On the meaning of “collision rate constants” for ion-molecule reactions: Association of hydrogen atoms with C6H5+ and small alkyl radicals with C7H7+ ions

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Highlights

  • Collision rate constants.

  • Ion-molecule reactions.

  • Physe space theory.

Abstract

Limiting high pressure rate constants for the association of H with C6H5+ and of CH3, C2H5, and n-C3H7 radicals with C7H7+ molecular ions are analyzed in terms of “rigidity factors” arising from the anisotropy of the interaction potentials and “collision rate constants” in the absence of anisotropy (the latter corresponding to phase space theory, PST). Model calculations based on ab initio potential energy surfaces show that the PST rate constants kPST(T) exceed collision rate constants from conventional ion-molecule capture theory (in this case given by Langevin rate constants kL). They can be represented by kPST(T) = kL + kh.sph. where kh.sph. denotes hard-sphere collision numbers with collision radii r0. The r0 derived from the modelled kPST(T) are related to properties of the interaction potentials. Applications to other ion-molecule reactions are proposed.

Introduction

It is common practice to compare experimental rate constants k of bimolecular ion-molecule reactions with “collision rate constants”. In the case of polar reactants, the latter are conveniently estimated by the Su-Chesnavich rate constant kS.Ch. , see ref. [1],kS.Ch./kL = 1 + 0.0967 x + 0.0950 x2 for x ≤ 2= 0.4764 x + 0.6200 for x ≥ 2where the Langevin rate constant kL is given bykL=2πq(α/μ)1/2α denotes the polarizability of the neutral reactant, q the charge of the ion, μ the reduced mass of the collision pair, and the parameter x is given by x = μD/(2αkB T)1/2 with the dipole moment μD of the neutral reactant. Eqs. (1) – (3) are the result of classical trajectory (CT) calculations for collision processes between two point masses in a long-range charge-induced + permanent neutral dipole potential. Rotational quantum effects at low temperatures result in a decrease of k to values smaller than kS.Ch.. Such effects are accounted for, e.g. by the statistical adiabatic channel model (SACM) [[2], [3], [4], [5], [6]], the average dipole orientation (ADO) [7,8] or the adiabatic capture centrifugal sudden approximation (ACCSA) [9,10], all essentially being equivalent [11,12]. Translational quantum effects become apparent at even lower temperatures, see e.g. ref. [13].

The comparison of eqs. (1) – (3) with experimental results showed a wide range of possibilities. Agreement with experiments, e.g., has been observed for reactions like H3+ + HCN (see the analysis of this reaction in refs. [2,4]). Limiting high-pressure rate constants for ion-molecule associations, which correspond to collisional control, in many cases are smaller than given by eqs. (1) – (3), see e.g. the modelling in ref. [14]. On the other hand, rate constants k can also exceed kS.Ch., particularly in reactions with ionic clusters, see refs. [[15], [16], [17], [18]], or in the attachment of electrons to neutral clusters [[19], [20], [21], [22], [23], [24]].

Rate constants, k, larger than the collision rate constants, kS.Ch., have been interpreted in different ways. “Finite-target size effects”, e.g., have been considered for reactions of ionic clusters or for the attachment of electrons to neutral clusters [[15], [16], [17], [18], [19], [20], [21], [22], [23], [24]]. Apart from contributions of quadrupole moments in addition to dipole moments of the neutral [5], influences of polarizabilities, dipole and quadrupole moments of the ions were also suggested for an explanation, see, e.g., ref. [15]; this accounts for changes of the potential between long-range (LR) and short-range (SR) electrostatic contributions. Alternatively, one may replace the SR electrostatic potential by a SR potential of valence character as discussed in the present article.

There are multiple influences of the potential on the collision rate constant k (if one understands by “collision” the occurrence of “capture”, i.e. the formation of an adduct able to react). There is first the influence of the minimum-energy path (MEP) potential V(r). In addition, the anisotropy of the potential is important. While a LR polarization potential often is nearly isotropic, the LR potential may also be anisotropic, e.g., in the case of a LR ion + permanent dipole interaction. The latter is accounted for in kS.Ch. [2]. The total (LR plus SR) anisotropy of the potential gives rise to “rigidity factors”, frigid, between zero and unity [14,25]. Anisotropy contributions to k are neglected in phase space theory (PST, see, e.g., ref. [26]). The aim of the present article is to investigate finite-target size effects in PST. As such effects are system-specific, we consider a series of reaction systems in order to obtain some generalizable results. We consider the example of high-pressure rate constants for association of small alkyl radicals to C7H7+ cations and their representation by PST, i.e. putting rigidity factors equal to unity. Likewise, we consider the association of hydrogen atoms to C6H5+ cations. We rely on SACM/CT calculations from refs. [[6], [14], [27]], employing SR valence/LR electrostatic switching potentials, in part based on ab initio calculations of the potential energy surfaces (PESs) such as described in refs. [28,29] and extended in the present work. The advantage of building on the results of refs. [6,14,27] lies in the fact that rigidity factors in this work have been calculated separately and rate constants kPST(T) for PST have been determined. In this way, the contribution of finite-target size effects to kPST(T) can directly be explored.

Section snippets

PST rate constants for association reactions in their high-pressure limits

We started with the n-C3H7 + C7H7+ → C10H14+ ion-molecule association system. We extended earlier ab initio calculations of the PES from refs. [[27], [29]]. In particular, the properties of the distance dependence and the anisotropy of the SR valence/LR polarization switching potential used in ref. [27] were controlled. Our calculations were made on the ωB97X-D/6–311++G(3df,3pd)//ωB97X-D/6–311++G(d,p) DFT level [30,31] of theory and compared with composite ab initio models such as G4MP2 as well

Conclusions

The present work has analyzed limiting high pressure rate constants for association processes of C6H5+ and C7H7+ ions with small radicals. After removing the “rigidity factors” frigid(T), which reduce the rate constants as a consequence of the anisotropy of the potential, the remaining PST rate constants kPST(T) all exceed collision rate constants from conventional ion-molecule capture theory. The added contributions correspond to hard-sphere collision numbers, being proportional to T1/2. The

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

N.S.S., A.A.V., and S.G.A. acknowledge support from the U.S. Air Force Office of Scientific Research under Grant AFOSR- 9RVCOR042.

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