Theoretical rate constant of methane oxidation from the conventional transition-state theory
Abstract
The potential energy surface for the first step of the methane oxidation CH_{4} + O_{2}➔CH_{3} + HO_{2} 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 pathIntroduction
Rate constant expressions in terms of temperature for \( C{H}_4+{O}_2\overset{k}{\to }C{H}_3+H{O}_2 \)
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 energy of the transition state (A⋯B⋯C)^{≠} 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]:
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.
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 (q_{A}, q_{B}). 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:
Parameters used in the LEPS potential energy surface of the CH_{4} + O_{2} → CH_{3} + HO_{2}
Parameter | CH_{3} − H | H − O_{2} | CH_{3} − O_{2} |
---|---|---|---|
D_{e} (kcal/mol) | 107.0^{a} | 54.6^{b} | 33.7^{d} |
\( {r}_e\ \left(\overset{{}^{\circ}}{A}\right) \) | 1.093^{a} | 0.971^{b} | 1.449^{d} |
ω_{e}(cm^{−1}) | – | 3436^{b} | 902^{e} |
\( \beta\ \left({\overset{{}^{\circ}}{\mathrm{A}}}^{-1}\right) \) | 1.83^{a} | 3.235^{c} | 3.156^{c} |
The partition functions used to solve Eq. 12
Molecule | Partition function | ||
---|---|---|---|
q_{t} (m^{−3}) | q_{r} (dimensionless) | q_{v} (dimensionless) | |
O_{2} | \( \frac{{\left(2\pi m{k}_BT\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{h^3} \) | \( \frac{8{\pi}^2I{k}_BT}{\sigma^{\prime }{h}^2} \) | \( \frac{1}{\left(1-{e}^{-\frac{hv}{k_BT}}\right)} \) |
CH_{4} | \( \frac{{\left(2\pi {m}_{C{H}_4}{k}_BT\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{h^3} \) | \( \frac{8{\pi}^2{\left(8\pi {I}_A{I}_B{I}_C\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left({k}_BT\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\sigma^{\prime \prime }{h}^3} \) | \( \frac{1}{\left(1-{e}^{-\raisebox{1ex}{$h{v}_1$}\!\left/ \!\raisebox{-1ex}{${k}_BT$}\right.}\right)\cdots \left(1-{e}^{-\raisebox{1ex}{$h{v}_4$}\!\left/ \!\raisebox{-1ex}{${k}_BT$}\right.}\right)} \) |
[CH_{3}···H···O_{2}]^{≠} | \( \frac{{\left(2\pi {m}_{ET}{k}_BT\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{h^3} \) | \( \frac{8{\pi}^2{\left(8{\pi}^3{I}_A{I}_B{I}_C\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left({k}_BT\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\sigma^{\prime \prime \prime }{h}^3} \) | \( \frac{1}{\left(1-{e}^{-\raisebox{1ex}{$h{v}_1$}\!\left/ \!\raisebox{-1ex}{${k}_BT$}\right.}\right)\cdots \left(1-{e}^{-\raisebox{1ex}{$h{v}_{14}$}\!\left/ \!\raisebox{-1ex}{${k}_BT$}\right.}\right)} \) |
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) |
H_{a} | (0,0.5041,0) | (0.1745,-0.0428,−0.8136) | (0,0,1.5) |
H_{b} | (−1.8954,0.1620,0) | (1.3427,1.1082,0.7154) | (0,1.1,0) |
H_{c} | (−1.3551,1.7024,0.9374) | (1.7328,1.1771,−1.0769) | (−0.9526,−0.55,0) |
H_{d} | (−1.3551,1.7024,−0.9374) | (0.1156,1.8243,−0.4395) | (0.9526,−0.55,0) |
O_{a} | (0.9556,−0.0407,0) | (0.0149,−0.7526,−0.0512) | (0,0,2.6) |
O_{b} | (0.6457,−1.3135,0) | (−1.1710,−0.6453,0.4750) | (0,1.31,2.6) |
Results and discussion
% 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 |
Degrees of freedom for reagents and transition state of Eq. 12
Molecule | CH_{4} | O_{2} | [CH_{3}···H···O_{2}]^{≠} |
---|---|---|---|
Translation | 3 | 3 | 3 |
Rotation | 3 | 2 | 3 |
Vibration | 9 | 1 | 15 |
Partition functions | \( {q}_t^3{q}_r^3{q}_v^9 \) | \( {q}_t^3{q}_r^2{q}_v \) | \( {q}_t^3{q}_r^3{q}_v^{14} \) |
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] | 10^{31}–10^{32} | 10^{1}–10^{2} | 10^{2}–10^{3} | 10^{0}–10^{1} | |
Methane | 10^{31}–10^{33} | 10^{1}–10^{2} | 10^{0}–10^{1} | ||
Oxygen | 10^{31}–10^{33} | 10^{1}–10^{2} | 10^{0}–10^{1} | ||
CCSD(T) | B3LYP | ||||
TS1 | 10^{32}–10^{33} | 10^{4}–10^{5} | 10^{1}–10^{5} | 10^{0}–10^{4} | |
TS2 | 10^{32}–10^{33} | 10^{4}–10^{5} | 10^{0}–10^{5} | 10^{0}–10^{4} | |
TS3 | 10^{32}–10^{33} | 10^{4}–10^{5} | 10^{0}–10^{5} | 10^{0}–10^{4} |
Ratio between each transition state rate constant value with CCSD(T) vibration frequencies and the Srinivasan rate constant formula (k_{TSi}/k_{S})
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 × 10^{1} | 6.19 | 1.21 × 10^{1} |
Ratio between each transition state rate constant value with B3LYP vibration frequencies and the Srinivasan rate constant formula (k_{TSi}/k_{S})
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
References
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