KINETICS OF THE HYDROGEN ABSTRACTION PAH + •CH3•C2H5 → PAH RADICAL + CH4C2H6 REACTION CLASS

ABSTRACT The hydrogen abstraction in polyaromatic hydrocarbons PAHs by a methyl/ethyl radical are predicted with an application of the reaction class transition state theory RC-TST to ca

ĐẠI HỌC QUỐC GIA TP HỒ CHÍ MINH TRƯỜNG ĐẠI HỌC BÁCH KHOA



LÊ KHÁNH AN

KINETICS OF THE HYDROGEN ABSTRACTION

CÔNG TRÌNH ĐƯỢC HOÀN THÀNH TẠI TRƯỜNG ĐẠI HỌC BÁCH KHOA –ĐHQG -HCM

Cán bộ hướng dẫn khoa học : PGS TS Huỳnh kim lâm

Cán bộ chấm nhận xét 1 : TS Lê thanh hưng

Cán bộ chấm nhận xét 2 : phạm trần nguyên nguyên

Luận văn thạc sĩ được bảo vệ tại Trường Đại học Bách Khoa, ĐHQG Tp HCM ngày 29 tháng 12 năm 2015

Thành phần Hội đồng đánh giá luận văn thạc sĩ gồm:

(Ghi rõ họ, tên, học hàm, học vị của Hội đồng chấm bảo vệ luận văn thạc sĩ)

ĐẠI HỌC QUỐC GIA TP.HCM

TRƯỜNG ĐẠI HỌC BÁCH KHOA

CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM

Độc lập - Tự do - Hạnh phúc

NHIỆM VỤ LUẬN VĂN THẠC SĨ

Họ tên học viên: Lê Khánh An MSHV:11054126

Ngày, tháng, năm sinh: Nơi sinh: Long An

Chuyên ngành: Công nghệ hóa học Mã số : 60520101

I TÊN ĐỀ TÀI: KINETICS OF THE HYDROGEN ABSTRACTION

PAH + •CH3/•C2H5 → PAH RADICAL + CH4/C2H6 REACTION CLASS: AN

APPLICATION OF THE REACTION CLASS TRANSITION STATE THEORY (RC-TST)

II NHIỆM VỤ VÀ NỘI DUNG:

- Thực hiện tính toán động học phản ứng PAH + •CH3/•C2H5 → PAH RADICAL +

CH4/C2H6 bằng phương pháp RC-TST

- Đánh giá sai số phương pháp

- thiết lập mô hình tính toán đáng tin cậy cho họ phản ứng nêu trên

III NGÀY GIAO NHIỆM VỤ : 06/07/2015

IV NGÀY HOÀN THÀNH NHIỆM VỤ: 04/12/2015

V CÁN BỘ HƯỚNG DẪN : PGS.TS Huỳnh Kim Lâm.

ABSTRACT

The hydrogen abstraction in polyaromatic hydrocarbons (PAHs) by a methyl/ethyl radical are predicted with an application of the reaction class transition state theory (RC-TST) to calculate the rate constants reaction We have derived all parameters for the RC-TST method for this reaction class from rate constants of 24 representative reactions, coupling with linear energy relationships (LERs), so that rate constants for any reaction in this class can be described from its reaction energy calculated at either the barrier height grouping (BHG) or BH&HLYP/cc-pVDZ level of theory The RC-TST/LER thermal rate constants for selected reactions are in good agreement with those available in the literature Detailed analyses of the results show that the RC-TST/LER and RC-TST/BHG method is an efficient method for accurately estimating rate constants for a large number of reactions in this class

Key words: Rate constant; Hydrogen abstraction; Reaction class; Poly aromatic hydrocarbon

combustion;

TÓM T ẮT

Trong nghiên cứu này, chúng tôi tiến hành tính toán hằng số phản ứng của Phản ứng tách Hydro trong nhóm các pylyaromatic hydrocarbon (PAHs) bới gốc tự do methyl/ethyl bằng cách áp dụng phương pháp RC-TST (reaction class transition state theory) Các phản ứng được phân chia theo

24 phản ứng đại diện được tính toán dựa trên việc áp dung phương pháp RC-TST kết hợp với LERs (linear energy relationships), với mục đích tính toán được hằng số ơhanr ứng cho bất cứ phản ứng nào dựa trên việc tính toán năng lượng phản ứng theo lý thuyết BHG BH&HLYP/cc-pVDZ Hằng số nhiệt phản ứng teo tính toán RC-TST/LER cho một số phản ứng cụ thể cũng cho thấy sự phù hợp cao với dự đoán lý thuyết Các kết quả phân tích số liệu chi tiết cho thấy việc sử dụng RC-TST/LER và RC-TST/BHG là phương pháp hiệu quả để tính toán chính xác hằng số phản ứng đối với họ phản ứng này

On the other hand, I would like to send my special thanks to Mr Mai Van Thanh Tam from Institute for Computational Science and Technology (ICST), my mentor, for profound support and giving me the good chance to conduct and have my research study completed

Words cannot express my respect, love and deep gratitude for my parents, my husband and my baby who are always stand by me, encourage, and share their great love to me

Finally, I deeply thank you who spend your precious time to read this document I really appreciate for being got your acceptance for me as the fruit of my labor

CONTENTS

ABSTRACTS……… i

TÓM TẮT……… ii

ACKNOWLEDGEMENTS……… iii

LIST OF FIGURES v

LIST OF TABLES vi

ACRONYMS AND ABBREVIATIONS… vii

CHAPTER 1 INTRODUCTION 1

CHAPTER 2 METHODOLOGY 6

2.1 Transition State Theory 6

2.2 Reaction Class Reaction Transition State Theory 7

2.3 Computational Details 7

2.3.1 Electronic Structure Calculations 9

2.3.2 Rate calculations 9

CHAPTER 3.RESULTS AND DISCUSSION 11

3.1Reference Reactions 11

3.2 Reaction Class Parameters 12

Potential Energy Factor 13

Reaction symmetry number Factor 15

Tunneling Factor 17

Partition Function Factor 18

HR factor 20

3.3 Prediction Rate Constants 23

3.4 Error Analyses 28

CHAPTER 4 CONCLUSIONS 31

References 32

Supplementary Material… 34

LIST OF FIGURES

Figure 1: List of Polycyclic Aromatic Hydrocarbons (PAH) for this study with their IUPAC

nomenclature and short notation in this study 4

Figure 2: Linear energy relationship between reaction energyC

Figure 3: Plots of the tunneling ratio factors, f κ as function of temperature for abstraction of

hydrogen from CH3 (solid line) and C2H5 (dotted line) radicals… 16

Figure 4: Hydrogen abstraction from PAH by methyl radical to illustrate the “α” and “β sites” of

hydrogen abstracted on different “benzene” sites of PAH… 17

Figure 5: Partition function factors, f Q versus temperature for some reactions listed in Table 1 in

the temperature range of 300 – 3000 K (Only for abstraction by CH3 with α sites of PAH) 19

Figure 6: Effect A of the hindered rotation treatment on the total rate constants for reactions R2

– R16 (only for CH3 abstraction) in the temperature range of 300 – 3000 K………20

Figure 7: Arrhenius plots of the calculated rate constants using RC-TST method for two

representative hydrogen abstraction reactions along with the available literature values,

Hemelsoet 2006 [20] and Mati 2007 [56]: (a) C10H8 + ·CH3 → CH4 + 1*C10H7 (R2 – β site); (b)

C10H8 + ·CH3 → CH4 + 2*C10H7 (R3 - α site) ………24

Figure 8: The same as Fig 6 but for rate constants of reactions with (a) C14Anth + ·CH3 → CH4

+ 1*C14Anth (R6 – β site) and (b) C14Anth + ·CH3 → CH4 + 2*C14Anth (R7 - α site) ………25

Figure 9: Mean absolute errors (%) of the total relative rate factors f(T) (Eq 2) and its

components, namely the tunneling (f κ ), partition function (fQ),potential energy (fV) and hindered

rotation (fHR) factors as functions of the temperature 28

Figure 10: Relative absolute deviations as functions of temp erature between rate constants

calculated from explicit full RC-TST calculations for all selected reactions………… ……… 29

ACRONYMS AND ABBREVIATIONS

Transition State Theory Variational Transition State Theory Zero Point Vibrational Energy

CHAPTER 1

INTRODUCTION

Polycyclic aromatic hydrocarbons (PAHs) have received many studies for organic molecules [1,2] They play an important role in the formation of combustion-generated particles such as soot, and their presence in atmospheric aerosols has been widely studied [3], they are also key intermediate products in coal conversion processes [4-7] They can increase as side products in steam cracking units used in the petrochemical industry for the production of light alkenes such as ethane and propene [8] The understanding of the formation of PAH molecules is very important for the efficient design of clean and practical combustion devices Specially, an essential requirement for reliable modeling of PAH growth is the availability of accurate kinetic parameters In the PAH growth processes, various classes of elementary reactions such as hydrogen abstraction, addition, cyclization, and dehydrogenation can be distinguished, and these reactions led to form a surface consisting of conjugated rings [9,10] The kinetic models often consist of thousands of elementary reactions, therefore it can be impractical model to carry out all calculations of the thermal rates for every single reaction Among the existing methods, the Transition State Theory (TST) [11] is the simplest and most cost-effective, it only requires geometries, energies, and vibrational frequencies of the reactants and transition states However, the large size of PAH molecules limit the use of the accurate quantum calculations to obtain such information The experimental kinetic data for the reactions involving the PAH species are generally not available, especially over an extended range because of these potentially complex reaction processes A common practice is to approximate the unknown kinetic parameters by

those for similar reactions A better approach is to employ Evan – Polanyi linear free-energy [12,13] relationship between the activation energies and bond dissociation energies or heat of reaction of similar reactions to estimate the unknown activation energy

In 2000, Truong [14] has introduced the concept of RC-TST into both electronic and dynamic calculations This approach recognizes that reactions in a given class have the same reactive moiety and their potential energy surfaces along the reaction coordinate are very similar The Reaction Class Transition State Theory (RC-TST) has showed that the relative rate constants for any reaction in given a class can be predicted from only its energy by the use of the linear energy relationship between the classical barrier height and reaction energies determined from a subset of reactions in a class There are some studies on combustion of PAHs, they occur many importantly elementary reactions such as H-abstraction by H-atom [15-19] or by methyl radical [20,21]; O-addition [22]; OH radical and HO2 radical,…Hydrogen-abstraction reactions are ubiquitous in chemistry and biology and have been studied in such diverse areas as cosmology, combustion science, and the polymer industry For example, the initiation step in coke formation [10], an industrially important side process of thermal hydrocarbon cracking, is hydrogen abstraction [15,16,19]

The aim of this study is to apply the RC-TST to study the H-abstraction by methyl/ethyl radical from different PAHs These results use to estimate the rate onstants of any arbitrary reaction belonging to this class reaction It is done by first deriving the expression for rate constant of the reference reaction with those in a small representative set of the class from

explicit ab initio kinetic calculations of rate constants for all reactions in this representative set

The assumption is that these correlation expressions are applicable to all reactions in the considered class In our knowledge, this assumption has shown to be valid [14,18,23-27] To

develop the RC-TST/LER parameters for H-abstraction reactions of PAHs by methyl/ethyl radical, 24 reactions were selected to form the representative set, Figure 1 In this study, we apply the RC-TST/LER method to study hydrogen abstraction by methyl/ethyl radicals from different PAH of only benzene sites (due to the methyl/ethyl groups being significantly steric effects in TSs of PAH with other sites which makes them not obey LER method [20]) of six-membered ring compounds (H-abstraction reactions from the five-membered rings expected to discuss separately in another study) In particular, two classes of PAH were considered; one consists of PAH + CH3 radical and other is PAH + C2H5 radical class

Figure 1: List of Polycyclic Aromatic Hydrocarbons (PAH) for this study with their IUPAC

nomenclature and short notation in this study Sites where hydrogen abstraction is considered are numbered with (A) for linear acenes and (B) for nonlinear acenes model The sites are also labeled on the basis of hydrogen-abstraction classifications ( for benzene – C6 sites, for phenanthrene–C14phen sites, for antracene – C14Anth sites and for tetraphene – C18Tp sites

defined as Hemelsoet et al [20])

1

Benzene (C 6 H 6 - C 6 ) Naphthalene (C 10 H 8 - C 10 )

Acenaphthalene (C 12 H 8 - C 12 )

14 10 14

Phenanthrene (C 14 H 10 - C 14 Phen) Pyrene (C 16 H 10 - C 16 Py) Acephenanthrylene (C 16 H 10 - C 16 Phen)

Aceanthrylene (C 16 H 10 - C 16 Anth) Cyclopenta[cd]pyrene (C 18 H 10 - C 18 )

2 1

3 2

1

3

4

2 3

6 7 8 9

1 10

1 2

3

4

8 9

1 2

3 4 5

6 7 8

9 10

1 2

3 4 5

1 12

2 3 4 5

6 7 8 9

10 11

(A)

(B)

2 3 4

5

9 10 11 12

Table 1: List of reactions in the training set

Class 1: hydrogen abstraction by CH3

CHAPTER 2

METHODOLOGY

2.1 Transition State Theory

A theory of the rates of elementary reactions which assumes a special type of equilibrium, having an equilibrium constant , to exist between reactants and activated complexes According to this theory the rate constant is given by:

where is the Boltzmann constant and is the Planck constant The rate constant can also be expressed as:

where , the entropy of activation, is the standard molar change of entropy when the activated complex is formed from reactants and , the enthalpy of activation, is the corresponding standard molar change of enthalpy The quantities (the energy of activation) and are not quite the same, the relationship between them depending on the type of reaction Also:

where , known as the Gibbs energy of activation, is the standard molar Gibbs energy change for the conversion of reactants into activated complex A plot of standard molar Gibbs energy against a reaction coordinate is known as a Gibbs-energy profile; such plots, unlike potential-energy profiles, are temperature-dependent In principle the equations above must be multiplied by a transmission coefficient, , which is the probability that an activated complex forms a particular set of products rather than reverting to reactants or forming alternative products It is to be emphasized that , and occurring in the former three equations are not ordinary thermodynamic quantities, since one degree of freedom

in the activated complex is ignored.[23-25,57]

2.2.Reaction Class Reaction Transition State Theory

As we mentioned above, the details of the RC-TST approach have been presented in many studies [14,18,23-35], we only discuss its main features here The RC-TST is based on the transition state theory framework within the reaction class approach in describing the expression for relative rate constants There is available potential energy information for the principal reaction, often the smallest reaction in a given class, the rate constants for any other reaction in this class can be estimated from its barrier height and reaction energy The rate constant of an

arbitrary reaction (denoted as ka) is proportional to the rate constant of a reference reaction

(denoted as kr) by a temperature-dependent function f(T)

The rate constants for reference reaction are often known experimentally or to be calculated

accurately from the first principles The key idea of the RC-TST method is the factor f(T) which

can be described into different components under the TST framework:

Where f σ , f κ , fQ, fV and fHR are the symmetry number, tunneling, partition function, potential

energy and hindered rotation factors, respectively These factors are simply the ratios of the corresponding components in the TST expression for the arbitrary and reference reactions:

Where σ is the reaction symmetry number (degeneracy path), κ(T) is the transmission coefficient

accounting for the quantum mechanical tunneling effects, Q‡ and R are the total partition function of the transition state and reactants, ∆V‡ is the classical reaction barrier height, T is the

temperature in Kelvin and kB is the Boltzmann constant, respectively The potential energy factor can be calculated by using the barrier heights of the arbitrary reaction and the reference reaction which can be obtained using the linear energy relationship (LER) similar to the well-known Evans – Polanyi linear free energy relationship [36-38] between classical barrier heights and reaction energies of reactions The rate constants of any reactions in hydrogen abstraction from PAHs by methyl radical can be predicted from LER approach only using the reaction energy

2.3 Computational Details

2.3.1 Electronic Structure Calculations

All the electronic structure calculations were carried out by using GAUSSIAN 09 package [39] Hybrid nonlocal density functional theory (DFT), particularly Becke’s half-and-half [40] (BH&H) nonlocal exchange and the Lee-Yang-Parr (LYP) [41] nonlocal correlation functional, has been found to be sufficiently accurate for predicting the transition state

properties, e.g., barrier height and vibrational frequency, for hydrogen abstraction reactions by a radical [23,25,26,42-45] Equilibrium geometries of reactants, transition states and products were optimized at BH&HLYP level of theory with the Dunning’s correlation-consistent polarized valence double-ζ basis set [3s2p1d/2s1p] denoted as cc-pVDZ [46] Vibrational frequencies

calculated at the same level of theory were employed for the zero point energy (ZPE) correction

with the use of a scaled factor of 0.9506 suggested by Merrick et al [47] for BH&HLYP

method Normal mode analysis was performed at each stationary point to ensure its characteristics, i.e., stable structure with no imaginary vibrational frequency and the transition state (TS) structure has one imaginary vibrational frequency Intrinsic reaction coordinate (IRC) [48] calculations are carried out to confirm that the transition states are connecting the right minima between the reactants and the products These information were used to derive the RC-TST factors

2.3.2 Rate calculations

High-pressure limits of thermal rate constants were calculated using both the classical Transition State Theory(TST) with Eckart tunneling method [49] for all reactions in the representative set with the temperature range of 300–3000 K The hindered internal rotation is explicitly treated in the most accurate manner, by direct solving of the 1-D Schrödinger equation using the MSMC code [50] The overall procedure is detailed elsewhere, for example in our previous studies [51,52] Thermal rate constants are calculated in the temperature range of 300-

3000 K, which is sufficient for combustion applications

classified as Hemelsoet et al [20] (see

Table 1) Reaction R1& R17 were extensively studied in our previous study [21], its rate constants calculated with the CVT/SCT method for the temperature range of 300–3000 K were proven to agree well with available experimental data The CVT/SCT derived rate expression from the more accurate CCSD(T)/CBS level of theory for the reference reaction is as follows [21]:

3.2 Reaction Class Parameters

This section describes how the RC-TST factors were derived using the representative reaction set

Potential-Energy Factor

The figure 2 shows the excellent relationship between the BH&HLYP reaction energies and BH&HLYP classical barrier height (Linear Energy Relationship (LER) method) for H abstraction by methyl/ethyl radicals in the representative set This linear fit was obtained using the least square fitting method and has the following expression for the two reaction classes, namely:

∆V ≠ = 0.5446x ∆E BH&HLYP + 15.483 (kcal/mol) (10)

The reaction energies and barrier heights for all reactions in the representative set are given explicitly in Table 2 The absolute deviations of reaction barrier heights between the LER and the direct DFT BH&HLYP/cc-pVDZ calculations are smaller than 0.2 kcal/mol (see Table 2) The mean absolute deviation of reaction barrier heights predicted from BH&HLYP reaction energies is0.07 kcal/mol, (the % mean absolute deviation - %MAD of the potential energy factor for all reactions over the temperature range of 300-3000K for both two approaches is reported in Table S8) Thus, eq (10) is expected to give a good estimation of reaction barrier heights reactions analyzed and then used to calculate the potential energy factor using eq (6) This is certainly an adequate level of accuracy for kinetic modeling with an acceptable confidence Note that in the RC-TST/LER methodology only the relative barrier height is needed To compute these relative values, the barrier heights of the reference reaction R1/R17 calculated at the same level of theory, i.e BH&HLYP/cc-pVDZ, are needed and have the value of 19.56/21.78 kcal/mol, respectively (see Table 2) For the BHG approach, the average barrier heights are 19.57/21.79 kcal/mol with the maximum deviation of 0.43 kcal/mol for R16 and the mean absolute deviation of 0.12 kcal/mol which is larger than that of LER method However, the key

advantage of this approach is that it does not require any additional information to estimate the rate constants

In conclusion, the barrier heights for any reaction in the title reaction class can be obtained by using either the LER method or BHG approach The estimated barrier height is then used to calculate the potential energy factor using eq 6 The performance for such estimations of both approaches is discussed

in the error analyses section

Table 2: Classical reaction energies, barrier heights, and absolute deviations between the

calculated barrier heights from BH&HLYP/cc-pVDZ and those from LER expressions and barrier height grouping (BHG) approach Zero-point energy correction is not included Energies

relationship using BH&HLYP/cc-pVDZ or from barrier height grouping.e Mean absolute deviation (MAD) between the LER, BHG approach and the directly calculated values

Figure 2: Linear energy relationship between reaction energy, ∆E, calculated at

BH&HLYP/cc-pVDZ level of theory and reaction barrier heights, ∆V ≠, calculated at BH&HLYP/cc-pVDZ level

of theory for the reactions listed in Table 1 (zero-point corrections are not included)

Reaction symmetry number factor

The reaction symmetry number factors f σ were calculated simply from the ratio of reaction symmetry numbers of the arbitrary and reference reactions using eq 3 and are listed in Table 3 The reaction symmetry number of a reaction is given by the number of symmetrically equivalent reaction paths The symmetry numbers for R1 and R17 are same values of 12 In any case, this

value can be easily determined from the molecular topology of the reactant, thus the symmetry number factor can be calculated exactly

Table 3: Calculated symmetry and tunneling factors

Reaction Sym no

factor, f σ

Tunneling ratio factor, fκ(T = 300 K)

Eckartb Fittingc Deviationd % Deviatione

Tunneling Factor

The tunneling factor f κ is the ratio of the transmission coefficient of the reaction Ra to that

of reaction Rp (R1/R17) Due to the cancellation of errors in calculations of the tunneling factors,

it was showed that the factor f κ can be appropriately estimated using the one-dimension Eckart method [53] The calculated results for the representative reaction set can be then fitted to an analytical expression It is known that the Eckart tunneling coefficient depends on both the barrier height and the imaginary frequency Because the barrier heights are grouped into two classes, namely the abstraction by methyl and ethyl radicals (see the potential energy factor section), and the imaginary frequency for these reactions at the same class are very similar (see Table S1 in the Supporting Information), for this reason the values of the tunneling coefficients for the reactions in the same class are expected to be similar Thus, the average value for the tunneling factors can be used for the whole group Simple expressions for the two tunneling factors for abstraction by methyl and ethyl radicals are obtained by fitting to the average calculated values are shown in Figure 3 and were fitted to analytical expressions as given below:

Figure 3: Plots of the tunneling ratio factors, f κ as function of temperature for abstraction of hydrogen from CH3 (solid line) and C2H5 (dotted line) radicals

Table 3 also lists the error analysis of the tunneling factors at 300 K The same tunneling factor expression can be reasonably assigned to different reactions in the same class with the largest unsigned deviation of 0.08 for R9, R11, R23 and the largest percentage deviation of 8.52% The mean unsigned deviation is 4.06%, compared to the direct Eckart calculation using reaction information from BH&HLYP/cc-pVDZ level At higher temperature, the tunneling contributions to the rate constants decrease and thus, as expected, the differences between the approximated values and the explicitly calculated ones also decrease; for instance, the mean unsigned deviation for all selected reactions is less than 1% at 500 K (c.f Table S3)

Partition Function Factor

H H

H

C

C

H H

to the difference of the reactive moiety for two positions on the considered “benzene sites” as well as the planarity of CH3 radical It is easily to find that this characteristic has more sufficient effect on the transition state structure abstracted by CH3 radical than that of C2H5 radical (c.f Figure S2) Thus, for the abstraction by C2H5, it should be not assigned into the small subclass From this consideration, the Class 1 with α and β sites includes the R3, R5, R7, R10, R11, R13,

R14, R16 and R2, R4, R6, R8, R9, R12, R15, respectively Note that the contributions from the

HR modes are treated separately, and they are not included in these partition factors The

temperature-dependent partition function factor is averaged over all reactions of a given subclass and expected not to depend strongly on temperature The partition function factors for the whole class, calculated over the temperature range of 300-3000 K, are plotted in Figure S2 and almost constant For the sake of simplicity, these values can be effectively fitted as:

f = (12c) for abstraction by C2H5 radical

Figure 5 depicts the temperature-dependent of fQ for some reactions (only for abstraction

by CH3 with α sites of PAH) reported in Table 1 This figure demonstrates that the ratios are nearly constant for the reactions in this subclass The percentage deviation for these simplicities

is 15.44%, compared to the direct calculation using reaction information from DFT theory at 300

K (c.f Table S5) At higher temperature, this deviation to the rate constants decreases; for example, the %MAD for all reactions is less than 12.5% at 1000 K (c.f Table S5)

Figure 5: Partition function factors, f Q versus temperature for some reactions listed in Table 1 in the temperature range of 300 – 3000 K (Only for abstraction by CH3 with α sites of PAH)

HR factor

In the hydrogen abstraction of PAHs, there are rotations of the alkyl (for example, CH3)

or alkanyl (CH2) groups along the C-C bond in the transition states, reactants (for C2H5 radical) and products (for C2H6) need to be treated as hindered rotors We used the approach proposed by

East et al [54] and Kilpatrick et al [55] The reaction class factor due to these hindered rotors is

a measure of the substituent effects on the rate constants from these hindered rotors relative to that of the reference reaction The effect of the hindered rotation treatment to total rate constants

can be seen in Figure 6 for CH3 abstraction, all data for HR factors can be found in Table S6 &

Figure S3 It can be easily seen from Figure 6 that the HR correction factors are dependent on

the temperature The average values at temperatures below 1000 K increase, whereas, for T >

1000 K the factor nearly become constant, with the value >1.5 and significant raise as the complexity of the C-ring

Figure 6: Effect of the hindered rotation treatment on the total rate constants for reactions R2 –

R16 (only for CH3 abstraction) in the temperature range of 300 – 3000 K

The hindered rotation factor is divided into a separate part for computing At the temperature range of 300-3000K, the plot of HR correction shows the result almost constant as the temperature increase, these value can be fitted by the following equation:

Table 4: Parameters and formulations of RC-TST method for the PAH + •CH3/•C2H5 → PAH

radical + CH4/C2H6 (C6H6 + •CH3/•C2H5 → C6H5• + CH4/C2H6 are the reference reactions)

k (T) = f σ × f κ (T) × fQ(T) × fV(T) × fHR(T)×kr(T); fV(T) = exp[-( ∆V ≠ - ∆V ≠ r)/kBT]

T is in Kelvin; ∆V ≠ and ∆E are in kcal/mol (zero-point energy correction is not included)

f σ Calculated explicitly from the symmetry of reactions (see Table 3)

1.000 66.595 exp− × −1.096×T , for abstraction by CH3 (Class 1)

0.999 75.211 exp− × −0.929×T , for abstraction by C2H5 (Class 2)

BHG 19.64 kcal/mol for abstraction by CH3 (Class 1)

21.79 kcal/mol for abstraction by C2H5 (Class 2)

∆V ≠ r 19.56 kcal/mola for abstraction by CH3 (Class 1)

21.78 kcal/mola for abstraction by C2H5(Class 2)

3.3 Prediction of Rate Constants

All the factors have established so far are the necessary parameters, namely, symmetry number factor, tunneling factor, partition function factor, correction factor for hindered rotations, and potential energy factor, for application of the RC-TST theory to predict rate constants for any reaction in the CH3/C2H5 + PAHs class By combining it with the linear relationship between the reaction energy and the reaction barrier height, only the reaction energy is needed, and it can

be obtained at the BH&HLYP/cc-pVDZ level of theory The procedure for calculating rate constants of an arbitrary reaction in this class is shown in Table 4 The symmetry number and tunneling factor had calculated previously in Table 3 Parts a–b of Fig 7-8 show the Arrhenius plots of the calculated rate constants using the RC-TST/LER method for the hydrogen abstraction reaction of naphthalene and anthracene at α and β carbon sites In the figures, the

“RC-TST exact” notation means that the reaction class factors were calculated explicitly within the TST/Eckart framework rather than using the approximate expressions listed in Table 4 The RC-TST/LER and RC-TST/BHG rate constants for the reactions agree very well with available data in the Note that the accuracy of the RC-TST rate constants depends on several factors At the fundamental level, it depends on the validity of the transition state theory approximations on which the RC-TST method is based and the semi-classical multidimensional tunneling approximations which are used for the reference (or principal) reaction In addition, it depends

on the accuracy of all approximations that were introduced so that explicit calculations of the transition state structure and frequency are not required The related errors will be referred to as systematic errors and are discussed below Figs 7a and 7b show Arrhenius plots of the calculated and available total rate constants for reactions (a) C10H8 + ·CH3 → CH4 + 1*C10H7

(R2 – β site); (b) C10H8 + ·CH3 → CH4 + 2*C10H7 (R3 - α site) This will provide some

information on the predictive capability of the RC-TST methodology The rate constants obtained by the RC-TST approach with both LER and BHG approximations are in good agreement with available data, particularly within the uncertainty of experimental as well as reviewed data which is given in Arrhenius form as

[for abstraction by CH3 with α sites.]

We referred to this approximation as statistical estimation Total rate constants calculated

by this approximation are also shown in Table 4.The accuracy of this approach is comparable to that of available data in the literature

A more systematic analysis of the efficiency of the RC-TST method would be to compare the RC-TST results with explicit theoretical calculations Comparisons between the calculated rate constants for a small number of reactions using both the RC-TST and full TST/Eckart methods provide additional information on the accuracy of the RC-TST method The difference between the LER and BHG is insignificant Relative errors are less than 20% for all reactions This is certainly an acceptable level of accuracy for reaction engineering purposes For most of the reactions, the unsigned relative errors are within 20%; thus it can be concluded that RC-TST can generally estimate thermal rate constants for reactions in this class when compared to those calculated explicitly using the TST/Eckart method

Figure 7: Arrhenius plots of the calculated rate constants using RC-TST method for two

representative hydrogen abstraction reactions along with the available literature values, Hemelsoet 2006 [20] and Mati 2007 [56]: (a) C10H8 + ·CH3→ CH4 + 1*C10H7 (R2 – β site); (b)

C10H8 + ·CH3→ CH4 + 2*C10H7 (R3 - α site)

Figure 8: The same as Fig 6 but for rate constants of reactions with (a) C14Anth + ·CH3→ CH4

+ 1*C14Anth (R6 – β site) and (b) C14Anth + ·CH3→ CH4 + 2*C14Anth (R7 - α site)

3.4 Error Analyses

In this section, the analysis is on the systematic errors in the factors in the RC-TST method Fig.9 a-b shows the averaged errors introduced by approximations in calculating tunneling, partition function, hindered rotation, and potential energy factors for reactions in the 2 fitting: BHG and LER The total error is affected by the errors in the approximations in individual factors introduced in the method Note that the symmetry number factor is exact The error in the partition function factor does include the error in the approximation for hindered rotation treatment The deviations between the approximated and the explicit TST/HR/Eckart calculations are calculated at each temperature for every reaction in the training set and then averaged over the whole class Of the individual factors, the error introduced by the partition function factor is the largest, roughly around 15% (c.f Table S7) for the whole temperature range The LER approach gives similar error in the potential energy and symmetric factor as the BHG method The overall performance of the BHG approach can be explained by cancellation of errors in the different approximations Thus, for this reaction class BHG is recommended since it does not required any additional information The total systematic errors for both LER and BHG approach less than 20% (c.f Figure 10) for the temperature range from 300 to 3000 K For the temperatures >2000 K, most of the reactions in this set (R2-R16), the absolute relative errors are within 60% In the low temperatures regime, five reactions have errors various and larger than 60% In the other group (R18-R24), at the temperature >1000K the errors show stable around 80%, whereas at lower temp, the error slightly higher – nearly 100% So, in general, it can be concluded that RC-TST can estimate thermal rate constants for reactions in this class are different between 2 groups CH3 (within 60%) and C2H5 (around 80%) when compared to those calculated explicitly using the TST/HR/Eckart method with the electronic information from BH&HLYP/cc-pVDZ level of theory As expected, these differences are only minor

Figure 9: Mean absolute errors (%) of the total relative rate factors f(T) (Eq 2) and its

components, namely the tunneling (f κ ), partition function (f Q ),potential energy (f V) and hindered

rotation (f HR) factors as functions of the temperature

Finally, an analysis on the systematic errors in different factors in the BHG/LER methods was performed These errors are from the use of fitted analytical expressions for the potential energy factor, tunneling factor, partition function factor and hindered rotation factor introduced

in the method The deviations/errors between the approximated and exact factors within the BHG framework are calculated at each temperature for every reaction in there presented set and then averaged over the whole class For the LER approach, error in the potential energy factor comes from the use of the LER expression: that of the tunneling factor, from using two equations (eqs 11a-b); that of the partition function factor, from using eqs 12a-c; and that of the HR factor from using eqs 13a-c Absolute errors averaged over all 22 reactions, R2-R16 and R18-R24, as functions of the temperature are plotted in Figure 10 Of the factors, the tunneling factor show the least temperature dependence for the whole temperature range whereas the HR factor show

an odd line compared to all the others The tunneling factor also introduced the smallest error of less than 5% in the low temperature regime and almost equal to 0 for T > 800 K The error from the potential energy factor (BHG) is largest for T < 500 K, but still < 20% for the whole temperature range For T > 1000 K, the total errors from both the LER and the BHG approach are almost constant For most cases, the total systematic errors due to the use of simple analytical expressions for different reaction class factors are less than 10% for the temperature range 300-3000K For the LER and BHG approaches, these errors are not to exceed 20% In general, the results show very good precision

Figure 10: Relative absolute deviations as functions of temperature between rate constants

calculated from explicit full TST calculations for all selected reactions: (a) From the TST/LER method where BH&HLYP/cc-pVDZ reaction energies were used for the LER; (b) From the RC-TST/BHG approach (solid lines for abstraction by CH3 and dotted lines for abstraction by C2H5)

RC-CHAPTER 4

CONCLUSIONS

This work has extended the application of the reaction class transition state theory TST) combined with the linear energy relationship (LER) and the barrier height grouping (BHG) approaches to the prediction of thermal rate constants for hydrogen abstraction reactions for the

(RC-CH3/C2H5 + PAHs class Combined with the rate constants expressions for the reference reactions, C6H6+CH3/C2H5, obtained from our previous study [21]; the RC-TST/LER, where only reaction energy is needed, and RC-TST/BHG, where no other information is required, are both found to be promising methods for predicting rate constants for any reaction in a given reaction class The error analysis indicates that when compared to explicit rate calculations, the averaged systematic errors in the calculated rate constants using either RC-TST/LER or RC-TST/BHG methods are less than 20% over the temperature range 300-3000 K In addition, it was found that the estimated rate constants using either LER or BHG approach are in good agreement with available data in the literature