Effect of Water and Formic Acid on ^·OH + CH₄ Reaction: An Ab Initio/DFT Study

Abstract

In this work, we used ab initio/DFT method coupled with statistical rate theory to answer the question of whether or not formic acid (HCOOH) and water molecules can catalyze the most important atmospheric and combustion prototype reaction, i.e., ^·OH (OH radical) + CH₄. The potential energy surface for ^·OH + CH₄ and ^·OH + CH₄ (+X) (X = HCOOH, H₂O) reactions were calculated using the combination of hybrid-density functional theory and coupled-cluster theory with Pople basis set [(CCSD(T)/ 6-311++G(3df,3pd)//M06-2X/6-311++G(3df,3pd)]. The results of this study show that the catalytic effect of HCOOH (FA) and water molecules on the ^·OH + CH₄ reaction has a major impact when the concentration of FA and H₂O is not included. In this situation the rate constants for the CH₄ + HO···HCOOH (3 × 10⁻⁹ cm³ molecule⁻¹ s⁻¹) reaction is ~10⁵ times and for CH₄ + H₂O···HO reaction (3 × 10⁻¹⁴ cm³ molecule⁻¹ s⁻¹ at 300 K) is ~20 times higher than ^·OH + CH₄ (~6 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹). However, the total effective rate constants, which include the concentration of both species in the kinetic calculation has no effect under atmospheric condition. As a result, the total effective reaction rate constants are smaller. The rate constants when taking the account of the FA and water for CH₄ + HO···HCOOH (4.1 × 10⁻²² cm³ molecule⁻¹ s⁻¹) is at least seven orders magnitude and for the CH₄ + H₂O···HO (7.6 × 10⁻¹⁷ cm³ molecule⁻¹ s⁻¹) is two orders magnitude smaller than ^·OH + CH₄ reaction. These results are also consistent with previous experimental and theoretical studies on similar reaction systems. This study helps to understand how FA and water molecules change the reaction kinetic under atmospheric conditions for ^·OH + CH₄ reaction.

1. Introduction

The tropospheric concentration of important greenhouse gases, i.e., methane (CH₄), is steadily increasing [1,2,3,4]. In the atmosphere, ^·OH (OH radical) plays a crucial role in removing the CH₄ [1,2,3,4]. The atmospheric lifetime of methane is due to loss by ^·OH is ∼12–17 years [1,2,3,4]. This reaction is also important in high-temperature combustion kinetics [5]. Due to large uncertainty in the rate constants data, many experimental measurements and theoretical studies have been performed [1,2,3,4,5]. To avoid the repetition of previous studies, only a few studies have been discussed here [1,2,3,4,5]. Various experimental measurements were used to estimate the rate constant for ^·OH + CH₄ reaction into the atmosphere condition and reported its atmospheric lifetime due to loss with the ^·OH [1,2,3,4]. Experimental measurement on ^·OH + CH₄ reaction was done by Vaghjian et al. [2] in the temperature range of 295 to 400 K. Due to the limited number of high temperature studies, Srinivasan et al. measured the rate constants of ^·OH + CH₄ using the reflected shock tube method [5]. The ^·OH + CH₄ reaction has been extensively studied by using various theoretical approaches [6,7,8,9]. Most of the theoretical studies have focused on calculating the rate constants using a high-level ab initio methods and statistical rate theory [6,7,8,9]. Ellingson et al. [9] proposed the rate constants for OH + CH₄ reaction and its ¹²C/¹³C kinetic isotopes effect using harmonic, hindered, and free rotor approaches. Their calculated rate constants were in good agreement with experimentally measurement values. Li and Gua calculated the thermal rate constants for isotopic effects for OH + CH₄ using UCCSD(T)-F12a/aug-cc-pVTZ and canonical variational transition state theory (CVT) using the free rotor approach [6,8]. In this study, we have re-calculated the rate constants for ^·OH + CH₄ using a similar level of theories to validate our theoretical approach and compare the rate constants for the effect of HCOOH/H₂O on ^·OH + CH₄ reaction.

It is believed that volcanic eruptions emitted various gases into the Earth’s atmosphere [10,11,12]. Toxic acids such as hydrochloric (HCl), formic acid (HCOOH), and sulfuric (H₂SO₄) acid in the atmosphere are produced from volcanic activities [10,11,12]. The source of HCOOH (FA) in atmospheric reactions has been extensively studied in recent years [10,11,12]. FA is mainly produced by oxidation of volatile organic compounds (VOCs), soil, burning of fossil fuels, and terrestrial vegetation [10,11,12]. FA also influences pH-dependent chemical reactions in clouds [11]. FA is one of the most abundant organic acid present in the atmosphere [10]. The atmospheric abundance of FA varies from 0.01 to 10 ppbv depending upon the altitudes [13,14,15,16,17,18,19]. FA can catalyze many important atmospheric reactions such as CH₃O + O₂ [13], ^·OH + HCl [14], H₂O + CH₂COO [15], HO₂ + Cl [16], H₂O + CH₂O [17], and ^·OH + CH₂O [19]. These studies suggest that FA can form ring types of complexes and transition states, which lower the barrier heights and increase the reaction rate constant under atmospheric conditions [13,14,15,16,17,18,19]. However, in realistic atmospheric conditions, the effective rate constants decrease and the catalytic effect does not favor the reaction [12,13,14,15,16,17,18,19]. To the best of our knowledge, the acidolysis of ^·OH + CH₄ by a FA molecule has not been studied. Therefore, we have investigated the effect of the FA molecule on the most important atmospheric and combustion prototype of reaction, i.e., ^·OH + CH₄. The focus of the present work is to explore the potential energy surface for the effect of FA on ^·OH + CH₄ and calculate the rate constants under atmospheric conditions.

The oxidation of CH₄ in the stratosphere is an important source of water vapor in this region. Over the last few years, many experimental and theoretical studies have been conducted on the catalytic effect of a single water molecule on many important atmospheric reactions, such as ^·OH + CH₂O, ^·OH + CH₂CH₂, ^·OH + CH₂NH, ^·OH + CH₃CHO, ^·OH + CH₃OH, OH + HCl, O₂ + CH₃O, and O₂ + CH₂OH [20,21,22,23,24,25,26,27,28,29,30,31,32]. Experimental measurements on some of the reactions, i.e., OH + CH₃OH (+H₂O) [20] and OH + CH₃CHO (+H₂O)] [22], have been performed. Jara et al. [20] and Vöhringer-Martinez [22] have measured reaction rate constants and suggested that the catalytic role of water molecules dominated at lower temperatures. Recently, Wu et al. [21], have proposed reaction kinetic data using a high-level ab initio method and an advanced kinetic model for ^·OH + CH₃OH. Their calculated rate constants were in good agreement with the experimentally measured values of Jara et al. [20]. Most of the theoretical studies suggest that a single water molecule formed a water-bounded ring type of pre-reactive complexes (PRCs) and transition states (TSs), which reduces the energy barrier but does not increase the rate constant under atmospheric conditions. In 2012, Thomsen et al [23]. proposed the effect of a single water molecule on ^·OH + CH₄ using ab initio/DFT method [23]. They suggested that the role of water molecules in OH + CH₄ reaction is less important under atmospheric conditions. To the best of our knowledge, the detailed kinetics analysis of this reaction is still not known, and the effect of temperature-dependent water concentration on ^·OH + CH₄ has not been reported. In our previous works, we proposed the catalytic effect of H₂O molecules in the atmospheric reactions, i.e., ^·OH + CH₂O, ^·OH + CH₂CH₂, ^·OH + CH₂NH, ^·OH + CH₃OH, CH₂NH + H₂O, and CH₂OH + O₂ reactions [25,26,27,28]. We have also suggested that the catalytic role of water is less important under atmospheric conditions.

Because of CH₄ long lifetime and a good model for the most important atmospheric and combustion reaction prototype molecule, the effect of FA and water molecules was investigated. The rate constant for the effect of FA and water molecules on ^·OH + CH₄ were investigated at different atmospheric concentrations of FA and water molecules at different temperatures. The results of this study may be useful for the benchmark performance for other higher chain alkanes. The comparison of reaction energies and rate constants with the literature values provides more confidence in our results.

2. Theoretical Methodology

2.1. Computational Method

All the electronic structure calculations were done with the Gaussian 09 suite of programs (Revision A.02, Gaussian, Inc., Wallingford, CT, USA, 2009) [33]. Optimized geometries for the species involved in ^·OH + CH₄, ^·OH + CH₄ (+HCOOH), and ^·OH + CH₄ (+H₂O) reactions were computed using M06-2X/6-311++G(3df,3pd) level [34] (see Table S1). The harmonic vibrational frequency of all the species was calculated to obtain the zero-point corrections (ZPE). The frequency analysis shows that all the reactants, complexes and products have all positive values whereas all the transition states (TSs) have a single imaginary (see Table S2). Intrinsic reaction coordinate (IRC) calculations were performed to confirm the identity of pre-reactive complex and post-reactive complex for every TS. The single point energies were calculated using CCSD(T)/6-311++G(3df,3pd) (CC) level of theory at optimized geometries of M06-2X/6-311++G(3df,3pd)(M06-2X). As discussed in the previous works, the combination of CC//M06-2X typically provides results that are accurate to ~1–2 kcal/mol [23,24,25,26]. Therefore, we believe the CC//M06-2X provides accurate energies//rotational vibrational parameters for ^·OH + CH₄, ^·OH + CH₄ (+HCOOH), and ^·OH + CH₄ (+H₂O) reactions. The ZPE, electronic energies, thermal correction, enthalpy correction, and free energy correction for all the species are tabulated in Table S3.

2.2. Chemical Kinetics Calculations

The temperature-dependent rate constants were calculated by Equations (1) and (2):

$$k^{GT}\left(T,s\right)=\Gamma L^{\ne }\times \frac{k_{B}T}{h}\frac{Q_{TS}^{\ne }\left(T,s\right)}{Q_R\left(T\right)}exp\left(-\frac{V_{MEP}\left(s\right)}{k_{B}T}\right)$$

(1)

$$k^{CVT}\left(T\right)=mi{n}_s k_{}^{GT}\left(T,s\right)=k_{}^{GT}\left(T,s^{CVT}(T\right)$$

(2)

where$k^{GT}\left(T,s\right)$ is generalized rate constants and $k^{CVT}\left(T\right)$ are the temperature dependent rate canonical variational transition state theory (CVT) rate constants, V_MEP is the barrier height without zero-point correction, $L^{\ne }$ is a reaction path degeneracy, h is Planck’s constant, k_B is the Boltzmann constant, and $Q_{TS}^{\ne }$ and $Q_R$ are the total partition functions for the transition state and the reactants, respectively, $\Gamma$ is the small curvature tunneling (SCT) correction as implemented in Polyrate [35]. The rate constants were calculated using a dual dynamic approach with CVT and the interpolated single point energies (ISPE) correction calculated using dual level direct dynamic approach CVT/SCT with interpolated single point energies (ISPE) as discussed in the reference [35,36]. PolyRate and GaussRate suite of programs were used to calculate the temperature-dependent bimolecular and unimolecular rate constants based on CVT/SCT (Table S4) approach [35,36].

The Multiwell Thermo code [37] was used calculate the equilibrium constant (K_eq) for the formation of complexes as given in Equation (3):

$$K_{eq}=\frac{Q_{RC}}{Q_R}exp\left(-\frac{E_{RC}-E_R}{k_{B}T}\right)$$

(3)

The equilibrium constants ($K_{eq}$) for the formation complexes calculated by Equation (3) were tabulated in Tables S5 and S6.

3. Results and Discussion

3.1. Reaction Pathways for OH + CH₄

The potential energy surface (PES) for ^·OH + CH₄ reaction is shown in Figure 1. The rotational vibrational parameters of CH₄, ^·OH and TS and products are given in Table S2. As shown in Figure 1, the reaction proceeds via the formation of a pre-reactive complex (RC₁) whose energy is higher than the reactants. This is due to fact that the orientation of H of ^·OH is toward the carbon atom of CH₄. The role of complex in ^·OH + CH₄ reaction is unimportant as suggested in earlier studies [8]. The barrier height for ^·OH + CH₄ is ~5.0 kcal/mol is in very good agreement with the value reported by earlier theoretical studies (5–6 kcal/mol) [6,7,8,9].

3.1.1. Reaction Pathways for ^·OH + CH₄ (+HCOOH)

As discussed in the earlier studies, [13,14,25,26,27,28], under true conditions, it is very unlikely that ^·OH, CH₄, and HCOOH collide simultaneously, therefore the probability of a termolecular reaction is negligible. It is expected that either a CH₄···HCOOH or ^·OH···HCOOH and CH₄···HO^· will form first, followed by an attack on this complex by third molecule ^·OH or CH₄ or HCOOH. In these three cases, the probable reactions are shown in Equations (4)–(6):

CH₄···HCOOH + OH →·CH₃ + (H₂O) + HCOOH

(4)

CH₄···HO + HCOOH →·CH₃ + (H₂O) + HCOOH

(5)

CH₄ + HCOOH···HO →·CH₃ + (H₂O) + HCOOH

(6)

The potential energy surface (PES) for the effect of FA on ^·OH + CH₄ reaction is shown in Figure 2. The optimized geometries and rotational vibrational parameters of CH₄, ^·OH, and TS and products are given in Tables S1 and S2. The calculated binding energy (BE) between ^·OH and FA1 (−3.85 kcal/mol) is in very good agreement with the value (−3.59 kcal/mol) reported previously [19]. The BE of FA2···OH (−2.83 kcal/mol) is 1 kcal/mol higher than the BE of FA11···OH (−3.85 kcal/mol). This is due to fact that the cis and trans orientation of FA. The trans form of FA with ^·OH leading more strong hydrogen bonding than the cis form. The B.E. of CH₄···OH (+0.3 kcal/mol) and CH₄···HCOOH (+0.3 kcal/mol) is very small compared to B.E. of HCOOH···OH, therefore, we have neglected these complexes. As shown in Figure 2, beginning with the CH₄ + HCOOH···OH, two three-body complexes PRC_FA-1 and PRC_FA-2 were formed due to the trans and cis orientation of FA (Figure 2). The relevant TS structures connected to these PRCs are shown in Figure 2. The other PRC_FA was also optimized and found that they are less stable than PRC_FA-1 and PRC_FA-2, therefore not included in the PES of CH₄ + ^·OH···HCOOH reaction (see Figure S1).

The BE of PRC_FA-1 (−6.96 kcal/mol) and PRC_FA-2 (−4.32 kcal/mol) is the result of C···H and O···H interactions. Starting from PRC_FA-1 and PRC_FA-2, we have identified two reaction pathways, i.e., hydrogen abstraction by ^·OH on two different orientations of FA. Transition state (TS_F1) corresponds to H-abstraction reaction from methane hydrogen via PRC_FA-1. The calculated barrier heights for this pathway (~3.76 kcal/mol), which are lower than the barrier height for ^·OH + CH₄ reaction (~5 kcal/mol). The transition state (TS_F2) corresponds to the H abstraction reaction from methane hydrogen via PRC_FA-2. The calculated barrier height TS_F2 (~ 5.32 kcal/mol) which leads to form a product FA-2 (cis-Hydrogen). The barrier height of TS_F1 (~4 kcal/mol,) is ~1 kcal/mol lower than the value of TS_F2 (~5 kcal/mol) expected to play a more important role in the kinetic calculations [13,14,19].

3.1.2. Reaction Pathways for ^·OH + CH₄ (+H₂O)

As suggested in the FA case, the simultaneous collisions of ^·OH, CH₄, and H₂O, are very unlikely, therefore the termolecular reaction probability is negligible under real conditions. It is expected that either a CH₄···H₂O or ^·OH···H₂O or CH₄···OH will form first, followed by an attack on this complex by the third molecule ^·OH or CH₄ or H₂O to this complex. In these three cases, the most probable reactions are given in Equations (7)–(9):

CH₄···H₂O + ^·OH → ^·CH₃ + (H₂O)₂

(7)

CH₄···HO^· + H₂O → ^·CH₃ + (H₂O)₂

(8)

CH₄ + H₂O···HO^· → ^·CH₃ + (H₂O)₂

(9)

Out of three, only one OH···H₂O (−4.1 kcal/mol) as a two-body complex is found to be more stable than CH₄···H₂O (−0.3 kcal/mol) and CH₄···OH (+0.3 kcal/mol), therefore the reactions (7) and (8) are not considered in the potential energy surface plot. The potential energy surface (PES) for the effect of water on ^·OH + CH₄ reaction is shown in Figure 3. As shown in Figure 3, beginning with the CH₄ + ^·OH···H₂O reaction, the three-body complex i.e., PRC_w was formed depending on the approach of the hydrogen atom in the ^·OH towards the water oxygen or the hydrogen of ^·OH (Figure 3). We tried different way of projecting OH and H₂O in the reaction, which led to two different PRC_ws, i.e., PRC_w-1 (−4.3 kcal/mol) and PRC_w-2 (−3.1 kcal/mol). The optimized structure is given in the supporting information Figure S1. As suggested in earlier work on a similar reaction system [21], only stable PRC_w was considered for the rate constant calculations. Therefore, we restricted our discussion to only stable PRC_w-1.

3.2. Enthalpies of Reactions (∆H_rxn (0 K))

The enthalpies (∆H(0 K)) relative to the reactants for ^·OH + CH₄, ^·OH + CH₄ (+HCOOH) and ^·OH + CH₄ (+H₂O) reactions are given in

Table 1. Enthalpies of reaction (∆H_rxn (0 K)) in kcal mol⁻¹ of ^·OH + CH₄ reaction.

·OH + CH4→	This Work	Literature a,b,c
·OH···CH4 (RC1)	0.30
HO···HCH3 (TS)	4.93
·CH3 + H2O	−14.4	−14.3 a, −13.9 b, −13.4 c
·OH + CH4 + HCOOH →	This Work
·OH···CH4···HCOOH (PRCFA1)	−6.96
·OH···CH4···HCOOH(PRCFA)	−4.32
·OH···HCH3···HCOOH (TSFA1)	−3.2
·OH···HCOOH···HCH3 (TSFA2)	+1.0
H2O···HCOOH···CH3 (Post-PRCFA1) H2O···HCOOH···CH3 (Post-PRCFA2) CH3 + H2O + FA	−17.4 −22.7 −14.4
·OH + CH4 +H2O→	This Work
·OH···CH4···H2O (PRCw)	−4.32
·OH···HCH3···H2O (TSw)	+1.6
H2O···H2O···CH3(Post-PRCw)	−18.42
·CH3 + 2H2O	−14.4

^a ATcT thermochemical data [38,39,40], ^b [9]; ^c [7].

. The calculated ∆H(0 K) of ^·OH + CH₄→CH₃ + H₂O (−14.4 kcal/mol) is in excellent agreement with the ATcT thermochemical data (−14.3 kcal/mol) base [38,39,40] and is in very good agreement with previous theoretical calculation (−13.9 kcal/mol) [7,9]. As shown in Figure 2, the formation of the complex in ^·OH···CH₄···HCOOH (PRC_FA1) and ^·OH···CH₄···HCOOH (PRC_FA2), are in trans- and cis orientation of HCOOH. Due to the different orientations of OH and HCOOH molecules, the BE of PRCF_A1 (−6.96 kcal/mol) and PRCF_A2 (−4.32 kcal/mol) are different (see Figure 2 and Table 1). The orientation of the H-atom of HCOOH makes a strong hydrogen bond between ^·OH···CH₄···HCOOH leading to more stable PRC_FA1. The calculated BE of ^·OH···CH₄···H₂O (PRC_w) (−4 kcal/mol) is lower than BE of PRC_FA (−6.96 kcal/mol) due to the presence of weak hydrogen bonds. It is clear from Table 1 that barrier height for the effect of FA and water molecules on ^·OH + CH₄ reaction is smaller than the barrier height of the free ^·OH + CH₄ reaction, which gives the indication of catalytic behavior of FA and water on ^·OH + CH₄ reaction.

3.3. Rate Constants

^·OH + CH₄ Reaction

The CVT/SCT rate constant for ^·OH + CH₄ reaction is calculated and tabulated in

Table 2. Comparison of calculated rate constants (cm³ molecule⁻¹ s⁻¹) and literature rate constants for the ^·OH + CH₄→CH₃ + H₂O.

Temp (K)	k-HO	k-FR	Exp [2,39]	Exp [5]	Theory [9]
200	1.7 × 10−16	4.6 × 10−16	3.6 × 10−16	3.7 × 10−16	1.5 × 10−16
225	3.4 × 10−16	8.9 × 10−16	9.5 × 10−16	9.5 × 10−16	4.2 × 10−16
250	7.0 × 10−16	1.8 × 10−15	2.0 × 10−15	2.1 × 10−15	9.8 × 10−16
275	1.3 × 10−15	3.5 × 10−15	3.9 × 10−15	4.0 × 10−15	2.0 × 10−15
300	2.3 × 10−15	6.1 × 10−15	6.6 × 10−15	6.9 × 10−15	3.8 × 10−15
325	3.9 × 10−15	1.0 × 10−14	1.0 × 10−14	1.1 × 10−14	6.4 × 10−15
350	6.1 × 10−15	1.6 × 10−14	1.5 × 10−14	1.8 × 10−14	1.0 × 10−14
375	9.2 × 10−15	2.4 × 10−14	2.2 × 10−14	2.6 × 10−14	1.6 × 10−14
400	1.3 × 10−14	3.5 × 10−14	2.9 × 10−14	3.6 × 10−14	2.3 × 10−14
k = ATn exp(−B/T)	A = 1.2 × 10−28 N = 5.9 B = 134	A = 3 × 10−29 N = 5.9 B = 134	A = 3 × 10−14 N = 0.67 B = 1575	A = 1.6 × 10−18 N = 2.1 B = 1231	A = 3.7 × 10−13 N = 2.3 B = 1377

and shown in Figure 4. The calculated value using harmonic oscillator (HO) approximation is a factor of ~2 lower than experimentally measured values over the entire temperature range (Table 2). The rate constant calculated using HO approximation (2.3 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹ at 300 K) is in very good agreement with the HO approximation of Ellingson et al [9] (2.3 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹ at 300 K).

As suggested by Ellingson et al. [9], the free-rotor approximation is the correct choice, which is consistent with our study. The calculated value using free-rotor approximation at 300 K (6.1 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹) is in excellent agreement with the experimentally measured [2,41] value (6.7 × 10⁻¹⁵ and 6.9 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹) and good agreement with theoretically calculated ones (4 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹ and 6 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹) [7,9]. Our value is in very good agreement over the entire temperature range with the experimentally measured values. The calculated value is also in very good agreement with Srinivasan et al. (Figure 4) [5]. The calculations of rate constant for ^·OH + CH₄ reaction provide more confidence in our theoretical approach CC//M06-2X with CVT/SCT method. To avoid the repetition of the ^·OH + CH₄ reaction, we have restricted our comparison to only a few experimental and theoretical values [2,5,7,9,41].

3.4. Formic Acid Assisted ^·OH + CH₄ Reaction

As suggested in the thermochemistry section, the binding energy of CH₄···HCOOH, and CH₄···OH smaller than that of HCOOH··· HO^· which will result in a shorter lifetime. Therefore, the formation of CH₄···HCOOH, CH₄···OH is almost negligible compared to HCOOH··· HO^· and only one channel, i.e., CH₄ + HCOOH··· HO^· is considered for the rate constants calculations.

$$\mathrm{OH}+\mathrm{HCOOH}···\underset{k_{-0f}}{\overset{k_{0f}}{\rightleftarrows }}\mathrm{HCOOH}···\mathrm{HO}+{\mathrm{CH}}_4\underset{k_{-1f}}{\overset{k_{1f}}{\rightleftarrows }}·\mathrm{HCOOH}···\mathrm{HO}···{\mathrm{CH}}_4\stackrel{k_{2f}}{\to }\mathrm{Products}$$

(10)

The temperature-dependent rate constants for the reactions of OH + CH₄ (+HCOOH) were calculated based on the high-pressure limit condition. Locating the TS of backward reaction, i.e., PRC_FA → ^·OH···HCOOH + CH₄ is complicated, in that case, to account for the presence of forward and backward reactions ($k_{FA}=K_{eq}{k}_{2f}$) equilibrium approach was used. This kinetic model is reasonably correct as discussed in our earlier studies [25,26,27,28]. The rate constants (s⁻¹) (k_2f) were computed using the CVT/SCT approach are presented in supporting information, Table S4. The equilibrium constants ($K_{eq}$) are shown in Tables S5 and S6. The rate constants $k_{2FA1}=K_{eq\left(FA1\right)}\times k_{TS1F1}$, $k_{2FA2}=K_{eq\left(FA2\right)}\times k_{TS2F{2}_{}}$ for ^·OH + CH₄ (+HCOOH) were calculated in the temperature range of 200 to 400 K and are shown in Figure 5. $k_{2FA1}$ are bimolecular rate constants of Pathway FA-1 and $k_{2FA2}$ are bimolecular rate constants for Pathway FA-2, where$K_{eq\left(FA1\right)}$ and $K_{eq\left(FA2\right)}$ are equilibrium constants for CH₄ + ^·OH···HCOOH →PRC_FA1 and CH₄ + ^·OH···HCOOH →PRC_FA2, reactions, respectively (see Table S6). The rate constant of Pathway FA-1 (3 × 10⁻⁹ cm³ molecule⁻¹ s⁻¹ at 300 K) is higher than Pathway FA-2 (6 × 10⁻¹³ cm³ molecule⁻¹ s⁻¹ at 300 K), which shows that Pathway FA-1 is more kinetically favorable than Pathway FA-2.

The rate constant of FA-assisted reaction is at least four to five orders of magnitude higher (1 × 10⁻⁹ to 6 × 10⁻¹² cm³ molecule⁻¹ s⁻¹) than the ^·OH + CH₄ (5 × 10⁻¹⁶ to 3.5 × 10⁻¹⁶ cm³ molecule⁻¹ s⁻¹) in the temperature range of 200 to 400 K. Our result is consistent with previous theoretical studies on similar reaction systems [13,14]. The calculations suggest that the catalytic effect of FA takes place if the concentration of FA is not included. As suggested earlier [25,26,27,28], the effective rate constant which include the concentration of FA is calculated by Equation (11):

$$k_{FA}^{eff}=\left[\left\{K_{eq\left(FA10\right)}\times k_{2FA1}^{CVT}\right\}+\left\{K_{eq\left(FA20\right)}\times k_{2FA2}^{CVT}\right\}\right]\times \left[\mathrm{HCOOH}\right]$$

(11)

where $K_{eq\left(FA10\right)}$ and $K_{eq\left(FA20\right)}$ are equilibrium constants of ^·OH+HCOOH →RC_4F1 and ^·OH+HCOOH →RC_4F2, reactions, respectively (see Table S5) and [HCOOH] is FA concentration used at 10 ppbv and 0.01 ppbv are based on previous studies [13,14]. The total effective rate constant for ^·OH + CH₄ (+HCOOH) (4 × 10⁻²² cm³ molecule⁻¹ s⁻¹ at 300 K) is ~7 order magnitude lower than ^·OH + CH₄ reaction (6 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹). This result is similar to FA-assisted reaction ^·OH + HCHO (+HCOOH): 7 × 10⁻²⁰ cm³ molecule⁻¹ s⁻¹ at 300 K) as reported in the earlier study [19]. The rate constants show positive temperature-dependent (Figure 5). The similar kind of behavior was observed in the literature on the effect of FA on atmospheric reactions [13].

3.5. Water-Assisted ^·OH + CH₄ Reaction

As discussed in the case of FA assisted reaction, only one CH₄ + ^·OH···H₂O channel is considered for the rate constants calculations. The other channels, i.e., the formation of CH₄···H₂O, CH₄···OH, are almost negligible.

The reaction pathways for the effect of a water molecule on ^·OH + CH₄ starting with ^·OH···H₂O are presented in Equation (12):

$${\mathrm{OH}+\mathrm{H}}_2\mathrm{O}\cdots \underset{k_{-0w}}{\overset{k_{0w}}{\rightleftarrows }}{\mathrm{H}}_2\mathrm{O}\cdots {\mathrm{HO}+\mathrm{CH}}_4\underset{k-1w}{\overset{k_{1w}}{\rightleftarrows }}{\mathrm{H}}_2\mathrm{O}\cdots \mathrm{HO}\cdots {\mathrm{CH}}_4\stackrel{k_{2w}}{\to }\mathrm{Products}$$

(12)

The rate constants (s⁻¹) for ^·OH + CH₄ (+H₂O) (PRC_w → TS → P) were tabulated in Table S4. The rate constants (cm³ molecule⁻¹ s⁻¹) were calculated using $k_{2w}=K_{eq\left(A\right)}\times k_{TSaw}$, where $K_{eq\left(A\right)}$ is in equilibriumconstants involved in the ^·OH···H₂O + CH₄ reaction. The rate constants in the temperature range of 200 K to 400 K for with and without water concentration are shown in Figure 6. The rate constants for CH₄ + H₂O···OH reaction (3 × 10⁻¹⁴ cm³ molecule⁻¹ s⁻¹ at 300 K) is ~20 times higher than ^·OH + CH₄ reaction (6 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹ at 300 K. As shown in Figure 6, the water-catalyzed reaction has higher rate constants than a water-free reaction in the entire temperature range 200–400 K.

As suggested in our earlier studies [25,26,27], the concentration of water at varying humidity levels must be used to estimate the accurate rate constant. Therefore, the correct expression to calculate the total effective rate constants is given by Equation (13)

$$k_w^{eff}=\left[\left\{K_{eq\left(2\right)}\times k_{2w}^{CVT}\right\}]\times [{\mathrm{H}}_2\mathrm{O}\right]$$

(13)

where $K_{eq\left(2\right)}$ are equilibrium constants of H₂O +^·OH →RC_2w reactions, [H₂O] is water concentration as discussed in earlier studies [21,27]. The effective rate constant (7 × 10⁻¹⁷ cm³ molecule⁻¹ s⁻¹ at 300 K) is factor of ~100 lower than the water-free OH + CH₄ reaction (~6 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹ at 300 K) at 100% humidity. The result is also consistent with similar atmospheric reactions, i.e., ^·OH + CH₂NH, ^·OH + CH₂CH₂, ^·OH + CH₂O, and OH + CH₃OH [25,26,27].

In order to gain correct insight into the effect of FA/water on ^·OH + CH₄ reaction, the free energy profiles for all the channels were computed and shown in supporting information Figure S2. Due to the high entropy of activation of (ΔS^‡) of FA/H₂O assisted reaction than free OH + CH₄ reaction, rate constants are higher. It is also important to point out that, in the gas phase reaction, the barrier height ΔE^‡ and ΔG^‡ are calculated as the energies difference between the transition state and those of the two-body complex, i.e., RC complex, while in the water-assisted and FA-assisted reaction, ΔE^‡ and ΔG^‡ are calculated from energies difference of transition state and termolecular complex i.e., PRCs. Thus, if step 0 is ignored and the rate constant is calculated as k_eff = K_eq × k₂ and the reaction rate constant is higher in the presence of FA/H₂O and catalytic effect is favorable for ^·OH + CH₄

The total rate constants for ^·OH + CH₄ and effective rate constants for ^·OH + CH₄ (+HCOOH) and ^·OH + CH₄ (+H₂O) reactions are tabulated in

Table 3. Calculated rate constants ( cm³ molecule⁻¹ s⁻¹) for the ^·OH + CH₄, ^·OH + CH₄ (+HCOOH) and ^·OH + CH₄ (+H₂O).

Temp (K)	kCH4+OH	keff FA	keffW
200	4.6 × 10−16	1.4 × 10−19	3.5 × 10−19
225	8.9 × 10−16	1.8 × 10−20	4.9 × 10−18
250	1.8 × 10−15	3.7 × 10−21	8.2 × 10−18
275	3.5 × 10−15	1.1 × 10−21	2.8 × 10−17
300	6.1 × 10−15	4.2 × 10−22	7.6 × 10−17
325	1.0 × 10−14	2.0 × 10−22	1.8 × 10−16
350	1.6 × 10−14	1.1 × 10−22	3.8 × 10−16
375	2.4 × 10−14	7.4 × 10−23	7.1 × 10−16
400	3.5 × 10−14	5.4 × 10−23	1.3 × 10−15
k = ATn exp(−B/T)	A = 3.0 × 10−29 N = 5.9 B = 134	A = 1.0 × 10−54 N = 9.7 B = −5872	A = 7.4 × 10−18 N = 2.0 B = 2717

and shown in Figure 7.

The result of this study suggests that the catalytic effect of FA and water takes place if the concentration of these molecules were not included, which is unrealistic. Therefore, to calculate the correct reaction rate constants we must include the concentration of HCOOH and H₂O in the final rate constant calculations. In that situation, the results explain that the total effective rate constants for systems ^·OH + CH₄ (+HCOOH) (~7 order) and ^·OH + CH₄ (+H₂O) (2 order) are magnitudes smaller than the free situation. The total effective rate constants ^·OH + CH₄ (+HCOOH) (~4 × 10⁻²² cm³ molecule⁻¹ s⁻¹ at 300 K) and ^·OH + CH₄ (+H₂O) (~7 × 10⁻¹⁷ cm³ molecule⁻¹ s⁻¹ at 300 K) are smaller than ^·OH + CH₄ (~6 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹ at 300 K). Similar results were reported in an earlier study [13,25,26,27,28]. It is clear from geometries of PRCs and TSs, which are different in OH + CH₄ (+HCOOH) than reaction ^·OH + CH₄(+H₂O) resulted in different computed enthalpies and rate constants. Because of that, the kinetics of ^·OH + CH₄ (+HCOOH) are quite different from those ^·OH + CH₄ (+H₂O) reaction systems. In the case of the FA, the rate constants show positive temperature dependence, and in the case of water, the rate constants show negative temperature dependence. This is due to water concentration varying greatly with temperature, whereas HCOOH concentration is nearly temperature independent.

In the case of FA-assisted reaction, the PRC_FA has the larger stability than that of PRC_W, which results in a larger equilibrium constant of CH₄ + HCOOH···HO (4 × 10⁻²¹ at 300 K) than CH₄ + H₂O···HO (1 × 10⁻²¹ at 300 K). The barrier height for FA-assisted reaction is lower than water-assisted reaction results larger rate constants for FA-assisted reaction (3 × 10⁻¹¹ cm³ molecule⁻¹ s⁻¹) than water-assisted reaction (3.3 × 10⁻¹⁴ cm³ molecule⁻¹ s⁻¹). However, the concentrations of the FA are much lower than H₂O in the atmosphere, which then makes the rate constants smaller for FA case (4.2 × 10⁻²² cm³ molecule⁻¹ s⁻¹ 300 K) than water (7.6 × 10⁻¹⁷ cm³ molecule⁻¹ s⁻¹ at 300 K).

In general, the effective rate constants of the FA and water-assisted reaction is smaller than the ^·OH + CH₄ reaction system. As a result, the catalytic effect of FA/H₂O on ^·OH + CH₄ reaction is of minor importance in gas-phase atmospheric reaction chemistry. This result is consistent with previously reported results on similar reaction system [13,25,26,27,28]. To understand the upper troposphere consequences of CH₄, we have calculated the lifetime of CH₄ at 250 K at an altitude of ∼11 km. The averaged concentration of [^·OH] ~ 1 × 10⁶ molecule cm⁻³ and the rate constant (2 × 10⁻¹⁵ cm³ molecule⁻¹ s⁻¹ at 250 K) were used to calculate the lifetime of CH₄ as τ = 1/{[OH] × k}. The calculated lifetime of CH₄ (16 years) is in good agreement with the value reported by Kulongoski et al. (~12–17 years) [42].

As discussed in this study, the rate constants for the effect of FA and water are almost negligible as compared to the naked reaction. It is important to mention that the comparison of effective rate constant with naked reaction does not provide the complete picture for the degradation mechanism of CH₄. Experimental measurement is required to validate the current study. Based on this study, we believe that the effect of FA and water on the formation of formaldehyde will even be slower. We believe the current finding provides better insights into the gas-phase catalytic activity of FA and water molecules on ^·OH + CH₄.

4. Conclusions

The effect of FA and water molecules on ^·OH + CH₄ was explored. The potential energy surfaces for OH + CH₄, CH₄ + HO··· HCOOH, and CH₄ + HO···H₂O have been explored using CC//M06-2X. The rate constants for these reactions were computed using CVT/SCT approach.

In the presence of FA, the two different channels of the hydrogen abstraction reaction were identified. In the case of the water reaction, only one reaction pathway was identified. Under the atmospheric condition, the kinetics of ^·OH + CH₄ (+HCOOH) is quite different from those of ^·OH + CH₄ (+H₂O). This difference is possibly due to the FA concentrations being much lower than H₂O. Our results demonstrate that a FA and water molecule has the potential to catalyze the gas-phase reaction if the atmospheric concentration of these species is not included in the kinetic calculations. However, as suggested, the total effective rate constant must include the concentration of FA and water. In that situation, the total rate is constants for the effect of FA and water than the ^·OH + CH₄ reaction is lower. The present study provides a comprehensive model of how the acidic nature of these catalysts affects the gas-phase reaction kinetics. The atmospheric degradation mechanism suggests that CH₃ can further react with O₂ molecules to form the formaldehyde under atmospheric and combustion conditions. The effect of FA and water molecules could be even slower in the formation of formaldehyde. These kinds of results are interesting and can be used to identify the effect of FA and water on higher chain alkane compounds.