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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 • Petra.Vereecken@UGent.be
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: Jonas.vanbelleghem@ugent.be
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 à 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.
4.Sabbe, M.K., et al., ChemPhysChem, 2010. 11(1): p. 195-
210.
5.Sabbe, M.K., et al., ChemPhysChem, 2010. 11(1): p. 195-
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: Jonas.vanbelleghem@ugent.be
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 à 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.
3.Sabbe, M.K., et al., ChemPhysChem, 2008. 9(1): p. 124-
140.
4.Sabbe, M.K., et al., ChemPhysChem, 2010. 11(1): p. 195-
210.
5.Sabbe, M.K., et al., ChemPhysChem, 2010. 11(1): p. 195-
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
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. ....................................................................................................... 112
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.
List of Figures vii
Green dots are simulation results obtained with Reaction Network 3, i.e. the network
generated with RMG 3.0. ....................................................................................................... 114
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
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. ....................................................................................................... 117
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
, 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
. ............................................................................................................................. 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.
Literature Review 11
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
Literature Review 12
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.
Literature Review 13
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
Literature Review 14
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
Literature Review 15
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
Literature Review 16
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:
Literature Review 17
[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
Literature Review 18
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]
Literature Review 19
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]
Literature Review 20
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
Literature Review 21
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,
Literature Review 22
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.
Literature Review 23
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.
Literature Review 24
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
Literature Review 25
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
Literature Review 26
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
Literature Review 27
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.
Literature Review 28
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.
Literature Review 29
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.
Literature Review 30
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) [-]
Literature Review 31
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
Literature Review 32
(
)
( ) (
) [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
Literature Review 33
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:
Literature Review 34
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
Literature Review 35
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
Literature Review 36
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]):
Literature Review 37
(
) (
)
( ) [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
Literature Review 38
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
Literature Review 39
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
Literature Review 40
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
Literature Review 41
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.
Literature Review 42
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.
Literature Review 43
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.
Literature Review 44
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
Literature Review 45
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.
Figure 2–7: Temperature dependence of
.
-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)
Literature Review 46
Figure 2–8: Temperature dependence of
.
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
Method selection to study recombination reactions 48
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
Method selection to study recombination reactions 49
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
Method selection to study recombination reactions 50
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
Method selection to study recombination reactions 51
[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
Method selection to study recombination reactions 52
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.
Method selection to study recombination reactions 53
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.
Method selection to study recombination reactions 54
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.
Method selection to study recombination reactions 55
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
Method selection to study recombination reactions 56
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
Method selection to study recombination reactions 57
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
Method selection to study recombination reactions 58
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
Method selection to study recombination reactions 59
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.
Method selection to study recombination reactions 60
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
Method selection to study recombination reactions 61
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
Method selection to study recombination reactions 62
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
Method selection to study recombination reactions 63
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.
Method selection to study recombination reactions 64
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
Recombination reactions involving hydrocarbons 66
Alkenes
Alkynes
5-rings
6-rings
Recombination reactions involving hydrocarbons 67
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
Recombination reactions involving hydrocarbons 68
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
Recombination reactions involving hydrocarbons 69
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
Recombination reactions involving hydrocarbons 70
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,
Recombination reactions involving hydrocarbons 71
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.
Recombination reactions involving hydrocarbons 72
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.
Recombination reactions involving hydrocarbons 73
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
Recombination reactions involving hydrocarbons 74
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
Recombination reactions involving hydrocarbons 75
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,
Recombination reactions involving hydrocarbons 76
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.
Recombination reactions involving hydrocarbons 77
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.
Recombination reactions involving hydrocarbons 78
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.
Recombination reactions involving hydrocarbons 79
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
Recombination reactions involving hydrocarbons 80
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
Recombination reactions involving hydrocarbons 81
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.
Recombination reactions involving hydrocarbons 82
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.
Recombination reactions involving hydrocarbons 83
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.
Recombination reactions involving hydrocarbons 84
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
Recombination reactions involving hydrocarbons 85
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
Recombination reactions involving hydrocarbons 86
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-
Recombination reactions involving hydrocarbons 87
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.
Recombination reactions involving hydrocarbons 88
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
Recombination reactions involving hydrocarbons 89
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
Recombination reactions involving hydrocarbons 90
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
Recombination reactions involving hydrocarbons 91
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
Determination of group additivity values 93
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
.
Determination of group additivity values 94
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
Determination of group additivity values 95
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
Determination of group additivity values 96
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.
Determination of group additivity values 97
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
Determination of group additivity values 98
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
Determination of group additivity values 99
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
.
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
.
Determination of group additivity values 100
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
.
Determination of group additivity values 101
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.
Determination of group additivity values 102
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
, 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.
Determination of group additivity values 103
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
.
Determination of group additivity values 104
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
, 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.
Determination of group additivity values 105
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
.
Determination of group additivity values 106
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
Modeling steam cracking of ethane and n-butane 108
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.
Modeling steam cracking of ethane and n-butane 109
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.
Modeling steam cracking of ethane and n-butane 110
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
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pre
dic
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ld [
-]
Experimental yield [-]
Ethane parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
Modeling steam cracking of ethane and n-butane 111
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 [
-]
Experimental yield [-]
H2 parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
Modeling steam cracking of ethane and n-butane 112
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
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ld [
-]
Experimental yield [-]
Ethyne parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
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 [
-]
Experimental yield [-]
Propene parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
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 [
-]
Experimental yield [-]
1,3-butadiene parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
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 [
-]
Experimental yield [-]
n-butane parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
Modeling steam cracking of ethane and n-butane 113
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.
Modeling steam cracking of ethane and n-butane 114
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
obtained with Reaction Network 3, i.e. the network generated with RMG 3.0.
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
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pre
dic
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ld [
-]
Experimental yield [-]
n-butane parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
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 [
-]
Experimental yield [-]
Propene parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
0
0,1
0,2
0,3
0 0,1 0,2 0,3
Ab
in
itio
pre
dic
ted
yie
ld [
-]
Experimental yield [-]
Ethene parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
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 [
-]
Experimental yield [-]
Methane parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
Modeling steam cracking of ethane and n-butane 115
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.
Modeling steam cracking of ethane and n-butane 116
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
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ld [
-]
Experimental yield [-]
Ethane parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
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 [
-]
Experimental yield [-]
1-butene parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
0
0,01
0,02
0,03
0,04
0,05
0 0,01 0,02 0,03 0,04 0,05
Ab
in
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pre
dic
ted
yie
ld [
-]
Experimental yield [-]
1,3-butadiene parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
Modeling steam cracking of ethane and n-butane 117
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
obtained with Reaction Network 3, i.e. the network generated with RMG 3.0.
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
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ld [
-]
Experimental yield [-]
H2 parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
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 [
-]
Experimental yield [-]
Ethyne parity plot
Reaction Network 1 Reaction Network 2 Reaction Network 3
Modeling steam cracking of ethane and n-butane 118
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
Conclusion and future work 120
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.
Conclusion and future work 121
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
Conclusion and future work 122
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
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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
Reaction network 129
H
H
H
H
H
H
H
Reaction network 130
H
H
H
H
H
H
H
H
H
H
Reaction network 131
H
H
H
H
H
Reaction network 132
H
H
H
Reaction network 133
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
W1bd calculations 135
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
W1bd calculations 136
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
W1bd calculations 137
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
W1bd calculations 138
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