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Theoretical rate constant of methane oxidation from the conventional transition-state theory

  • Claudia Aranda
  • Arlette Richaud
  • Francisco MéndezEmail author
  • Armando Domínguez
Original Paper
Part of the following topical collections:
  1. International Conference on Systems and Processes in Physics, Chemistry and Biology (ICSPPCB-2018) in honor of Professor Pratim K. Chattaraj on his sixtieth birthday

Abstract

The potential energy surface for the first step of the methane oxidation CH4 + O2➔CH3 + HO2 was studied using the London-Eyring-Polanyi-Sato equation (LEPS) and the conventional transition-state theory (CTST). The calculated activation energy and rate constant values were in good agreement with the experimental and theoretical values reported in the literature using the shock tube technique and coupled cluster method respectively. The rate equation from CTST, although simple, provides good results to study the H-shift between methane and the oxygen molecules.

Keywords

Methane Oxidation Potential energy surface Reaction path 

Introduction

Methane (CH4) plays critical roles in atmospheric and combustion chemistry. It is the second most abundant anthropogenic greenhouse gas in the atmosphere after carbon dioxide (CO2), and it is the main constituent in natural gas (over 90%) that is used as a fuel to obtain energy by combustion [1, 2]. The enthalpy of combustion of methane is the most important property used in the determination of the calorific value of natural gas [3]. Therefore, the combustion of methane (Eq. 1) has been widely studied experimentally and theoretically (shock tube technique, molecular dynamics simulations, quantum mechanical calculations, etc.) [4, 5, 6].
$$ C{H}_4+2{O}_2\to C{O}_2+2{H}_2O $$
(1)
Previous studies have revealed that the first step of the mechanisms of combustion begins with the shift of a hydrogen atom from methane to the oxygen molecule with the concomitant formation of methyl and perhydroxyl radicals (Eq. 2) [6, 7, 8, 9, 10, 11].
$$ C{H}_4+{O}_2\overset{k}{\to }C{H}_3+H{O}_2 $$
(2)
In order to determine the rate constant k of the chemical reaction in Eq. 2, Srinivasan et al. [6] made ab initio calculations to characterize the key points on the potential energy surface for the abstraction from the triplet state (reagents, transition state (TS), and products) along with a variational transition state theory (VTST) treatment of the kinetics. Their calculations were done with the open shell, unrestricted, coupled cluster method, CCSD(T), using the Dunning basis sets (aug-cc-pvdz and aug-cc-pvtz). They found that the reaction in Eq. 2 takes place through a transition state and an intermediate (see Fig. 1). The relative energy of the transition state with respect to the reagents was 229.4 kJ mol−1, whereas the experimental value was 231.86 ± 0.29 kJ mol−1. Table 1 shows some expressions of the rate constant in terms of the temperature for the reaction of Eq. 2. The values are in molecular units; multiplication by Avogadro number (6.022 × 1023 mol−1) gives molar units [6, 12, 13, 14, 15, 16].
Fig. 1

Methane oxidation reaction path [6]

Table 1

Rate constant expressions in terms of temperature for \( C{H}_4+{O}_2\overset{k}{\to }C{H}_3+H{O}_2 \)

k (cm3 molecule−1 s−1)

1.33 × 10−10exp(−28,200 K/T)a [12]

1.42 × 10−17 T2exp(−26,197 K/T)a [13]

6.59 × 10−11exp(−28,597 K/T)a [14, 15]

8.1 × 10−19 T2.5exp(−26,370 K/T)a [16]

3.37 × 10−19 T2.745exp(−26,041 K/T)b [6]

aExperimental range 200–2000 K

bExperimental range 1655–1822 K

In this work we studied the first step of methane oxidation (Eq. 2) using the conventional transition-state theory (CTST) and potential energy surface (PES) calculated with the London-Eyring-Polanyi-Sato (LEPS) equation [17]. The energy and rate constant values calculated by CTST for the methane oxidation were in good agreement with the experimental and theoretical values reported in the literature using the shock tube technique and coupled cluster method respectively. The CTST supports the chemical kinetic theory, and it is relative easy to apply; other theories of rates need much more effort and quantity of time by several orders of magnitude. In this case the relative ease of use of CTST allowed us a great saving of computing time compared to the coupled cluster method. Although the CTST was developed for a three-body interaction, it was possible to apply it to the methane oxidation reaction involving more than three bodies in the transition state.

The LEPS equation and the CTST have been discussed widely elsewhere [17]. Briefly, the interaction between reagents can be viewed as a three-body problem and the PES, in which potential energy is plotted in terms of bond distances and angles [18]. In the substitution reaction three atoms are involved:
$$ A+B-C\to {\left(A\cdots B\cdots C\right)}^{\ne}\to A-B+C $$
(3)

The energy of the transition state (ABC) corresponds to the energies of the three diatomic molecules AB, BC, and AC, if they were isolated, and is described in the LEP formulation (Eq. 4) in terms of coulomb (Q) and interchange (J) energies [17, 19]:

$$ {E}_{LEP}={Q}_{AB}+{Q}_{BC}+{Q}_{AC}\pm {\left\{\frac{1}{2}\left[{\left({J}_{AB}-{J}_{BC}\right)}^2+{\left({J}_{BC}-{J}_{AC}\right)}^2+{\left({J}_{AC}-{J}_{AB}\right)}^2\right]\right\}}^{\frac{1}{2}} $$
(4)

Using the Heitler-London equations for the attractive and repulsive curve for a diatomic molecule (Eq. 5), the Morse equation for the attractive potential energy curve for a diatomic molecule (Eq. 6), an antibonding states function (Eq. 7), and an adjustable constant (K) to replace the overlap integral (S), Sato [20] proposed empirical approximations for Q and J for a diatomic molecule A − B shown in Eqs. 8 and 9.

$$ {E}_{HL}=\frac{Q_{AB}\pm {J}_{AB}}{1+{S}^2} $$
(5)
$$ {E}_M={D}_e\left\{\mathit{\exp}\left[-2\beta \left(R-{R}_e\right)\right]-2\mathit{\exp}\left[-\beta \left(R-{R}_e\right)\right]\right\} $$
(6)
$$ {E}_{aM}={D}_e/2\left\{\mathit{\exp}\left[-2\beta \left(R-{R}_e\right)\right]+2\mathit{\exp}\left[-\beta \left(R-{R}_e\right)\right]\right\} $$
(7)
$$ {Q}_{AB}=\frac{1}{2}\left\{{E}_{AB}^M\left(1+{K}_{AB}\right)+{E}_{AB}^{aM}\left(1-{K}_{AB}\right)\right\} $$
(8)
$$ {J}_{AB}=\frac{1}{2}\left\{{E}_{AB}^M\left(1+{K}_{AB}\right)-{E}_{AB}^{aM}\left(1-{K}_{AB}\right)\right\} $$
(9)
where De is the classical dissociation energy, Re is the internuclear distance at equilibrium, β is an interaction constant, and R is the internuclear distance. The Sato correction to the LEP (LEPS) formulation is [20, 21].
$$ {E}_{LEPS}=\frac{Q_{AB}}{1+{K}_{AB}}+\frac{Q_{BC}}{1+{K}_{BC}}+\frac{Q_{AC}}{1+{K}_{AC}}-{\left\{\frac{1}{2}\left[{\left(\frac{J_{AB}}{1+{K}_{AB}}-\frac{J_{BC}}{1+{K}_{BC}}\right)}^2+{\left(\frac{J_{BC}}{1+{K}_{BC}}-\frac{J_{AC}}{1+{K}_{AC}}\right)}^2+{\left(\frac{J_{AC}}{1+{K}_{AC}}-\frac{J_{AB}}{1+{K}_{AB}}\right)}^2\right]\right\}}^{\frac{1}{2}} $$
(10)
The activation energy (EC) is obtained from Eqs. 610, and EC is used in the Arrhenius formula to obtain the rate constant value (Eqs. 11 and 12):
$$ k= Aexp\left(-\raisebox{1ex}{${E}_C$}\!\left/ \!\raisebox{-1ex}{$R\mathrm{T}$}\right.\right) $$
(11)
$$ k=\frac{k_BT}{h}\ \frac{q_{\ne }}{q_A{q}_B}\mathit{\exp}\left(-\raisebox{1ex}{${E}_C$}\!\left/ \!\raisebox{-1ex}{$R\mathrm{T}$}\right.\right) $$
(12)

In order to calculate the pre-exponential factor value (Eqs. 11 and 12), it is necessary to know the partition functions for the transition state (q) and reagents (qA, qB). The partition functions depend on the temperature and the energy levels of the system, and then, the accuracy of the rate constant highly depends on the partition functions values. Note that only partition functions for translation, rotation, and vibration are considered [17].

Methods

Equation 13 shows the corresponding transition state for the system, Eq. 2:

$$ C{H}_4+{O}_2\to {\left[C{H}_3\cdots H\cdots {O}_2\right]}^{\ne } $$
(13)
In order to calculate the energy in terms of the internuclear distance R (Eqs. 67), we used the parameters of LEPS PES for De, Re and β reported in the literature, see Table 2 [22, 23, 24, 25]. To calculate the β parameter (cm−1), we used [19]: β = (2π2/Deh)1/2ωe, where c (speed of light) in m s−1, μ (reduced mass) in kg, De (bond dissociation energy) in m−1 (1 cm−1 = 1.1963 × 10−2 kJ mol−1), h (Planck constant) in J s, and ωe (vibrational frequency) in cm−1. Commonly, β is expressed in Å-1 in the Morse equation. Then, we adjusted the distance H3C − H given by Srinivasan et al. (2007) from 1.1 to 1.12 Å, and found that the angle H3C − H − OO is equal to 165.67°. The LEPS PES (Eq. 10) was built to localize the transition state energy (E), and the activation energy is obtained from EC = E − ER.
Table 2

Parameters used in the LEPS potential energy surface of the CH4 + O2 → CH3 + HO2

Parameter

CH3 − H

H − O2

CH3 − O2

De (kcal/mol)

107.0a

54.6b

33.7d

\( {r}_e\ \left(\overset{{}^{\circ}}{A}\right) \)

1.093a

0.971b

1.449d

ωe(cm−1)

3436b

902e

\( \beta\ \left({\overset{{}^{\circ}}{\mathrm{A}}}^{-1}\right) \)

1.83a

3.235c

3.156c

aTaken from ref. [22]

bTaken from ref. [23]

cCalculated by us

dTaken from ref. [24]

eTaken from ref. [25]

Three partition functions (q, qA and qB) are required for each participant species to find the pre-exponential factor (Eq. 11). The determination of translation and vibration partition functions is trivial for all species, see Table 3. However, because methane and the transition state are nonlinear molecules, it is more complicated to calculate their rotation partition functions due to their inertia moments IA,IB,IC, see Table 3. The inertia moments were obtained from the Knox’s approximation [26] by means of a 3 × 3 determinant of inertial moment products (Eq. 14).
$$ {I}_A{I}_B{I}_C=\left|\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ {}-{I}_{xy}& {I}_{yy}& -{I}_{yz}\\ {}-{I}_{xz}& -{I}_{yz}& {I}_{zz}\end{array}\right| $$
(14)
Table 3

The partition functions used to solve Eq. 12

Molecule

Partition function

qt (m−3)

qr (dimensionless)

qv (dimensionless)

O2

\( \frac{{\left(2\pi m{k}_BT\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{h^3} \)

\( \frac{8{\pi}^2I{k}_BT}{\sigma^{\prime }{h}^2} \)

\( \frac{1}{\left(1-{e}^{-\frac{hv}{k_BT}}\right)} \)

CH4

\( \frac{{\left(2\pi {m}_{C{H}_4}{k}_BT\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{h^3} \)

\( \frac{8{\pi}^2{\left(8\pi {I}_A{I}_B{I}_C\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left({k}_BT\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\sigma^{\prime \prime }{h}^3} \)

\( \frac{1}{\left(1-{e}^{-\raisebox{1ex}{$h{v}_1$}\!\left/ \!\raisebox{-1ex}{${k}_BT$}\right.}\right)\cdots \left(1-{e}^{-\raisebox{1ex}{$h{v}_4$}\!\left/ \!\raisebox{-1ex}{${k}_BT$}\right.}\right)} \)

[CH3···H···O2]

\( \frac{{\left(2\pi {m}_{ET}{k}_BT\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{h^3} \)

\( \frac{8{\pi}^2{\left(8{\pi}^3{I}_A{I}_B{I}_C\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left({k}_BT\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\sigma^{\prime \prime \prime }{h}^3} \)

\( \frac{1}{\left(1-{e}^{-\raisebox{1ex}{$h{v}_1$}\!\left/ \!\raisebox{-1ex}{${k}_BT$}\right.}\right)\cdots \left(1-{e}^{-\raisebox{1ex}{$h{v}_{14}$}\!\left/ \!\raisebox{-1ex}{${k}_BT$}\right.}\right)} \)

\( {\displaystyle \begin{array}{l}{\sigma}^{\prime }=2,{\sigma}^{\prime \prime }=12,{\sigma}^{\prime \prime \prime }=1\\ {}\end{array}} \)

First, to obtain inertia moments, we obtain Cartesian coordinates for methane through spherical coordinates [27]. Then, three different transition state (TS) geometries (see Fig. 2 and Table 4) were considered to study the CTST sensibility: the first-one (TS1) is the geometry reported in ref. [6], obtained at the CCSD(T)/aug-cc-pvdz level of theory, the second-one (TS2) is the geometry calculated for us at the B3LYP/6-311+G(3df,2p)(5d,7f) level of theory using Gaussian 09 [28], and the third-one (TS3), a naive proposal geometry. The CTST calculations were performed with Matlab 7 [29].
Fig. 2

Schematic representation of transition state geometry of Eq. 13 under three distributions: i) TS1, from CCSD(T) method, ii) TS2, from B3LYP method, iii) TS3, a naive proposal geometry

Table 4

Cartesian coordinates of transition states utilized for the determination of the rate constant of Eq. 2: i) TS1, from the CCSD(T) method, ii) TS2, from the B3LYP functional calculated for us, and iii) TS3, a naive proposal geometry

Atom

TS1 (Å)

TS2 (Å)

TS3 (Å)

C

(−1.3674,1.1270,0)

(0.9805,1.1860,−0.2960)

(0,0,0)

Ha

(0,0.5041,0)

(0.1745,-0.0428,−0.8136)

(0,0,1.5)

Hb

(−1.8954,0.1620,0)

(1.3427,1.1082,0.7154)

(0,1.1,0)

Hc

(−1.3551,1.7024,0.9374)

(1.7328,1.1771,−1.0769)

(−0.9526,−0.55,0)

Hd

(−1.3551,1.7024,−0.9374)

(0.1156,1.8243,−0.4395)

(0.9526,−0.55,0)

Oa

(0.9556,−0.0407,0)

(0.0149,−0.7526,−0.0512)

(0,0,2.6)

Ob

(0.6457,−1.3135,0)

(−1.1710,−0.6453,0.4750)

(0,1.31,2.6)

Results and discussion

According to Eq. 10 the shape of the LEPS surface around the transition state depends strongly on the value of K. The usual way of selecting K is to choose the value which gives the best agreement with the experimental activation energy for a particular reaction. We found that the values of KAB = 0.25, KBC =  − 0.5, and KAC =  − 0.7 give the best LEPS potential energy surface and transition state properties (see Fig. 3). The basin along the \( {R}_{CH_3-{O}_2} \) axis can be attribute to the small values of bond dissociation energy (De) of the systems with oxygen (H − O2 and CH3 − O2). Negative values for K have been previously reported by Ree et al. [30]. The negative value cannot attribute to the tunneling factors because quantum tunneling plays a prominent role in hydrogen migration reactions at low temperature and not for higher temperatures of the reaction (200–2000 K) [6]. We calculated energy values for ELEPS (Eq. 10), and found that in \( {r}_{CH_3-H}=1.5\ \overset{{}^{\circ}}{A} \) and \( {r}_{H-{O}_2}=1.12\ \overset{{}^{\circ}}{A} \) (in accordance with Srinivasan et al.), \( {E}_{LEPS}^{\ne }=-213.62\ \mathrm{kJ}\ {\mathrm{mol}}^{-1} \), and consequently EC = 234.07 kJ mol−1 with only 2.21 kJ mol−1 of difference with the experimental value (231.86 kJ mol−1).
Fig. 3

LEPS’s potential energy surface for CH4 + O2 → CH3 + HO2 with KAB = 0.25, KBC =  − 0.5, and KAC =  − 0.7 . Note that the transition state is in the col. E =  − 213.62 kJ mol−1, and consequently, EC = 234.07 kJ mol−1

To calculate the rate constant from the partition functions via Eq. 12, we compared the CCSD(T) and B3LYP vibration frequencies, and observed that the values from B3LYP are higher than CCSD(T) in most cases, see Table 5 and Fig. 4, but in six of the highest frequencies the percentage error is ≤5%, only one is higher than 50%, and the rest of the frequencies are >10% and < 40%. Reminding that it is necessary to know the degrees of freedom for all partition functions, the degrees of freedom for each system are reported in Table 6. According to Eq. 12, when methane and oxygen become the transition state, the system loses three translational and two rotational degrees of freedom, but gains four vibration degrees of freedom, see Eq. 15.
$$ \frac{q_{\ne }}{q_{C{H}_4}{q}_{O_2}}=\frac{q_t^3{q}_r^3{q}_v^{14}}{\left({q}_t^3{q}_r^3{q}_v^9\right)\left({q}_t^3{q}_r^2{q}_v^1\right)}=\frac{q_v^4}{q_t^3{q}_r^2} $$
(15)
Table 5

% Error between CCSD(T) [6] and B3LYP vibrational wavenumbers

Frequency number

CCSD(T)

B3LYP

% Error

1

56

63

13

2

148

200

35

3

282

384

36

4

484

670

38

5

551

732

33

6

872

1041

19

7

1042

1176

13

8

1234

1301

5

9

1415

1398

1

10

1419

1448

2

11

1449

2461

70

12

3066

3069

~0

13

3236

3216

1

14

3255

3266

~0

15

1816i

1206i

34

Fig. 4

Differences between CCSD(T) and B3LYP vibrational frequencies values for transition state

Table 6

Degrees of freedom for reagents and transition state of Eq. 12

Molecule

CH4

O2

[CH3···H···O2]

Translation

3

3

3

Rotation

3

2

3

Vibration

9

1

15

Partition functions

\( {q}_t^3{q}_r^3{q}_v^9 \)

\( {q}_t^3{q}_r^2{q}_v \)

\( {q}_t^3{q}_r^3{q}_v^{14} \)

The rate constant determination for each geometry of the transition states considered was calculated by Eq. 11, at 200–2000 K range. We used experimental EC (231.86 kJ mol−1). Table 7 shows the order of magnitude of the partition functions for each species. There are differences between our partition functions calculations and the standard range of values given by Laidler [17]. The translation partition functions of methane, oxygen, and the transition state are higher than Laidler’s upper threshold (LU) for T ≥ 1900 K, T ≥ 1000 K, and T ≥ 700 K, respectively, see Fig. 5. The rotation partition functions for oxygen and methane are in the Laidler’s upper range (LU) [17], but transition state values are highest, see also Fig. 6. Moreover, the rotational partition functions for TS2 (B3LYP geometry) and TS3 (naive geometry) have a similar behavior to that of TS1 (CCSD(T) geometry). The vibration partition functions for methane and oxygen are in the range of Laidler’s values; however, the transition state values are larger than the Laidler ones when T ≥ 600 K for CCSD(T) and T ≥ 700 K for B3LYP (Fig. 7).
Table 7

Translational, rotational, and vibrational partition functions values for the species of Eq. 12 for 200 ≤ T ≤ 2000 K. The vibration partition function of transition state was calculated with CCDS(T) and B3LYP vibrational wavenumbers. The typical values compiled by Laidler are also included [17]

Molecule

Motion

Translation (m−3)

Rotation

Vibration

Linear molecule

Nonlinear molecule

Laidler [17]

1031–1032

101–102

102–103

100–101

Methane

1031–1033

 

101–102

100–101

Oxygen

1031–1033

101–102

 

100–101

    

CCSD(T)

B3LYP

TS1

1032–1033

 

104–105

101–105

100–104

TS2

1032–1033

 

104–105

100–105

100–104

TS3

1032–1033

 

104–105

100–105

100–104

Fig. 5

Evolution of translational partition functions for methane, oxygen, and transition state. Laidler’s range is indicated as LU (upper) and LL (lower)

Fig. 6

Rotational partition functions behavior by the whole species of Eq. 12. Laidler’s range is indicated as LU (upper) and LL (lower)

Fig. 7

Comparison of vibrational partition function for transition state under CCSD(T) and B3LYP vibrational frequencies. Laidler’s range is indicated as LU (upper) and LL (lower)

Figure 8 shows the rate constant behavior of the Srinivasan formula, kS = 3.37 × 10−19 T2.745exp(−26,041/T), including the experimental range (ER) (1655–1822 K), and those for kTS1, kTS2 and kTS3 calculated with the vibration values from the CCSD(T) method. Note that formulas obtained from TS1-TS3 have very close behavior to that of the Srinivasan formula after the ER: i) 1840–1890 K for TS1, ii) 1860–1900 K for TS2, iii) 1850–1890 K for TS3, see Table 8. Using the B3LYP vibration values, kTS1-kTS3 converges to kS near the ER (1655–1822 K), see Fig. 9 and Table 9. Regarding the influence of transition state partition functions on the behavior of k, the translation and rotation partition functions have the same trend (Figs. 5 and 6), but vibration rotation functions are different for CCSD(T) and B3LYP (Fig. 7). Furthermore, for k neither CCSD(T) behavior nor B3LYP are capable of following the trend of the Srinivasan formula below 1200 K, although it must be remembered that its validity is in the range 1655–1822 K, where CTST converges.
Fig. 8

Comparison of rate constant values obtained with the Srinivasan rate constant formula and the CTST ones for different geometries of transition state employing CCSD(T) vibrational frequencies: i) TS1, from CCSD(T) method, ii) TS2, B3LYP method, iii) TS3 a naive proposal geometry

Table 8

Ratio between each transition state rate constant value with CCSD(T) vibration frequencies and the Srinivasan rate constant formula (kTSi/kS)

Temperature(K)

k TS1 /k S

k TS2 /k S

k TS3 /k S

1650

2.00 × 10−4

8.74 × 10−5

1.70 × 10−4

1700

2.05 × 10−3

8.97 × 10−4

1.75 × 10−3

1750

2.00 × 10−2

8.76 × 10−3

1.71 × 10−2

1800

1.86 × 10−1

8.16 × 10−2

1.59 × 10−1

1810

2.90 × 10−1

1.27 × 10−1

2.47 × 10−1

1820

4.49 × 10−1

1.97 × 10−1

3.83 × 10−1

1830

6.96 × 10−1

3.05 × 10−1

5.94 × 10−1

1840

1.08

4.71 × 10−1

9.18 × 10−1

1850

1.66

7.26 × 10−1

1.42

1860

2.56

1.12

2.18

1870

3.93

1.72

3.35

1880

6.03

2.64

5.15

1890

9.24

4.05

7.89

1900

1.41 × 101

6.19

1.21 × 101

Fig. 9

Comparison of rate constant values obtained with the Srinivasan rate constant formula and the CTST ones for different geometries of transition state employing B3LYP vibrational frequencies: i) TS1, from CCSD(T) method, ii) TS2, B3LYP method, iii) TS3 a naive proposal geometry

Table 9

Ratio between each transition state rate constant value with B3LYP vibration frequencies and the Srinivasan rate constant formula (kTSi/kS)

Temperature(K)

k TS1 /k S

k TS2 /k S

k TS3 /k S

1650

7.30 × 10−14

3.20 × 10−14

6.23 × 10−14

1700

6.02 × 10−13

2.64 × 10−13

5.14 × 10−13

1750

4.77 × 10−12

2.09 × 10−12

4.07 × 10−12

1800

3.64 × 10−11

1.59 × 10−11

3.11 × 10−11

1810

5.44 × 10−11

2.38 × 10−11

4.64 × 10−11

1820

8.12 × 10−11

3.56 × 10−11

6.93 × 10−11

1830

1.21 × 10−10

5.30 × 10−11

1.03 × 10−10

1840

1.80 × 10−10

7.89 × 10−11

1.54 × 10−10

1850

2.68 × 10−10

1.17 × 10−10

2.28 × 10−10

1860

3.97 × 10−10

1.74 × 10−10

3.39 × 10−10

1870

5.88 × 10−10

2.58 × 10−10

5.02 × 10−10

1880

8.71 × 10−10

3.81 × 10−10

7.43 × 10−10

1890

1.29 × 10−9

5.63 × 10−10

1.10 × 10−9

1900

1.90 × 10−9

8.31 × 10−10

1.62 × 10−9

Conclusions

Conventional transition-state theory (CTST) and the London-Eyring-Polanyi-Sato (LEPS) equation were used to obtain the potential energy surface (PES), activation energy, and the rate constant of the first step of methane oxidation. The energy and rate constant values calculated were in good agreement with the experimental and theoretical values reported in the literature using the shock tube technique and coupled cluster method respectively. The accuracy of the rate constant highly depends on the partition functions, and their values depend on the temperature, the energy levels of the system, the accuracy of inertia moments, and the vibration frequencies. Calculation of the partition functions is easy for the linear molecules (oxygen molecule), but for nonlinear molecules (methane and transition state) it depends mostly on the rotational partition function. The sensitivity of the rate constant with the transition state geometry was analyzed using three geometries for the transition state: two geometries constructed from ab initio calculations (CCSD (T) and B3LYP) and another simple geometry based on the chemical sense. The dependency of the rate constant in terms of temperature followed the same trend. Therefore, CTST methodology provided a simple and fast vision of the transition state at high temperatures (>1500 K) in the gas phase.

Notes

Acknowledgments

The authors would like to thank CONACYT-México for scholarship number 207214 and 18053 and 163234 CONACYT-México project grant.

Supplementary material

894_2018_3829_MOESM1_ESM.docx (149 kb)
ESM 1 (DOCX 149 kb)

References

  1. 1.
    Aul CJ, Metcalfe WK, Burke SM, Curran HJ, Petersen EL (2013) Ignition and kinetic modeling of methane and ethane fuel blends with oxygen: a design of experiments approach. Combust Flame 160:1153–1167.  https://doi.org/10.1016/j.combustflame.2013.01.019 CrossRefGoogle Scholar
  2. 2.
    El Merhubi H, Kéromnès A, Catalano G, Lefort B, Le Moyne L (2016) A high pressure experimental and numerical study of methane ignition. Fuel 177:164–172.  https://doi.org/10.1016/j.fuel.2016.03.016 CrossRefGoogle Scholar
  3. 3.
    Dale A, Lythall C, Aucott J, Sayer C (2002) High precision calorimetry to determine the enthalpy of combustion of methane. Thermochim Acta 382(1-2):47–54.  https://doi.org/10.1016/S0040-6031(01)00735-3 CrossRefGoogle Scholar
  4. 4.
    Chenoweth K, Van Duin ACT, Goddard III WA (2008) ReaxFF reactive force field for molecular dynamics simulations of hydrocarbon oxidation. J Phys Chem A 112:1040–1053.  https://doi.org/10.1021/jp709896w CrossRefGoogle Scholar
  5. 5.
    Page AJ, Moghtaderi B (2009) Molecular dynamics simulation of the low-temperature partial oxidation of CH4. J Phys Chem A 113(8):1539–1547.  https://doi.org/10.1021/jp809576k CrossRefGoogle Scholar
  6. 6.
    Srinivasan NK, Michael JV, Harding LB, Klippenstein SJ (2007) Experimental and theoretical rate constants for CH4+O2→CH3+HO2. Combust Flame 149:104–111.  https://doi.org/10.1016/j.combustflame.2006.12.010 CrossRefGoogle Scholar
  7. 7.
    Rasmussen CL, Jakobsen JG, Glarborg P (2008) Experimental measurements and kinetic modeling of CH4/O2 and CH4/C2H6/O2conversion at high pressure. In J Chem Kinet 40(12):778–807.  https://doi.org/10.1002/kin.20352 CrossRefGoogle Scholar
  8. 8.
    Mai TV-T, Duong MV, Le XT, Huynh LK, Ratkiewicz A (2014) Direct ab initio dynamics calculations of thermal rate constantsfor the CH4 + O2= CH3 + HO2 reaction. Struct Chem 25:1495–1503.  https://doi.org/10.1007/s11224-014-0426-2 CrossRefGoogle Scholar
  9. 9.
    Giménez-López J, Millera A, Bilbao R, Alzueta MU (2015) Experimental and kinetic modeling study of the oxy-fuel oxidation of natural gas, CH4 and C2H6. Fuel 160:404–412.  https://doi.org/10.1016/j.fuel.2015.07.087 CrossRefGoogle Scholar
  10. 10.
    Hashemi H, Christensen JM, Gersen S, Levinsky H, Klippenstein SJ, Glarborg P (2016) High-pressure oxidation of methane. Combust Flame 172:349–364.  https://doi.org/10.1016/j.combustflame.2016.07.016 CrossRefGoogle Scholar
  11. 11.
    Ryu S-O, Shin KS, Hwang SM (2017) Determination of the rate coefficients of the CH4 + O2➔HO2 + CH3 and HCO + O2➔HO2 + CO reactions at high temperatures. Bull Kor Chem Soc 38:228–236.  https://doi.org/10.1002/bkcs.11070 CrossRefGoogle Scholar
  12. 12.
    Skinner GB, Lifshitz A, Scheller K, Burcat A (1972) Kinetics of methane oxidation. J Chem Phys 56(8):3853–3861.  https://doi.org/10.1063/1.1677790 CrossRefGoogle Scholar
  13. 13.
    Shaw R (1978) Semi-empirical extrapolation and estimation of rate constants for abstraction of H from methane by H, O, HO and O2. J Phys Chem Ref Data 7(3):1179–1190.  https://doi.org/10.1063/1.555577 CrossRefGoogle Scholar
  14. 14.
    Tsang W, Hampson RF (1986) Chemical kinetic data vase for combustion chemistry. Part I. Methane and related compounds. J Phys Chem Ref Data 15(3):1087–1279.  https://doi.org/10.1063/1.555759 CrossRefGoogle Scholar
  15. 15.
    Baulch DL, Cobos CJ, Cox RA, Esser C, Frank P, Just T, Kerr JA, Pilling MJ, Troe J, Walker RW, Warnatz J (1992) Evaluated kinetic data for combustion modeling. J Phys Chem Ref Data 21(3):411–734.  https://doi.org/10.1063/1.555908 CrossRefGoogle Scholar
  16. 16.
    Baulch DL, Bowman CT, Cobos CJ, Cox RA, Just T, Kerr JA, Pilling MJ, Stocker D, Troe J, Tsang W, Walker RW, Warnatz J (2005) Evaluated kinetic data for combustion modeling: supplement II. J Phys Chem Ref Data 34:757–1397.  https://doi.org/10.1063/1.1748524 CrossRefGoogle Scholar
  17. 17.
    Laidler KJ (1987) Chemical kinetics, 3rd edn. Harper Collins, New YorkGoogle Scholar
  18. 18.
    Galindo Hernández F, Méndez Ruiz F (2003) Determinación de la energía de activación para la reacción de H+H2 mediante el cálculo de superficies de energía potencial. Rev Mex Fis 49(3):264–270Google Scholar
  19. 19.
    Moss SJ, Coady CJ (1983) Potential-energy surfaces and transition-state theory. J Chem Educ 60(6):455–461.  https://doi.org/10.1021/ed060p455 CrossRefGoogle Scholar
  20. 20.
    Sato S (1955) On a new method of drawing the potential energy surface. J Chem Phys 23:592–593.  https://doi.org/10.1063/1.1742043 CrossRefGoogle Scholar
  21. 21.
    Wang X, Ben-Nun M, Levine RD (1995) Peripheral dynamics of the Cl + CH4 → HCl + CH3 reaction. Chem Phys 197:1–17.  https://doi.org/10.1016/0301-0104(95)00134-A CrossRefGoogle Scholar
  22. 22.
    Liu Y, Liu Z, Lv G, Jiang L, Sun J (2006) Product polarization distribution: Stereodynamics of the reactions Cl+CH4→HCl+CH3 and Cl+CD4→DCl+CD3. Chem Phys Lett 423:157–164.  https://doi.org/10.1016/j.cplett.2006.03.059 CrossRefGoogle Scholar
  23. 23.
    Lemon WJ, Hase WL (1987) A potential energy function for the hydroperoxyl radical. J Phys Chem 91(6):1596–1602.  https://doi.org/10.1021/j100290a061 CrossRefGoogle Scholar
  24. 24.
    Zhu R, Hsu C-C, Lin MC (2001) Ab initio study of the CH3 + O2 reaction: kinetics, mechanism and product branching probabilities. J Chem Phys 115(1):195–203.  https://doi.org/10.1063/1.1376128 CrossRefGoogle Scholar
  25. 25.
    Ase P, Bock W, Snelson A (1986) Alkylperoxy and alkyl radicals. 1. Infrared spectra of CH3O2 and CH3O4CH3 and the ultraviolet photolysis of CH3O2 in argon+oxygen matrices. J Phys Chem 90(10):2099–2109.  https://doi.org/10.1021/j100401a024 CrossRefGoogle Scholar
  26. 26.
    Knox JH (1971) Molecular partition functions. Wiley-Interscience, LondonGoogle Scholar
  27. 27.
    Kalman D (1982) Dot products, spherical coordinates, and 109°. Int J Math Educ Sci Technol 13(4):493–494.  https://doi.org/10.1080/0020739820130412 CrossRefGoogle Scholar
  28. 28.
    Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson GA, Nakatsuji H, Caricato M, Li X, Hratchian HP, Izmaylov AF, Bloino J, Zheng G, Sonnenberg JL, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, Montgomery JA, Peralta JE, Ogliaro F, Bearpark M, Heyd JJ, Brothers E, Kudin KN, Staroverov VN, Keith T, Kobayashi R, Normand J, Raghavachari K, Rendell A, Burant JC, Iyengar SS, Tomasi J, Cossi M, Rega N, Millam JM, Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Martin RL, Morokuma K, Zakrzewski VG, Voth GA, Salvador P, Dannenberg JJ, Dapprich S, Daniels AD, Farkas O, Foresman JB, Ortiz JV, Cioslowski J, Fox DJ (2010) Gaussian 09 revision B.01. Gaussian Inc, WallingfordGoogle Scholar
  29. 29.
    MATLAB version 7.0.0. R14 (2004) The MathWorks Inc., NatickGoogle Scholar
  30. 30.
    Ree J, Kim YH, Shin HK (2007) Classical trajectory study of the formation of XeH and XeCl+ in the Xe++HCl collision. J Chem Phys 127(5):054304-1–054304-13.  https://doi.org/10.1063/1.2751499 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Claudia Aranda
    • 1
  • Arlette Richaud
    • 1
  • Francisco Méndez
    • 1
    Email author
  • Armando Domínguez
    • 1
  1. 1.Departamento de Química, División de Ciencias Básicas e IngenieríaUniversidad Autónoma Metropolitana-IztapalapaMéxico, D.F.México

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