3.1. Potential energy surfaces and reaction mechanisms
The geometries of the reactants and products for the C2H5 and n-CnH2n+1OH (n = 1 - 4) reactions are shown in Figs. 1 (for n = 1, 2) and S1 (for n = 3, 4) in the ESI; those of the transition states are in Figs. 2 (for n = 1, 2) and S2 (for n = 3, 4). The potential energy surfaces for the title reactions are shown in Figs. 3 (for n = 1, 2) and S3 (for n = 3, 4) in the ESI, and heats of reaction for the C2H5 + CnH2n+1OH (n = 1, 2) reactions are shown in Table 1.
Table 1
Comparison of Heats of reaction for the C2H5 + CnH2n+1OH (n = 1, 2) reactions calculated at the CCSD(T)//B3LYP /6-311++G(3df,2p) level of theory with available data [54–58].
Reactions | ΔrH°0 (kcal/mol) | ΔrH°298 (kcal/mol) |
Predicted | Literature* | Predicted | Literature* |
CH3OH + C2H5 |
PR1 (CH3O + C2H6) | 3.2 | 4.4 ± 1.1 | 2.7 | 4.1 ± 1.1 |
PR2 (CH2OH + C2H6) | -4.6 | -5.0 ± 0.8 | -4.9 | -5.0 ± 0.8 |
PR3 (CH3OC2H5 + H) | 17.7 | 19.3 ± 0.6 | 17.4 | 19.5 ± 0.6 |
PR4 (C2H5OH + CH3) | -2.4 | -2.1 ± 0.7 | -2.4 | -1.9 ± 0.7 |
PR5 (C3H8 + OH) | 3.1 | 3.1 ± 0.7 | 2.8 | 3.1 ± 0.7 |
PR6 (n-C3H7OH + H) | 9.8 | 10.3 ± 0.6 | 9.6 | 10.3 ± 0.6 |
C2H5OH + C2H5 |
PR7 (C2H5O + C2H6) | 3.0 | 4.0 ± 2.1 | 2.8 | 3.9 ± 2.1 |
PR8 (CH3CHOH + C2H6) | -5.8 | --- | -6.1 | --- |
PR9 (CH2CH2OH + C2H6) | 1.5 | --- | 1.5 | --- |
PR10 (C2H5OC2H5 + H) | 17.3 | 18.6 ± 1.1 | 17.3 | 19.4 ± 1.1 |
PR11 (C2H5OH + C2H5) | 0.0 | 0.0 | 0.0 | 0.0 |
PR12 (n-C4H10 + OH) | 5.8 | 5.7 ± 1.3 | 5.7 | 6.1 ± 1.3 |
PR13 (s-C4H9OH + H) | 8.0 | --- | 7.9 | --- |
PR14 (n-C4H9OH + H) | 12.5 | --- | 12.5 | --- |
PR15 (C3H8 + CH2OH) | -1.6 | -1.9 ± 1.3 | -1.9 | -1.9 ± 1.3 |
PR16 (n-C3H7OH + CH3) | 0.6 | 1.0 ± 1.1 | 0.8 | 1.3 ± 1.1 |
*The experimental values employed in the calculations are obtained based on the heats of formation at 0K and 298K for the corresponding species from the literature [54–58]. |
The reaction of C 2 H 5 + CH3OH:
The PES in Figure 1a reveals that the C2H5 radical can abstract H atoms in the OH and CH3 groups giving PR1 (CH3O + C2H6) and PR2 (CH2OH + C2H6), or substitute atoms and groups such as H, OH, CH3 in the CH3OH molecule giving PR3 (CH3OC2H5 + H) – PR6 (n-C3H7OH + H). The substitution channels should be very minor because of having significantly higher barrier energies of ~ 30 kcal/mol.
Formation of PR1 (CH 3 O + C 2 H 6 ) and PR2 (CH 2 OH + C 2 H 6 ):
These channels can occur from CH3OH when the C2H5 radical abstracts H atoms in the OH, and CH3 groups via TS1 (C2H5...H...OCH3) and TS2 (C2H5...H...CH2OH) giving PR1 (CH3O + C2H6) and PR2 (CH2OH + C2H6), respectively, as clearly shown in Fig. 1. The geometries of CH3OH, C2H6, and CH3O optimized at the B3LYP/6-311++G(3df,2p) level agree well with the experimental values (see Fig. 1). For example, the O-H, C-O, and C-H bond lengths of CH3OH computed in this work are 0.960, 1.421, and 1.088 Å, which is in good agreement with the experimental value of 0.956, 1.427, and 1.096 Å, respectively [10]. For the transition states TS1 and TS2, there are no experimental results but their geometries agree well with the previous theoretical studies. For example, the length of the broken O…H bond in TS1 calculated in this work, 1.253 Å, is close to 1.2479 Å for the CH3 + CH3OH reaction by Jodkowski et al.[25] The imaginary frequencies are 1524i cm−1 for TS1 and 1715i cm−1 for TS2 computed at the B3LYP/6-311++G(3df,2p) level corresponding migrations of the H atoms from O and C of the CH3OH molecule to the C2H5 radical. The relative energies for TS1 and TS2 predicted at the CCSD(T)//B3LYP/6-311++G(3df,2p) level of theory in this work are 13.0 and 14.4 kcal/mol which are in good agreement with 13.6 and 14.0 kcal/mol, respectively, corresponding to the barrier energies for H-abstractions of the CH3 + CH3OH reaction calculated at the G2 level of theory reported by Jodkowski et al. [25]. In addition, we have also compared the predicted heats of reaction (ΔrH°) for the channels with available experimental data at both 0 and 298 K (see Table 1). The heat of formation for each channel is evaluated with the heats of formation of CH3OH (-190,12 ± 0,6 kJ/mol), C2H5, CH3O (28,4 ± 2,1 kJ/mol), CH2OH (-10,7 ± 0,7 kJ/mol) and C2H6 (-68,38 ± 0,4 kJ/mol) species from the available experimental data [54–57]. The heats of reaction for channels giving the PR1 and PR2 predicted at the CCSD(T)//B3LYP/6-311++G(3df,2p) level to be 3.2 and -4.6 kcal/mol at 0 K; 2.7 and -4.9 kcal/mol at 298 K, respectively. These values are in good agreement with the experimental values of 4.4 ± 1.1 and -5.0 ± 0.8 kcal/mol at 0 K and 4.1 ± 1.1 and -5.0 ± 0.8 kcal/mol at 298 K, respectively.
Formation of PR3 (CH 3 OC 2 H 5 + H), PR4 (C2H5OH + CH3), PR5 (C3H8 + OH), and PR6 (n-C3H7OH + H):
The products can be formed when the C2H5 radical substitutes the H atoms, OH, and CH3 groups in the CH3OH molecule via TS3, TS4, TS5, and TS6, respectively (see Fig. 1a). The geometries of these transition states are also close to those in the previous studies. For example, the length of the breaking (C2H5…O) and forming (CH3…O) bond lengths of TS4 in this work, 1.845 and 1.860 Å, are close to 1.847 and 1.862 Å for the CH3 + C2H5OH by Lin et al. [21]. Our predicted heats of reaction for channels giving the PR3, PR4, PR5 and PR6, 17.7, -2.4, 3.1 and 9.8 kcal/mol at 0 K; 17.4, -2.4, 2.8 and 9.6 kcal/mol at 298 K, respectively, are reasonable agreement with the experimental values of 19.3 ± 0.6, -2.1 ± 0.7, 3.1 ± 0.7 and 10.3 ± 0.6 kcal/mol at 0 K; 19.5 ± 0.6, -1.9 ± 0.7, 3.1 ± 0.7 and 10.3 ± 0.6 kcal/mol at 298 K, respectively (see Table 1). It is noticed that these barrier energies (TS3 – TS6) are much higher (> 27 kcal/mol) than those for the H-abstractions (TS1 and TS2); this picture agrees with the results for the CH3 + C2H5OH reaction by Lin et al. [21]. It is obvious that the channels giving PR3 – PR6 should be kinetically unimportant because of having very high barrier energies at TS3 – TS6.
The reaction of C 2 H 5 + C2H5OH:
Figure 1b reveals that the C2H5 radical can abstract H atoms in the OH, CH2, and CH3 groups giving PR7 (C2H5O + C2H6, 3.0 kcal/mol), PR8 (CH3CHOH + C2H6, -5.5 kcal/mol), and PR9 (CH2CH2OH + C2H6, 1.5 kcal/mol) via transition states at TS7, TS8, and TS9, respectively. The geometries of the ethanol molecule and transition states predicted at the B3LYP/6-311++G(3df,2p) level of theory agree well with the previous studies (see Figures 1 and 2). For example, in the TS7 – TS9, the broken O…H, C…H (in CH2 group), and C…H (in CH3 group) are elongated by 31%, 21%, and 25% while the corresponding values for the CH3 + C2H5OH reaction are 26%, 17%, and 22%, respectively by Lin et al. [21]. The relative energies of TS7 (13.0 kcal/mol), TS8 (12.6 kcal/mol), and TS8 (16.7 kcal/mol) are close to the corresponding values of 13.2, 12.0, and 16.0 kcal/mol for the CH3 + C2H5OH reaction predicted at the G2M(RCC2) level of theory. It is noticed that among H- abstractions of the o-H (OH group), α-H (CH2 group), and β-H (CH3 group), the last channel has higher barrier energy of about 3.7 kcal/mol suggesting that this channel has a small contribution at the normal temperature. This picture is also in reasonable agreement with the result of Lin et al for the CH3 + C2H5OH reaction [21]. In addition, the C2H5 radical can substitute H atom and OH, CH3, and C2H5 groups in the C2H5OH molecule via transition states at TS10 – TS16 giving PR10 (C2H5OC2H5 + H) – PR16 (n-C4H9OH + CH3). However, these substitution channels are unfavorable on account of the high barriers at the TS10 – TS16 (41.5 – 57.0 kcal/mol) (see Figure 3b).
The reaction of C 2 H 5 + n/i-C3H7OH:
For the C2H5 + n-C3H7OH system, there are four reaction channels as clearly shown in Figure S2a: the C2H5 radical can abstract o-H (in the OH group), α-H (in the CH2 group), β-H (in the CH2 group) or γ-H (in the CH3 group) giving PR17 (n-C3H7O + C2H6, 3.0 kcal/mol), PR18 (CH3CH2CHOH + C2H6, -5.4 kcal/mol), PR19 (CH3CHCH2OH + C2H6, -1.2 kcal/mol), and PR20 (CH2CH2CH2OH + C2H6, 0.4 kcal/mol) via transition states TS17, TS18, TS19, and TS20, respectively. Figure S2a also shows that among the energy barriers, the value of TS18 corresponding to the α-H abstraction is lowest with an activation energy of 12.1 kcal/mol compared to 12.9 kcal/mol at TS18 (for o-H abstraction), 14.0 kcal/mol at TS19 (for γ-H abstraction) and 15.8 kcal/mol at TS20 (for δ-H abstraction) suggesting the α-H abstraction is the major channel for the C2H5 + n-C3H7OH reaction. For C2H5 + i-C3H7OH reaction, the C2H5 radical can abstract o-H (in the OH group), α-H (in the CH group), β-H (in the CH3 groups) giving PR21 (i-C3H7O + C2H6, 4.8 kcal/mol), PR22 (CH3C(CH3)OH + C2H6, -6.6 kcal/mol) and PR23 (CH2CH(CH3)OH + C2H6, 1.9 kcal/mol) via TS21, TS22, and TS23, respectively (see Figure S2b). It is noticed that the energy barrier for the α-H abstraction (TS21) is only 10.4 kcal/mol which is the lowest energy barrier among the C2H5 + CnH2n+1OH (n = 1 - 4) reactions because of the effect of the two CH3 groups and OH group in i-C3H7OH molecule. This barrier is also lower ~ 2 – 6 kcal/mol than the others at TS22 (12.7 kcal/mol) and TS23 (16.4 kcal/mol) suggesting that the α-H abstraction channel should be the major channel in the C2H5 + i-C3H7OH reaction.
The reaction of C 2 H 5 + n/i/s/t-C4H9OH:
For the C2H5 + n/i/s/t-C4H9OH systems, we calculate with a smaller basic set at the CCSD(T)//B3LYP/6-311+G(d,p) level of theory because of their large systems. For the C2H5 + t-C4H9OH reaction, Figure S2 shows that there are two channels including o-H and β-H abstractions in which the former has lower barrier energy of 12.8 (TS38) than that of 15.4 kcal/mol (TS39) for the latter. The barrier energy for o-H abstraction channel (12.8 kcal/mol) for the C2H5 + t-C4H9OH reaction predicted at the CCSD(T)//B3LYP/6-311+G(d,p) level of theory agree well with those of 12.9 and 13.0 kcal/mol for the C2H5 + CnH2n+1OH (n = 2,3) reactions, respectively, predicted at the CCSD(T)//B3LYP/6-311++G (3df,2p) level of theory. The barrier energy of the H abstraction from the CH3 group for the t-C4H9OH (TS39, 15.4 kcal/mol) also agrees well with those for n/i/s-C4H9OH with the values of 15.6-16.0 kcal/mol (see Fig. 3S in the ESI). It is noticed that TS39 is the unique species in the title reactions optimized at the BHandHLYP/6-311+G(d,p) level because we cannot find it at the B3LYP/6-311+G(d,p) level of theory. Its relative energy, therefore, is calculated from energies of TS39, t-C4H9OH, and C2H5 species at the CCSD(T)//BHandHLYP/6-311+G(d,p) level. The C2H5 + i-C4H9OH reaction has four channels including o-H, α-H, β-H, and γ-H abstractions in which the lowest energy barrier is the β-H abstraction (TS31, 11.7 kcal/mol) because of the effect of the two CH3 groups in the i-C4H9OH molecule. Energy barriers for the o-H, α-H, and γ-H abstraction channels in the C2H5 + i-C4H9OH reaction, 12.6, 12.3, and 15.6 kcal/mol at TS29, TS30, and TS32, respectively, also agree well with the corresponding values at 12.9, 12.1, and 15.8 kcal/mol for n-C3H7OH + C2H5 reaction as discussed above. For the reactions of the C2H5 radical with n/s-C4H9OH, Figure S2 shows that there are five channels in which the o-H and α-H abstraction channels have lowest the barrier energies. The o-H abstractions occur via TS24 and TS33 lying 12.6 above the reactants for both n-C4H9OH and s-C4H9OH, respectively. The α-H abstractions are slightly lower than those for o-H abstractions with relative energies of 11.9 and 11.0 kcal/mol for n-C4H9OH and s-C4H9OH, respectively while the values for the other channels are 13.8 – 16.0 kcal/mol. It is clear that the difference between the barrier energies is small suggesting that all of the channels can contribute at high temperatures.
3.2. Rate constant calculations
Based on the PES and molecular parameters of all the related species in the C2H5 + CnH2n+1OH (n = 1 - 4) reactions calculated by both the CCSD(T) and B3LYP methods as discussed above, we calculated the individual and total rate constants with TST considering Eckart tunneling effect in the temperature range of 300 – 2500 K; the results are presented in Fig. 4 for the C2H5 + CnH2n+1OH (n = 1, 2) reactions and in Fig. S4 in the ESI for the C2H5 + CnH2n+1OH (n = 3, 4) reactions. The branching ratios calculated from the individual and total rate constants for each channel are presented in Figs. 5 and S5 for the reactions of C2H5 with CnH2n+1OH (n = 1, 2) and CnH2n+1OH (n = 3, 4), respectively.
It can be seen in Fig. 4 and S4 that all the rate constants have positive temperature dependence reflecting the fact that each channel has prominent and distinct barrier energy. For the C2H5 + CH3OH reaction, the rate constants for the substitution reactions via TS3 – TS6 is very much smaller than those for H-abstraction via TS1 and TS2 for all the considered temperatures (see Figure 5a). For example, the values for the channels via TS1 and TS3 are 4.06 ⋅ 10−16 and 1.70 ⋅ 10−24 (cm3 molecule−1 s−1) at 1000 K, respectively. Therefore, the branching ratios for the channels via TS3 – TS6 are very small with the highest value is only ~4 ⋅ 10−3 for the channel via TS4 at 2500 K. This is because the barrier energies for these channels are much higher (> 27 kcal/mol) than those for the H-abstraction channels as discussed above. As a result, the H-abstraction channels giving PR1 (CH3O + C2H6) and PR2 (CH2OH + C2H6) should be the major channels while the substitution channels giving PR3 – PR6 should be ignored in the C2H5 + CH3OH reaction. Our kinetics results show that the branching ratios for PR1 and PR2 channels are not much different in the considered temperatures; the values for PR1 are from 0.62 at 300 K to 0.32 at 2500 K. The temperature dependence of the total rate constant for the C2H5 + CH3OH reaction can be expressed, in terms of three-parameters fits of the form ATn exp(-E/T), as follows:
k(T) = 1.31 × 1040 T8.69 exp(-2264.4 K/T) (T = 300 – 600 K)
k(T) = 2.11 × 1025 T3.94 exp(-5105.8 K/T) (T = 600 – 2500 K)
For the C2H5 + CnH2n+1OH (n = 2 - 4) reactions, we did not calculate the rate constants for the substitution channels because of high barrier energies as discussed above. For the reaction of C2H5OH, the β-H abstraction channel via TS9 are about 2 orders lower than those for α-H and o-H abstraction channels via TS7 and TS8 at 300 K agreeing with the fact that the barrier energy at TS9 (16.7 kcal/mol) is higher than those at TS7 (13.0 kcal/mol) and TS8 (12.6 kcal/mol). However, the branching ratios for the three pair products PR7 (C2H5O + C2H6), PR8 (CH3CHOH + C2H6), and PR9 (CH2CH2OH + C2H6, 1.5 kcal/mol) via TS7, TS8, and TS9 are almost the same at high temperatures; at 2500 K, the values for PR7, PR8, and PR9 are 0.31, 0.39, and 0.30, respectively. For the reactions of n/i-C3H7OH and n/s-C4H9OH, one can see that the α-H abstraction channels are dominant at 300 K due to the low barrier energies as discussed above. But their branching ratios decrease when temperature increases; all the products via H-abstraction channels, therefore, become competitive. For example, at 2500 K, for the reaction of n-C3H7OH, the branching ratios of o-H, α-H, β-H, and γ-H abstraction channels are 0.19, 0.36, 0.27, and 0.18, respectively. For the reaction of i-C4H9OH, Fig. 5S show that the β-H abstraction channel is dominated at all the considered temperature range of 300-2500K due to its lowest barrier energy. Its branching ratio also decreases when temperature increases but it is still highest at 2500K with the value of 0.431. For the reactions of t-C4H9OH, there are two H abstraction channels at o-H and β-H; the first channel is dominated at 300K with the branching ratio value of 0.761 but decreases to 0.297 at 2500K.
For convenient modeling applications, the temperature dependence of the individual reaction rate constants for the title reactions given in terms of three parameters fits the form ATnexp(-E/T) expressions in the temperature range of 300 - 1000 K are summarized in Table II.