Péter
Szabó
*ab and
György
Lendvay
*cd
aDepartment of Chemistry, KU Leuven, Celestijnenlaan, 200F, Leuven, 3001, Belgium. E-mail: peter88szabo@gmail.com
bRoyal Belgian Institute for Space Aeronomy (BIRA-IASB), Avenue Circulaire 3, Brussels, 1180, Belgium
cInstitute of Materials and Environmental Chemistry, HUN-REN Research Centre for Natural Sciences, Magyar tudósok krt. 2., H-1117 Budapest, Hungary. E-mail: lendvay.gyorgy@ttk.hu
dCenter for Natural Sciences, Faculty of Engineering, University of Pannonia, Egyetem u. 10, Veszprém, 8200, Hungary
First published on 15th March 2024
The rate coefficient for two deuterium substituted isotopologues of reaction CH3 + HBr → CH4 + Br has been determined using the quasiclassical trajectory (QCT) method. We used the analytical potential energy surface (PES) fitted to high-level ab initio points in earlier work. The PES exhibits a pre-reaction van der Waals complex and a submerged potential barrier. The rate coefficients of the deuterated isotopologue reactions, similarly to the pure-protium isotopologue, show significant deviation from the Arrhenius law, namely, the activation energy is negative below about 600 K and positive above it: k[CH3 + DBr] = 1.35 × 10−11exp(− 2472/T) + 5.85 × 10−13
exp(335/T) and k[CD3 + HBr] = 2.73 × 10−11
exp(− 2739/T) + 1.46 × 10−12
exp(363/T). The CH3 + DBr reaction is slower by a factor of 1.8, whereas CD3 + HBr isotopologue is faster by a factor of 1.4 compared to the HBr + CH3 system across a wide temperature range. The isotope effects are interpreted in terms of the properties of various regions of the PES. Quantum state-resolved simulations revealed that the reaction of CH3 with HBr becomes slower when any of the vibrational modes of the methyl radical is excited. This contradicts the assumption that vibrational excitation of methyl radicals enhances its reactivity, which is of historical importance: this assumption was used as an argument against the existence of negative activation energy in a decade-long controversy in the 1980s and 1990s.
CH3 + DBr → CH3D + Br | (R1) |
CH3 + HBr → CH4 + Br | (R2) |
![]() | ||
Fig. 1 Potential energy profile of CH3 + HBr reaction. The energy levels of the stationary points characterizing the Czakó–Góger–Szabó–Lendvay PES13 are given in kJ mol−1. The zero-point energy corrected energies are shown in parentheses. The S.P. denotes the saddle point. |
Among the numerous arguments on the possible experimental errors behind negative activation energies, a particularly interesting objection was that the alkyl, including methyl radicals, when generated by laser flash photolysis, are vibrationally excited when formed and in the experiments there was not enough time for collisional relaxation. Remaining excited, they presumably reacted faster than the thermalized ones, discrediting the negative activation energy. This idea was raised and discussed again even as recently as 2014 at the Gas Kinetics Symposium in Szeged, Hungary. Although by now it is well established that the rate coefficients determined by the modern direct methods are reliable, we found it challenging to check whether vibrationally excited reactants react faster.
Reactions between alkane (R) and halogen atoms (X): RH + X → R + HX have been and continue to be a central subject of study in reaction dynamics over the past several decades.17–26 The PES of this reaction exhibits potential wells that may influence the reaction dynamics. The R + HX reactions where X = F, Cl, O are characterized by a large potential barrier19 which separates a pre-reactive complex from the product side. However, when X = Br, the reaction pathway features a van der Waals well along with a submerged barrier that lies in the entrance channel below the reactant's energy level. Understanding such reactions, and how the submerged barrier affects the dynamics, is far from complete.
Earlier, we performed quasiclassical trajectory (QCT) calculations on a reliable full-dimensional PES13 for reaction (R2) and obtained a number of interesting conclusions on the dynamics of the reaction.13,27–29 First, the experimental results were very well reproduced. Second, the excitation function, diverging at very low collision energy with decreasing Ecoll changes course and at higher collision energies increases with growing Ecoll. This suggests that the activation energy can be positive at higher temperatures. This expectation was confirmed in a combined experimental and theoretical work.29 Namely, reaction (R2) does have negative activation energy below about 500 K, but the rate coefficients pass minimum at about 650K, and start to increase with increasing temperature, producing a V-shaped Arrhenius plot. The experimental activation energy changes from −1.82 kJ mol−1 at 200 K to 13.5 kJ mol−1 at 1000 K. This confirms that the activation energy of (R2) is negative, but only at low temperatures, and a chance is given to researchers favoring positive Ea. An interesting consequence can be noticed when one follows the reverse path of the second-law method and calculates the activation energy of the reverse counterpart to (R2), the CH4 + Br reaction (−R2) from the reaction enthalpy and the measured, temperature-dependent activation energy of (R2). The reaction enthalpy and even its temperature dependence can now be calculated accurately from tabulated heats of formation that are reliable. The temperature dependence of the heats of formation of the four species involved, and so the reaction enthalpy calculated from tabulated data is mild29 (it changes non-monotonically, by less than 5 kJ mol−1 in the studied temperature range). As a result, similarly to the activation energy for (R2), that for (−R2) must also increase by about 20 kJ mol−1 (from about 70 kJ mol−1 at 200 K to about 90 kJ mol−1 at 1000 K), being about 71.2 kJ mol−1 at room temperature. The value used in the thermochemical calculations in the 1990s, 73.9 kJ mol−1 (considered as temperature-independent) is pretty close to this expectation, explaining why the alkyl heats of formation derived from the direct experiments were correct.
In this work, we intend to calculate the temperature dependence of the rate coefficients for reaction (R1) and evaluate the relationship between the experimental results of the two groups mentioned above. In addition, since NvKDW9 as well as Donaldson and Leone30 measured the rate coefficient for the
CD3 + HBr → CD3H + Br | (R3) |
![]() | ||
Fig. 2 Calculated (QCT) rate coefficients for the CH3 + DBr (filled symbols), CH3 + HBr (open symbols) and CD3 + HBr (crossed symbols) reactions. The sources of the experimental data: NvKDW = Nicovich et al., ref. 9, D&L = Donaldson and Leone, ref. 20, Bedjanian, ref. 19, Gac&al, ref. 5. |
That the V shape of the Arrhenius plot for (R2) predicted by the QCT calculations is correct is demonstrated by the excellent experiment-QCT agreement. This agreement extends to (R1) in the low-temperature regime where data from NvKDW9 are available. Note that similarly to (R2), for CH3 + DBr, all rate coefficients are in a narrow range, here between 1 × 10−12 and 2 × 10−12 cm3 molecule−1 s−1; we consider the capability of measuring a structured curve within this narrow range as a remarkable experimental achievement. For reaction R3, both of the room-temperature experiments9,30 provided somewhat larger rate coefficients than for the pure-protium reaction. The measured point of Donaldson and Leone et al.30 is very close to the nearby QCT points; it is roughly 1.5 times larger than the rate coefficient measured by Nicovich et al. The agreement is not as good between the QCT curve and the points reported by Gac et al. for reaction (R1).
Although the QCT curve runs between some of the experimental points, the latter, even considering their large scatter, cannot be claimed to follow the same tendency.
The Arrhenius plots for reactions (R1)–(R3) are well separated. The temperature dependence of the rate coefficients of the reactions involving deuterated reactants ((R1) and (R3)) follows the same tendency as the pure-protium (R2). At very high temperatures for reaction (R1) and elsewhere for (R3), only the QCT results are available. The good agreement with the experiments where the latter exist and the similarity to the Arrhenius plot of (R2) encourages us to consider the calculated results satisfactorily accurate. The curves are roughly parallel, and the location of the rate coefficient minima is close (around 550 and 600 K). The fastest is reaction (R3), the slowest is (R1).
The V shape of the Arrhenius plots means that at low temperatures the activation energy is negative, which by now became a widely accepted fact. That the activation energy at high temperatures is positive was first demonstrated experimentally by Bedjanian on (R2), whose measured rate coefficients agree very well with the QCT results. One can see that the other isotopologues also follow (R2) in that the activation energy at high temperatures is positive and exceeds the absolute value of the negative Ea below 400 K. In Fig. 2 (see also Fig. S1, ESI†) it is visible that for reaction (R1), the high-temperature points measured by Gac et al.5 look as if they formed a continuation of the lower-temperature data points of NvKDW9 However, the continuation of the straight Arrhenius line matching the latter data set to the region above 600 K does not follow the curvature of the QCT Arrhenius plot.
So, while it would be appealing to consider the two experimental data sets as consistent, one can conclude that the points measured using the early VLPP apparatus underestimate the rate coefficients for (R1) in that region and definitely do not follow the trend calculated by the QCT method which we consider realistic. As a result, even if the scatter of the Gac et al. measurements allowed the derivation of reasonable Arrhenius parameters, they would not be appropriate for thermochemical purposes, including the determination of the enthalpy of formation of the methyl radical, for which they were used by Gac et al.5 The activation energy in the 600 K–1000 K range, according to the QCT calculations for (R1) is roughly 8 kJ mol−1, exceeding even the upper uncertainty limit set by Benson and coworkers.1,2,11 Should an activation energy measured in this range combined with that for the reverse reaction – whose activation energy is also temperature-dependent – measured in a different temperature range, the resulting second-law heats of formation could become meaningless.
The rate coefficients of the CH3 + DBr and the CD3 + HBr isotopologue are smaller and larger, respectively than those of the CH3 + HBr variant. The experimental HBr/DBr kinetic isotope effect can be estimated to be around 1.7 from the data measured by NvKDW9 The corresponding QCT rate coefficient ratio on average is about 1.8. The remarkable rate lowering due to the substitution of the transferred H-atom by D, however, is not because of the less efficient tunnel effect as one could expect. Here the barrier is very flat (the imaginary frequency is 303i cm−1 for the CH3–H–Br isotopologue) and penetrating it would not make an appreciable advantage for the atom transfer.
However, a dynamical effect connected to mass change does well explain the rate reduction. The preliminaries to this reasoning are as follows. We have seen earlier27 that for reaction (R2), the vibrational amplitude of the breaking bond's vibration has a critical role in determining the rate coefficient. The reason for this is that the shape of the potential energy surface is such that when the HBr molecule arrives at the neighborhood of CH3 in the H–Br vibrational phase when the bond is stretched significantly, an attraction arises between the two reactants. One can interpret this effect as if the attacking radical pulled the H-atom away from Br when seeing the H–Br bond halfway broken. This “dynamically induced attraction” has been manifested in the experiments of Smith and coworkers44 on the reaction of H atoms with water molecules excited by 4 stretch vibrational quanta as a huge, about 9 Å effective collision diameter. The effect was interpreted in QCT calculations44–48 and was seen as extreme rate enhancement in QCT simulations on other simple H-atom-transfer reactions involving reactants in which the breaking H–X bond is vibrationally excited, such as H + HF49 and F + H2 and H + HCl.50 In one of our recent papers27 we have shown that in the CH3 + HBr reaction, the amplitude of the H–Br oscillation is so large already in the vibrational ground state, that this kind of attraction arises whenever the H–Br oscillation is at the outer turning point and the CH3 reactant is close enough. Notably, when the H–Br vibrational amplitude is reduced by, for example, decreasing the vibrational excitation artificially below the zero-point (feasible in QCT simulations by setting the quantum number to non-integer negative values between –0.5 and 0) or by increasing the mass of the atom being transferred, the reaction can be frozen because the “dynamically induced attraction” will be inefficient or nil. The H/D isotope substitution in HBr involves such an amplitude reduction in the vibrational ground state, because the zero-point energy of DBr is smaller than that of HBr. The consequence is a sizeable primary isotope effect. The inverse secondary isotope effect arising when the H-atoms of the CH3 reactant are replaced by D can be traced back to a completely different reason: it is a steric effect, acting in conjuction with the zero-point vibration. Fig. 3 shows the potential energy of the HBr molecule pointing toward the carbon atom placed in various locations around the radical. One can see that the approach in the plane of the radical, which is set perpendicular to the plane of the figure, is not favorable. There is a “repulsive” ring around the edge of the radical, while its faces are attractive. When the umbrella mode oscillates and the out-of-plane angle changes, the side of the radical toward which the C–H bonds are bent becomes covered by a repulsive hemisphere, while the attraction on the other remains essentially the same. This makes one side of the methyl radical unattractive when the umbrella oscillation arrives to the outer turning point. This occurs alternately for the two sides of the radical, impeding the approach of the two reactants. At large amplitude, both sides are covered by a repulsive shell periodically. Thus, large-amplitude umbrella vibration is not favorable for the reaction. When the H atom is replaced by deuterium, the umbrella bending frequency and thus the zero-point energy is smaller. Consequently, the vibrational amplitude is reduced, just as we sketched in connection with the isotope replacement in HBr. In (R3), because the deviation of the CD3 umbrella bending angle from its equilibrium value remains small, the shielding effect is less efficient than with CH3. This explains the speed-up of (R3) wrt. (R2). In summary, the amplitude reduction upon H/D substitution in CH3 is also a zero-point energy effect: it occurs because the zero-point energy corresponding to the umbrella bending mode of CD3 is smaller than that of CH3.
The largest effect is caused by the excitation of the umbrella mode: the amplitude increases by so much due to excitation to the second energy level that the reactivity decreases by almost a factor of two below 500 K and somewhat less at higher temperatures. The second and third umbrella quanta cause further reactivity reduction, but the effect is less spectacular.
Among the remaining modes, the excitation of the symmetric stretching causes the smallest rate reduction; that induced by the amplitude increase of the in-plane bending or of the asymmetric stretch is larger and commensurable with each other. The reason for the adverse effect of the umbrella excitation is easy to find: it is the same that makes CH3 less reactive than CD3. The amplitude of the umbrella vibration increases with each vibrational quantum, inducing more and more efficient shielding of the reactive faces of the radical. There is probably a similar reason behind the effects of the other modes (Table 1).
v umb = 1 | v umb = 2 | v umb = 3 | v bend = 1 | v sstr = 1 | v astr = 1 |
---|---|---|---|---|---|
5.8 | 11.6 | 17.4 | 17.0 | 37.0 | 39.2 |
In addition to the decreasing reactivity, the activation energy at low temperatures is getting less and less negative with increasing umbrella excitation of the methyl radical. In this temperature range, the reaction rate is predominantly influenced by the capture of the reactants in the van der Waals well, induced by their attraction. It is known that the slower the reactants approach each other, the more efficient capture is. In thermal systems, this appears as a negative activation energy increasing in magnitude when the temperature decreases see the black dots in Fig. 3. When the attraction is reduced for some reason, for example, by the appearance of the little repulsive range when the umbrella amplitude is large (see Fig. 3), then its efficiency in bringing the reactants together will be smaller or cease, thus reducing the magnitude and the rate of increase of the activation energy with decreasing temperature.
The results on the effect of vibrational excitation of the methyl radical on its reactivity allow us to comment on the argument presented by Dobis and Benson11 about larger reaction rates due to unsatisfactory relaxation of the radicals after being generated in high-internal-energy states. We have seen that excitation of any CH3 mode is adverse to the reactions. As a result, incomplete vibrational relaxation would be unfavorable for the reaction, instead of enhancing the rate.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3cp05610d |
This journal is © the Owner Societies 2024 |