Introduction

Mercury (Hg) is a volatile heavy metal pollutant [1] with trace amounts in coal [2], and it is one of the sources of global atmospheric pollutants which causes serious damage to the environment because the large amount of coal combustion releases mercury in the flue gas. Many efforts have been devoted to investigating the atmospheric chemistry of mercury and its transformation via the oxidation reactions of Hg(0) with reactive species [3]. For instance, Peterson and coworkers have systematically studied the reactions of Hg with a series of small halogen-containing molecules including Cl2, Br2, BrCl, ClO, BrO, I2, IBr, ICl and IO [4,5,6]. Subsequently, they theoretically investigated the collision-induced dissociation and recombination of Hg and Br atoms [7]. Auzmendi-Murua and Bozzelli paid much attention on the emissions of gaseous Hg from combustion sources and their control [8], and they used a mechanism to explain the gas-phase Hg conversion in H2, O2, chloro C1-hydrocarbon, and NOx combustion effluent. Then they reported that Hg present in the flue gas in coal burning can be oxidized by the addition of halogens (Cl, Br, I), and they proposed the detailed reaction mechanism and discussed the influence of different air–fuel equivalence ratios [9]. Wilcox and coworkers performed theoretical work in predicting rate constants for Hg oxidation reactions by hydrogen chloride and chloride [10,11,12], and for the decomposition of HgCl2 [13] that may occur in the flue gases of coal combustion [14]. Dibble and coworkers also extensively explored the oxidation of Hg in the global atmosphere [15,16,17,18,19,20]. Saiz-Lopez et al. suggested that photoreduction of gaseous oxidized Hg(II) affects the global atmospheric Hg speciation, transport and deposition [21]. Different types of Mercury bonding were investigated by Cremer and coworkers, and their bond dissociation energies were reported which can provide useful data for the chemistry of Hg and reactions of elemental Hg in the atmosphere [22].

Besides Hg, the flue gas generated from coal combustion contains many compounds including the fluorine content [23, 24] due to the combustion of high fluorine-containing coals. Hydrogen fluoride (HF) can be generated during the thermal treatment of coals [25, 26] and it is an extremely toxic, corrosive gas at room temperature and normal pressure [27]. To know the fluorine gas species in flue gas, the pyrolysis of fluoroborate residue and proportions of three main fluorine gases (i.e., SiF4, BF3, and HF) at different temperatures was reported by Feng and coworkers [28], and they suggested that the content of HF increases continuously with the temperature, meaning that HF is an important form during coal processing F. Previous study showed that the halogen acids like HCl [29], HBr and HI have important effects on mercury conversion [2] and especially HBr and HI can oxidize more than 85% of the gas phase mercury at a low concentration of 2 ppm. The reactions of other species like chlorine [10], bromine [30], iodine [5], chlorine hydroxide [31], bromine hydroxide [5, 6], nitrogen oxides (NOx) and sulfur oxides (SOx) [9] with Hg would also occur and these reactions have also been studied. Therefore, the trace amount Hg could react with other components in the flue gas like HF. It stimulates our great interest in uncovering the influence of HF on the oxidation reaction of Hg and the detailed mechanism.

To shed light on the reaction processes of HF and Hg and the mechanism, we aim at using quantum chemical calculations to provide the information about the reactant, transition state, intermediate and product in the chemical reaction. It is shown that quantum chemical calculations are powerful in explaining the reaction mechanism [32]. Previously, several theoretical studies have been devoted to the reaction mechanism between mercury and chlorine containing gases or HF during coal combustion. For instance, Liu et al. theoretically investigated the reaction of hydrochloric acid (i.e., HCl) with Hg and elaborated the reaction mechanism [33]. Subsequently, Liu and co-workers investigated the reaction mechanism of Hg and HF during coal combustion and obtained the reaction kinetic parameters [34]. Gao and co-workers considered the effects of temperature and pressure on the Hg/HF reaction in coal combustion [35]. Noticeably, in these previous studies, they did not considered the initial reaction of HF (or HCl) and Hg to form the complex of HF∙∙∙Hg (or HCl∙∙∙Hg) because the strong electronegativity of F (or Cl) atom. Accordingly, in the present work, quantum chemical calculations were employed to thoroughly explore the oxidation reaction processes of Hg by HF and the reaction mechanism, the geometries of the transition states and intermediates in the reaction processes will be discussed, the detailed mechanism and the kinetics of the reactions will be illustrated.

Computational details

The geometric structures of Hg, HF, transition state, intermediate and product were optimized using the M06-L functional [36, 37] corrected with the Grimme’s dispersion (D3) [38] because of the importance of dispersion interaction in weak interaction system [39,40,41], and the 6–311 +  + G(2d,2p) basis set was used for the H and F atoms while the Hg atom was treated using the relativistic ECP60MDF pseudopotential of the Stuttgart group [42] together with cc-pwCVnZ-PP [43] valence basis set. The counterpoise method proposed by Boys and Bernardi [44] was used to consider the basis set superposition error (BSSE) for the complex formed in the reaction. Vibrational frequency calculations were conducted at the same level of theory to ensure that the optimized reactants, intermediates and product have no imaginary frequencies while the transition state has only one imaginary frequency. Intrinsic reaction coordinate (IRC) [45] calculations at the above level of theory were carried out to confirm that the transition state obtained was true. All the calculations were performed via the Gaussian16 program [46]. Quantum theory of atoms in molecules (QTAIM [47,48,49]) was employed to analyze the bond critical points (BCPs) in the complex and intermediate using the Multiwfn software [50]. The relative Gibbs free energy (i.e., ΔG) is calculated as the difference between the Gibbs free energy of the complex (or the transition state or intermediate or product) and the sum of Gibbs free energies of the reactants. The interaction energy (i.e., ΔE) of the complex is calculated as the energy difference between the total energy of the complex (or the intermediate) and the sum of energies of its components. The interaction energy of the complex after considering BSSE is termed as ΔEcorr, which is calculated as the sum of ΔE and EBSSE (i.e., ΔEcorr = ΔE + EBSSE).

The reaction occurs between Hg and HF at the different temperatures from room temperature to 1000 K was investigated, which involves the following reactions (i.e., reaction formulas Eqs. 1 and 2):

$$\text{HF}+\text{Hg}\to \text{H}+\text{HgF}$$
(1)
$$\text{HF}+\text{HgF}\to \text{H}+{\text{HgF}}_{2}$$
(2)

In terms of transition state theory, the parameters including the reaction rate constant k, pre-exponential factor A and activation energy Ea were calculated to understand the reaction tendency.

$$k=A{T}^{n}{e}^{\left(-\frac{{E}_{a}}{RT}\right)}$$
(3)
$$A=\frac{{k}_{B}T}{h}\frac{{Q}^{\ne }}{\Pi {Q}_{B}}$$
(4)

For Eqs. 3 and 4, n is the temperature index, R is the gas constant, T is the temperature, kB is Boltzmann’s constant, h is Planck’s constant, Q≠ is the partition function of the transition state, ΠQB is the continued product of the partition functions (QB) of all reaction species B. The KiSThelP [51] software was used for the data processing and analysis in obtaining the reaction rate constant k and pre-exponential factor A.

Results and discussion

The reaction between Hg and HF first goes through the reaction formula (Eq. 1), the oxidation of Hg by one HF molecule enables Hg to react with F atom of HF as HgF, and then HgF reacts with the other HF molecule to form HgF2 via the reaction formula (Eq. 2). The calculated results show that the oxidation reaction between Hg and HF goes through an intermediate in which HF interacts with Hg to form a complex HF∙∙∙Hg, and then a transition state H∙∙∙F∙∙∙Hg forms due to the broken of H–F bond, the further approaching of F atom to Hg atom results in the bonding between F and Hg atom, so the processes can be represented as:\(\text{HF}+\text{Hg}\to \text{HF}\bullet \bullet \bullet \text{Hg }\to \text{H}\bullet \bullet \bullet \text{F}\bullet \bullet \bullet \text{Hg }\to \text{H}+\text{HgF}\). The subsequent reaction between HgF and the other HF molecule undergoes the formations of the intermediate HF∙∙∙HgF, the transition state H∙∙∙F∙∙∙HgF and the final product HgF2, so it can be represented as: \(\text{HF}+\text{HgF}\to \text{ HF}\bullet \bullet \bullet \text{HgF}\to \text{ H}\bullet \bullet \bullet \text{F}\bullet \bullet \bullet \text{HgF}\to \text{H}+{\text{HgF}}_{2}\). Herein the reaction processes occur at 298 K and the effect of the temperature from 298 to 1000 K on the reaction were explored and discussed in detailed.

With regard to the reaction between Hg and HF at each temperature (298–1000 K), the Gibbs free energy profiles was depicted in Figs. 1 and S1, and the optimized structures of the complex, transition state and intermediate were depicted in Fig. 2. It can be seen that both the complex HF∙∙∙Hg and the transition state H∙∙∙F∙∙∙Hg form in the reaction of Hg and HF at each temperature, and it is different from the previous result in which only one transition state but not any intermediate was observed [34, 35]. The interaction energy (i.e., ΔE1) between HF and Hg for the complex HF∙∙∙Hg is − 0.46 kcal/mol or − 0.37 kcal/mol after considering BSSE at each temperature, meaning the formation of weak interaction between HF and Hg before the oxidation reaction. QTAIM analysis shows that the electron density (i.e., ρ) and its Laplacian (i.e., ∇2ρ) at the BCP of F∙∙∙Hg interaction are 0.0032 and 0.0136 a.u., respectively. These values are out of the ranges of the criteria for the existence of hydrogen bond (i.e., ρ and ∇2ρ should be within the ranges of 0.002–0.035 and 0.024–0.139 a.u., respectively [52]). The result indicates that the F∙∙∙Hg interaction in the complex HF∙∙∙Hg is van der Waals interaction. The positive (or negative) electronic energy density (i.e., H) value at the BCP is indicative of electrostatic (or covalent) dominant interaction [53], so the positive H value (H = 0.0007 a.u.) for F∙∙∙Hg interaction in the complex HF∙∙∙Hg suggests it is of mainly electrostatic character. To form the complex HF∙∙∙Hg, it needs to absorb an energy ranging from 3.1 to 11.1 kcal/mol, while the energy barrier for the conversion from the complex HF∙∙∙Hg to the transition state H∙∙∙F∙∙∙Hg is in the range of 126.6–132.2 kcal/mol, which is decreased with the increase of the temperature (Fig. 1). It is found that the geometries of the reactant, complex, transition state, intermediate and product are not influenced by the temperature. At each temperature (298–1000 K), the H–F bond length and F∙∙∙Hg distance in the complex HF∙∙∙Hg are 0.9177 and 3.6234 Å (Fig. 2), respectively. The elongated H-F bond length in the complex HF∙∙∙Hg relative to that in HF molecule at each temperature should be ascribed to the formation of F∙∙∙Hg weak interaction. It is observed that the H∙∙∙F distance in the transition state (i.e., TS) H∙∙∙F∙∙∙Hg is 2.5434 Å (Fig. 2), suggesting that the H-F bond is broken. As a result, the intermediate HgF is produced.

Fig. 1
figure 1

Gibbs free energy profiles for the reactions of Hg and HF at 298 K, 500 K and 1000 K

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Fig. 2
figure 2

The optimized structures of the complex, transition state and intermediate for the reaction of Hg and HF (The units of distances and bond angles are in Å and degree)

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Subsequently, HgF interacts with the other HF molecule to form the intermediate HF∙∙∙HgF, which is not observed in the previous work [34, 35]. The interaction energy (i.e., ΔE2) between HF and HgF for the intermediate HF∙∙∙HgF is − 1.93 kcal/mol or − 1.75 kcal/mol after considering BSSE, meaning that HF can interact with HgF through weak attractive interaction. The electron density ρ and its Laplacian ∇2ρ at the BCP of F∙∙∙Hg interaction in the intermediate HF∙∙∙HgF from QTAIM analysis are 0.0086 and 0.0304 a.u., respectively. Obviously, these values within the ranges of the criteria for the existence of hydrogen bond, and they are even larger than 0.002 and 0.024 a.u., respectively. It suggests that the F∙∙∙Hg interaction in the intermediate HF∙∙∙HgF is a strong attractive interaction. The electronic energy density (i.e., H) at the BCP of F∙∙∙Hg interaction in the intermediate HF∙∙∙HgF is 0.0006 a.u., meaning the F∙∙∙Hg interaction is of mainly electrostatic character. The process in the formation of the intermediate HF∙∙∙HgF needs to absorb an energy in the range of 3.4–13.5 kcal/mol from 298 to 1000 K (Figs. 3 and S2). Then the F atom of HF in the intermediate HF∙∙∙HgF approaches to HgF, resulting in the formation of the transition state (i.e., TS) H∙∙∙F∙∙∙HgF. The process from HF∙∙∙HgF to H∙∙∙F∙∙∙HgF needs to overcome an energy barrier ranging from 47.0 to 57.5 kcal/mol from 298 to 1000 K (Fig. 3). In the intermediate HF∙∙∙HgF, the F∙∙∙Hg distance is 3.1521 Å, while the H–F and Hg–F bond lengths are 0.9193 and 2.1038 Å, respectively (Fig. 4). In the transition state H∙∙∙F∙∙∙HgF, the H∙∙∙F and F∙∙∙Hg distances are 1.3500 and 2.2801 Å, respectively. Due to the bonding between F and Hg atoms, the final product HgF2 is yielded. The Hg-F bond in HgF2 (ca. 1.9472 Å) is shorter than that in the intermediate HgF (i.e., 2.0988 Å).

Fig. 3
figure 3

Gibbs free energy profiles for the reactions of HF and HgF at 298 K, 500 K and 1000 K

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Fig. 4
figure 4

The optimized structures of the intermediate, transition state and product for the reaction of HF and HgF (The units of distances and bond angles are in Å and degree)

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In terms of the above calculations, the kinetic parameters for the reactions between HF and Hg to form HgF (i.e., the reaction occurring in the form of formula (Eq. 1)) and then HgF2 (i.e., the reaction occurring in the form of formula (Eq. 2)) at different temperatures were calculated using the KiSThelP software. As outlined in Table 1, the reaction rate constant k gradually increases with the rise of the temperature for the reaction between HF and Hg in the formation of HgF. For the reactions between HF and HgF to form HgF2, it can be seen from Table 2 that the pre-exponential factor A varies slightly with the rise of the temperature, and the reaction rate constant k gradually increases with the rise of the temperature. To obtain the expression of reaction rate, the relationship between log10k and 103/T is explored. As depicted in Fig. 5a and b, the linear relationship between log10k and 103/T is observed, and the fitting reaction kinetic parameters including pre-exponential factor A and activation energy Ea can be derived which are summarized in Table 3. Consequently, the expression of reaction rate k can be written as well, as shown in Table 3. It is expected that these kinetic parameters would be helpful for the investigation of the reaction involving Hg and other gas in flue gas.

Table 1 Kinetic parameters (A, unit in cm3 mol−1 s−1; Ea, unit in kJ mol−1; k, unit in cm3 mol−1 s−1) for the reactions between HF and Hg to form HgF at different temperatures (T, unit in K)
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Table 2 Kinetic parameters (A, unit in cm3 mol−1 s−1; Ea, unit in kJ mol−1; k, unit in cm3 mol−1 s−1) for the reactions between HF and HgF to form HgF2 at different temperatures (T, unit in K)
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Fig. 5
figure 5

The relationships a between log10k and 103/T and b among log10k, 103/T and and log10T for the reaction between HF and Hg to form HgF and then HgF2

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Table 3 Fitting reaction kinetic parameters (A, unit in cm3 mol−1 s−1; n, dimensionless constant; Ea, unit in kJ mol−1) for the reactions between HF and Hg to form HgF and then HgF2, and the expression of reaction rate (k, unit in cm3 mol−1 s−1)
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Conclusions

In this work, the reactions of Hg with hydrogen fluoride (HF) at different temperatures were theoretically investigated and the detailed mechanisms were explored, and the results indicate that the reactions can be divided into two steps. In the step one, Hg was firstly oxidized by one HF molecule and forms HgF. In this reaction processes, Hg first interacts with HF to form a complex HF∙∙∙Hg, and then the approaching of F atom of HF to Hg leads to a transition state H∙∙∙F∙∙∙Hg which is accompanied by the broken of H–F bond, and subsequently, the bonding between F and Hg atom causes the formation of HgF. Afterwards the other HF molecule takes part in and reacts with HgF to form HgF2. In this step, HgF and the other HF molecule first interacts with each other to form the intermediate HF∙∙∙HgF, and then the approaching of F atom of HF to Hg atom of HgF results in the transition state H∙∙∙F∙∙∙HgF, and the final product HgF2 is produced due to the bond formed between F and Hg atom. The temperature has a significant effect on the reaction process but has little effect on the geometries of the reactant, complex, intermediate, transition state and product. Based on the above results, we further calculated the kinetic parameters including the pre-exponential factor A, activation energy Ea and reaction rate k at different temperatures, and the linear fitting further provides us the expressions of reaction rates k for the reactions between HF and Hg to form HgF and then HgF2. The results presented here would be insightful into the understanding of the chemical reactions of Hg and HF, the mechanisms and the kinetics.