New insights in atmospheric acid-catalyzed gas phase hydrolysis of formaldehyde: a theoretical study

Abstract

The gas phase hydrolysis of HCHO catalyzed via nitric acid and acetic acid, the typical atmospheric acids has been theoretically investigated using M06-2X, CCSD(T), and CCSD(T)-F12A theoretical methods using the 6-311++G(d,p), aug-cc-pVTZ, and VTZ-F12 basis sets and utilizing transition state theory. Our studies predict that when the HNO₃ or CH₃COOH and HCHO⋯H₂O act as reactants, the reactions occur in one step, whereas the reactions of HNO₃⋯H₂O or CH₃COOH⋯H₂O with HCHO proceed via a two-step mechanism. Our results also show that the free energy barrier of the gas phase hydrolysis of HCHO assisted by HNO₃ or CH₃COOH is reduced to 13.95 or 14.27 kcal mol⁻¹ relative to the respective pre-reactive complex from 40.23 kcal mol⁻¹ in the naked HCHO + H₂O reaction. The calculated kinetic data suggests that the HCHO + HNO₃⋯H₂O entrance channel with a two-step mechanism is 1.84–2.76 times faster than HNO₃ + HCHO⋯H₂O with a one-step mechanism, whereas the HCHO⋯H₂O + CH₃COOH entrance path is significantly more favorable than that of HCHO + CH₃COOH⋯H₂O, in the temperature range of 200–300 K. The reaction rates of the gas phase hydrolysis of HCHO catalyzed by HNO₃ or CH₃COOH are much slower than that of the gas phase reaction of HCHO with an OH radical, which demonstrates that the contributions of both catalytic reactions are of minor importance for the sink of HCHO in gas-phase atmospheric chemistry. However, the new findings in this investigation are not only of great necessity and importance for elucidating the gas phase hydrolysis of formaldehyde, but are also of great interest for understanding the importance of other carbonyl compounds in the atmosphere.

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1. Introduction

Aldehydes emitted into the troposphere as primary pollutants from the partial oxidation of natural gas and other alkane-based hydrocarbon fuels^2,3 play an important role in the troposphere.¹ Aldehydes have received scientific and regulatory attention due to their potential adverse health effects on humans as well as their important roles in atmospheric photochemical reactions.^4,5 Formaldehyde as a prototypical aldehyde molecule is the most abundant carbonyl compound in the atmosphere, with concentration levels up to 150 ppb in air-polluted areas,⁶ which is released from the oxidation of biogenic and anthropogenic hydrocarbons.⁷ HCHO is of special concern because it is involved in numerous key atmospheric chemistry processes.^4,8–10 HCHO is also an important source of OH and HO₂ radicals.^7,11 Consequently, HCHO is a key component for understanding of the oxidizing capacity of the atmosphere. Also, formaldehyde makes significant contributions to the formation and growth of secondary organic aerosols (SOA).¹² Furthermore, the aqueous phase chemistry of aldehydes is of great importance in the formation of SOA¹³ because the products of aldehyde hydrolysis act as potential seed molecules for aerosol growth. Therefore, the atmospheric sinks of formaldehyde are of particular concern for fully elucidating the atmospheric processes involving formaldehyde.

Previous investigations have suggested that the major sink processes of HCHO in gas-phase atmospheric chemistry are photolysis^14,15 and reaction with the hydroxyl radical,¹⁶ which produces HO_x radicals.^7,17 The atmospheric lifetime of formaldehyde resulting from its major removal processes is on the order of a few hours in the troposphere.¹⁸ Recently, of great interest is gas phase hydrolysis of HCHO with atmospheric acids as catalysts,^19,20 which could present an additional HCHO sink pathway. The previous studies^21–24 were focused on the hydrolysis of HCHO with the assistance of a water cluster. However, the reaction of HCHO with a single water molecule does not take place in the gas phase due to the presence of a remarkably high barrier.^25,26 Nevertheless, recent experiments^27,28 by Vaida and co-workers on methylglyoxal and ketene have shown that the hydrolysis of aldehydes could occur in the gas phase under conditions corresponding to a water-restricted environment. Very recently, some studies also demonstrated that atmospheric acids can significantly lower the barrier of certain gas-phase reactions,^29–38 and acid catalysis could play an important role in promoting hydrolysis reactions.^29,32,34,39 However, the reaction mechanisms and kinetics of hydrolysis of HCHO are still limited. Therefore, systematic investigation of the hydrolysis of HCHO is of great necessity and importance for elucidating the environmental effects of HCHO.

In the present work, we consider nitric acid as well as acetic acid as catalysts of the gas phase hydrolysis of formaldehyde to result in the formation of methylene glycol and fully estimate the importance of gas phase hydrolysis of HCHO catalyzed by these atmospheric acids. Nitric acid is chosen because the reaction between formaldehyde and nitric acid in solution is likely to take place,⁴⁰ which reflects that formaldehyde plays an important role in the atmosphere for influencing the partitioning between HNO₃ and “odd nitrogen” (NO_x = NO + NO₂). Acetic acid is a typical carboxylic acid in the atmosphere, which is of similar importance as formic acid.⁴¹ Moreover, computational studies have shown that the energy barrier of hydrolysis of formaldehyde in the gas phase is significantly lowered when formic acid acts as a catalyst.²⁰ However, the kinetic data has not been estimated to judge the importance of gas phase hydrolysis of HCHO in the atmosphere. The aim of the present study is to verify whether nitric acid and acetic acid can effectively catalyze the gas phase hydrolysis of formaldehyde and to provide insight into the detailed mechanisms and kinetics of the gas phase hydrolysis of HCHO catalyzed by atmospheric acids. Therefore, the present investigation has wide applications in the understanding of the formation of methylene glycol and gas phase hydrolysis of other carbonyl compounds in the atmosphere.

2. Computational methods

In the present study, geometrical structures of all stationary points are optimized using the M06-2X⁴² method in conjunction with the 6-311++G(d,p)^43,44 basis set, which is performed on the Gaussian 09 program package.⁴⁵ At the same level, the harmonic vibrational frequency calculations establish the nature of the stationary points by confirming that the minimum energy structures have all positive vibrational frequencies and the transition states have only one imaginary frequency. The M06-2X functional has been shown to be sufficiently reliable for predicting geometries and frequencies of the stationary points in the literature.^19,46,47 Furthermore, intrinsic reaction coordinate (IRC) calculations⁴⁸ are also carried out to unambiguously verify that the given transition state connects with the desired reactants and products. To refine the relative energies of the various stationary points, high level single-point energy calculations are executed using the CCSD(T)-F12A theoretical method^49,50 with the VTZ-F12 basis set,⁵¹ based on the M06-2X/6-311++G(d,p) optimized geometries, which is done using the Molpro software.⁵² The simple and efficient CCSD(T)-F12A theoretical method has great improvements in basis set convergence for a wide variety of applications.⁵⁰ Moreover, the results obtained at the CCSD(T)-F12A/VTZ-F12 level are better than that of the conventional CCSD(T)/aug-cc-pV5Z theoretical method.⁵³ In order to determine the reliability of single-determinant-based methods, we have checked the T1 diagnostic⁵⁴ of the CCSD wave function, which is given in Table S1 (ESI†). The T1 diagnostic values of the closed-shell stationary points are less than 0.02, herein, revealing that the CCSD wave functions in this study are reliable. In addition, the single point energy calculations have also been performed at the CCSD(T)⁵⁵/aug-cc-pVTZ^56,57 level of theory to compare the accuracy of the conventional CCSD(T)/aug-cc-pVTZ method. All the calculated reaction profiles are reported in terms of Gibbs free energies at 298 K and 1 atm to account for the entropic effects associated with the complexation of the formaldehyde, water and catalyst. Finally, the rate constant for every elementary step is calculated using conventional transition state theory with Eckart tunneling correction⁵⁸ at the CCSD(T)-F12A/VTZ-F12//M06-2X/6-311++G(d,p) level by means of The Rate Code.⁵⁹ Conventional transition state theory has been widely utilized in studying atmospheric reactions in the literature.^35,60–63

3. Results and discussion

The calculated relative energies (designated by ΔE at 0 K), enthalpies (designated by ΔH at 298 K), and the Gibbs free energies (designated by ΔG at 298 K) of all species for reactions HCHO + OH assisted by HNO₃ and CH₃COOH are listed in Table 1.

Table 1 The enthalpies, Gibbs free energies, and relative energies of all species for the reactions HCHO + H₂O + HNO₃ and HCHO + H₂O + CH₃COOH with zero-point energy correction included (in kcal mol⁻¹)

Species	ΔH^a	ΔG^a	ΔE^a	ΔH^b	ΔG^b	ΔE^b
a ΔH, ΔG, and ΔE are obtained at the CCSD(T)-F12A/VTZ-F12//M06-2X/6-311++G(d,p) level of theory.b ΔH, ΔG, and ΔE are calculated at the CCSD(T)/aug-cc-pVTZ//M06-2X/6-311++G(d,p) level of theory.
HCHO + H2O + HNO3	0.00	0.00	0.00	0.00	0.00	0.00
HCHO⋯H2O + HNO3	−1.75	4.91	−1.67	−1.84	4.81	−1.77
HCHO + HNO3⋯H2O	−8.65	0.00	−8.18	−8.87	−0.22	−8.41
CR1a	−13.50	5.14	−12.93	−14.36	4.28	−13.78
TS1a	−9.45	9.10	−8.97	−10.15	8.40	−9.67
CR1b	−12.99	5.50	−12.46	−13.71	4.79	−13.18
TS1b	−2.58	19.45	−0.42	−3.46	18.56	−1.30
CP1	−21.24	0.23	−19.36	−21.87	−0.40	−19.99
H2C(OH)2 + HNO3	−10.16	0.69	−8.36	−10.08	0.77	−8.27
HCHO + H2O + CH3COOH	0.00	0.00	0.00	0.00	0.00	0.00
HCHO⋯H2O + CH3COOH	−1.75	4.91	−1.67	−1.84	4.81	−1.77
HCHO + CH3COOH⋯H2O	−8.56	0.53	−7.97	−8.70	0.40	−8.11
CR2a	−10.98	8.05	−10.43	−11.66	7.37	−11.11
TS2a	−9.84	9.08	−9.10	−10.48	8.44	−9.74
CR2b	−13.61	4.80	−13.08	−14.31	4.09	−13.79
TS2b	−3.63	19.07	−1.26	−4.54	18.16	−2.16
CP2	−22.07	−0.58	−20.21	−22.60	−1.11	−20.75
H2C(OH)2 + CH3COOH	−10.16	0.69	−8.36	−10.08	0.77	−8.27

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3.1 The hydrolysis of HCHO catalyzed by HNO₃

When nitric acid as a catalyst was introduced into the reaction of HCHO with H₂O, two possible entrance channels, HCHO⋯H₂O + HNO₃ and HCHO + HNO₃⋯H₂O, were considered, which are presented in Fig. 1 at the CCSD(T)-F12A/VTZ-F12//M06-2X/6-311++G(d,p) level. The formaldehyde–water complex has been widely studied in the literature.^64–66 The binding free energy of HCHO⋯H₂O herein is calculated to be 4.91 kcal mol⁻¹, which is higher than that of the reported values⁶⁵ due to the fact that the complex corresponding to the entry channel is not a global minimum but a local minimum. The structure and thermodynamics values of the global minimum (HCHO⋯H₂O_global) are given in Fig. S1 and Table S2 (ESI†). The binding free energy of HCHO⋯H₂O_global is 3.99 kcal mol⁻¹, which shows good agreement with the reported value of 3.60 kcal mol⁻¹ (ref. 65) at the CCSD(T)/6-311++G(d,p)//MP2/6-311++G(d,p) level. In addition, it is noted that the effects of basis sets CCSD(T)-F12A/VTZ-F12 and CCSD(T)/aug-cc-pVTZ are not obvious in the HCHO⋯H₂O complex because the difference between the two methods is only about 0.1 kcal mol⁻¹ as shown in Table 1.

Fig. 1 Free energy profile for the HCHO + H₂O + HNO₃ → H₂C(OH)₂ + HNO₃ reaction calculated at the CCSD(T)-F12A/VTZ-F12//M06-2X/6-311++G(d,p) level of theory (in kcal mol⁻¹).

When the HNO₃ and the formed HCHO⋯H₂O complex act as reactants, the reaction proceeds via the pre-reactive complex CR1b (Fig. 1) with a computed binding free energy of 5.50 kcal mol⁻¹ at the CCSD(T)-F12A/VTZ-F12//M06-2X/6-311++G(d,p) level relative to the free reactants formaldehyde, water, and nitric acid as shown in Table 1. In complex CR1b, it is noted that the difference between CCSD(T)-F12A/VTZ-F12 and CCSD(T)/aug-cc-pVTZ is about 0.7 kcal mol⁻¹. Therefore, the discussion below is based on the CCSD(T)-F12A/VTZ-F12//M06-2X/6-311++G(d,p) level. The pre-reactive complex CR1b is a ternary complex, which is held together by two hydrogen bondings and one van der Waals interaction. The first hydrogen bonding is formed between one oxygen atom of nitric acid and one hydrogen atom of water. The second hydrogen bonding is formed between the hydrogen atom of HNO₃ and the oxygen atom of HCHO, whereas the van der Waals interaction takes place between the oxygen atom of water and the carbon atom of HCHO. Starting from the complex CR1b, the reaction undergoes the transition state TS1b prior to the formation of post-reactive complex CP1, and subsequently forms methylene glycol and nitric acid, where nitric acid is released. In the transition state TS1b, the hydrogen atom from the nitric acid is transferred to the oxygen atom of the HCHO moiety, the hydrogen atom of water is migrated to the nitric acid moiety, and the hydroxy of water is added to the carbon atom of the HCHO moiety. The free energy barrier height of the hydrolysis of HCHO with the assistance of nitric acid is 13.95 kcal mol⁻¹ relative to the complex CR1b (HCHO⋯H₂O⋯HNO₃), which is about 26.28 kcal mol⁻¹ or 12.52 kcal mol⁻¹ lower than those of the corresponding reaction¹⁹ HCHO with water or water dimer, respectively. Thus, nitric acid is a strong catalyst to facilitate the formation of methylene glycol via its ability to reduce the barrier for the hydrolysis of formaldehyde. It is worth mentioning that the free energy barrier for the HNO₃-assisted HCHO hydrolysis reaction is 5.70 kcal mol⁻¹ higher than that of the H₂SO₄-assisted reaction,¹⁹ while the calculated energy barrier of 12.04 kcal mol⁻¹ for HNO₃-catalyzed reaction is 1.24 kcal mol⁻¹ above that for the HCOOH-assisted reaction.²⁰ The complex CP1 is stabilized by two hydrogen-bonded interactions, which is about 0.46 kcal mol⁻¹ more stable than the products.

The water molecule interaction with nitric acid leads to the formation of the HNO₃⋯H₂O complex, which has been extensively investigated in the literature.^67–69 The binding energy of our most stable computed HNO₃⋯H₂O complex is −8.18 kcal mol⁻¹, which is consistent with the reported values of −8.45 kcal mol⁻¹ (ref. 68) and −8.1 kcal mol⁻¹.⁶⁹ The reaction of HCHO + HNO₃⋯H₂O entry channel proceeds through the formation of the pre-reactive complex CR1a and the transition state TS1a before the formation of complex CR1b, and the subsequent unimolecular conversion to nitric acid and methylene glycol, which is similar to the nitric acid-catalyzed hydrolysis of SO₃.⁷⁰ The complex CR1a is computed to be 5.14 kcal mol⁻¹ above the reactants HCHO + HNO₃⋯H₂O in Table 1. The complex CR1a which is stabilized via two hydrogen bonds and one van der Waals interaction is rearranged into its isomer CR1b through TS1a with a the barrier of about 3.96 kcal mol⁻¹ relative to the complex CR1a. Note that CR1a is more stable than CR1b by 0.36 kcal mol⁻¹. In addition, the free energy of TS1b lies 14.54 kcal mol⁻¹ above the reactants HCHO⋯H₂O + HNO₃ and 19.45 kcal mol⁻¹ above the reactants HCHO + HNO₃⋯H₂O, which indicates the entrance channel relative to the HCHO⋯H₂O + HNO₃ reactants is more favored than the HCHO + HNO₃⋯H₂O entry pathway. However, due to the stronger binding of the HNO₃⋯H₂O complex, its concentration of HNO₃⋯H₂O complex is higher than that of the HCHO⋯H₂O complex in the atmosphere. Therefore, further kinetic studies are necessary to determine which entry channel is dominant in the hydrolysis of HCHO catalyzed by nitric acid.

3.2 The hydrolysis of HCHO catalyzed by acetic acid

The free energy profile for the hydrolysis of HCHO catalyzed by CH₃COOH is depicted in Fig. 2. The optimized geometries of all the stationary points are also shown in Fig. 2. The calculated results show that the hydrolysis of HCHO in the presence of CH₃COOH has similar features with the HNO₃-assisted reaction. From Fig. 2, there are still two entry pathways through which the CH₃COOH-assisted hydrolysis of HCHO can take place. The reactants could be either the HCHO⋯H₂O complex and CH₃COOH or HCHO and the CH₃COOH⋯H₂O complex. The binding free energy of the CH₃COOH⋯H₂O complex at CCSD(T)-F12A/VTZ-F12//M06-2X/6-311++G(d,p) level is 0.53 kcal mol⁻¹, which indicates that the formation of the CH₃COOH⋯H₂O complex is 4.38 kcal mol⁻¹ more favorable than that of HCHO⋯H₂O complex.

Fig. 2 Free energy profile for the HCHO + H₂O + CH₃COOH → H₂C(OH)₂ + CH₃COOH reaction calculated at the CCSD(T)-F12A/VTZ-F12//M06-2X/6-311++G(d,p) level of theory (in kcal mol⁻¹).

Regarding the reaction of HCHO with the CH₃COOH⋯H₂O complex, the pre-reactive complex CR2a is formed with free energy of about 7.52 kcal mol⁻¹ with respect to the HCHO + CH₃COOH⋯H₂O. The HCHO + CH₃COOH⋯H₂O reaction occurs via transition state TS2a to form a complex CR2b (HCHO⋯H₂O⋯CH₃COOH) from CR2a with a barrier of 1.03 kcal mol⁻¹ relative to the pre-reactive complex CR2a. This step involves a geometric rearrangement that plays a crucial role in the HCHO + CH₃COOH⋯H₂O reaction. CR2b is better stabilized than CR2a by 3.25 kcal mol⁻¹. When CH₃COOH and the formed HCHO⋯H₂O complex are regarded as reactants, the CR2b that is firstly formed undergoes unimolecular isomerization via an eight-member ring cyclic transition state TS2b to form the post-reactive complex, H₂C(OH)₂⋯CH₃COOH, in the exit channel before the release of the products H₂C(OH)₂ and CH₃COOH. In the process, the C–O single bond of the C–O–H in CH₃COOH is converted to a C=O double bond as the hydrogen atom is transferred to the oxygen atom of HCHO, the C=O double bond of the CH₃COOH moiety becomes a single bond upon accepting the hydrogen atom from H₂O, and simultaneously the OH group in water is added to the carbon atom in HCHO as characterized in TS2b of Fig. 2. The free energy barrier for the HCHO⋯H₂O⋯CH₃COOH → H₂C(OH)₂⋯CH₃COOH step is 14.27 kcal mol⁻¹, which is 0.32 kcal mol⁻¹ higher than that of the corresponding step of the HNO₃-catalyzed hydrolysis of HCHO. It is seen in Fig. 2 that the barrier between the HCHO⋯H₂O + CH₃COOH reactants and the transition state TS2b is 14.16 kcal mol⁻¹. Therefore, the acetic acid-catalyzed reaction is expected to be an energetically favorable mechanism for the formation of methylene glycol. In addition, the free energy barrier of TS2b is 18.54 kcal mol⁻¹ relative to the HCHO + CH₃COOH⋯H₂O reactants. Thus, the barrier associated with the HCHO + CH₃COOH⋯H₂O channel is computed to be higher than that for the HCHO⋯H₂O + CH₃COOH pathway. This is on account of the stronger binding free energy of the CH₃COOH⋯H₂O complex as compared to that of the HCHO⋯H₂O dimer.

The hydrolysis of HCHO with the aid of formic acid is also reinvestigated at the same level mentioned above to compare the catalytic abilities of different acids. The relative energies and the corresponding thermodynamic values of all stationary points are provided in Table S3 (ESI†). The corresponding free energy profile is displayed in Fig. S2 (ESI†). From Fig. S2,† there are still two entrance channels. One entrance channel involving the HCHO⋯H₂O + HCOOH reactants proceeds via a one-step process, which is consistent with the reports in the literature.²⁰ The second is that starting with the HCHO + HCOOH⋯H₂O entry channel, the reaction is found to occur by a stepwise mechanism that needs to go through two transition states prior to the formation of H₂C(OH)₂, which is different than the previous reports.²⁰ When the HCOOH⋯H₂O complex is attacked by HCHO, the CR3a is produced with the formation of hydrogen bond between the oxygen atom in H₂O and the carbon atom in HCHO. Then, the next step leads to the formation of the common complex of two entry channels, CR3b, which is more stable than CR3a by 3.86 kcal mol⁻¹. The process occurs through the TS3a transition state, which lies 0.55 kcal mol⁻¹ above CR3a. We also find that the free energy barrier of the HCOOH-assisted reaction is 13.92 kcal mol⁻¹ and nearly identical to that of the HNO₃-assisted reaction. Herein, we revise the mechanism of the HCOOH-assisted reaction,²⁰ which is also found in the HNO₃ and CH₃COOH catalyzed reactions discussed above. In addition, the hydrolysis of glyoxal, (HCO)₂ catalyzed by nitric acid or acetic acid is also studied theoretically at the M06-2X level in conjunction with the 6-311++G(d,p) basis set, and the schematic free energy diagrams are given in Fig. S3 and S4 (ESI†). Our research shows that the hydrolysis of glyoxal catalyzed via the atmospheric acids presents a similar reaction mechanism to the hydrolysis of formaldehyde. Therefore, the new mechanism in this study has not only provided a clear understanding of the rate constant of the corresponding channel but also provides new insight into the gas phase hydrolysis of various carbonyl compounds catalyzed by atmospheric acids.

3.3 Kinetics and potential atmospheric impact

Given the ability of HNO₃ and CH₃COOH to lower the barrier for HCHO hydrolysis and their potential to play an important role in the formation of atmospheric H₂C(OH)₂, the rate constant is calculated using conventional transition-state theory to examine the possible atmospheric impacts of the reactions investigated in this work. The computed data are summarized in Table 2. For the case of formaldehyde hydrolysis catalyzed by a single nitric acid molecule, HNO₃ can participate in the atmospheric hydrolysis of HCHO either by directly colliding with a HCHO⋯H₂O complex or by having a HCHO molecule collide with a HNO₃⋯H₂O complex. The two pathways can be expressed as

HCHO + H₂O + HNO₃ ⇌ HCHO⋯H₂O + HNO₃ (1)

HCHO⋯H₂O + HNO₃ ⇌ HCHO⋯H₂O⋯HNO₃ (2)

HCHO⋯H₂O⋯HNO₃ → H₂C(OH)₂⋯HNO₃ (3)

and

HCHO + H₂O + HNO₃ ⇌ HCHO + HNO₃⋯H₂O (4)

HCHO + HNO₃⋯H₂O ⇌ HCHO⋯H₂O⋯HNO₃ (5)

HCHO⋯H₂O⋯HNO₃ → H₂C(OH)₂⋯HNO₃ (6)

Table 2 Calculated equilibrium constants (molecules per cm³) for the formation of the complexes HCHO⋯H₂O, HNO₃⋯H₂O and CH₃COOH⋯H₂O, the rate constants (cm³ mol⁻¹ s⁻¹) for the individual reaction pathway and the ratio of rate expressions of the two channels for each catalytic reaction for the temperature range 200−300 K

Reaction	200 K	220 K	240 K	260 K	280 K	298 K	300 K
KHCHO⋯H2O	6.75 × 10^−23	4.69 × 10^−23	3.53 × 10^−23	2.81 × 10^−23	2.35 × 10^−23	2.06 × 10^−23	2.04 × 10^−23
KHNO3⋯H2O	3.73 × 10^−17	5.52 × 10^−18	1.14 × 10^−18	3.02 × 10^−19	9.78 × 10^−20	4.07 × 10^−20	3.71 × 10^−20
KCH3CHOOH⋯H2O	1.40 × 10^−17	2.13 × 10^−18	4.48 × 10^−19	1.21 × 10^−19	3.97 × 10^−20	1.66 × 10^−20	1.52 × 10^−20
k1	7.80 × 10^−18	7.91 × 10^−18	8.00 × 10^−18	8.07 × 10^−18	8.12 × 10^−18	8.16 × 10^−18	8.15 × 10^−18
k^′1	3.90 × 10^−23	1.67 × 10^−22	5.62 × 10^−22	1.57 × 10^−21	3.82 × 10^−21	7.67 × 10^−21	8.24 × 10^−21
k2	3.50 × 10^−17	2.83 × 10^−17	2.36 × 10^−17	2.02 × 10^−17	1.75 × 10^−17	1.57 × 10^−17	1.55 × 10^−17
k^′2	8.01 × 10^−26	5.37 × 10^−25	2.63 × 10^−24	1.01 × 10^−23	3.23 × 10^−23	8.02 × 10^−23	8.82 × 10^−23
Ratio1	2.76	2.48	2.26	2.09	1.96	1.85	1.84
Ratio2	4.74 × 10^−4	8.61 × 10^−4	1.42 × 10^−3	2.16 × 10^−3	3.10 × 10^−3	4.13 × 10^−3	4.25 × 10^−3

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Assuming that the pre-reactive complex is in equilibrium with the reactants and according to the steady-state conditions, similar to the formation of H₂SO₄,⁷¹ the rate constant of the HCHO⋯H₂O + HNO₃ pathway can be represented as

where K_HCHO⋯H2O stands for the equilibrium constant of the formation of the starting HCHO⋯H₂O reactant complex from isolated HCHO and H₂O, k₂ and k₋₂ denote the forward and reverse rate coefficients for step (2), and [HCHO], [H₂O], and [HNO₃] are the typical tropospheric concentrations of HCHO, H₂O and HNO₃, respectively. For the path associated with the HCHO + HNO₃⋯H₂O reactants, analogous to eqn (7), the rate expression for H₂C(OH)₂ formation can be written as

Because the existence of TS1a makes the kinetics of the reaction channel more complicated, the k₆ is computed using eqn (9) according to the unified statistical model.⁷²

Moreover, the and are regarded as equilibrium constants for reaction steps 2 and 5 denoted as K^′₂ and K^′₅, respectively. In order to explore that which pathway is more feasible and important, the ratio of the rate expressions for the two pathways can be written as follows:

When the concentrations of these species in the atmosphere are not considered and the HCHO⋯H₂O + HNO₃ and HCHO + HNO₃⋯H₂O are treated as the reactants, the rate constants of the two reactions expressed as k₁ and k^′₁ can be calculated from the following equation:

k₁ = K^′₂k₃ (11)

k^′₁ = K^′₅k₆ (12)

Consequently, ratio1 can be compactly rewritten as

As shown in Table 2, ratio1 ranges from 2.76 to 1.84 between 200 and 300 K. Although the rate constant k^′₁ of the reaction HCHO + HNO₃⋯H₂O is smaller than the other by 3–5 orders of magnitude due to the higher barrier, HCHO hydrolysis via the HCHO + HNO₃⋯H₂O pathway is faster than that of the HCHO⋯H₂O + HNO₃ reaction because the formation of the HNO₃⋯H₂O complex is more favorable than the HCHO⋯H₂O complex since the equilibrium constants of HNO₃⋯H₂O is larger than that of HCHO⋯H₂O by 3–6 orders of magnitude.

Analogous to the HNO₃ catalytic reaction discussed above, there are two main pathways for the reaction of HCHO + H₂O assisted by CH₃COOH, which are shown as

HCHO + H₂O + CH₃COOH ⇌ HCHO⋯H₂O + CH₃COOH (14)

HCHO⋯H₂O + CH₃COOH ⇌ HCHO⋯H₂O⋯CH₃COOH (15)

HCHO⋯H₂O⋯CH₃COOH → H₂C(OH)₂⋯CH₃COOH (16)

and

HCHO + H₂O + CH₃COOH ⇌ HCHO + CH₃COOH⋯H₂O (17)

HCHO + CH₃COOH⋯H₂O ⇌ HCHO⋯H₂O⋯CH₃COOH (18)

HCHO⋯H₂O⋯CH₃COOH → H₂C(OH)₂⋯CH₃COOH (19)

If we apply a kinetic analysis to the CH₃COOH-assisted mechanism that is similar to that applied to HNO₃, the ratio of the rate expressions for the two pathways of the CH₃COOH-catalyzed reaction can be theoretically obtained using the formula

From Table 2, we estimate that ratio2 is 4.74 × 10⁻⁴ at 200 K, and increases to 4.25 × 10⁻³ at 300 K. The rate constant k₂ of the reaction HCHO⋯H₂O + CH₃COOH is far larger than the other by 6–9 orders of magnitude, which exceeds the effect of the equilibrium constants of HCHO⋯H₂O and CH₃COOH⋯H₂O. Therefore, the hydrolysis of HCHO via the stepwise reaction pathway is slower than the other by 3–4 orders of magnitude. For the HCOOH-assisted hydrolysis of HCHO, the kinetic results are listed in Table S4 (ESI†). Our calculations predict that the HCOOH-catalyzed hydrolysis of HCHO occurring through the HCHO + HCOOH⋯H₂O reaction channel is much slower than that of the HCHO⋯H₂O + HCOOH channel, given that ratio3 is 6.66 × 10⁻⁵–1.52 × 10⁻³ in the temperature range of 200–300 K.

To compare the catalytic effects of nitric, acetic and formic acids on the HCHO + H₂O reaction under atmospheric conditions, the rates of the dominated channel of each catalytic reaction are compare. The ratio of the rates of the HCHO⋯H₂O + CH₃COOH and HCHO + HNO₃⋯H₂O paths are expressed in eqn (21).

The atmospheric concentrations of nitric acid⁷³ and carboxylic acids⁷⁴ are 1.16 × 10¹⁰ and 1 × 10¹¹ molecules per cm³. Then, taken together, the three factors appearing in eqn (21) combine to give a ratio of the rates which ranges between 8.9 and 14.0, in favor of the CH₃COOH-catalyzed reaction. As for HCHO⋯H₂O + HCOOH and HCHO⋯H₂O + CH₃COOH pathways, the results indicate that the HCOOH-catalyzed reaction is 2.0–3.6 times faster than that of CH₃COOH as a catalyst. In addition, the hydrolysis of formaldehyde catalyzed by H₂SO₄ (ref. 19) is much faster than those of the HNO₃, HCOOH, and CH₃COOH-catalyzed reactions, indicating that H₂SO₄ has the strongest catalytic effect on the gas phase hydrolysis of HCHO.

The gas phase concentration of water⁶⁵ is about 3.97 × 10¹⁷ molecules per cm³ in the atmosphere, corresponding to 50% of relative humidity. In addition, considering the rate constant of the hydrolysis of HCHO with the assistance of water,¹⁹ and the equilibrium constants for the formation of the complex H₂O⋯H₂O,²⁹ this simple relative rate analysis suggests that the hydrolysis of HCHO catalyzed by HNO₃ or CH₃COOH is more favorable and much faster than the water-assisted hydrolysis. At 298 K, the rate constants of the reactions for the HCHO⋯H₂O complex with nitric acid and acetic acid are 8.16 × 10⁻¹⁸ and 1.57 × 10⁻¹⁷ cm³ per molecule per s, which is 10⁻⁶ and 10⁻⁵ times lower than that of the reaction of HCHO + OH (about 10⁻¹² cm³ mol⁻¹ s⁻¹).^75,76 Moreover, a very small amount of HCHO⋯H₂O complex can be formed under atmospheric conditions. For example, the HCHO⋯H₂O complex accounts for only 0.024% of the original HCHO concentration at 298 K.⁷⁷ It is noted that the rate constants of the path associated the HCHO + HNO₃⋯H₂O reactants vary from 3.90 × 10⁻²³ to 8.24 × 10⁻²¹ cm³ mol⁻¹ s⁻¹ with the temperature range of 200 to 300 K. Although the path is approximately 2 times faster than the other and the concentration of HNO₃ is more than that of the OH radical (10⁶ molecules per cm³) in the atmosphere, the reaction of HCHO with the formed HNO₃⋯H₂O complex is far slower than the reaction with OH due to the low reaction rate. Analogously, starting from the HCHO + CH₃COOH⋯H₂O reactants, the reaction is very slow and cannot compete with the reaction HCHO + OH. As a consequence, the hydrolytic process of formaldehyde catalyzed by nitric acid or acetic acid is of minor importance for the sink of HCHO in gas phase atmospheric chemistry. However, the present findings are of atmospheric importance as the results provide new insights into how methylene glycol might be formed in the atmosphere, which is significant for understanding the growth of SOA in water-restricted environments.

4. Conclusions

In this work, the gas phase hydrolysis of formaldehyde catalyzed by nitric acid or acetic acid to form H₂C(OH)₂ is theoretically investigated at the CCSD(T)-F12A/VTZ-F12//M06-2X/6-311++G(d,p) and CCSD(T)/aug-cc-pVTZ//M06-2X/6-311++G(d,p) levels. The conclusions obtained in this investigation are summarized as follows:

(1) As for both catalytic reactions, there are two entry channels, respectively. Starting with HCHO⋯H₂O + HNO₃ or CH₃COOH reactants, the reactions correspond to one-step processes. Regarding the HCHO + HNO₃⋯H₂O and HCHO + CH₃COOH⋯H₂O channels, the reactions begin with a complex pre-reactive region with two hydrogen-bonded complexes, which are connected through the transition states TS1a or TS2a with the reorganization of the hydrogen bonds. This geometrical rearrangement is necessary to prepare the subsequent HCHO⋯H₂O⋯HNO₃/CH₃COOH → H₂C(OH)₂⋯HNO₃/CH₃COOH unimolecular isomerization. Therefore, the reactions HNO₃⋯H₂O and CH₃COOH⋯H₂O with HCHO occur by a two-step mechanism responsible for the formation of H₂C(OH)₂.

(2) For the HNO₃- and CH₃COOH-catalyzed HCHO hydrolysis, the free energy barrier heights for the step involving unimolecular isomerization of the collision complex, HCHO⋯H₂O⋯HNO₃/CH₃COOH → H₂C(OH)₂⋯HNO₃/CH₃COOH, are 13.95 and 14.27 kcal mol⁻¹, respectively, which is greatly lower than the 40.23 and 26.47 kcal mol⁻¹ barriers¹⁹ for the corresponding step in the naked reaction and the water-assisted reaction. Therefore, our results show that HNO₃ and CH₃COOH have an important catalytic effect and, through their ability to facilitate intermolecular hydrogen atom transfer, are considerably more efficient than an equal number of water molecules as a catalyst in the hydrolysis process.

(3) The relative rate analysis suggests that for the HNO₃-catalyzed reaction, the path associated with the HCHO + HNO₃⋯H₂O reactants is 2.76–1.84 times faster than the other, while for the CH₃COOH-catalyzed reaction, the HCHO + CH₃COOH⋯H₂O pathway is slower than the other by 3–4 orders of magnitude over the temperature range of 200–300 K. In addition, the catalytic effect of HCOOH for gas phase hydrolysis of formaldehyde is slightly larger than the HNO₃ and CH₃COOH. Furthermore, the reaction rates are much smaller for the hydrolysis of HCHO in the presence of HNO₃ or CH₃COOH than that of the reaction HCHO + OH, which means that the contributions of both catalytic reactions are of no account for the degradation of HCHO.

This present study presents a new mechanism for understanding the formation of methylene glycol via HNO₃ or CH₃COOH catalysis. Moreover, the new mechanisms have wide applications in investigating the gas phase hydrolysis of various carbonyl compounds responsible for the formation of atmospheric diols. Therefore, the present results could also be of great importance in elucidating and understanding other atmospheric oxidation processes with acids as catalysts.

Acknowledgements

The authors thank Prof. Ellen Mitchell from Bridgewater College for improving the language and re-writing the manuscript. This work is supported by the National Natural Science Foundation of China (Grant no. 41165007), the Science and Technology Foundation of Guizhou Province, China (no. [2011]2107 and [2012]2189), the Science and Technology Foundation of Guizhou Province & Guizhou Minzu University (no. [2014]7380), and the Open Research Fund of Key Laboratory of Atmospheric Composition and Optical Radiation, Chinese Academy of Sciences, China (JJ1107).

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Footnotes

† Electronic supplementary information (ESI) available: The T1 diagnostic values of all species for the reactions HCHO + H₂O catalyzed by HNO₃, CH₃COOH and HCOOH are listed in Table S1. The enthalpies, Gibbs free energies, and relative energies of the various configurations for the HCHO⋯H₂O, HNO₃⋯H₂O, CH₃COOH⋯H₂O, HCHO⋯H₂O⋯HNO₃, and HCHO⋯H₂O⋯CH₃COOH complexes are shown in Table S2, and the optimized structures of these complexes are displayed in Fig. S1. The enthalpies, Gibbs free energies, and relative energies of all species for the reaction HCHO + H₂O + HCOOH are summarized in Table S3 while and the free energy profile for the reaction is shown in Fig. S2. The free energy profiles for the reactions (HCO)₂ + H₂O catalyzed by HNO₃ and CH₃COOH are depicted in Fig. S3 and S4. Table S4 reports the rate constants of the formic acid-catalyzed hydrolysis of HCHO. The Cartesian coordinates, vibrational frequencies, and total energies of all species for the reactions HCHO + H₂O + HNO₃, HCHO + H₂O + CH₃COOH and HCHO + H₂O + HCOOH are provided in Table S5. The atomic Cartesian coordinates of the various configurations for the HCHO⋯H₂O, HNO₃⋯H₂O, CH₃COOH⋯H₂O, HCHO⋯H₂O⋯HNO₃, and HCHO⋯H₂O⋯CH₃COOH complexes and all species for the reactions (HCO)₂ + H₂O + HNO₃ and (HCO)₂ + H₂O + CH₃COOH are given in Tables S6 and S7. See DOI: 10.1039/c5ra04118j

‡ Fang-Yu Liu and Xing-Feng Tan contributed equally.

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