained 2 electrons and 2 orbitals for the reaction in

which 1-cyclopenten-4-yl and 1,4-cyclopentadien-1-yl are involved, 4 electrons and 4 orbitals

for the recombination of cyclopenten-3-yl and cyclohexen-3-yl with hydrogen and 6 electrons

and 6 orbitals for the reaction of 2,5-cyclohexadien-1-yl and 2,4-cyclopentadien-1-yl with

hydrogen. For the latter, the active space is depicted on Figure 4–6. The Arrhenius parameters

of the 6 reactions are listed in Table 4−5. Experimental data could only been found on the

NIST Chemical Database for one of the six studied reactions. The comparison with this

experimental rate coefficient is also presented in Table 4−5. The CVTST data are in good

agreement with the experimental data, i.e. the ratio kCVTST/kexpvaries from 0.8 at 300 K to 0.94

at 1000 K.

Figure 4–6: Orbitals present spanning the active space of the multi-reference

calculations. Top and middle: the orbitals that make up the conjugated π system.

Bottom: bonding and anti-bonding orbital of the breaking σ bond.


In Table 4–3, the ratios at 300 K and 1000 K are presented between the rate coefficient for

scission in the cyclic compound and the corresponding rate coefficient in the noncyclic

analogue.

Table 4–3: Ratio of the scission rate for 5- or 6-membered rings to the scission rate

of the alkane or alkene analogue.

Reaction Equivalent reation ratio of scission rate coefficient

300 K 1000 K

CHH +

CHH +

2.67 5.55

CHH +

CH +

2.56 0.58

CHH +

CHH +

2.61E-04 1.98

CH +

CH +

1.47E-06 0.02

CH+H

CH

H +

35.10 13.67

CH+H

CHH +

33.37 86.13

For two reactions the ratio of the scission rate coefficients shows a strong temperature

dependency. The corresponding reactios are indicated by an italic font. This temperature

dependence is due to a large difference in activation energy. For the scission of a diallylic C–

H bond in 2,4-cylcopentadiene, the activation energy is 32 kJ mol-1

higher than the activation

energy for scission of a diallylic hydrogen atom in 1,4-pentadiene. This illustrates that

resonance effects need to be treated carefully. The resonance effect present in 2,4-

cyclopentadien-1-yl radical is different than the resonance effect observed for the 1,4-

pentadien-3-yl radical: in the former, the conjugated π-system forms a closed system as can be

seen on Figure 4–6. In 1,4-pentadien-3-yl the two double bonds interfere with each other

through the orbital in which the radical is present. The other reaction, 1,4-cyclopentadiene →

H• +1,4-cyclopentadien-1-yl, has an activation energy that is 35 kJ.mol-1

higher than for

scission in propene leading to a secondary vinyl and hydrogen radical. This is due to

geometrical constraints.

In Table 4–4, the ratio kcyclic/knoncyclic analogue is shown for recombination reactions, both at 300

and 1000 K.


Table 4–4: Comparison of reaction rates for recombination reactions involving 5-

or 6- membered ring radicals to reaction rates for recombination of the alkyl or

alkenyl equivalent.

Reaction Equivalent reaction

ratio of recombination rate

coefficient

300 K 1000 K

CHH +

CHH +

0.20 0.19

CHH +

CH +

3.34 2.16

CHH +

CHH +

0.08 0.07

CH +

CH +

0.72 0.70

CH+H

CH

H +

0.54 0.87

CH+H

CHH +

1.27 1.50

For recombination reactions it is seen from Table 4–4 that much better agreement is obtained.

Also the temperature dependence is much less pronounced compared to scission reactions.

The largest deviations are seen for recombination involving the 2,4-cyclopentadien-1-yl

radical for which the ratio kcyclic/knoncyclic analogue amounts to a factor 0.08.


Table 4–5: Results of the CVTST calculations for the scission of alkylic, vinylic, allylic and propargylic C–H bonds (second and

third column). Comparison with most recent review values (fifth and sixth column) or most recent experimental data (seventh,

eighth, ninth and tenth column) available from NIST.

Reaction A [s-1

] Ea[kJ/mol]

kcalc/knist

review experimental

300K 1000K 300K 1000K 300K 1000K

Alkanes

CH4 CH3H +

1986TSA/HAM1087 1989STE/SMI923-945 1985DEA4600-4608

2.97E+16 436.92 2.79 6.21 5.02 5.39 7.70 4.21

CH2H +

1989STE/LAR25-31 1985DEA4600-4608

2.75E+16 422.36 - - 0.24 0.28 0.02 0.54

CHH +

1985DEA4600-4608

7.35E+15 410.33 - - 163.74 3.95 - -

CH +

3.37E+15 403.02 - - - - - -

Alkenes

CHH +

1985DEA4600-4608

1.55E+16 464.80 - - 5.35 1.86 - -

CH +

1986NAR/NIE281

3.47E+15 452.28 - - 1.09E-07 - - -

CHH +

2.16E+15 379.93 - - - - - -

CH +

6.12E+15 425.86 - - - - - -

CH2H +

1991TSA221-273 1996BAR/MAR829-847 1986NAR/NIE281

1.16E+15 363.36 0.46 0.51 2,19E-08 1 3,72E-07

2 93.10


CHH +

1985DEA4600-4608

2.44E+14 346.01 - - 0.30 0.39 - -

CHH +

3.55E+23 310.96 - - - - - -

Alkynes

CH2H +

2.35E+16 381.67 - - - - - -

CHH +

5.24E+15 368.78 - - - - - -

Ring structures

CHH +

1.98E+15 348.69 - - - - - -

CHH +

2.24E+15 405.08 - - - - - -

CHH +

3.59E+15 342.9 - - - - - -

CH +

3.46E+15 485.70 - - - - - -

CH+H

1985DEA4600-4608

2.34E+15 342.63 - - 0.80 0.94 - -

CH+H

4.82E+15 314.37 - - - - - -

1 valid at 762

2 valid at 811


4.3. Recombination reactions of carbon centered radicals

Rate coefficients were calculated for 15 reactions involving C–C bond scission with formation

of both alkyl, vinylic, allyl, and alkynyl radicals.

4.3.1 Alkanes

Four rate coefficients have been calculated for the scission of a C–C bond in alkanes, i.e. the

scission of ethane, propane, isobutane and neopentane resulting in the formation of methyl

and methyl, ethyl, iso-propyl and tert-butyl, respectively. The CASSCF calculations are

performed using an active space of 2 electrons and 2 orbitals. The resulting CVTST Arrhenius

parameters are presented in Table 4–6.

For scission reactions resulting in a methyl and alkyl radicals, an adjacent methyl ligand has a

less pronounced influence on the reaction barrier compared with scission of C–H bonds: the

activation energy decreases with only 5 kJ mol-1

whereas for scission reactions for C–H bond

scissions the barrier was reduced up to 14 kJ mol-1

.

For all four studied scission reactions, experimental data could be retrieved from the NIST

Chemical Kinetics Database. In Table 4–6, the kCVTST/kexp is represented for the most recent

review values and the two most recent experimental values. It is noted that the agreement with

experimental data is good for C–C bond scission of ethane, propane and isobutane: the ratio of

the calculated rate coefficient to the experimental rate coefficient is mostly within a factor 2.

However, for neopentane larger deviations from the experimental values are observed: the

calculated rate coefficient underestimates the experimental values with a factor 10.

On NIST Chemical Kinetics Database, rate coefficients for the recombination reactions of

methyl with ethyl, iso-propyl and tert-butyl are available. A comparison between the

calculated recombination rate coefficient with experimental values and rate coefficients

reported by Klippenstein et al.13

for these three reactions is presented on Figure 4–7 to Figure

4–9.


Figure 4–7: The results of the CVTST calculations for the rate coefficient for the

recombination of a methyl with an ethyl radical are presented together with

experimental and theoretical data.

Figure 4–7 shows that the temperature dependence for the rate coefficient of the

recombination of methyl with ethyl in the low temperature area is slightly too strong

compared with the data reported by Klippenstein et al. However, at higher temperatures, the

temperature dependence is less pronounced. The ratio of the CVTST rate coefficient with the

Klippenstein rate coefficient varies from 1.9 at 300 K to 1.15 at 1000 K. Good agreement is

also obtained with the review data of.54

The CVTST rate coefficient overestimates the other

experimental data on average with a factor 2.


Figure 4–8: Representation of calculated, theoretical and experimental rate

coefficients for the recombination reaction of methyl with iso-propyl.

On Figure 4–8, it is seen that the calculated rate coefficient for the recombination of iso-

propyl with methyl underestimates the experimental rate coefficients and theoretical rate

coefficient obtained by Klippenstein et al.13

. The ratio of the CVTST rate coefficient to the

experimental and theoretical rate coefficients amounts, on average, to a factor 0.3 over the

entire temperature range. In particular for this reaction it is clear that large discrepancies exist

between the experimental rate coefficients for the forward and reverse reaction. Although rate

coefficients for the scission reaction are overestimated by a factor 10 at lower temperatures,

the rate coefficients for the reverse reaction are underestimated. As the W1bd

thermochemistry corresponds well with the NIST values, the error has to be caused by

experimental errors.


Figure 4–9: Depiction of the results for the CVTST rate coefficient for the

recombination of a methyl and tert-butyl radical together with experimental data

and data calculated by Klippenstein et al.13

The ratio of the CVTST rate coefficient for the recombination of methyl with tert-butyl to the

experimental one varies from 0.5 at 300 K to 0.2 at 1000 K(see Figure 4–9). The agreement

with the rate coefficient reported by Klippenstein et al.13

is good, varying from 1.12 at 300 K

to 0.7 at 1000 K.

4.3.2 Alkenes

4.3.2.1 Scission of vinylic C-C bond

Rate coefficients for the scission of a vinylic C–C bond are calculated for the scission of

propene, isobutene, 1,2-butadiene and 2-methyl-1,3-butadiene. An active space consisting of 2

electrons and 2 orbitals were used to study scission in propene and isobutene, 6 electrons and

6 orbitals were used for 1,2-butadiene and 2-methyl-1,3-butadiene (see discussion in section

4.2.2.1) The resulting rate coefficients are reported in Table 4–6.

The activation energy for the scission of propene increases with 50 kJ mol-1

compared with

the scission of an ethane molecule into two methyl fragments. This is due to the fact that the

forming radical on the vinyl fragment is centered on a sp hybridized carbon atom. Resonance

present in the TS for C–C bond scission in 1,2-butadiene and 2-methyl-1,3-butadiene lowers


the activation energy with 75 kJ mol-1

and 40 kJ mol-1

respectively. These are approximately

the same values obtained for C–H bond scission in propadiene and 1,3-butadiene,

respectively. The additional methyl group present in isobutene lowers the activation barrier

with 10 kJ mol-1

. This is two times more than observed for an additional methyl group for the

C–C bond scission of alkanes.

For the scission of propene, experimental data are available from the NIST Chemical Kinetics

Database (see Table 4–6). The ratio of the calculated rate coefficient to the review rate

coefficient varies over several orders of magnitude. However, when compared with the

experimental data of Dean et al.69

the ratio varies from 0.34 to 0.86.

Experimental rate coefficients for the recombination of methyl with vinyl 70

or 1,2-propadien-

3-yl71

are reported. The ratio of kCVTST/kexp for the former amounts to 0.61 at 298 K, however,

data is lacking at higher temperatures. For the second reaction, this ratio varies from 0.85 at

800 K to 0.93 at 1000 K.

4.3.2.2 Scission of allylic C-C bond

Three rate coefficients have been calculated for the scission of allylic C–C bonds, i.e. scission

in 1-butene, 3-methyl-1-butene and 3-methyl-1,4-pentadiene. The active space for the former

two contained 4 electrons and 4 orbitals. For the latter one, 2 additional electrons and orbitals

are included in the active space. The results are summarized in Table 4–6.

The adjacent double bond in 1-butene lowers the activation energy by 65 kJ mol-1

compared

with the scission of the C–C bond in ethane. This decrease is of the same magnitude as was

observed for the scission of propene into a hydrogen radical and an allyl radical. An additional

methyl group lowers the activation energy with 5 kJ mol-1

. The activation energy for the

scission the diallylic C–C bond in 3-methyl-1,4-pentadiene is lowered with 45 kJ mol-1

compared with scission in 1-butene. This decrease is again 20 kJ mol-1

lower than observed

when including one single double bond (see section 4.3.2.2).

For the three scission reactions, experimental data have been retrieved, see Table 4–6. The

CVTST rate coefficients for the scission in 1-butene locate between the experimental rate

coefficients obtained from the NIST database: the ratio kCVTST/kexp is lower than 1 for the data

reported by Dean69

and above 1 for the data obtained by Trenwith72

. The similarity of the

calculated rate coefficient for the scission of 3-methyl-1-butene with experimental data is


acceptable: varying from 0.16 to 0.19. The ratio of the kCVTST/kexp for the scission in 3-methyl-

1,4-pentadiene varies from 4.55 at 653 K to 3.75 at 718 K.

To the best of our knowledge, experimental rate coefficients are only available for the

recombination of methyl with allyl. The comparison is shown in Figure 4–10. The

temperature dependence is too strong compared with the review rate coefficient from NIST: at

lower temperatures, the CVTST rate coefficient overestimates the review value by a factor 15,

however, in the higher temperature area, the ratio is lowered to 2.4.

Figure 4–10: Comparison of the CVTST rate coefficient for the recombination of a

methyl and allyl radical with experimental data reported by Tsang 73

.

4.3.3 Alkynes

Two C–C bond scission reactions in alkynes have been considered, i.e. the scission of 1-

butyne into methyl and the 1-propyn-3-yl radical and the scission of 3-methyl-1-butyne into

methyl and the 1-butyn-3-yl radical. The CASSCF calculations contained 6 electrons and 6

orbitals in the active space. The results of the CVTST calculations are summarized in Table

4–6.

The adjacent triple bond lowers the activation energy by 50 kJ mol-1

compared with the

scission of ethane into two methyl radicals. The activation energy remains however15 kJ mol-


1 higher than the one obtained for scission of an allylic C–C bond. The additional methyl

group present during scission of 3-methyl-1-butyne lowers the activation energy with an

additional 6 kJ mol-1

compared with the scission of 1-butyne into methyl and 1-propyn-3-yl

radical.

Experimental scission rate coefficients for the two reactions could be retrieved. The calculated

rate coefficient for the scission of 1-butyne is intermediate to the available experimental data

(see Table 4–6). The ratio of kCVTST/kexp for the scission of 3-methyl-1-butyne is smaller than

the reported uncertainty factor 2. It is noted that this reaction has a stereogenic center in the

TS.

4.3.4 Ring structures

The two scission reactions considered here are (a) the scission of methylcyclopentene leading

to methyl and cyclopentenyl and (b) 5-methyl-1,3-cyclopentadiene leading to methyl and 2,4-

cyclopentadien-1-yl. The active space calculations contained 4 electrons and 4 orbitals and 6

electrons and 6 orbitals, respectively. The CVTST rate coefficients are listed in Table 4–6.

For the scission in methylcyclopentene it is observed that the Arrhenius parameters are in

close agreement with the Arrhenius parameters obtained for the reaction 3-methyl-1-butene →

methyl + 1-buten-3-yl. The activation energies correspond within 2 kJ.mol-1

. For the other

reaction, i.e. the scission of 5-methyl-1,3-cyclopentadiene, larger differences are observed

between the activation energies of this reaction and the scission of the equivalent alkene: the

activation barrier is 26 kJ mol-1

higher. It is noted that this difference in activation energy was

also observed when the rate coefficient of the scission of a diallylic C–H bond in

cyclopentadiene was compared with the scission in 1,4-pentadiene.


Table 4–6: CVTST results for the scission of alkylic, vinylic, allylic and propargylic C–C bonds (second and third column).

Comparison with most recent review values (fifth and sixth column) or most recent experimental data (seventh, eighth, ninth and

tenth column) available from NIST.

Reaction A [s-1

] Ea[kJ/mol] kcalc/knist

review experimental

300K 1000K 300K 1000K 300K 1000K

alkanes

CH3CH3 +

1994BAU/COB847-1033 2005OEH/DAV1119-1127 1989STE/LAR25-31

1.01E+17 370.46 2.88 0.87 1,79 1 0.92 18.86 2.22

CH2CH3 +

1988TSA887 2005OEH/DAV1119-1127 1994BEL/PER313-328

2.63E+17 365.42 0.85 1.47 1,62 2 0.90 0,83

3

CHCH3 +

1990TSA1-68 1989TSA71-86

5.02E+17 361.06 10.40 2.04 2,22 4 2.04 - -

CH3 + C

1980PAC/WIM2221 1978PAC/WIM593

6.98E+16 355.54

0.11 5

0.07

6

alkenes

CHCH3 +

1991TSA221-273 1986NAR/NIE281 1985DEA4600-4608

1.12E+17 420.51 9E-04 0.09 4.8E-07

0.34 0.86

CCH3 +

1.20E+17 410.71 - - - - - -

CH3 + CH

1.28E+16 334.70 - - - - - -

CCH3 +

1.15E+17 379.80

CH2CH3 +

1985DEA4600-4608 1970TRE2805-2811


6.57E+15 305.23 - - 0.63 0.64 1.11 7 1.23

8

CHCH3 +

1970TRE2805-2811

9.92E+15 299.62 - - 0.16 9 0.19

10 - -

CHCH3 +

1982TRE3131

1.16E+15 260.06 4.55 11

3.75 12

alkynes

1985DEA4600-4608 1982TRE/WRI2337

CH2CH3 +

1.02E+17 321.10 - - 3.87 11.75 0.14

13 0.17

14

1981NGU/KIN3130

CHCH3 +

1.86E+17 315.56 - - 1.12 15

1.27 - -

ring structures

CHCH3 +

2.71E+16 297.41 - - - - - -

CHCH3 +

2.30E+16 285.76 - - - -

-

1 valid for 700 K

2 valid for 600 K

3 valid for 320 K

4 valid for 713 K

5 valid for 823 K

6 valid for 821 K


7 valid for 689 K

8 valid for 760 K

9 valid at 685 K

10 valid at 740 K

11 valid at 653 K

12 valid at 716 K

13 valid at 652 K

14 valid at 731 K

15 valid at 940 K


4.4. Conclusions

The rate coefficients for scission and recombination reactions calculated with canonical

variational transition state theory are generally within a factor five of experimental data. The

agreement with data reported by Klippenstein is even better, and particularly at higher

temperatures the deviations are limited to a factor 2. At lower temperatures the CVTST and

Klippenstein data agree within a factor 5. The good agreement with the experimental and

Klippenstein data allows to conclude that the implementation of CVTST used in this master

thesis permits to calculate rate coefficients of scission or recombination reactions quite

accurately, certainly at higher temperatures.

In particular for bond scission reactions, the activation energies are strongly related to the

bond dissociation enthalpy. As expected it is observed that the activation energies decrease in

the following order: vinylic bond > alkylic bond > propargylic bond > allylic bond > diallylic

bond. Due to hyperconjugation an adjacent methyl ligand lowers the bond scission activation

energy with 5 kJ mol-1

for C–C scission and with approximately 14 kJ mol-1

for C–H scission.

When resonance effects are involved, it is necessary to calculate the rate coefficient for each

new type of resonance effect as resonance effects not act additively. This was, e.g., observed

for the lowering in activation energy when one or two double bonds stabilize the radical.

For recombination reactions in which 5- or 6-membered ring radicals are involved, it is

possible to make a reliable estimate for the rate coefficients based on the rate coefficients for

the scission in the alkane or alkene counterpart.

Determination of group additivity values 92

Chapter 5: Determination of group

additivity values

In this chapter, group additivity values to model the Arrhenius parameters for recombination

reactions are determined. Based on experimental data, it was found that a group additivity

scheme for recombination reactions can work using only primary contributions, except for

recombination reactions in which bulky fragments are present (see section 2.4.3). For this type

of recombination reactions, it seems necessary to include next nearest neighbor interaction

corrections.

As discussed in section 2.4.2 of this work, it is required to determine the single-event pre-

exponential factor. This pre-exponential factor is obtained after dividing the pre-exponential

factor by the number of single-events which are defined as:

∏

∏

[5–1]

number of single events

number of optical isomers in the transition state

number of optical isomers in the reactant j

number of symmetry in reactant j, this is the product of the internal and

external symmetry numbers

number of symmetry in the transition state, this is the product of internal and

external symmetry numbers

Once the single event pre-exponential factor is determined, the ΔGAV0’s for this pre-

exponential factor and the activation energy are determined as:

[5–2]

[5–3]

In section 2.4.3, it was illustrated that the ∆GAV°’s are temperature dependent, in particular at

lower temperatures. The study here focuses on temperatures of relevance for steam cracking


of hydrocarbons. Therefore Arrhenius expressions were fitted to the recombination rate

coefficients in the temperature interval ranging from 700 K to 1000 K.

5.1. ΔGAV0’s for recombination reactions of hydrogen

centered and carbon centered radicals

Arrhenius parameters were determined by fitting to the ab initio determined rate coefficients

at 700, 800, 900 and 1000 K. The linear regressions were performed using the Solver package

in Excel. The average R² value is 0.987 with a standard deviation of 0.005. The results can be

found in Table 5–1.

Most reactions have rate coefficients that decrease with increasing temperature, leading to

negative activation energies. It can be seen that an adjacent methyl group generally increases

the activation energy for recombination with a few kJ mol-1

, with exception for the reaction H•

+ CH2=CHC•HCH3. Similar activation energies and pre-exponential factors are obtained for

recombinations leading to vinylic C–H bonds. The single-event pre-exponential factor for

these recombination reactions range around 6 107 m

3 mol

-1 s

-1 and the activation energy

amount to ±2 kJ mol-1

.

Derivation of the ∆GAV°’s is straightforward. For example, to determine

(C–

(Cd)(H)2) the activation energy of the reference reaction H• + methyl needs to be subtracted

from the activation energy obtained for the training set reaction H• + allyl. Doing so, one

obtains

(C–(Cd)(H)2)= –3.2 – (–2.3)= –0.9 kJ mol-1

.


Table 5–1:

’s and

’s of the rate coefficients for the

recombination of a hydrogen and carbon centered radical.

reaction primary group ne Ã

[m3 mol

-1 s

-1]

Ea

[kJ mol-1

] ∆GAV

0Ea ∆GAV

0log(Ã)

alkanes

CH3H +

2 1.42E+08 -2.28

CH2H +

2 6.01E+07 1.26 3.54 -0.37

CHH +

1 2.42E+07 3.68 5.96 -0.77

CH +

1 2.11E+07 3.16 5.44 -0.83

alkenes

CHH +

1 9.19E+07 1.98 4.26 -0.19

CH +

1 6.32E+07 2.35 4.62 -0.35

CHH +

1 3.73E+07 -0.01 2.27 -0.58

CH +

1 6.34E+07 3.61 5.89 -0.35

CH2H +

2 5.52E+07 -3.20 -0.92 -0.41

CHH +

1 3.01E+07 -5.79 -3.51 -0.67

CHH +

2 3.84E+07 -2.01 0.27 -0.57

alkynes

CH2H +

2 1.70E+08 -2.60 -0.32 0.08

CHH +

1 5.86E+07 -1.62 0.66 -0.39

5-rings

CHH +

1 1.72E+07 -4.67 -2.40 -0.92

CHH +

2 1.80E+07 0.52 2.80 -0.90

CHH +

2 1.89E+06 -4.76 -2.48 -1.88

CH +

1 4.18E+07 1.76 4.04 -0.53

6-rings

CH+H

1 2.80E+07 -5.33 -3.06 -0.71

CH+H

1 1.19E+08 -1.76 0.52 -0.08


5.2. ΔGAV0’s for recombination reactions of two carbon

centered radicals

The ∆GAV°’s were derived similarly as for C–H recombinations. The average R² value is

0.993 with a standard deviation of 0.001.

Table 5–2:

’s and

’s of the rate coefficients for the

recombination of a methyl and carbon centered radical.

reaction primary group ne Ã

[m3 mol

-1 s

-1]

Ea

[kJ mol-1

] ΔGAV

0Ea ΔGAV

0log(Ã)

alkanes

CH3CH3 +

2 5.44E+06 -8.33

CH2CH3 +

4 4.02E+06 -4.86 3.47 -0.13

CHCH3 +

2 1.61E+06 -2.27 6.06 -0.53

CH3 + C

2 1.47E+06 -4.13 4.21 -0.57

alkenes

CHCH3 +

2 7.00E+06 -3.87 4.47 0.11

CCH3 +

2 5.57E+06 -3.91 4.42 0.01

CH3 + CH

2 1.24E+06 2.46 10.79 -0.64

CCH3 +

2 3.64E+06 -7.00 1.33 -0.17

CH2CH3 +

4 2.73E+06 -7.90 0.43 -0.30

CHCH3 +

2 1.40E+06 -5.28 3.05 -0.59

CHCH3 +

4 2.47E+06 -3.85 4.48 -0.34

Alkynes

CH2CH3 +

4 5.35E+06 -10.00 -1.66 -0.01

CHCH3 +

2 2.60E+06 -7.29 1.04 -0.32

5-rings

CHCH3 +

4 3.00E+05 -9.32 -0.99 -1.26

CHCH3 +

4 5.08E+04 -11.42 -3.09 -2.03


On Figure 5–1 the

’s of the recombination of a hydrogen radical with a carbon

centered radical are plotted as function of

’s of the recombination of two carbon

centered radicals. The dashed lines indicate deviations on the

’s of +2 and -2 kJ mol-1

,

respectively. A difference of ± 2 kJ mol-1

corresponds with a deviation on the rate coefficients

of less than 30% at 1000 K. It is seen that only a few points deviate significantly from the

bisection line, mainly for reactions involving alkenes. Due to the limited timeframe of this

thesis no further attention was paid to them but these can be caused by convergence problems

that occurred during the CASSCF calculations involving methyl recombinations.


Figure 5–1: The

’s of the recombination reactions involving a hydrogen

and a carbon centered radical as function of the

’s determined from the

rate coefficients for recombination of a methyl and carbon centered radical.

On Figure 5–2 a parity plot is shown between the ’s derived for C–C and C–H

recombination reactions. The dashed lines enclose an area which would result in a factor two

on the pre-exponential factors.

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8 10

ΔG

AV

0E

a C

-H r

eco

mb

ina

tio

n

ΔGAV0Ea C-C recombination

Alkanes Alkenes Alkynes 5-rings


Figure 5–2: the

’s determined from the rate coefficients for

recombination of a hydrogen radical and carbon centered radical as function of

the

’s obtained from the rate coefficients for the recombination of

methyl with a carbon centered radical.

Figure 5–1 and Figure 5–2 show that the ∆GAV°’s for C–C and C–H recombination reactions

are in good agreement. This result is quite unexpected as it should be noted that the number of

transition state modes, which have a large effect on the rate coefficient, is different. It shows

that this effect can be captured by proper selection of reference reactions.

It can hence be concluded that it is possible to obtain the rate coefficient for recombination of

a hydrogen radical with a carbon centered radical from the rate coefficients for recombination

of two carbon centered radicals, and the activation energy and pre-exponential factor of the

reference reaction, i.e. the recombination of methyl with hydrogen. This result can now be

extrapolated to recombination reactions involving hydroxyl, sulfanyl, methoxy and

methylsulfanyl radicals. This is illustrated in the following paragraphs.

-2.1

-1.6

-1.1

-0.6

-0.1

-2.1 -1.6 -1.1 -0.6 -0.1

ΔG

AV

0lo

g(Ã

) C

-H r

ecom

bin

ati

on

ΔGAV0log(Ã) C-C recombination

Alkanes Alkenes Alkynes 5-rings


5.3. Group additive modeling of recombination reactions

involving oxygen compounds

In Table 5–3 the ’s determined for recombination reaction of a hydrogen with a carbon

centered radical are used to estimate the kinetic parameters, i.e. the pre-exponential factor and

the activation energy, for recombination reactions involving hydroxyl radicals. The reactions

considered are the recombination of a hydroxyl radical with the smallest radical fragment

corresponding to the groups which were also present in Figure 5–1 and Figure 5–2 The

reaction rate at 1000 K is also indicated.

The use of the ’s presented in Table 5-1 is illustrated for the recombination reaction of

t-butyl with HO•. The CVTST Arrhenius parameters for the recombination reaction C

•H3 +

HO• are 8.49 10

7 m

3 mol

-1 s

-1 for the pre-exponential factor and –2.69 kJ mol

-1 for the

activation energy. From Most reactions have rate coefficients that decrease with increasing

temperature, leading to negative activation energies. It can be seen that an adjacent methyl

group generally increases the activation energy for recombination with a few kJ mol-1

, with

exception for the reaction H• + CH2=CHC

•HCH3. Similar activation energies and pre-

exponential factors are obtained for recombinations leading to vinylic C–H bonds. The single-

event pre-exponential factor for these recombination reactions range around 6 107 m

3 mol

-1 s

-1

and the activation energy amount to ±2 kJ mol-1

.


(C–



obtains

(C–(Cd)(H)2)= –3.2 – (–2.3)= –0.9 kJ mol-1

.


Table 5–1, it can be seen that the group additivity values for Ea and log(Ã) amount for the C-

(C)3 group to +5.44 kJ mol-1

and -0.83, respectively. The activation energy for the

recombination reaction t-butyl + HO• hence is: –2.69 + 5.44 = 2.75 kJ mol

-1. The pre-

exponential factor is calculated as 10log(8.49E+7)–0.83–log(2)

= 6.28 106.

Table 5–3: Arrhenius parameters for the recombination of a hydroxyl and carbon

centered radical based on the ’s determined for the recombination of

hydrogen with carbon centered radicals.

reaction A [m3 mol

-1 s

-1] Ea [kJ mol

-1] k(1000K) [m³ mol

-1 s

-1]

alkanes

CH3 + OH

8.49E+07 -2.69 1.17E+08

CH2 + OH 3.59E+07 0.84 3.24E+07

CH + OH

7.22E+06 3.26 4.87E+06

C + OH 6.28E+06

2.75 4.51E+06

alkenes

CH + OH

2.74E+07 1.57 2.27E+07

C + OH

1.89E+07 1.93 1.50E+07

CH + OH

1.12E+07 -0.42 1.18E+07

C+ OH

1.89E+07 3.20 1.29E+07

CH2+ OH

3.30E+07 -3.6 5.09E+07

CH+ OH

8.99E+06 -6.21 1.90E+07

CH + OH

2.29E+07 -2.43 3.07E+07

Alkynes

CH2+ OH

1.01E+08

-3.0 1.46E+08

CH+ OH

3.46E+07 -2.03 4.42E+07

5-rings

CH+ OH

5.15E+06 -5.09 9.49E+06

CH+ OH

1.12E+06 -5.20 2.09E+06

It is noted that the reaction rate at 1000 K for the recombination of hydroxyl with ethyl is in

quantitative agreement with the experimental reaction rates reported for the recombination of

hydroxyl with iso-butyl74

or n-propyl75

radical, i.e. 2.41 10+7

m3 mol-1

s-1

.


In Table 5–4 GA estimates are presented for recombination reactions involving a hydrocarbon

radical and the methoxy radical.

Table 5–4: recombination rate coefficients for recombination of methoxy radical

with a carbon centered radical based on the ’s determined for the

recombinations of a hydrogen radical and a carbon centered radical.

reaction A [m3 mol

-1 s

-1] Ea [kJ mol

-1] k(1000 K) [m³ mol

-1 s

-1]

alkanes

H + O

4.01E+07 6.08 1.93E+07

CH3 + O 3.07E+06 0.03 3.06E+06

CH2 + O

1.30E+06 3.56 8.45E+05

CH + O

2.61E+05 5.99 1.27E+05

C + O

2.27E+5 5.47 1.18E+5

alkenes

CH + O

9.91E+05 4.29 5.92E+05

C + O

6.82E+5 4.65 3.90E+05

CH + O

4.03E+05 2.30 3.06E+05

C+ O

6.84E+5 5.92 3.35+05

CH2+ O

1.19E+06 -0.90 1.33E+06

CH+ O

3.25E+05 -3.48 9.88E+05

CH + O

8.27E+05 0.30 7.89E+05

Alkynes

CH2+ O

3.66E+6 -0.30 3.79E+06

CH+ O

6.25E+05 0.69 1.15E+06

5-rings

CH+ O

1.86E+05 -2.37 2.47E+05

CH+ O

4.04E+04 -2.45 5.43E+04

In Table 5–5, the estimated rate coefficients for recombination reactions involving CH3O• at

1000 K obtained with the GA model are compared with experimental data.


Table 5–5: Comparison of rate coefficients obtained with the GA model with the

experimental data from NIST.

Reaction Reference uncertainty kexp(1000 K) [m³ mol-1

s-1

] kGA/kexp

O + CH3 1968TSA/HAM1087 5 1.21E+07 0.25

CH2O +

1988TSA887 3 9.64E+06 0.09

O +CH2

1990TSA1-68 3 6.02E+06 0.14

CHO +

1988TSA887 5 6.02E+06 0.02

CO +

1990TSA1-68 3 9.03E+06 0.01

The rate coefficient for the recombination of methoxy with methyl is within the reported

uncertainty range. The other rate coefficients underestimate the reported review coefficients

up to two orders of magnitude. However, the reported rate coefficients have never been

determined experimentally, but were obtained using the geometric mean rule or were

estimated based on experimental data for analogue reactions.

5.4. Group additive modeling of recombination reactions

involving sulfur compounds

Also for recombination reactions involving sulfur compounds, estimates can be made using

the ∆GAV°’s determined for the recombination of hydrogen with carbon centered radicals. To

do so, additional CVTST rate coefficients were calculated for the recombination reaction HS•

+ H• → H2S and CH3S

• + H

• → CH3SH. The GA estimated rate coefficients are presented in

Table 5–6 and Table 5–7, respectively for recombination reactions involving HS• and CH3S

•

radicals.

Table 5–6: Arrhenius parameters for the recombination of a sulfanyl radical and a carbon

centered radical based on the ’s presented in Most reactions have rate coefficients that

decrease with increasing temperature, leading to negative activation energies. It can be seen

that an adjacent methyl group generally increases the activation energy for recombination

with a few kJ mol-1


•HCH3. Similar activation

energies and pre-exponential factors are obtained for recombinations leading to vinylic C–H

bonds. The single-event pre-exponential factor for these recombination reactions range around

6 107 m

3 mol

-1 s


-1.



(C–



obtains

(C–(Cd)(H)2)= –3.2 – (–2.3)= –0.9 kJ mol-1

.


Table 5–1.

reaction A [m3 mol

-1 s

-1] Ea [kJ mol

-1] k(1000 K) [m³ mol

-1 s

-1]

alkanes

H + SH

1.92E+09 1.8 1.55E+09

CH3 + SH

1.47E+08 1.2 1.27E+08

CH2 + SH

6.22E+07 4.8 3.51E+07

CH + SH

1.25E+07 7.2 5.27E+06

C + SH

1.09E+07 6.7 4.88E+06

alkenes

CH + SH

4.76E+07 5.5 2.46E+07

C + SH

3.27E+07 5.8 1.62E+07

CH + SH

3.28E+07 7.1 1.39E+07

C+ SH

1.93E+07 3.5 1.27E+07

CH2+ SH

5.71E+07 0.3 5.51E+07

CH+ SH

3.12E+07 -2.3 4.10E+07

CH + SH

3.97E+07 1.5 3.32E+07

Alkynes

CH2+ SH

1.75E+08 0.9 1.57E+08

CH+ SH

5.99E+07 1.9 4.78E+07

5-rings

CH+ SH

8.92E+06 -1.2 1.03E+07

CH+ SH

1.94E+06 -1.3 2.26E+06

Table 5–7: Arrhenius parameters for the recombination of a methylsulfanyl radical and a

carbon centered radical based on the ’s presented in Most reactions have rate

coefficients that decrease with increasing temperature, leading to negative activation energies.

It can be seen that an adjacent methyl group generally increases the activation energy for

recombination with a few kJ mol-1


•HCH3.

Similar activation energies and pre-exponential factors are obtained for recombinations

leading to vinylic C–H bonds. The single-event pre-exponential factor for these recombination

reactions range around 6 107 m

3 mol

-1 s


-1.



(C–



obtains

(C–(Cd)(H)2)= –3.2 – (–2.3)= –0.9 kJ mol-1

.


Table 5–1.

reaction A [m3 mol

-1 s

-1] Ea [kJ mol

-1] k(1000 K) [m³ mol

-1 s

-1]

alkanes

H + S

3.14E+08 7.3 1.31E+08

+ SCH3 2.40E+07 1.2 2.07E+07

CH2 + S

1.02E+07 4.8 5.73E+06

CH + S

2.04E+06 7.2 8.61E+05

C + S

1.78E+06 6.7 7.97E+05

alkenes

CH + S

7.77E+06 5.5 4.01E+06

C + S

5.34E+06 5.8 2.64E+06

CH + S

5.35E+06 7.1 2.28E+06

C+ S

3.16E+06 3.5 2.08E+06

CH2+ S

9.33E+06 0.3 9.00E+06

CH+ S

5.09E+06 -2.3 6.70E+06

CH + S

6.48E+06 1.5 5.42E+06

Alkynes

CH2+ S

2.86E+07 0.9 2.57E+07

CH+ S

9.79E+06 1.9 7.80E+06

5-rings

CH+ S

1.46E+06 -1.2 1.68E+06

CH+ S

3.17E+05 -1.3 3.68E+05

Experimental data for recombination reactions involving sulfur radicals are scarce. Shum and

Benson76

estimated rate coefficient for the recombination reaction C•H3 + HS

• → CH3SH of

107 m

3 mol

-1 s

-1 at 700 K. This agrees within a factor 3.5 with the predicted value using the

Arrhenius parameters presented in Table 5–6.

Modeling steam cracking of ethane and n-butane 107

Chapter 6: Modeling steam cracking of

ethane and n-butane

In this chapter, experiments performed with the steam cracking pilot plant at the LCT, are

simulated. This was done to demonstrate the validity of the rate coefficients and group

additivity method of Chapter 4 and 5, respectively. Sabbe et al. were the first to construct an

ab initio based reaction network for the steam cracking of C2- to C4- hydrocarbons.1 As the

authors claim that their network yields fairly good agreement with the experimental data, their

network has been used as reference.

6.1. Reactor modeling

The pilot plant consist of three parts: a feed section, the furnace with the suspended reactor

coil and the analysis section. The reactor coil is for all the experiments the same and is made

of Incoloy 800H and measures 23.14 m in length and 1 cm in diameter. These dimensions are

chosen to achieve fast turbulent mixing in the coil at the applied feed flow rates.1 The small

diameter and fast turbulent mixing limit radial temperature, pressure and concentration

gradients and, hence, a 1D reactor model suffices. During the experiments, the temperature

and pressure are measured along the reactor tube which are used for the simulation. This

makes that only the continuity equation has to be integrated for simulations.

6.2. Reaction networks

Three reaction networks are considered. The first reaction network is a reaction network that

was previously developed at the LCT.1 This reaction network focuses on the three reaction

families that play a major role during steam cracking of hydrocarbons,1 i.e. (i) carbon-carbon

and carbon-hydrogen bond scission and the reverse recombination reactions, (ii) hydrogen

abstraction reactions which can be intramolecular and intermolecular and (iii) radical addition

to olefins and the reverse β scission which can also be intermolecular and intramolecular. The

constructed network consists of 1512 reversible reactions between 129 species, i.e. 92 radical

and 37 molecular species with maximally eight carbon atoms. 1302 of these 1512 reactions

are hydrogen abstraction reactions. The remaining reactions are 90 radical

recombination/bond scissions and 120 radical addition/β scissions [ref naar First principles

based simulation of Ethane steam cracking]. The network also includes intramolecular


reactions, with four intramolecular H-abstractions and 12 intramolecular radical additions [ref

naar First principles based simulation of Ethane steam cracking]. Thermochemical data for the

major components were obtained from CBS-QB3 calculations.

Rate coefficients for hydrogen abstraction reactions and radical addition reactions were

obtained from group additive models. For the third reaction family, i.e. recombination of

radicals, no group additive model had been constructed yet that could have been used to

estimate rate coefficients for this type of reactions. Rate coefficients for these reactions were

taken from calculations performed by Klippenstein et al.13-15

However, the reported rate

coefficients were not sufficient to cover the 90 recombination reactions present in this

network. To predict rate coefficients for reactions lacking calculated data use was made of the

geometric mean rule or rate coefficients were assumed to be equal to the rate coefficients for

structurally similar reactions.1 This reaction network will be referred to as Reaction Network

1.

The second reaction network considered here departs from the previously described reaction

network in this that the rate coefficients for the recombination reactions are substituted by rate

coefficients that are estimated by using the group additive model that is developed in Chapter

5 of this master thesis. This reaction network will be referred to as Reaction Network 2.

The third reaction network is a new network generated by RMG 3.0 . The thermodynamics

database used to generate the network contains all calculated W1bd data, covering all major

components. The recently derived group additivity schemes for hydrogen abstractions and

addition reactions and the group additivity scheme for recombination reactions developed in

this master thesis, were used to populate the kinetics library. The network is generated for the

thermal decomposition of ethane at 1000 K and a pressure of 2 bar. A target conversion of

0.65 was used and a tolerance of 0.01. The tolerance was chosen sufficiently small so that the

generated reaction network can also be applied to study the thermal decomposition of

propane, butane and mixtures of C4- components. The generated network contains over 3000

reactions, among which more than 2000 hydrogen abstraction reactions. As two of the main

decomposition reactions for the thermal decomposition of ethane (CH3 + CH3 → C2H6 and

C2H5 → C2H4 + H) were proven to be in the fall-off regime, their rate coefficients were

lowered with a factor 0.7 in accordance with the work of Saeys et al.77

This reaction network

will be referred to as Reaction Network 3.


6.3. Steam cracking of ethane

In order to demonstrate the validity of the generated networks for the steam cracking of

ethane, 7 pilot experiments are simulated. The most important features of the experimental

conditions are listed in Table 6–1.

Table 6–1: Experimental conditions during the ethane cracking experiments. The

HC feed is the hydrocarbon feed and is in g s-1, the steam dilution δ is in g g-1,

CIT and COT stand for coil inlet and outlet temperature and are in °C, the Max

Temp is the maximum temperature observed along the reaction tube and is in °C,

CIP and COP stand for coil inlet and outlet pressure and are in bar.

experimentHC feed

[g/s]δ [g/g]

CIT

[°C]

COT

[°C]

Max Temp

[°C]

CIP

[bar]

COP

[bar]

ethane

conversion [-]

1 1.05 0.284 263 310 789 2.29 1.90 0.18

2 0.565 0.601 244 234 753 2.18 1.94 0.10

3 0.565 0.598 247 236 754 2.18 1.92 0.10

4 0.775 0.269 233 263 801 2.19 1.92 0.24

5 0.806 0.267 234 272 831 2.19 1.90 0.49

6 0.806 0.269 237 272 829 2.19 1.90 0.48

7 0.806 0.267 239 279 850 2.19 1.90 0.61

The experimental conditions span a whole range of hydrocarbon feed flow rates and steam

dilutions. There is a difference of 100 °C between the maximum observed temperatures and,

as a consequence, the ethane conversion ranges from 10 to 60 %.

On Figure 6–1 to Figure 6–3 the parity plots of the product yields are depicted. The axes are

in mass fraction and the dashed lines represent a 10% deviation on the experimental values.


Figure 6–1: Parity plots for the two main components during ethane steam

cracking. Red dots are simulation results obtained with Reaction Network 1 of

Sabbe et al. Orange dots are simulation results obtained with Reaction Network 2.

This is the network of Sabbe et al. in which the rate coefficients for

recombinations are substituted by estimates based on the GA scheme developed in

this work. Green dots are simulation results obtained with Reaction Network 3,

i.e. the network generated with RMG 3.0.

Figure 6–1 shows the yields of the main components, i.e. ethane and ethene. The ethane yield

is generally well predicted, although there is a tendency to underestimate the ethane yield,

meaning that all three reaction networks overestimate the ethane conversion. The reaction

network of Sabbe et al (Reaction Network 1) reproduces the experimental ethane yield within

10% of the experimental values. Modifying the rate coefficients for the recombination

reactions to the values obtained in this work slightly worsen the agreement with experiment.

However Reaction Network 3 outperforms the two previous ones and succeeds to reproduce

the ethane yield within a few wt%.

As all three reaction networks overestimate the ethane conversion, it is expected that all three

of them will overestimate the ethene yield. This can also be seen from Figure 6–1. At low

ethene yields, the estimates are outside the 10% deviation range. As for ethane, best

agreement between simulation and experiment is obtained with reaction network 3.

Figure 6–2 shows parity plots of the products with yields up to 5 wt%. i.e. hydrogen and

methane.

0

0,2

0,4

0,6

0,8

1

0 0,2 0,4 0,6 0,8 1

Ab

in

itio

pre

dic

ted

yie

ld [

-]

Experimental yield [-]

Ethane parity plot

Reaction Network 1 Reaction Network 2 Reaction Network 3


Figure 6–2: Parity plots for dihydrogen and methane. Red dots are simulation

results obtained with Reaction Network 1 of Sabbe et al. Orange dots are

simulation results obtained with Reaction Network 2. This is the network of Sabbe

et al. in which the rate coefficients for recombinations are substituted by estimates

based on the GA scheme developed in this work. Green dots are simulation results

obtained with Reaction Network 3, i.e. the network generated with RMG 3.0.

The dihydrogen yields are overestimated with all the reaction networks. Reaction Network 3

outperforms the other two as it predicts the H2 yields just within 10 % of the experimental

values, while Reaction Network 1 and 2 are outside this area. Reaction Network 1 is slightly

better than Reaction Network 2.

Although ethane conversions are overestimated, methane yields are generally underestimated.

Sabbe et al. state that this result is probably attributable by faulty thermochemical data for the

methyl radical 1 as experimental and CBS-QB3 enthalpies of formation deviate by 3 kJ mol

-1.

Reaction Network 3 performs again considerably better than the other two: the predicted

methane yields are within 10% of the experimental values which is not the case for Reaction

Network 1 and 2. Reaction Network 2 predicts the methane yields better than Reaction

Network 1.

On Figure 6–3, the parity plots of the products with minor yields are presented, i.e. ethyne,

propene, 1,3-butadiene and n-butane.

0

0,01

0,02

0,03

0,04

0 0,01 0,02 0,03 0,04

Ab

in

itio

pre

dic

ted

yie

ld [

-]


H2 parity plot



Figure 6–3: Parity plots of products with minor yields. Red dots are simulation

results obtained with Reaction Network 1 of Sabbe et al. Orange dots are

simulation results obtained with Reaction Network 2. This is the network of Sabbe

et al. in which the rate coefficients for recombinations are substituted by estimates

based on the GA scheme developed in this work. Green dots are simulation results

obtained with Reaction Network 3, i.e. This is the network generated with RMG

3.0.

From Figure 6–3, it can be concluded that the reaction networks are not capable to capture the

ethyne yields accurately. This is probably caused by pressure dependence effects. This is the

only product for which Reaction Network 3 performs considerably worse than the other two

reaction networks. Reaction Network 1 and 2 yield similar ethyne yields which are a factor 2

higher than experimentally observed.

Propene yields are overestimated by Reaction Network 3 and underestimated by Reaction

Network 1 and 2. Reaction Network 3 performs slightly better than the other two. This is

0

0,005

0,01

0,015

0 0,005 0,01 0,015

Ab

in

itio

pre

dic

ted

yie

ld [

-]


Ethyne parity plot


0

0,005

0,01

0,015

0,02

0 0,005 0,01 0,015 0,02

Ab

in

itio

pre

dic

ted

yie

ld [

-]


Propene parity plot


0

0,005

0,01

0,015

0,02

0 0,005 0,01 0,015 0,02

Ab

in

itio

pre

dic

ted

yie

ld [

-]


1,3-butadiene parity plot


0

0,002

0,004

0,006

0,008

0,01

0 0,002 0,004 0,006 0,008 0,01

Ab

in

itio

pre

dic

ted

yie

ld [

-]


n-butane parity plot



probably related to the higher methane yield predicted by this network: reaction paths to

propene involve recombination reactions or addition reactions with methyl.

The 1,3-butadiene yields are overestimated by all three reaction networks. Reaction Network

1 performs best, but still overestimates the observed values considerably, i.e. with more than

50%.

n-butane yields are considerably better predicted by Reaction Network 3. Reaction Network 1

and 2 underestimate the n-butane yields. Reaction Network 2 is slightly better than reaction

network 1 as all the red dots corresponding with the simulation results of Reaction Network 1

are somewhat shifted to higher values.

6.4. Steam cracking of n-butane

Four pilot experiments are simulated with the three reaction networks. The experimental

conditions for the four experiments are listed in Table 6–2.

Table 6–2: Experimental conditions during the steam cracking of n-butane. The

HC feed is the hydrocarbon feed and is in g s-1

, the steam dilution δ is in g g-1

, CIT

and COT stand for coil inlet and outlet temperature and are in °C, the Max Temp

is the maximum temperature observed along the reactor tube and is in °C, CIP

and COP stand for coil inlet and outlet pressure and are in bar.

expHC feed

[g/s]δ [g/g]

CIT

[°C]

COT

[°C]

Max Temp

[°C]

CIP

[bar]

COP

[bar]

n-butane

conversion [-]

1 0.84 0.992 547 423 770 2.43 1.94 0.43

2 0.83 1.002 543 436 798 2.48 2.00 0.62

3 0.82 1.022 533 449 828 2.48 1.98 0.81

4 0.83 1.018 529 465 856 2.52 1.99 0.94

The hydrocarbon feed flow rates and steam dilutions do not vary as much as was the case for

steam cracking of ethane. However the difference between the maximum observed

temperatures along the reactor coil amounts to 90 °C corresponding with a n-butane

conversion ranging from 43 % to 94 %.

Parity plots between experimental and simulated results are presented on Figure 6–4 to Figure

6–6. The axes are in mass fraction. The dashed lines indicate deviations of 10% on the

experimental values.


Figure 6–4: Parity plots of the four main products. Red dots are simulation results

obtained with Reaction Network 1 of Sabbe et al. Orange dots are simulation

results obtained with Reaction Network 2. This is the network of Sabbe et al. in

which the rate coefficients for recombinations are substituted by estimates based

on the GA scheme developed in this work. Green dots are simulation results


Figure 6–4 presents the partity plots of the most important product, i.e. n-butane, propene,

ethene and methane.

At low conversion, i.e. high n-butane yields, Reaction Network 1 and 3 perform equally well,

Reaction Network 2 is slightly worse, although it still predicts the n-butane yields within 10%.

At higher conversions, i.e. low n-butane yields, Reaction Network 1 and 2 both overestimate

the n-butane yields by roughly 4wt% while Reaction Network 3 succeeds to reproduce the

experimental data within 2 wt%. As no sensitivity analysis was performed on the reaction

0

0,2

0,4

0,6

0 0,2 0,4 0,6

Ab

in

itio

pre

dic

ted

yie

ld [

-]


n-butane parity plot


0

0,05

0,1

0,15

0,2

0,25

0 0,05 0,1 0,15 0,2 0,25

Ab

in

itio

pre

dic

ted

yie

ld [

-]


Propene parity plot


0

0,1

0,2

0,3

0 0,1 0,2 0,3

Ab

in

itio

pre

dic

ted

yie

ld [

-]


Ethene parity plot


0

0,05

0,1

0,15

0,2

0 0,05 0,1 0,15 0,2

Ab

in

itio

pre

dic

ted

yie

ld [

-]


Methane parity plot



network and as the rate coefficients of most important reactions are actually in relative good

agreement it is not easy to pinpoint the reason of this deviation. However, it was noticed that

the thermochemistry of the 1-butyl and 2-butyl radical in the network derived by Sabbe et al.

might need some revision. During thermal cracking of n-butane, the three main decomposition

products are propene, ethene and methane. For propene and ethene, the best results are

obtained with Reaction Network 3, reproducing the experimental yields within, on average, 4

and 2 wt% respectively. . For Reaction Network 1 and Reaction Network 2 the overestimation

of the propene yield is somewhat troublesome as these networks simultaneously

underestimate the n-butane conversion. It seems that those two networks hence largely

overestimate the selectivity to propene.

The yields of methane are best predicted by Reaction Network 1 and 2, they perform for all

the simulated experiments evenly well. Reaction Network 3 systematically overestimates the

methane yield. Nevertheless, it still predicts the methane yields fairly as the estimated yields

are just reproduced within 10% of the experimental values. The simultaneous overestimation

of the methane and propene yield might point towards a too fast β-scission in the 2-butyl

radical. Furthermore, these observations deviate from those obtained for the steam cracking of

ethane, where methane yields were generally underestimated.

Figure 6–5 depicts the parity plots of products that are formed between 1 to 5 wt%, i.e.

ethane, 1-butene, 2-butene and 1,3-butadiene.


Figure 6–5: Parity plots of the products with yields between the 1 and 5 wt%. Red

dots are simulation results obtained with Reaction Network 1 of Sabbe et al.

Orange dots are simulation results obtained with Reaction Network 2. This is the

network of Sabbe et al. in which the rate coefficients for recombinations are

substituted by estimates based on the GA scheme developed in this work. Green

dots are simulation results obtained with Reaction Network 3, i.e. the network

generated with RMG 3.0.

Once more, best agreement between experiment and simulations is obtained with the RMG

network, i.e. Reaction Network 3. In particular 1,3-butadiene, 1-butene and 2-butene yields

are reproduced much more accurate. For ethane, it is observed that all reaction networks

systematically underestimate the ethane yield, i.e. within a factor of two.

Figure 6–6 presents the parity plots of products that are produced with less than 1 wt%, i.e.

hydrogen, ethyne, propane and propyne.

0

0,01

0,02

0,03

0,04

0,05

0 0,01 0,02 0,03 0,04 0,05

Ab

in

itio

pre

dic

ted

yie

ld [

-]


Ethane parity plot


0

0,005

0,01

0,015

0,02

0 0,005 0,01 0,015 0,02

Ab

in

itio

pre

dic

ted

yie

ld [

-]


1-butene parity plot


0

0,01

0,02

0,03

0,04

0,05

0 0,01 0,02 0,03 0,04 0,05

Ab

in

itio

pre

dic

ted

yie

ld [

-]


1,3-butadiene parity plot



Figure 6–6:Parity plots of the minor products. Red dots are simulation results

obtained with Reaction Network 1 of Sabbe et al. Orange dots are simulation

results obtained with Reaction Network 2. This is the network of Sabbe et al. in

which the rate coefficients for recombinations are substituted by estimates based

on the GA scheme developed in this work. Green dots are simulation results


Dihydrogen yields are underestimated by 10% (this corresponds with ±0.2 wt%) by the three

studied reaction networks. This result deviates from what was observed for ethane. For ethane

cracking, dihydrogen yields were overestimated by 10%.

Not one reaction network succeeds to capture the experimentally observed ethyne yields.

Reaction Network 3 performs the worst, overestimating the experimentally observed yields by

a factor 3. Reaction Network 1 and 2 perform equally bad, i.e. the ethyne yield is

0

0,005

0,01

0,015

0 0,005 0,01 0,015

Ab

in

itio

pre

dic

ted

yie

ld [

-]


H2 parity plot


0

0,005

0,01

0,015

0,02

0 0,005 0,01 0,015 0,02

Ab

in

itio

pre

dic

ted

yie

ld [

-]


Ethyne parity plot



overestimated by a factor 2. These observations are in line with the observations made for the

steam cracking of ethane .

Reaction Network 3 predicts the experimental propane yields considerably better than

Reaction Network 1 and 2. The latter two fail to predict the propane yields acceptably.

Propyne is systematically underestimated by Reaction Network 1 and 2. Simulations with

Reaction Network 3 do not yield propyne as propyne is not present in the reaction network

6.5. Conclusions

In this chapter, the steam cracking of ethane and n-butane is discussed and simulations were

performed using three different reaction networks. For steam cracking of ethane, 7

experiments, carried out with the pilot plant at the LCT, were simulated. For n-butane

cracking 4 experiments were considered. The experiments were chosen to cover a wide range

of conversion.

The three reactions networks studied were: (a) the original network derived by Sabbe et al.,

(b) the original network of Sabbe et al. in which rate coefficients for recombination reactions

were modified and (c) a newly generated network with RMG. The latter network deviates

from the first as thermochemical data was obtained using W1bd and the rate coefficients for

recombination reactions were obtained using the GA method presented in Chapter 5 of this

work.

Generally, it is observed that the RMG network outperforms the other two in reproducing

experimental conversions and yields of the important steam cracking products. The other two

networks perform similar. In particular for ethane, all three networks overestimate the ethane

conversion. This can be attributed to different effects. One of them can be the neglect of

disproportionation reactions. By adding for example the reaction C2H5 +C2H5 → C2H4 + C2H6

(rate coefficient obtained from the NIST computational database), the ethane conversion

drops with 0.3 wt%. Adding more disproportionation reactions can bring the experimental and

simulated results even closer to each other. A second effect might be pressure dependence. In

particular for smaller compounds, pressure dependence will play an important effect and

might be the main reason why the acetylene yields are strongly overestimated. For the steam

cracking of n-butane good agreement is observed with experimental data, in particular with

the RMG network. Experimental conversions are reproduced within a few wt%.

Conclusion and future work 119

Chapter 7: Conclusion and future

work

The primary goal of this master thesis was to obtain accurate rate coefficients for

recombination reactions involving hydrocarbons and O- and S-containing compounds.

Recombination reactions and the reverse bond scissions play an important role in many

radical processes and an accurate treatment of these reactions is hence required in order to be

able to perform reliable simulations of these processes. In order to model this reaction family,

the accuracy of a group additivity model for Arrhenius parameters was assessed.

The literature available on this topic has been reviewed. It is stressed that multi-reference ab

initio methods are required to grasp the energetic effects along the reaction coordinate. More

specifically it was decided to use CASSCF calculations for geometry optimization along the

reaction coordinate and to carry out CASMP2 calculations on the CASSCF geometries in

order to include dynamical electron correlation. Based on the literature survey three transition

state theories (TST’s) were selected in order to assess their accuracy and computational

feasibility. These three theories are the Gorin model, canonical variational TST and flexible

TST; The Gorin model is one of the simplest models out there and allows to express the rate

coefficient as a simple analytic formula containing polarizabilities and ionization potentials of

the reacting compounds. Within CVTST the reaction coordinate is scanned for a minimum k.

The implementation of CVTST is based on previous work performed at the LCT. Compared

to CVTST, FTST allows for a more detailed description of the transition state modes.

However, to do so, a full potential energy surface needs to be mapped which is computational

expensive.

Before starting any calculations, the validity of a group additive method used to calculate rate

coefficients is assessed based on experimental data, obtained from the NIST Chemical

Kinetics Database. An experimental training set containing 8 reactions was used to reproduce

10 rate coefficients. It was illustrated that a group additivity model for Arrhenius parameters

succeeds to predict the rate coefficient within a factor 2. The ∆GAV°’s show to be

temperature dependent which can be attributed to the loose transition state shifting to shorter


bond lengths at higher temperatures. For bulky fragments, next nearest neighbor interaction

corrections should be taken into account.

A small level of theory study was conducted based on calculations performed on five reaction:

H• + C

•H3, H

• +C

•2H5, H

• + CH3O

•, C

•H3 + C

•H3 and C

•H3+O

•H. These five reactions were

chosen as they have been studied experimentally and theoretically by Klippenstein et al. It

was opted to include two reactions involving oxygen compounds in order to check the validity

of the three methods for heterogeneous element containing reactions. No sulfur reactions were

included due to the fact that no data are available to compare with. Based on the level of

theory study, it was concluded that the implementation of CVTST used in this master thesis

allows to calculate rate coefficients with a satisfactorily accuracy at a reduced computational

cost compared to FTST.

CVTST rate coefficients have been calculated for 34 recombination reactions involving

hydrogen and methyl. Both recombination reactions involving vinylic, allylic, propargylic and

cyclic radicals were studied. The calculated set of reactions was mainly chosen to cover the

recombination reactions occurring in the recently developed network for steam cracking of

ethane. The general agreement between experimental and calculated data are good as most

experimental data are reproduced within a factor 3. This work also illustrates that Arrhenius

parameters for recombination reactions involving ring-structures correspond fairly well with

those obtained for the noncyclic analogue reaction.

’s were determined from the Arrhenius parameters obtained for the recombination

reactions with both methyl and hydrogen radicals. This led to two sets of almost identical

’s. Deviations are generally restricted to 2 kJ mol-1

for the activation energy and 0.3

for log(A). This illustrates once more the general applicability of the GA scheme. Rate rules to

determine rate coefficients for recombination reactions involving radicals containing O and S

have been formulated.


In order to validate the group additivity method, the ’s were used to determine all the

recombination reactions present in the extensive reaction network used to model the steam

cracking process. The rate coefficients were introduced in the network of Sabbe et al. Next to

these two reaction networks, a new network was generated using RMG. Pilot experiments for

steam cracking of ethane and n-butane were simulated with these three reaction networks. For

both feedstock, i.e. ethane or n-butane, the RMG network outperforms the other two.

7.1. Future work

As this is one of the first attempts to fully grasp recombination reactions, further

improvements can be made.

First of all, after applying the GA method to experimental rate coefficients it was concluded

that for the recombination reactions involving very bulky fragments, non-nearest-neighbor

interaction corrections have to be taken into account. In this work, recombination of such

bulky fragments have not been studied with CVTST due to the limited time frame. However,

the time consuming, W1bd calculations for some of these reactions are already performed.

The influence of the CASMP2 cut-off value for configuration states has to be further

investigated. When comparing the

’s for some recombination reactions of hydrogen

with an alkenyl radical differed by more than 2 kJ mol-1

with the

’s calculated for the

corresponding recombination reactions of methyl with alkenyl radicals. It is possible that this

is caused by the MP2 corrections. The Gaussian package uses a standard cut-off value of 0.01

during the CASMP2 calculation. This indicates that configuration state functions contributing

less than 1% to the total wave function are not accounted for. In particular for calculations

with larger active spaces lower cut-off values are required to improve the smoothness of the

calculated energy profile

The ’s were used to calculate the rate coefficient for recombination reactions involving

other radicals like H•,

•OH,

•SH, CH3O

• and CH3S

•. However more rate coefficients for

recombination reactions involving those radicals need to be calculated as experimental data is

lacking to validate this.

The temperature range in which the rate coefficients have been determined goes from 300 K

to 1000 K. However, for combustion applications, the upper limit can be too low. To extent

the GA model to higher temperatures, the rate coefficients will have to be calculated at higher


temperatures. Furthermore, it is expected that at temperatures above 1000 K the presented

implementation of CVTST performs better as the transition state will shift to shorter distances

where internal modes can be treated more accurately as harmonic oscillators.

References 123

References

1. Sabbe, M.K., et al., First Principles-based Simulation of Ethane Steam Cracking.

AIChE J., 2011. 57(2): p. 482-496.

2. Sabbe, M.K., Ab Initio Based Kinetic Modeling for the Simulation of Industrial

Chemical Processes, in Faculty of engineering science: Laboratory for Chemical

Technology, 2009, Ghent University: Ghent.

3. A., V., The Thermochemistry and Decomposition Mechanism of Organosulfur and

Organosphosphorus Compounds, in Faculty of Engineering: Laboratory for Chemical


4. Broadbelt, L.J., S.M. Stark, and M.T. Klein, Computer-Generated Pyrolysis Modeling

- On-The-Fly Generation of Species, Reactions, and Rates. Industrial & Engineering

Chemistry Research, 1994. 33(4): p. 790-799.

5. Susnow, R.G., et al., Rate-based construction of kinetic models for complex systems.

Journal of Physical Chemistry A, 1997. 101(20): p. 3731-3740.

6. Sabbe, M.K., et al., Group additive values for the gas phase standard enthalpy of

formation of hydrocarbons and hydrocarbon radicals. Journal of Physical Chemistry

A, 2005. 109(33): p. 7466-7480.

7. Sabbe, M.K., et al., First principles based Group additive values for the gas phase

standard entropy and heat capacity of hydrocarbons and hydrocarbon radicals.

Journal of Physical Chemistry A, 2008. 112(47): p. 12235-12251.

8. Sabbe, M.K., et al., Carbon-centered radical addition and beta-scission reactions:

Modeling of activation energies and pre-exponential factors. ChemPhysChem, 2008.

9(1): p. 124-140.

9. Sabbe, M.K., et al., Hydrogen Radical Additions to Unsaturated Hydrocarbons and

the Reverse beta-Scission Reactions: Modeling of Activation Energies and Pre-

Exponential Factors. ChemPhysChem, 2010. 11(1): p. 195-210.

10. Sabbe, M.K., Hydrogen abstractions by H radicals: modeling of activation energies

and pre-exponential factors, 2009.

11. Sabbe, M.K., et al., Modeling the influence of resonance stabilization on the kinetics

of hydrogen abstractions. Physical Chemistry Chemical Physics, 2010. 12(6): p. 1278-

1298.

12. Vandeputte, A.G., et al., Theoretical Study of the Thermodynamics and Kinetics of

Hydrogen Abstractions from Hydrocarbons. Journal of Physical Chemistry A, 2007.

111(46): p. 11771-11786.

13. Klippenstein, S.J., Y. Georgievskii, and L.B. Harding, Predictive theory for the

combination kinetics of two alkyl radicals. Physical Chemistry Chemical Physics,

2006. 8(10): p. 1133-1147.

14. Harding, L.B., S.J. Klippenstein, and Y. Georgievskii, On the combination reactions

of hydrogen atoms with resonance-stabilized hydrocarbon radicals. Journal of

Physical Chemistry A, 2007. 111(19): p. 3789-3801.

15. Harding, L.B., Y. Georgievskii, and S.J. Klippenstein, Predictive theory for hydrogen

atom - Hydrocarbon radical association kinetics. Journal of Physical Chemistry A,

2005. 109(21): p. 4646-4656.

References 124

16. Bajus, M. and J. Baxa, Coke Formation during the Pyrolysis of Hydrocarbons in the

Presence of Sulfur-Compounds. Collection of Czechoslovak Chemical

Communications, 1985. 50(12): p. 2903-2909.

17. NIST Chemical Kinetics Database Standard Reference Database 17, Version 7.0 (Web

Version), Release 1.6.5, Data Version 2012.02 2011; Available from:

http://kinetics.nist.gov/kinetics/index.jsp.

18. Vandeputte, A.G., M.F. Reyniers, and G.B. Marin, Theoretical Study of the Thermal

Decomposition of Dimethyl Disulfide. Journal of Physical Chemistry A, 2010.

114(39): p. 10531-10549.

19. Klippenstein, S.J., A Bond Length Reaction Coordinate for Unimolecular Reactions .2.

Microcanonical and Canonical Implementations with Application to the Dissociation

of Ncno. Journal of Chemical Physics, 1991. 94(10): p. 6469-6482.

20. Klippenstein, S.J. and R.A. Marcus, High-Pressure Rate Constants for Unimolecular

Dissociation Free-Radical Recombination - Determination of the Quantum Correction

Via Quantum Monte-Carlo Path Integration. Journal of Chemical Physics, 1987.

87(6): p. 3410-3417.

21. Sabbe, M.K., et al., Hydrogen radical addition to unsaturated hydrocarbons and

reverse beta-scission reactions: modeling of activation energies and pre-exponential

factors. ChemPhysChem, 2010. 11(1): p. 195-210.

22. Cramer, C., - Essentials of Computational Chemistry: Theories and Models, Second

Edition. 2004. 618.

23. Bultinck, P., Kwantumchemie, 2008. p. 182.

24. Waroquier, M. and V.S. V., Moleculaire modellering, 2010.

25. Plesset, M.S., - Note on an Approximation Treatment for Many-Electron Systems.

1934. - 46(- 7): p. - 622.

26. Andersson, K., et al., Second-order perturbation theory with a CASSCF reference

function. J. Phys. Chem., 1990. 94(14): p. 5483- 5488.

27. Abrams, M.L. and C.D. Sherrill, An assessment of the accuracy of multireference

configuration interaction (MRCI) and complete-active-space second-order

perturbation theory (CASPT2) for breaking bonds to hydrogen. Journal of Physical

Chemistry A, 2003. 107(29): p. 5611-5616.

28. Gaussian 09 keywords. 2011.

29. Golubeva, A.A., et al., Performance of the Spin-Flip and Multireference Methods for

Bond Breaking in Hydrocarbons: A Benchmark Study. The Journal of Physical

Chemistry A, 2007. 111(50): p. 13264-13271.

30. Klippenstein, S.J. and L.B. Harding, A direct transition state theory based study of

methyl radical recombination kinetics. Journal of Physical Chemistry A, 1999.

103(47): p. 9388-9398.

31. Klippenstein, S.J., Y. Georgievskii, and L.B. Harding, A theoretical analysis of the

CH3+Hreaction: Isotope effects, the high-pressure limit, and transition state

recrossing. Proceedings of the Combustion Institute, 2002. 29: p. 1229-1236.

32. Klippenstein, S.J. and L.B. Harding, A theoretical study of the kinetics of C2H3+H.

Physical Chemistry Chemical Physics, 1999. 1(6): p. 989-997.

33. Vleeschouwer, F.D., Ab initio groepsadditieve waarden voor de standaard molaire

entropie en warmtecapaciteit voor koolwaterstoffen en koolwaterstofradicalen, in

Faculty of Engineering: Laboratory for Chemical Technology, 2006, Ghent

University: Ghent.

34. Laidler, K., - Chemical Kinetics (3rd Edition). 1987.

35. Gilbert, R.G. and S.C. Smith, Theory of unimolecular and recombination reactions.

1990, Oxford: Blackwell scientific publications.

http://kinetics.nist.gov/kinetics/index.jsp

References 125

36. Sumathi, R. and W.H. Green, A priori rate constants for kinetic modeling. Theoretical

Chemistry Accounts, 2002. 108(4): p. 187-213.

37. Laidler, K.J., Chemical kinetics. 1987, New York: Harper & Row.

38. Fernandez-Ramos, A., et al., Modeling the kinetics of bimolecular reactions. Chemical

Reviews, 2006. 106(11): p. 4518-4584.

39. Li, H., Z. Chen, and M.B. Huang, CASPT2 investigation of ethane dissociation and

methyl recombination using canonical variational transition state theory. International

Journal of Chemical Kinetics, 2008. 40(4): p. 161-173.

40. Klippenstein, S.J., Variational Optimizations in the Rice-Ramsberger-Kassel-Marcus

Theory Calculations for Unimolecular Dissociations with No Reverse Barrier. Journal

of Chemical Physics, 1992. 96(1): p. 367-371.

41. Robertson, S., A.F. Wagner, and D.M. Wardlaw, Flexible transition state theory for a

variable reaction coordinate: Analytical expressions and an application. Journal of


42. Georgievskii, Y. and S.J. Klippenstein, Transition state theory for multichannel

addition reactions: Multifaceted dividing surfaces. Journal of Physical Chemistry A,

2003. 107(46): p. 9776-9781.

43. M. J. Frisch, G.W.T., H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G.

Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li,

H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M.

Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O.

Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M.

Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J.

Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M.

Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C.

Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R.

Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski,

G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B.

Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian 09, Revision A.02, in

Gaussian, 2009: Wallingford CT.

44. Benson, S.W. and J.H. Buss, Additivity rules for the estimation of molecular

properties. Thermodynamic properties. Journal of Chemical Physics, 1958. 29(9): p.

546-561.

45. Evans, M.G. and M. Polanyi, Further Considerations on the Thermodynamics of

Chemical Equilibria and Reaction Rates. Proc.Roy.Soc.A, 1936. 154: p. 1333-1360.

46. Saeys, M., et al., Ab initio group contribution method for activation energies for

radical additions. AIChE Journal, 2004. 50(2): p. 426-444.

47. Saeys, M., et al., Ab initio group contribution method for activation energies of

hydrogen abstraction reactions. ChemPhysChem, 2006. 7(1): p. 188-199.

48. Sumathi, R., H.H. Carstensen, and W.H. Green, Reaction rate prediction via group

additivity Part 1: H abstraction from alkanes by H and CH3. Journal of Physical

Chemistry A, 2001. 105(28): p. 6910-6925.

49. Sendt, K. and B.S. Haynes, Quantum chemical and RRKM calculations of reactions in

the H/S/O system. Proceedings of the Combustion Institute, 2007. 31: p. 257-265.

50. Benassi, R. and F. Taddei, A theoretical ab initio approach to the S-S bond breaking

process in hydrogen disulfide and in its radical anion. Journal of Physical Chemistry

A, 1998. 102(30): p. 6173-6180.

51. Grabowy, J.A.D. and P.M. Mayer, Entropy changes in the dissociation of proton-

bound complexes: A variational RRKM study (vol 109, pg 23, 2005). Journal of


References 126

52. NIST Computational Chemistry Comparison and Benchmark Database, Standard

Reference Database 101, Release 12. 2005; Available from:

http://srdata.nist.gov/cccbdb/.

53. Brouwer, A.M. and R. Wilbrandt, Vibrational Spectra of N,N-Dimethylaniline and Its

Radical Cation. An Interpretation Based on Quantum Chemical Calculations. The

Journal of Physical Chemistry, 1996. 100(23): p. 9678-9688.

54. Baulch, D.L., et al., Evaluated Kinetic Data for Combustion Modeling Supplement-I.

Journal of Physical and Chemical Reference Data, 1994. 23(6): p. 847-1033.

55. Forst, W., Microcanonical variational theory of radical recombination by inversion of

interpolated partition function, with examples: methyl + hydrogen atom, methyl +

methyl. The Journal of Physical Chemistry, 1991. 95(9): p. 3612-3620.

56. Tsang, W., Rate constants for the decomposition and formation of simple alkanes over

extended temperature and pressure ranges. Combust. Flame, 1989. 78: p. 15.

57. Pilling, M.J., Association reactions of atoms and radicals. Int. J. Chem. Kinet., 1989.

21: p. 24.

58. Sillesen, A.R., E.; Pagsberg, P., Kinetics of the reactions H+C2H4→C2H5,

H+C2H5→2CH3 and CH3+C2H5→products studied by pulse radiolysis combined

with infrared diode laser spectroscopy. Chem. Phys. Lett., 1993. 201: p. 6.

59. Jasper, A.W., et al., Kinetics of the reaction of methyl radical with hydroxyl radical

and methanol decomposition. Journal of Physical Chemistry A, 2007. 111(19): p.

3932-3950.

60. Dobe, S., T. Berces, and I. Szilagyi, Kinetics of the Reaction between Methoxyl

Radicals and Hydrogen-Atoms. Journal of the Chemical Society-Faraday Transactions,

1991. 87(15): p. 2331-2336.

61. Tschuiko.E, Critical Bond Length in Radical Combination and Unimolecular

Dissociation Reactions. Journal of Physical Chemistry, 1968. 72(3): p. 1009-&.

62. Georgievskii, Y. and S.J. Klippenstein, Variable reaction coordinate transition state

theory: Analytic results and application to the C2H3+H -> C2H4 reaction. Journal of

Chemical Physics, 2003. 118(12): p. 5442-5455.

63. Vandeputte, A.G., Thermal cracking of hydrocarbons: ab initio kinetic parameters for

hydrogen abstractions, in Faculty of Engineering: Laboratory for Chemical


64. Munk, J., et al., Spectrokinetic studies of i-C3H7 and i-C3H7O2 radicals. Chemical

Physics Letters, 1986. 132(4–5): p. 417-421.

65. Warnatz, J. and W.C. Gardiner, Jr., Rate coefficients in the C/H/O system, in

Combustion Chemistry, 1984, Springer-Verlag: New York.

66. Fernandez-Ramos, A., et al., Symmetry numbers and chemical reaction rates.

Theoretical Chemistry Accounts, 2007. 118(4): p. 813-826.

67. Barbe, P., et al., Kinetics and modeling of the thermal reaction of propene at 800 K .1.

Pure propene. International Journal of Chemical Kinetics, 1996. 28(11): p. 829-847.

68. Naroznik, M. and J. Niedzielski, Propylene Photolysis at 6.7 Ev - Calculation of the

Quantum Yields for the Secondary Processes. Journal of Photochemistry, 1986. 32(3):

p. 281-292.

69. Dean, A.M., Predictions of Pressure and Temperature Effects Upon Radical-Addition

and Recombination Reactions. Journal of Physical Chemistry, 1985. 89(21): p. 4600-

4608.

70. Laufer, A.H. and A. Fahr, Reactions and kinetics of unsaturated C-2 hydrocarbon

radicals. Chemical Reviews, 2004. 104(6): p. 2813-2832.

71. Faravelli, T., et al., An experimental and kinetic modeling study of propyne and allene

oxidation. Proceedings of the Combustion Institute, 2000. 28: p. 2601-2608.

http://srdata.nist.gov/cccbdb/

References 127

72. Trenwith, A.B., Dissociation of 1-Butene, 3-Methyl-1-Butene, and of 3,3-Dimethyl-1-

Butene and Resonance Energy of Allyl, Methyl Allyl and Dimethyl Allyl Radicals.

Transactions of the Faraday Society, 1970. 66(575): p. 2805-&.

73. Tsang, W., Chemical Kinetic Data Base for Combustion Chemistry: Part V. Propene.


74. Tsang, W., Chemical Kinetic Data-Base for Combustion Chemistry .4. Isobutane.


75. Tsang, W., Chemical Kinetic Data-Base for Combustion Chemistry .3. Propane.


76. Shum, L.G.S. and S.W. Benson, The Pyrolysis of Dimethyl Sulfide, Kinetics and

Mechanism. International Journal of Chemical Kinetics, 1985. 17(7): p. 749-761.

77. Sun, W.J. and M. Saeys, Construction of an Ab Initio Kinetic Model for Industrial

Ethane Pyrolysis. Aiche Journal, 2011. 57(9): p. 2458-2471.

Reaction network 128

Appendix A: Reaction network

Table A - 1: 90 recombination reactions present in the extensive steam cracking

network.

Reaction Group

H

H

H

H

H

H

H

H


H

H

H

H

H

H

H


H

H

H

H

H

H

H

H

H

H


H

H

H

H

H


H

H

H

W1bd calculations 134

Appendix B: W1bd calculations

Table B - 1: Ground state energy, ZPVE and enthalpy at 298K.

compound electronic energy

(hartree) ZVPE (hartree)

ΔfH°(298K) W1BD

(kJ/mol)

alkanes

methane -40.522991 0.043921 -76.40

ethane -79.842435 0.073264 -88.42

propane -119.165164 0.101465 -110.91

butane -158.487996 0.129445 -132.60

2-methylpropane -158.490302 0.129050 -141.14

2,2-dimethylpropane -197.816665 0.156361 -175.22

alkenes

ethene -78.604776 0.050154 49.94

propene -117.931797 0.078242 15.84

propadiene -116.680718 0.054123 187.22

1-butene -157.254120 0.106459 -4.78

2-methyl-1-propene -157.260049 0.105810 -22.28

3-methyl-1-buten -196.578943 0.133964 -33.23

1,2-butadiene -156.006062 0.082418 158.59

1,3-butadiene -156.026357 0.083701 107.62

2-methylbuta-2,3-diene -195.354132 0.111526 71.44

1,4-pentadiene -195.342943 0.111461 103.27

3-methyl-1,4-pentadiene -234.667040 0.139237 75.10

alkynes

propyne -116.682904 0.054657 183.29

1-butyne -156.005025 0.083161 163.18

3-methylbut-1yn -195.329441 0.110778 135.27

5-rings

cyclopentene -195.372494 0.114633 27.79

1,3-cyclopentadiene -194.148798 0.090930 127.93

3-methylcyclopentene -234.696999 0.142175 -0.38

5-methyl-1,3-pentadiene -233.472183 0.118849 103.74

6-rings

cyclohexene -234.702585 0.143656 -12.57

1,4-cyclohexadiene -233.474474 0.120025 100.01

radicals

alkyls

methyl -39.843598 0.029229 144.73

ethyl -79.168686 0.058146 116.15

iso-propyl -118.495732 0.086231 81.57

tert-butyl -157.823940 0.114269 46.29

alkenyls

vinyl -77.916275 0.035847 295.67


sec-vinyl -117.248204 0.064199 249.66

propadienyl -116.024379 0.040353 350.49

allyl -117.280416 0.065015 165.92

but-1-en-3-yl -156.607306 0.092627 131.95

3-methylbut-1en-3-yl -195.934963 0.120133 95.02

buta-1,3-dien-2-yl -155.352089 0.069013 315.29

penta-1,4-dien-3-yl -194.710474 0.098410 201.59

alkynyls

prop-1-yn-3-yl -116.024378 0.040354 350.49

but-1-yn-3-yl -155.351155 0.068639 317.73

3-methylbyt-1yn-3-yl -194.679410 0.096211 280.33

5-rings

cyclopenten-3-yl -194.726280 0.100812 163.69

cyclopent-1-en-4-yl -194.705931 0.099796 214.73

cylcopenta-2,4-dien-1-yl -193.504850 0.076589 258.24

cyclopenta-1,4-dien-1-yl -193.453653 0.078329 395.38

6-rings

cyclohexa-2,5-dien-1-yl -232.841902 0.106865 201.22

Table B - 2: Entropy at 298 K and symmetry number

compound S° W1BD (298K) (J/mol/K) Symmetry

alkanes

methane 186.05 12

ethane 228.94 6

propane 270.95 2

butane 310.98 2

2-methylpropane 295.57 3

2.2-dimethylpropane 328.78 1

Alkenes

ethene 218.91 4

propene 265.74 1

propadiene 254.29 1

1-butene 308.17 1

2-methyl-1-propene 298.68 1

3-methyl-1-buten 351.44 1

1.2-butadiene 291.54 1

1.3-butadiene 277.06 2

2-methylbuta-2.3-diene 313.31 1

1.4-pentadiene 338.71 1

3-methyl-1.4-pentadiene 345.17 1

alkynes

propyne 247.36 3

1-butyne 290.02 1

3-methylbut-1yn 319.87 1


5-rings

cyclopentene 291.51 1

1.3-cyclopentadiene 273.70 2

3-methylcyclopentene 323.08 1

5-methyl-1.3-pentadiene 309.62 1

6-rings

cyclohexene 310.02 1

1.4-cyclohexadiene 306.39 1

radicals

alkyls

methyl 194.57 6

ethyl 247.59 1

iso-propyl 296.93 1

tert-butyl 319.79 3

alkenyls

vinyl 233.66 1

sec-vinyl 273.23 1

propadienyl 260.07 1

allyl 254.29 1

but-1-en-3-yl 301.36 1

3-methylbut-1en-3-yl 338.24 1

buta-1.3-dien-2-yl 291.29 1

penta-1.4-dien-3-yl 308.06 2

alkynyls

prop-1-yn-3-yl 254.32 2

but-1-yn-3-yl 302.34 1

3-methylbut-1yn-3-yl 336.11 2

5-rings

cyclopenten-3-yl 296.90 1

cyclopent-1-en-4-yl 291.02 2

cylcopenta-2.4-dien-1-yl 301.82 2

cyclopenta-1.4-dien-1-yl 284.36 1

6-rings

cyclohexa-2.5-dien-1-yl 306.81 1

Table B - 3: Heat capacity as function of temperature

cp [kJ mol

-1 K

-1]

compound 300 K 400 K 500 K 600 K 700 K 800 K 900 K 1000 K 1500K

alkanes

methane 35.60 40.30 46.05 51.86 57.34 62.41 67.03 71.20 86.01

ethane 52.34 64.73 76.97 88.06 97.91 106.64 114.38 121.22 144.81

propane 73.69 93.05 111.16 127.02 140.79 152.83 163.39 172.65 204.33

butane 98.93 123.16 146.12 166.31 183.83 199.10 212.47 224.17 263.96

2-methylpropane 97.38 123.65 147.44 167.82 185.26 200.37 213.54 225.06 264.32


2,2-dimethylpropane 122.83 156.35 185.84 210.61 231.52 249.46 265.04 278.64 324.89

alkenes

ethene 42.65 52.45 61.79 69.92 76.93 83.03 88.37 93.07 109.30

propene 63.86 79.50 94.10 106.83 117.83 127.39 135.75 143.07 168.03

propadiene 58.47 71.23 82.14 91.19 98.84 105.41 111.13 116.14 133.27

1-butene 86.56 108.54 128.60 145.85 160.64 173.43 184.57 194.30 227.36

2-methyl-1-propene 87.99 109.27 128.70 145.61 160.25 173.00 184.14 193.90 227.14

3-methyl-1-buten 113.15 140.65 165.57 186.89 205.10 220.83 234.52 246.48 287.15

1,2-butadiene 79.05 96.84 112.87 126.64 138.44 148.65 157.54 165.29 191.54

1,3-butadiene 77.34 100.24 119.87 135.31 147.43 157.19 165.31 172.20 195.17

2-methylbuta-2,3-diene 102.63 131.53 155.42 174.24 189.34 201.88 212.57 221.83 253.33

1,4-pentadiene 102.18 125.31 146.68 165.05 180.71 194.19 205.87 216.04 250.44

3-methyl-1,4-pentadiene 126.46 156.89 183.81 206.53 225.71 242.13 256.33 268.69 310.49

alkynes

propyne 60.46 72.06 82.11 90.69 98.09 104.57 110.28 115.30 132.66

1-butyne 81.21 99.73 115.61 128.94 140.27 150.06 158.61 166.09 191.70

3-methylbut-1yn 105.38 130.49 151.84 169.58 184.52 197.35 208.51 218.25 251.49

5-rings

cyclopentene 81.54 111.97 139.09 161.63 180.29 195.92 209.17 220.49 257.54

1,3-cyclopentadiene 75.23 102.41 125.43 143.92 158.84 171.14 181.45 190.21 218.78

3-methylcyclopentene 105.82 142.24 174.52 201.35 223.59 242.29 258.19 271.83 316.78

5-methyl-1,3-pentadiene 98.74 131.96 160.36 183.33 202.00 217.46 230.50 241.62 278.13

6-rings

cyclohexene 100.50 137.54 170.89 198.94 222.35 242.06 258.80 273.11 319.82

1,4-cyclohexadiene 94.05 126.95 155.85 179.82 199.63 216.21 230.22 242.15 280.89

radicals

alkyls

methyl 39.32 42.44 45.51 48.42 51.20 53.86 56.39 58.76 67.89

ethyl 51.20 61.69 71.74 80.66 88.51 95.44 101.60 107.05 126.02

iso-propyl 67.48 84.03 100.16 114.53 127.07 138.03 147.62 156.01 184.47

tert-butyl 89.44 110.34 131.43 150.46 167.17 181.78 194.55 205.68 243.18

alkenyls

vinyl 43.66 51.46 58.21 63.86 68.68 72.87 76.57 79.84 91.23

sec-vinyl 63.06 75.79 87.78 98.33 107.51 115.52 122.53 128.66 149.47

propadienyl 62.77 72.34 79.75 85.71 90.73 95.09 98.94 102.36 114.47

allyl 62.45 78.54 92.16 103.29 112.55 120.41 127.21 133.13 153.40

but-1-en-3-yl 83.23 103.88 122.58 138.52 152.03 163.61 173.63 182.35 211.82

3-methylbut-1en-3-yl 103.78 130.06 154.12 174.84 192.53 207.75 220.95 232.43 271.02

buta-1,3-dien-2-yl 73.61 98.12 118.60 134.92 148.01 158.73 167.68 175.25 199.74

penta-1,4-dien-3-yl 95.43 121.14 143.17 161.23 176.12 188.64 199.32 208.53 239.31

alkynyls

prop-1-yn-3-yl 62.78 72.35 79.75 85.71 90.73 95.09 98.94 102.36 114.47

but-1-yn-3-yl 83.23 103.88 122.58 138.52 152.03 163.61 173.63 182.35 211.82

3-methylbyt-1yn-3-yl 100.92 121.06 139.43 155.36 169.10 181.01 191.38 200.43 231.05

5-rings

cyclopenten-3-yl 82.11 110.88 135.66 155.80 172.21 185.84 197.33 207.12 239.15


cyclopent-1-en-4-yl 84.26 112.45 136.81 156.70 173.00 186.57 198.04 207.82 239.71

cylcopenta-2,4-dien-1-yl 79.34 103.41 123.17 138.68 151.01 161.08 169.50 176.64 200.09

cyclopenta-1,4-dien-1-yl 73.61 98.12 118.60 134.92 148.01 158.73 167.68 175.25 199.74

6-rings

cyclohexa-2,5-dien-1-yl 92.45 124.31 151.37 173.27 191.05 205.73 218.05 228.49 262.31

Kinetic modeling of scission and recombination...

Documents

Transcript of Kinetic modeling of scission and recombination...