Advances in low temperature gas-phase kinetics

Abstract

Rate constants for elementary gas-phase reactions were first measured reliably following the development of pulsed photolysis and flow methods in the 1960's. These techniques have continued to be employed as kinetics experiments have been performed at lower and lower temperatures. Sub-ambient temperatures are reached either by cryogenic methods or by using expansion techniques. In this article, we review the possibilities and limitations of these cooling techniques and the results that have been obtained. These efforts have been driven both by the desire to understand the fundamental factors that control the rates of chemical reactions and also by the wish to provide rate constants that can be used in models of complex environments, such as planetary atmospheres and the interstellar medium. In this review, some emphasis is given to the CRESU (Cinétique de Réaction en Ecoulement Supersonique Uniforme) method which has now been used to determine rate constants for many elementary reactions, including those between neutral species, as well as ion–molecule reactions. This method has provided rate constants for a limited number of reactions to below 10 K. Major efforts are now being made to go to still lower temperatures and we describe some of the results obtained at these very low temperatures in the last section of this article.

1. Introduction

There has been continuous progress in the experimental and theoretical study of elementary reactions over the past half century and especially at sub-ambient temperatures over the last two decades. As so often in science, this progress has been driven simultaneously by advances in experimental techniques and by the wish to provide kinetic data for incorporation into reaction schemes (‘models’) seeking to explain chemical change in complex environments. In addition, there is a strong desire to understand the factors that determine the magnitude of rate constants for particular reactions and how they vary with temperature, which might then make it possible to predict reliable rate constants for some reactions that are difficult to study experimentally – at least over the complete temperature range of interest.

To illustrate how these factors work, we briefly survey advances that occurred in the 1960s and 1970s. It was during this period that the pulsed photolysis and flow techniques (see below) were developed to a stage where they generally provided reliable values of the rate constants for the reactions being studied. These two general methods were applied particularly to the kinetics of reactions that are important in the chemistry of the Earth's atmosphere – such as those of the reactions of the hydroxyl (OH) radical. Up to about 1970, most studies were performed at ‘room temperature’. In the early 1970s, Smith and Zellner¹ described how pulsed photolysis experiments, employing cryogenic cooling, could provide rate constants at temperatures down to 210 K – a temperature close to the lowest found in the Earth's atmosphere. At much the same time, Kaufman's group reported rate constants over a very similar temperature range obtained using the discharge-flow technique.² Recent evaluations of rate constants for the purposes of atmospheric modelling (http://www.iupac-kinetic.ch.cam.ac.uk) cite rate constants for most reactions at temperatures down to about 200 K.

More recently, similar influences have stimulated experiments on the kinetics of elementary reactions at temperatures down to as low as ca. 10–20 K. Now the stimulation comes from the observation of molecules in the dark ‘cold cores’ of dense interstellar clouds,³ and the inference that there is a rich chemistry occurring in these astronomical objects, despite the very low temperatures (ca. 10 K) and low densities (ca. 10⁴ cm⁻³) which characterise this environment. Although much of this chemistry is driven by ion–molecule reactions, it is evident that reactions involving neutral free radicals also play a role.⁴ Moreover, for such reactions to occur at significant rates at these very low temperatures, they must take place over a potential energy surface (PES) which exhibits no barrier along the minimum energy path leading from reactants to products. Consequently, the main factor which leads to an activation energy is absent and it is necessary to investigate what other factors determine the value of the rate constants, and their temperature-dependence, for such reactions.⁵

The present article focuses on the measurement and theory for elementary reactions between neutral species that are rapid in the range of temperature between 300 and 10 K. This temperature range can be divided into two: (i) down to ca. 196 K, the temperature of ‘dry-ice’, which can be employed to provide cryogenic cooling, and which is coincidentally close to the lowest temperature found in the Earth's atmosphere – though lower temperatures are found in the atmospheres of other planets (and their satellites) in the solar system,⁶ and (ii) from 196 K to 10 K, where the use of cryogenic cooling techniques becomes more difficult and expansion methods are more commonly applied to create low temperatures.

Finally, it is important to recognise the increasing amount of information that is available for processes at ‘ultra-low temperatures’ (say, ≤1 K). Special techniques are required to cool molecules to such temperatures and to study collisional processes, including chemical reactions. Moreover, because at these temperatures the de Broglie wavelength associated with the relative motion of the reactants is at least of the same order as the range of intermolecular forces, special theoretical methods are needed to treat such processes. We shall provide some limited information about such experiments at ultralow temperatures in Section 4 of this article. Before that, in Section 2, we describe the experiments that have been performed to provide kinetic information about reactions at low temperatures, and then, in Section 3, we shall review some of the results obtained at low temperatures and how these results can be understood in terms of transition state theory (TST). Finally, in Section 5, we shall briefly try to highlight a few areas where our knowledge is deficient, in the hope that these gaps will be filled in the near future.

2. Experimental methods

The majority of experiments on the kinetics of elementary reactions, including those at low temperatures, use one of two general methods. These methods were reviewed by one of the present authors (IWMS) in ref. 6, but, for completeness, a briefer description will be given here.

In the first kind of experiment, atomic or molecular free radicals (e.g. OH) are created by pulsed photolysis – or occasionally pulsed radiolysis – of a suitable precursor (e.g. H₂O, HNO₃ or H₂O₂) using ultraviolet light, either from a flash lamp or nowadays more often from a pulsed laser. The initial concentration of these radicals is much less than that of the co-reactant (e.g. HCl) which is present in the gas mixture at a measured concentration which is much greater than the initial concentration of the radical; that is, in studies of OH + HCl, [OH]_t=0 ≪ [HCl]. The decay in the radical concentration as reaction occurs is observed spectroscopically by resonance absorption, resonance fluorescence, or now generally by time-delayed laser-induced fluorescence (LIF) techniques using a tuneable dye laser. Since pseudo-first-order kinetics pertain, it is not necessary to know the absolute concentration of the radical. The analysis of a single experiment yields a first-order rate constant (k_1st), and the second-order rate constant for the reaction can be obtained by observing how k_1st changes as the concentration of the co-reactant is varied in a series of separate experiments.

In the second kind of kinetics experiment, the reacting mixture flows along a tubular reactor. Radicals (or ions) are created at the upstream end of the reactor and variations in their relative concentration are measured at a point downstream, as different concentrations of the co-reactant are added,⁷ usually through an injector that can be moved along the axis of the flow tube.⁸ In the simplest version of such experiments, performed under so-called ‘plug-flow’ conditions on reactions between neutral species, the total pressure is ca. 5 Torr and it is easy to convert the distance between the injection and detection points to a ‘reaction time’, over which both reactants have been in the flow, and hence to derive rate constants. Molina and co-workers have developed a turbulent flow reactor which can be operated at much higher pressures.⁹ In some cases, the flow may have to be modelled to extract the desired kinetic information. The major sources of rate constants for reactions between ions and neutral molecules have been (i) ‘flowing afterglow’ (FA) experiments, and (ii) the development of such experiments, generally referred to by the acronym FA-SIFT (flowing afterglow selected ion flow tube), in which particular ions are created and selected upstream of the main flow reactor and then injected into the main flow.¹⁰

In the remainder of this section, we shall describe the methods that have been employed to perform kinetics experiments at temperatures down to ca. 10 K. Again one of two general techniques can be employed to access these temperatures. Firstly, the reactor and the gases that it contains can be cryogenically cooled; for example, to ca. 80 K using liquid nitrogen, or to a few K using liquid helium. Secondly, the gas mixture can be cooled by expansion methods: either using Laval nozzles, which in a CRESU† apparatus can achieve temperatures down to ca. 10 K, or in ‘free jets’ where more drastic cooling is possible but kinetic information is hard to obtain because of rapid changes in the gas density and temperature as the flow evolves. These latter methods have the advantage that the cold gas does not come into contact with surfaces so that condensation is restricted.

2(a) Cryogenic cooling

The temperature-dependence of the rate constants for many elementary reactions has been studied by varying the temperature of the reaction cell. Sub-ambient temperatures can be reached using a variety of coolants/cryogens. An obvious danger is that the molecular components of the gas mixture (i.e., the radical precursor and the co-reactant may condense on the walls of the reactor). In the worst scenario, condensation of the radical precursor may lead to very low (or zero) signals from the radical. Even partial condensation of the co-reactant means that it is impossible to convert the observed decay coefficients (k_1st) reliably to the second-order rate constants for the reaction between the radical and the co-reactant.

In a number of studies, one of the present authors (IWMS) and his colleagues circumvented these difficulties by flowing the gas mixture (radical precursor + co-reactant + diluent) through the cooled reaction cell at a rate which was slow enough to ensure adequate cooling but fast enough to prevent most of the radical precursor diffusing to the vessel walls and freezing-out, so that sufficient precursor was present to yield acceptable signals from the photolytically generated radicals. Of course, the lowest temperature of these experiments was restricted by the requirement that no co-reactant was removed from the gas-phase. In this way, rate constants were obtained for the reactions between the following species (with the lowest temperature achieved in brackets): CN + O₂ (99 K),¹¹ CN + NO (99 K),¹² OH + HCl (138 K),¹³ OH + CH₄ (178 K),¹³ OH + C₂H₆ (138 K),¹³ OH + N (103 K), OH + O (158 K), CH + N (216 K), OH + CO (80 K). Table 2 in ref. 6 provides more details of these pulsed photolysis experiments and a similar series undertaken in the laboratory of Ravishankara in Boulder, Colorado.

The trick of balancing the rate of loss of a species by diffusion and condensation with the rate at which the species is added to a cooled reaction cell has been exploited by De Lucia and co-workers in experiments at very low temperatures. In their experiments, excess He or H₂ is cooled to as low as 1 K in some cases. The condensable molecular test gas is then added continuously to the cell through a warm capillary tube to a pressure typically in the range 1 to 30 mTorr. Microwave spectra of the molecular gas at a steady-state concentration can then be recorded. Besides spectroscopic data, these experiments yield pressure-broadening parameters which can be related to the rate at which collisions with the diluent gas transfer the test gas between rotational levels. Experiments of this kind have been carried out on CO colliding with He at temperatures down to 1 K,¹⁴ and with H₂ at T ≥ 8 K.¹⁵ Other experiments have been reported for a variety of test gases – NO, DCl, NH₃, H₂S and CH₃F, with He and H₂ as collision partners¹⁶ – and on HCO⁺ collisions with H₂ at 40 and 77 K.¹⁷

Discharge-flow experiments on reactions between neutral species have frequently been conducted in jacketed flow tubes allowing the temperature to be lowered quite routinely to ca. 200 K. The most impressive low temperature experiments of this kind have been those conducted by Clarke et al.¹⁸ in the very large flow tube (10 m long, 12.36 cm internal diameter) at Harvard. The apparatus is cooled for almost 2 hours with a steady stream of liquid N₂. Once the low temperature has stabilised, the liquid N₂ flow is replaced by one of N₂ gas. The temperature is continuously monitored as it rises, and at various times an experiment is performed; during the time taken to carry out an experiment the increase in temperature is quite small. This method enabled Clarke et al. to obtain rate constants for the reaction between OH radicals and ethane, propane and cyclopropane at temperatures down to 180 K.¹⁸

There is special interest in the reactions of ions with neutral species at low temperatures, since they are the principal driving force behind the chemistry in the cold cores of dark interstellar clouds. Rate constants for such reactions have been obtained using ion cyclotron resonance (ICR) and by flowing afterglow (FA) methods.¹⁹ The ICR method has largely been limited to measurements close to room temperature.²⁰ In the FA method, first developed by Ferguson and co-workers,²¹ ions are generated at the upstream end of a flow reactor, and changes in the relative concentration of ions are measured downstream as neutral co-reactants are introduced into the flow through one or more injection ports. As with the discharge-flow method, jacketed versions of the FA apparatus were soon developed and permitted kinetic studies between 82 and 600 K.^21b,c Other important developments have included: (i) the addition of an upstream quadrupole mass filter allowing the ion to be selected, yielding the flowing afterglow selected ion flow tube (FA-SIFT) technique,²² (ii) the combination of a FA reactor with discharge sources of neutral atoms,¹⁰ and (iii) the modification of the apparatus to allow the kinetics of negative, rather than positive, ions to be studied.²³

The lowest temperatures reached in flow experiments on ion–molecule experiments, employed a drift tube (DT), rather than an FA apparatus. In a DT, ions are injected into an essentially static neutral gas, and move along a tubular reactor under the influence of an electric field. Bohringer and Arnold²⁴ cryogenically cooled a DT using not only liquid nitrogen but also liquid helium. This method enabled them to study the reaction of He⁺ ions with H₂ – a reaction of potential importance in dense interstellar clouds – at temperatures down to 18 K. The reaction was found to proceed by both bimolecular and termolecular mechanisms, but with rather small rate constants. At the lowest temperatures, the termolecular reaction dominated and it was only possible to give upper limits to the bimolecular charge exchange reactions to He + H + H⁺ and He + H₂⁺. Given that the termolecular reaction will be negligible under the low density conditions in dense interstellar clouds, it was concluded that the main fate of He⁺ ions in dense ISCs might well be reaction with CO to yield C⁺ ions.

Of course, DT experiments make use of the charge on the ions which means that they, unlike neutral species, can be manipulated in an electric field. The appropriate deployment of fields also means that ions can be stored in ‘ion traps’ for long times. Early experiments of this kind with a cryogenically cooled Penning ion trap enabled rate constants to be measured to temperatures as low as 11 K.²⁵ Moreover, as ions can be stored for long times, it was possible to measure rate constants for the slow reactions of He⁺ + H₂ at temperatures between 10 and 40 K.²⁶ Rate constants for both the dissociative charge exchange channel to He + H + H⁺ and the non-dissociative charge exchange to He + H₂⁺ were obtained and the branching ratio between these two channels was determined.

More recently, most ion trapping experiments have used the 22-pole ion trap developed by Gerlich.²⁷ In these traps, it is possible to store ions for “many minutes, even hours”.²⁸ This means that slow reactions can be studied. For the study of reactions, the neutral target gas is added to a number density of ca. 10⁹–10¹⁵ cm⁻³. Of course, it is necessary that this neutral gas does not condense during the time that reaction occurs, so the method has been applied particularly to reactions of H₂ and its isotopologues. Trapping experiments have been employed to study the kinetics of radiative association, in which a species formed by the combination of an ion with a neutral molecule is stabilised by the emission of radiation; for example, in the reaction of C⁺ ions with H₂, radiative association starts to dominate collisionally stabilised (‘three-body’) association at H₂ densities below ca. 10¹² cm⁻³. Recent applications of cooled ion traps include: (i) the study of the reactions of CH⁺, CH₄⁺ and CH₅⁺ with slow H atoms and H₂,²⁹ and (ii) the study of H⁻ + H → H₂ + e⁻.³⁰

2(b) Expansion techniques

The spectroscopic study of cold species in supersonic ‘free jet’ expansions has established itself as a major technique in physical chemistry.³¹ In such experiments, a pure gas or generally one diluted in an inert ‘carrier’ such as helium is expanded from a high pressure reservoir (>1 bar) through a small orifice (diameter <1 mm) into a low pressure chamber producing a gas-phase sample with a temperature of ca. 1 K.‡ Pulsed nozzles have proved especially useful, as the pumping requirements are then relatively modest and the resultant pulses of gas can be matched to the light from a pulsed laser. This cooling technique has been applied, intra alia: (i) to cause rotational cooling and thereby greatly simplify the spectra of large molecules,³² and (ii) to generate species, bound weakly by hydrogen bonds or van der Waals forces, in order to study their spectra and hence characterise the intermolecular forces.³³

However, the gas in the free jet is not a promising medium in which to perform kinetics experiments, since the density and temperature both decrease rapidly at increasing distances from the nozzle exit – until the density has fallen so far that further collisions do not occur. Despite such difficulties, M. A. Smith and co-workers have used free jets to obtain rate constants for a number of bimolecular and termolecular ion–molecule reactions at temperatures below 5 K,³⁴ including (i) the reactions of C⁺ ions with O₂, NO and N₂O³⁵ and reactions of NO⁺ ions,³⁶ and (ii) the three-body association reactions of N₂⁺ + 2N₂³⁷ and Ar⁺ + 2Ar.³⁸

More uniform supersonic flows, but not the extreme temperatures found in free jets, can be obtained using Laval nozzles and they have been used in a range of kinetics experiments over the past 25 years. Axisymmetric, convergent–divergent, Laval nozzles can be designed to produce different temperatures in the collimated gas flow produced downstream from the nozzle. The molecular density in this gas flow is typically in the range 10¹⁶–10¹⁷ cm⁻³, so that collisions are sufficiently frequent to ensure that a single temperature defines the distributions of both relative translational velocities and rotational level populations.³⁹ Of course, if a species is created in the flow – for example, by photolysis of a precursor – its distributions over its rovibrational levels will not initially correspond to the translational temperature of the main gas and they may take some significant time to equilibrate with those of other degrees of freedom.⁴⁰

The earliest kinetic experiments using Laval nozzles were performed by B. R. Rowe and co-workers in a very large rarefied wind tunnel in Meudon (France) that had originally been built for aerodynamic studies.⁴¹ This facility had a huge pumping capacity (1.44 × 10⁵ m³ h⁻¹) over a wide, but low, pressure range: 10⁻⁴–10⁻¹ mbar. Rowe realised the potential of the low temperature flows from Laval nozzles for kinetics experiments. The equipment was first designed to examine the kinetics of ion–molecule reactions. Downstream from the exit of the Laval nozzle, the gas jet was crossed with an electron beam that could ionise either the main He carrier gas or any other molecular gas that had been admitted to the reservoir upstream of the Laval nozzle. Changes in the concentration of ions as reaction occurs with a co-reactant were monitored mass spectrometrically downstream from the exit of the Laval nozzle.⁴²

In these early CRESU experiments, as in later measurements, several Laval nozzles were constructed in order to generate different temperatures in the gas flows (ca. 20–160 K).⁴³ To access an even lower temperature, a nozzle was manufactured which, with the reservoir upstream from the nozzle, could be cooled in liquid nitrogen, making it possible to reach a temperature as low as 8 K in the jet downstream from the nozzle.⁴⁴ In the 1980s, this CRESU apparatus was deployed to measure rate constants for a number of reactions involving cations with non-polar⁴³ and polar⁴⁵ molecules. A summary of the ion–molecule reactions for which rate constants had been measured in a CRESU apparatus up to 2000 has been given by Smith and Rowe,⁴⁶ and a comparison between the temperature-dependence of the rate constants for some reactions involving non-polar and polar neutral species is given in Fig. 2 of ref. 19. Those for reactions between ions and molecules with small or zero dipole moments show little or no dependence on temperature, as predicted by the simple capture model generally attributed to Langevin⁴⁷ (see below), whereas the rate constants for reactions of ions with polar molecules increase to low temperature.⁴⁵

In 1993, Rowe moved to the University of Rennes I and took the opportunity to build a CRESU apparatus designed specifically for kinetics experiments. This apparatus had a large pumping capacity (1.95 × 10⁴ m³ h⁻¹), but one lower than that in Meudon. Also, at about this time, Rowe began to collaborate with one of the authors (IWMS) of the present article who had realised that the rate constants for rapid reactions between neutral species could be measured at low temperatures in an adapted CRESU apparatus. In these experiments, pulsed laser photolysis is used to generate free radicals along the central core of the flow and time-delayed LIF is generally employed to observe the decrease in their concentration as reaction occurs with a selected co-reactant that is present in the gas flow at a known concentration. Repeating these measurements with different concentrations of the co-reactant present then enables a second-order rate constant for the reaction to be derived. The first measurements of this kind were made on the reactions of CN radicals with O₂ (at T ≥ 13 K) and with NH₃ (at T ≥ 25 K).⁴⁸

In the mid-1990s, a second continuous flow CRESU apparatus was built at the University of Birmingham UK, which had a pumping capacity of 3.31 × 10⁴ m³ h⁻¹. In 2003, this apparatus was moved to the University of Rennes I, which is now home to both the existing continuous high flow CRESU apparatuses. In these apparatuses, the uniformity of the central core of the flow is maintained for several tens of centimetres, corresponding to a reaction time§ of a few hundred microseconds.

Table 1 summarises the neutral–neutral reactions that have been studied using these two continuous flow CRESU apparatuses, including information about the photolytic precursor for the radical reactant and the lowest temperature at which the rate constants have been measured. In general, two factors limit the lowest temperature which can be obtained: (i) clustering of the co-reactant in the low temperature flow must be avoided, and (ii) the concentration of the co-reactant in the reservoir must be sufficiently low that it does not condense if the reservoir is cryogenically cooled. In addition, it is necessary that sufficient co-reactant can be included in the flow to ensure that the pseudo-first-order rate constant for the reaction under investigation ≥10⁴ s⁻¹.

Table 1 Summary of neutral–neutral reactions studied in continuous fast flow CRESU apparatuses. The number after the formula of the co-reactant indicates the lowest temperature (in K) at which a rate coefficient has been obtained^a

Radical	Precursor	Observation	Co-reactant (minimum T/K)
a Further information, including data obtained in experiments employing pulsed Laval nozzles is given in ref. 42.
CN	(a) PLP of NCNO at 583 nm,	LIF on the (0,0) band of the B^2Σ^+–X^2Σ^+ system at 388 nm	O2(13), NH3(25), C2H2(25), C2H4(25), C2H6(25),CH3C≡CH(15), H2C=C=CH2(15)
	(b) PLP of ICN at 266 nm
OH	PLP of H2O2 at 266 nm	LIF on the (0,0) band of the A^2Σ^+–X^2Π system at 308 nm	HBr(23), O(^3P) (39), O2(55), but-1-ene(23), (E)-but-2-ene(23), (Z)-but-2-ene(23)
CH	Multiple photon PLP of CHBr3 at 266 nm	LIF on the (0,0) band of the A^2Δ^+–X^2Π system at 430 nm	O2 (13), NO(13), D2(13), NH3(23), CH4(23), C2H2(23), C2H4(23), C2H6(23), CH3C≡CH(77), H2C=C=CH2(77)
C2H	PLP of C2H2 at 193 nm	Chemiluminescence from CH(A^2Δ) formed in the C2H + O2 reaction	O2(15), C2H2(15), C2H4(15), C3H6(15), CH3C≡CH(63), H2C=C=CH2(63)
C2(^1Σg^+)	PLP of C2Cl4 at 193nm	LIF on the (0,0) band of the D^1Σu^+–X^1Σg^+ system at 231 nm	CH4(24), C2H2(24), C2H4(24), C2H6(24), C3H8(24)
C2(^3Πu)	PLP of C2Cl4 at 193nm	LIF on the (0,0) band of the d^3Πg–a^3Πu system	CH4(24), C2H2(24), C2H4(24), C2H6(24), C3H8(24)
C4H	PLP of HC4H at 248 nm	LIF on the [J] band of the system from the ground state to the B^2Πi3v5(Σ^+) at 408.3 nm	C2H6(39), C3H8(39), C4H10(39), C2H4(39), C3H6(39), but-1-ene (39) C2H2(39), HCCCH3(39), HCCC2H5(39), H2C=C=CH2(39), H2C=CHCH=CH2(39),
C(^3PJ)	PLP of C3O2 at 193 nm^2	(a) Chemiluminescence from CH(A^2Δ) formed in the C(^3P) + NO2 reaction	O2(15), NO(15), C2H2(15), C2H4(15), C2H6(15), CH3C≡CH(15), H2C=C=CH2(15)
		(b) LIF on the line at 165.69 nm
Al(^2PJ)	Multiple photon PLP of Al(CH3)3 at 266 nm	LIF on the Al transitions at 394.40 nm (^2S1/2–^2P1/2) and 396.15 nm (^2S1/2–^2P3/2)	O2(23)
Si(^3PJ)	Multiple photon PLP of Si(CH3)4 at 266 nm	LIF on ^3P1–^3P0 atomic transition at 251.43 nm	O2(15), NO(15), C2H2(15), C2H4(15)
B(^2PJ)	PLP of B(CH3O)3 at 266 nm	LIF on ^2S–PJ atomic transitions at 249.77 and 249.68 nm	O2(24), C2H2(23), C2H4(23)
O(^3PJ)	PLP of NO2 at 355 nm	Chemiluminescence from NO2	1-C4H8(23), iso-C4H8(27), (E)-C4H8(27), (Z)-C4H8(27),
S(^1D)	PLP of CS2 at 193 nm	LIF on the ^2D0–^2D0 line at 166.67 nm	CH4(23), C2H2(23), C2H4(23)

The large size of a continuous CRESU apparatus has apparently inhibited their widespread deployment. Other limitations to the CRESU method relate to the high gas flows which make it difficult (i) to use expensive co-reactants – for example, deuterated species – and (ii) to generate useful concentrations of radical species using electrical discharges. This has led to the development of equipment that utilise Laval nozzles to create low temperature gas flows, but which reduces the overall gas throughput.

The first development along these lines was the design of an apparatus yielding pulsed uniform supersonic expansions. This type of apparatus was first described by Atkinson and Smith⁴⁹ and they used it to measure rate constants for the association reaction of OH with NO at temperatures down to 90 K.⁵⁰ The apparatus had a pumping capacity of 170 m³ h⁻¹, about two orders-of-magnitude less than that in the continuous CRESU apparatuses in Rennes and Birmingham. Using this pulsed Laval apparatus, kinetics experiments have been performed on the reactions of NH with NO and with various hydrocarbons,⁵¹ of OH with HBr,⁵² and of OH with propene and isoprene.⁵³ The lowest temperature achieved in these experiments was 53 K.

Similar pulsed Laval nozzle apparatuses to that described by Atkinson and Smith have been built by the groups of Abel (Gottingen, and later Leipzig),⁵⁴ of Leone (Colorado, and later Berkeley),⁵⁵ and of Heard (Leeds).⁵⁶ The experiments in Abel's group, which access temperatures down to about 60 K, have mainly focused on examining the effect on reaction rates under situations where the co-reactant can form hydrogen-bonded complexes with water.⁵⁴ Although it has been demonstrated that the rates of some reactions are increased when complexes are formed with single molecules of H₂O, it has not proved possible to quantify this effect because of uncertainty in the fraction of co-reactant species that are present as complexes.

The latest experiments from Leone's laboratory reach 74 K.⁵⁵ His low temperature experiments have mainly focused on reactions of OH and C₂H radicals. An important development has been the marriage of a pulsed Laval nozzle apparatus with product detection via single photon ionisation of the products.⁵⁷ This has enabled isomer-specific identification of the products of the reaction of C₂H with 1-butyne⁵⁵ and the measurement of product branching ratios from the reaction of C₂H radicals with ethene and propene at 79 K.⁵⁸

An alternative method of obtaining low temperatures via expansion through a Laval nozzle, but one which avoids the large space and pumping requirements of a full-scale CRESU apparatus, has been successfully pursued in Bordeaux.⁵⁹ Bergeat and co-workers have constructed a more modest CRESU apparatus with a pumping speed of <10³ m³ h⁻¹. They first deployed three Laval nozzles that gave temperatures in the gas flow of 170, 128 and 77 K,⁵⁹ though later experiments reached as low as 48 K.^60,61

One advantage of the smaller scale CRESU apparatus is that it is possible to generate significant concentrations of radical atoms using a microwave discharge. Recent work from Bordeaux has concentrated on reactions of N atoms. Rate constants have been measured for the reaction, N + NO → N₂ + O, essentially using a flow method, where changes in concentrations of the N atoms with [N] ≪ [NO], were observed by resonance fluorescence in the vacuum ultraviolet (120.0 nm).⁶⁰ These measurements extended the temperatures for which the rate constants for this well-studied reaction have been measured down to 48 K. These results for N + NO were then used as the basis for measurements of rate constants for the reaction between N atoms and OH radicals.⁶² In essence, the relative rates of the two reactions, N with NO and N with OH, were measured for the same (unknown) concentration of N atoms and rate constants for the N + OH reaction derived from the relative rate measurements and the previously measured rate constants for N + NO. These reactions are considered in further detail in the next section of this article.

3. Rate constants for reactions at low temperatures

In this section, we provide a selective review of the results of kinetic measurements on elementary reactions at low temperatures. Necessarily, we shall be concerned with reactions that are rapid¶ at low temperatures, and which are therefore characterised by potential energy surfaces (PESs) with little or no barrier along the minimum energy path (MEP) leading from reactants to products. A class of reaction where many individual reactions proceed along MEPs with no barrier|| are those between ions and neutral molecules. Such reactions have been extensively studied using methods described in the previous section and are important in low temperature environments, such as those found in the cold cores of dense interstellar clouds. There is a large kinetic data base for such reactions (e.g. Ancich⁶³), although for many reactions rate constants are only available at room temperature. It is with ion–molecule reactions that we start this part of our review.

3(a) Ion–neutral reactions

For many – but not all – ion–molecule reactions, there is no activation barrier to inhibit reaction. Rather the rate constant is determined by the ability of the long-range attraction between the reactants to bring them into close contact. The simplest treatment of this ‘capture’ process applies if the dominant forces arise between the charge on the ion and the electric dipole which that charge induces on the neutral species.⁴⁷ If the polarisability (α) of the molecule is independent of orientation and the ion carries a single charge (positive or negative, but of magnitude e), then a simple treatment (in c.g.s. units),⁴⁷ generally referred to as the Langevin model, yields the following expression for the rate constant for capture between two reactants with a collisional reduced mass μ:

k_L = 2πe(α/μ)^1/2 (3.1)

Rate constants for a number of simple ion–molecule reactions (e.g. He⁺ + N₂, He⁺ + O₂, and N⁺ + CH₄) have been measured at low temperatures both in the original CRESU apparatus in Meudon and in the second generation apparatus in Rennes.⁶⁴ The results are in excellent agreement with one another, the rate constants are independent of temperature, as predicted by eqn (3.1), and their absolute values are in fair agreement with the prediction of the Langevin formula.

Rowe and co-workers have also made kinetic measurements designed to test whether the rate constants for ion–molecule reactions are influenced by the presence of a dipole moment⁴³ or a quadrupole moment⁶⁵ on the neutral reactant. For reactions between ions and polar neutrals, the rate constants increase below room temperature, and Wakelam et al.⁶⁶ recommend that one can estimate rate constants based on the work of Su and Chesnavich.⁶⁷ The ratio of the required rate coefficient (k_D) to the value (k_L) predicted by eqn (3.1) depends on a parameter x:

and formulae are given for the ratio k_D/k_L in terms of x.⁶⁶ Using quasiclassical trajectories, Maergoiz et al.⁶⁸ have shown that these formulae work well for neutral polar species that are linear, symmetric tops, or asymmetric tops. To examine if the presence of a quadrupole moment on the molecular reactant significantly changes the rate constant for an ion–molecule reaction, Rebrion et al.⁶⁵ have determined rate constants for the reactions of He⁺, C⁺ and N⁺ with C₆F₆ and c-C₆H₁₂ at 27 and 68 K in CRESU experiments and at 297 K in a FA-SIFT apparatus. Despite the presence of large quadrupole moments in C₆F₆ and c-C₆H₁₂, the measured rate constants were close to those predicted by eqn (3.1) and were invariant with temperature.

Because of their importance in interstellar chemistry,⁶⁹ and the scarcity of experimental data, the reactions of H₃⁺ ions with O(³P) atoms and with CO molecules have been examined theoretically by Klippenstein et al.⁷⁰ using a detailed transition state approach. For H₃⁺ + O(³P), as well as the ion-induced dipole term, the long-range potential also includes an isotropic ion–quadrupole term, and it is necessary to treat the interplay of the spin–orbit coupling with the ion–quadrupole interaction. At 300 K, the value of the rate constant calculated by Klippenstein et al. is close to the value of 1.2 × 10⁻⁹ cm³ s⁻¹ measured by Milligan and McEwan.⁷¹ It is predicted to increase by a factor of 1.4 as the temperature falls to 30 K, partly as a result of the ion–quadrupole capture rate constant having a T^−1/6 dependence and partly because the partition function for O(³P) decreases as the temperature is lowered, so that the population of the reactive spin–orbit states increases. For H₃⁺ + CO, the analysis of Klippenstein et al.⁷⁰ indicates that between 10 and 400 K the ion–dipole, ion–quadrupole and charge-induced dipole interactions are all important factors in the long-range potential and, therefore, in determining the magnitude and temperature-dependence of the rate constant for capture. The calculated rate constant for the overall reaction (that is, to both H₂ + HCO⁺ and H₂ + HOC⁺) rises from 2.2 × 10⁻⁹ cm³ s⁻¹at 300 K to 2.7 × 10⁻⁹ cm³ s⁻¹at 30 K. This room temperature value is in reasonable agreement with several measurements (cited in ref. 70), but, at 30 K, the only measured result⁷² is about half the calculated value.

To summarise: rate constants for ion–molecule reactions have been measured for many cation–molecule reactions, including those between ions and neutral atoms.⁷³ Stimulated, in part, by the discovery of anions in the interstellar medium, measurements are now being made on reactions between anions and neutral species.⁷⁴ It remains the case that many of the experimental measurements of rate constants for ion–molecule reactions have only been made at room temperature. However, as we have tried to demonstrate, the magnitude of the rate constants are frequently determined by ‘capture’ – and this can generally be confirmed or otherwise by room temperature experiments. If the rate constants obtained at room temperature are close to those predicted by the Langevin treatment, then reliable methods do exist to extrapolate these rate constants to low temperature.

3(b) Radical–neutral molecule reactions

The existence, or otherwise, of barriers on PESs for reactions between electrically neutral species depend, to first approximation, on the chemical nature of the reactants: for example, one may ask whether one or both of them are saturated molecules, unsaturated molecules, free radicals? Reactions between molecules, in which all the electrons are paired up, have high barriers with the result that they are very slow even at 300 K. The reactions of radicals with saturated molecules – e.g. of OH radicals with alkanes – generally have small barriers, whilst the reactions of radicals with unsaturated molecules may exhibit no ‘real’ barrier though there may be a ‘submerged’ barrier on the MEPs for such reactions. The term ‘real’ barrier is used here where there is a maximum along the MEP which has a higher energy than that of the separated reactants, whereas ‘submerged’ barrier is used when there is a maximum on the MEP but the energy at this maximum is lower than that associated with the separated reactants. When two radicals interact, multiple PESs arise. Usually, for the lowest PES, electrons of different m_s pair up to form a bond and the PES falls monotonically as the two species approach.

In the event that there is no real barrier, the rate constant for a reaction between neutral species may be determined by ‘capture’; that is, by the ability of the long-range potential between the reactants to bring them into close contact. In the case of two neutral species, the electrostatic forces that arise at long-range differ in two important respects from those where one of the species is an ion: first, they are weaker and shorter-range; for example, the forces between a point charge and a polarisable species vary as R⁻⁴, whereas dispersion forces vary as R⁻⁶ (where R is the separation between the two particles); second, directional forces, like those between two dipoles are likely to play a more significant role than in ion–molecule reactions.

Georgievskii and Klippenstein⁷⁵ have proposed a ‘long-range transition state theory’, which can estimate the rate constant for mutual capture of two neutral species. The intermolecular forces are weaker and act over a shorter-range than those for ion–neutral species, and directional forces derived, for example, from dipole–dipole forces, dipole–quadrupole forces, etc. are often pre-dominant. The theory is variational, it is implemented both at an energy (E) and angular momentum (J) resolved level, and is given the acronym μJ-VTST. For each combination of E and J, it is necessary to calculate an average rate coefficient for different reactant orientations – which can be done using Monte Carlo methods. By comparing their calculated results with low temperature experimental data, Georgievskii and Klippenstein show that this long-range TST provides upper limits to the true low temperature rate constants for a fairly large sample of neutral–neutral reactions. The reasons why the true rate constants are usually lower than the capture rate constants are explored in the following sections.

Reactions of simple free radicals, atomic or molecular, with molecules, both saturated and unsaturated, play an important role in atmospheric and combustion chemistry, and have been widely studied, both experimentally and theoretically. When the co-reactant is a saturated molecule, the rate constant is generally (but not always, see below) characterized by a small positive activation energy, usually less than ca. 50 kJ mol⁻¹. One result of this behavior is that it can be difficult to measure the rate constants down to very low, or even low, temperatures.

As examples of such reactions, we take the reactions of OH radicals with CH₄,¹³ C₂H₆,¹³ HCl¹³ and HBr. Reactions of hydroxyl radicals have been extensively studied because of their crucial importance in atmospheric chemistry. In each case, reaction proceeds by abstraction of an H-atom yielding H₂O and a new radical. The first three of these reactions have been studied by the PLP-LIF method in a cryogenically cooled reaction cell;¹³ the lowest temperatures at which rate constants were obtained by Sharkey and Smith¹³ were 178 K for OH + CH₄, and 138 K for OH + C₂H₆ and OH + HCl. These results are shown in Fig. 1, together with the results of similar measurements down to 212 K made by Ravishankara and co-workers.⁷⁶ Both sets of experiments employed LIF to observe relative concentrations of OH. A difficulty with the low temperature experiments was that the fluorescence from OH(A²Σ⁺) was heavily quenched by the large concentrations of co-reactant that had to be present to bring the pseudo-first-order rate constants into a conveniently measurable range.

Fig. 1 Arrhenius plots of measured rate constants for reactions of OH radicals with HCl, CH₄ and C₂H₆, including data obtained using cryogenic cooling: (a) OH + HCl: ○ from Sharkey and Smith,¹³ ■ from Battin-Leclerc et al.;^77c (b) OH + CH₄: ○ from Sharkey and Smith,¹³ ◆ Gierczak et al.;^77b (c) OH + C₂H₆: ○ from Sharkey and Smith,¹³ ▲ from Gierczak et al.^77b

The rate constant for OH + HBr is approximately 14 times greater than that for OH + HCl at 298 K,⁷⁷ and it shows little dependence on temperature between 200 and 400 K. Encouraged by these observations, Sims et al.⁷⁸ performed CRESU experiments on this reaction down to 23 K and found that the rate constant increased monotonically as the temperature was lowered below ca. 200 K. The observed negative temperature dependence at lower temperatures is well predicted by a simple formula deduced from quantum scattering calculations employing the rotating bond approximation.⁷⁷ Later experiments in Rennes and in Tucson came to a consensus view of the temperature-dependence of the rate constant for this reaction.⁷⁹

Two interesting examples of reactions involving saturated molecules are those of NH₃ with CN and CH radicals. The rate constants for the CN + NH₃ reaction show a steep negative temperature dependence between 295 and 25 K;⁴⁸ those for CH+NH₃ increase from 295 K to about 100 K but then decrease slowly to about 23 K, the lowest temperature in these experiments.⁸⁰ These two reactions have been analysed theoretically by Talbi and Smith⁸¹ and by Blitz et al.,⁸² using the two transition state model of Klippenstein and co-workers.⁸³Ab initio calculations demonstrate that both these reactions proceed via the initial, and transitory, formation of a complex. In the case of CN + NH₃, the attractive forces between the reactants arise largely through the interaction of the two strong dipoles; in the case of CH + NH₃, a dative bond forms arising from the lone electron pair on NH₃ donating into the empty orbital on the CH radical. The next stage of these reactions involves progress over a submerged barrier. The results of transition state calculations show excellent agreement with the experimental results for CN + NH₃⁸⁰ and fair agreement for CH + NH₃.⁸¹

A majority of CRESU experiments on radical–molecule reactions have been carried out on reactions where the neutral co-reactant is an unsaturated molecule (see Table 1), reflecting the fact that many of these reactions are ‘barrier-less’ and therefore remain rapid at low temperatures. The results obtained for these reactions before 2006 were reviewed and analysed by Smith et al.⁸⁴ Based on ideas previously deployed by Donahue et al.,⁸⁵ and still earlier by Cvetanovic,⁸⁶ they correlated the low temperature rate constants for a number of reactions with the values of (I.E. − E.A.) where I.E. is the ionisation energy of the unsaturated molecular reactant and E.A. is the electron affinity of the radical reactant. They found that, if (I.E. − E.A.) for a particular reaction was less than ca. 8.75 eV, the rate constant at low temperature (ca. 25 K) was ≥10⁻¹¹ cm³ s⁻¹. This gave a semi-empirical basis for predicting low temperature rate constants for reactions between radicals and unsaturated molecules.

Table 2 Association reactions studied in continuous fast flow CRESU apparatuses. For each reaction, the table gives the range of temperature over which low pressure rate coefficients have been measured, and the parameters k(298) and n which describe the variation of the rate coefficients with temperature according to the expression: k(T)/cm³ molecule⁻¹ s⁻¹ = k(298 K) (298/T)⁻ⁿ [M]

Reaction	Range of (T/K)	k(298 K)/cm^3 molecule^−1 s^−1	M	n
OH + NO	23–295	4.5 × 10^−31	Ar	2.6
OH + O2	55.9–79.2	4.2 × 10^−34	Ar	3.5
CH + H2	13–295	5.2 × 10^−30	Ar	1.6
CH + N2	53–295	1.6 × 10^−31	Ar	2.2
CH + CO	53–295	4.1 × 10^−30	Ar	2.1

In order to examine further the hypothesis of Smith et al.,⁸⁴ rate constants were measured in a CRESU apparatus at temperatures down to ca. 23 K for the reactions of O(³P) atoms (E.A. = 1.461 eV) with a range of alkenes which give rise to values of (I.E. − E.A.) between 9.05 eV for ethene to 7.64 eV for trans-butene. The experimental data were accompanied by the results of theoretical calculations by Georgievskii and Klippenstein. High quality ab initio calculations showed that for all these reactions there are two transition states on the MEP leading to formation of the chemically bound hydroxyl alkyl radicals: one ‘outer’ transition state whose location depended on E and J and one ‘inner’ transition state between the van der Waals' minimum and the inter-reactant separation where chemical forces start to act. Applying the ‘two transition state theory’ that had been developed earlier, the calculated rate constants were shown to be in good agreement with the measured rate constants, as demonstrated in Fig. 2. In general, it appears that μJ-VTST, along with high quality ab initio calculations of the MEP at long- and medium-range separation of the reactants, enable low temperature rate constants to be estimated with good accuracy.

Fig. 2 Rate coefficients for the reactions of O(³P) atoms with alkenes:⁹³ (1),▲ cis-butene; (2),■ trans-butene; (3),● iso-butene; (4),★ 1-butene; (5),◆ propene; (6) ethene. The points show experimental results obtained at low temperatures using the CRESU technique, the dashed lines the results from calculations using the two transition state approach of Georgievskii and Klippenstein. The solid lines towards the right of the diagram represent the Arrhenius expressions recommended by Cvetanovic.⁸⁶

The reactions of O(³P) atoms with alkenes proceed by initial addition followed by re-arrangement of the resultant complex and fragmentation to one or more sets of products. The overall rate is determined by the first step – and it is this rate constant which is measured in the CRESU experiments. When OH radicals react with alkenes, the first step is again addition of the radical – to create a hydroxyl-alkyl radical – but now, rather than re-arrangement and fragmentation, the adduct will either decompose back to reactants or be stabilised by a collision with a ‘third-body’. Consequently, the second-order rate constant will depend on the total gas density. Moreover, this behaviour will change with temperature: as the temperature is lowered, the rate constant will approach the high pressure limit at lower total pressures. Recognising these facts and that the range of total gas densities is limited in a CRESU apparatus, when Sims et al.⁸⁷ measured rate constants for the reactions of OH with alkenes, they chose, as co-reactants, three butenes, but-1-ene, (Z)-but-2-ene and (E)-but-2-ene, in the confident expectation that the kinetics for these reactions at low temperatures would be in the high pressure limit. The measured rate constants for reaction with the three butenes were very similar in size and all showed a negative dependence on temperature. Vakhtin et al.⁸⁸ have employed a pulsed Laval nozzle apparatus of the kind described in Section 2(b) to measure rate constants for the reaction of OH radicals with propene and 1-butene at 103 K. Their result for OH + 1-butene was in excellent agreement with that which Sims et al.⁸⁷ obtained in a continuous flow CRESU apparatus and the rate constant for OH + propene was similar to that for OH + 1-butene, suggesting that the kinetics were in the high pressure limit under the conditions of these measurement.

3(c) Radical–radical reactions

The interaction between two free radicals generally gives rise to multiple PESs. The symmetries of these surfaces are given by the well-known correlation rules.⁸⁹ In general, the potential energy of one or more of these PESs falls monotonically as the two radicals approach and electrons on the individual radicals pair up to form a chemical bond. The overall result of a collision between a pair of radicals on this PES is then likely to lead initially to the formation of an internally energised adduct or complex, the fate of which will primarily depend on the subsequent form of the PES. So, if there is no exothermic pathway from the potential energy well associated with formation of a bond between the radicals, then any reaction is likely to require collisional or radiative stabilisation of the initially formed energised adduct and lead to association of the radicals, but, if there is an exothermic exit from the PES, reaction can lead to two product species. An example of the first type of reaction is OH + NO (+M) → HONO (+M);** an example of the second type CN + O₂ → NCO + O. In what is referred to as the high pressure limit, the rate constants for association reactions correspond to those for initial formation of the adduct and, in those cases, as in radical–radical reactions leading to bimolecular products, the magnitude of the rate constants and their temperature-dependence are likely to be determined by ‘capture’ on the attractive potential between the reactants.

As the previous paragraph makes clear, reactions between radicals frequently do not involve any potential energy barrier and are therefore likely to occur rapidly at low temperature. However, when the radical reactants are both ‘unstable’ and therefore have to be generated ‘on the fly’ there are real experimental difficulties, since not only do two radicals have to be generated but it must also be possible to measure or estimate the absolute concentration of one or other radical. These problems are magnified if the experiment is performed in a CRESU apparatus. On the other hand, if one of the reactants is NO or O₂, which might be referred to as ‘stable radicals’, the problems are reduced. Thus, rate constants have been determined for the association reactions: CN + NO (down to 99 K, using cryogenic cooling),⁹⁰ OH + NO (down to 23 K in CRESU experiments)⁹¹ and OH + O₂ (55.9–99.8 K in CRESU experiments).⁹² In each of these cases, reliable low pressure rate constants have been obtained but deriving high pressure rate constants requires a long extrapolation. The low pressure rate constants are generally expressed in the form: k⁰(T) = A(T/298)⁻ⁿ. The values of A and n for the three reactions that have just been identified, and some other association reactions, and the range of temperature over which these expressions hold are listed in Table 2.

The CRESU experiments on the association of OH with O₂ are especially interesting since they allowed the strength of the weak HO–OO bond to be determined.⁹³ Between 87.4 and 99.8 K, because the association reaction did not go to completion at long times, the OH concentrations approached non-zero values, allowing equilibrium constants for OH + O₂ = HOOO to be estimated. Using expressions for the equilibrium constant from classical and statistical thermodynamics, and values of partition functions and standard entropies calculated from spectroscopic data, the dissociation energy was found to be (12.3 ± 0.3) kJ mol⁻¹.⁹³

The reaction between CN and O₂ was one of the first two reactions studied using the continuous flow CRESU technique and experiments were performed at temperatures down to 13 K.⁴⁸ The principal products are NCO + O, though about 20% of the reaction proceeds to CO + NO,⁹⁴ probably via a ‘roaming mechanism’.⁹⁵ The overall rate constant shows a strong negative temperature-dependence below 295 K. The value of the rate constant measured at 13 K is half the value estimated via long-range TST.⁷⁵ These facts suggest that the overall rate is influenced by ‘bottlenecks’ at shorter separations than those determined by the long-range potential. The reaction has been examined by Klippenstein and Kim⁹⁶ using a variational version of statistical RRKM theory that was originally designed to treat ‘barrierless’ association reactions. They used ab initio electronic structure calculations to define the variation of the potential energy, as the NC–OO separation was varied between 1.7 and 3.0 Å with the NCO and COO bending angles optimised at each separation. The conserved modes, such as the CN and OO vibrations, and the transitional modes, such as those which correlate with the CN and OO rotations, were treated separately. The partition functions for the former motions were treated in the standard manner for harmonic oscillators, whereas those for the transitional modes were estimated via the corresponding phase space integrals. The reaction was assumed to occur on the lower doublet PES but not on the quartet PES which correlates with CN(²Σ⁺) + O₂(³Σ_g⁻).

The calculations of Klippenstein and Kim reproduce the strong negative temperature-dependence of k(T) at temperatures below 295 K, and the essentially constant value at higher temperatures. Their analysis also makes it clear that the dynamics of this reaction – as for the association reactions discussed earlier – are determined at inter-reactant separations where chemical forces are important. Qualitatively, this behaviour is attributable to the fact that as the separation between the reactants decreases, although the potential energy decreases, the separation between the energy levels associated with the transitional modes (those orthogonal to the MEP) widen and consequently the minimum in the number of states for selected E and J moves to shorter inter-reactant separation. As E increases, the location of the minimum number of states moves to shorter NC–OO distances and the thermally averaged rate constant decreases as the temperature increases.

In some senses, the simplest radical–radical reactions are those between an atomic radical and a diatomic radical. A system that is experimentally tractable – since it involves the ‘stable’ free radical, NO – is the reaction of N(⁴S) with NO. Rate constants for this reaction have been measured several times,⁷⁶ usually by the discharge-flow method. Using cryogenic cooling, experiments have been performed at temperatures down 196 K⁹⁷ and 213 K.⁹⁸ Bergeat et al.⁶⁰ have extended the temperature range over which rate constants for this reaction have been measured down to 48 K using their small CRESU reactor (see above, Section 2(b)). Their results are consistent with those reported for temperatures around 300 K, and show a mild negative temperature-dependence, with the rate constant increasing from (3.3 ± 0.2) × 10⁻¹¹ cm³ s⁻¹ at 211 K to (5.8 ± 0.3) × 10⁻¹¹ cm³ s⁻¹ at 48 K.

Direct efforts to measure rate constants for several reactions of this type involving two unstable free radicals were first made in the early 1980s, on the reactions N + OH,⁹⁹ O + OH,⁹⁸ D + OH¹⁰⁰ and N + CN.¹⁰¹ These experiments combined aspects of the discharge-flow and pulsed photolysis methods: the atomic reactants were generated using discharge-flow methods, and had their concentrations determined by well-established gas titration techniques; the diatomic radicals were generated by photolysis of a suitable precursor and the kinetic decays of relative concentrations of the diatomic radical were observed using resonance fluorescence or LIF. The experiments reported in ref. 98 and 99 were performed over a range of temperature. The measurements on N + OH and O + OH were extended in later experiments¹⁰² down to 103 K and 158 K, respectively. The rate constants derived from these studies are displayed in Fig. 3.

Fig. 3 Rate constants for some radical atom + OH reactions. N + OH: (a) N + OH: ● from Howard and Smith;¹⁰⁰ ▲ from Smith and Stewart;¹⁰² ■ from Daranlot et al.¹⁰⁴ (b) O + OH: ● from Howard and Smith;¹⁰⁰ ▲ from Smith and Stewart;¹⁰² ■ from Carty et al.¹⁰³

The reactions of OH with N and O atoms have both been examined to still lower temperatures using CRESU apparatuses. The O + OH reaction was studied in a fast flow continuous CRESU apparatus by Carty et al.¹⁰³ They used an excimer laser operating at 157 nm to simultaneously photolyse O₂, to generate O atoms, and H₂O, to produce a much smaller concentration of OH. The concentration of atomic oxygen was estimated from the absorption cross-section of O₂ and the fluence from the laser and the kinetic decays of OH were observed using time-delayed LIF. The results are compared with those obtained by Smith and Stewart¹⁰² in Fig. 3. At the temperature where the two sets of results overlap, the rate constants derived from the CRESU experiments are ca. 60% of those from the earlier experiments of Smith and Stewart.¹⁰²

Experiments on the N + OH reaction have been performed at several temperatures down to 58 K, in the small CRESU apparatus in Bordeaux.¹⁰⁴ In these experiments, the difficulty of determining the concentration of N atoms in the fast flowing gas was circumvented by performing relative rate measurements on the reactions of N atoms with OH and with NO. This was achieved by passing nitrogen gas through a microwave discharge in order to dissociate a fraction of the N₂ and generate N atoms. The gas was passed through the selected Laval nozzle and OH radicals were then produced by pulsed laser photolysis of a small concentration of H₂O₂ that was included in the main gas flow. The reactions N + OH → NO + H and N + NO → N₂ + O then took place and LIF was used to follow the variation in the OH and NO concentrations as a function of time. Analysis of these signals yield pseudo-first-order rate constants for the two reactions and for the same N-atom concentration in both cases, so that the ratio of these first-order constants correspond to the ratio of the two second-order rate constants, and the second-order rate constant for N + OH can be derived as that for N + NO had already been evaluated through the experiments described above.⁶⁰ The rate constants for N + OH obtained from these experiments are compared in Fig. 3 with those measured by Smith and Stewart.¹⁰¹ Clearly, these two sets of data show significant differences – though we would not describe them as “large”. Nevertheless, the comparison does suggest that there are unidentified and significant sources of error in one or other of the two sets of measurements.

More recently, the Bordeaux group have reported measurements of the rate constants for the reaction: N + CN → N₂ + C relative to those for the N + NO reaction. These results are reported in a paper¹⁰⁵ that also reports modelling calculations designed to investigate the partitioning of elemental nitrogen in dense interstellar clouds. The formation of N₂ occurs through the agency of four radical–radical reactions involving N atoms: N + OH, N + NO, N + CH and N + CN. As yet, rate constants for the N + CH reaction have only been measured, in experiments using cryogenic cooling, at temperatures down to 216 K.¹⁰⁶ As discussed above, the rate constants for the other reactions have been determined at lower temperatures though not yet below 50 K. Unusually, the rate constant for the N + CN reaction increases with temperature from 56 K to 296 K. This probably means that the reaction is determined predominantly by the ‘outer’ or long-range transition state with only a slight influence by an ‘inner’ transition state.

4. Towards absolute zero

The emphasis in Section 2 has been on the measurement of rate constants at temperatures down to about 10 K. The term ‘rate constant’ implies that the rate of reaction has been measured by monitoring the loss of a reactant – or more rarely the growth of a reaction product – under conditions where the reactants are thermally equilibrated; that is, the distributions over translational and internal degrees of freedom are consistent with the Maxwell–Boltzmann laws. In CRESU experiments, which have figured prominently in Section 2, this condition is generally satisfied because the chemically active species, i.e., reactants and products, are usually heavily diluted in the inert ‘carrier gas’. The lowest temperature achieved in such experiments is 5.8 K in experiments on the reaction of S(¹D) atoms with H₂.¹⁰⁷ As the results recorded in Table 1 show, more often the lowest achievable temperature is higher, because it is frequently limited either by the creation of weakly bound dimers (and possibly higher oligomers) in the gas expansion or by condensation of the molecular co-reactant in the cold reservoir upstream from the Laval nozzle.

In the last two decades, strenuous – and ingenious – efforts have been made to generate lower temperatures and colder molecules for a variety of applications in chemical physics. In this field of ‘ultracold chemistry’, it now usual to refer to temperatures between 1 K and 1 mK as cold conditions and temperatures below 1 mK as ultracold conditions.¹⁰⁸ A number of ingenious methods have been employed to reach these extreme and unnatural†† conditions. Two books,^109,110 several reviews^111–113 and dedicated issues of journals^114,115 have been devoted to this topic within the past few years. These publications have largely been devoted to descriptions of the experimental methods that have been deployed to attain extremely low temperatures or to prepare slow-moving molecules,‡‡ rather than to the multitude of physical phenomena that may be explored utilising the samples of cold molecules which these methods might provide.

Amongst the targets which might be identified, and which are of special significance within the context of the present article, is the exploration of chemical reactions under these extreme conditions, where (i) the de Broglie wavelength associated with the relative motion of the potential reactants is greater than the size of the reactants (or equivalently the extent of their interaction), and (ii) the collisional angular momentum becomes severely constrained, eventually reaching the limit where l = 0 for potentially reactive collisions; that is, only s-wave scattering occurs. Under such conditions, it is clear that quantum effects will be important. However, as yet, there have been few examples of kinetic studies under extreme conditions, which might therefore link up with the studies that I have discussed earlier in this article. Accordingly, in this section I shall confine myself to only a few comments that are relevant to studies of bimolecular collisions under cold or ultracold conditions.

Clearly, for the study of bimolecular reactions, it is necessary that the laboratory speed of both reactants must be low – and yet collisions must occur. This requirement suggests the use of experiments which employ two low velocity molecular beams – or one such beam and one cold gas ensemble. Meijer and his group in the Fritz Haber Institute in Berlin have been in the forefront of developing methods that generate slowly moving molecular beams of polar molecules by exploiting the Stark effect, and this technique is described by Schnell and Meijer¹¹² – and in several other reviews and papers. One difficulty, besides that of generating one or two samples of slow-moving molecules, is that the flux of molecules in a low velocity beam is very low and therefore the number of collisions per second in the volume where two such beams cross is only ca. 10³, which requires a very sensitive method to detect the scattered products of the collisions.

Meijer and co-workers have pioneered the use of Stark decelerators to generate beams of slow-moving molecules. They are based on the principle that in an electric field the energies of the internal states of a polar molecule are shifted. Molecules can possess a positive Stark effect (that is, such molecules occupy low-field-seeking states) or a negative Stark effect; and these include the ground state of molecules. When a bunch of polar molecules enters an electric field (transverse to the direction of the beam), their potential energy is altered and there is a corresponding change in their kinetic energy. However, such changes are quite small, typically ca. 2% of the translational energy of the molecules. Consequently, the Stark decelerators employed to reduce the translational energy of the packets of molecules in a beam, typically employ a sequence of about 100 electrode sets through which the pulsed beam passes. As the bunch of molecules approaches a given electrode set, a potential is applied which decelerates the molecules. The potential on the electrodes must be rapidly switched so that the effect of each pair is to decelerate the bunch of molecules as it approaches a given set of electrodes, but does not accelerate the molecules as the packet of molecules recedes from a given electrode set.

Although the slow-moving beams generated using Stark decelerators have been used in a number of elegant experiments on a variety of molecules,§§ they have not, as yet, been used to study reactive collision. An alternative, and somewhat simpler, method of generating a slow-energy beam has been developed by Rempe and co-workers.^116,117 In this method, rather then produce slow-moving molecules directly, the coldest fraction of molecules from a cold thermal beam are filtered out using an electrostatic quadrupole. The parallel electrodes of the quadrupole curve through a right-angle and the voltage is adjusted so that only the slowest-moving molecules follow the bend. For H₂CO and ND₃, the fluxes were determined to be ca. 10⁹–10¹⁰ s⁻¹ with a longitudinal temperature of a few K.^116,117

A cold beam source of the type developed by Rempe and co-workers has been employed to study reactive collisions between laser-cooled Ca⁺ ions and velocity-selected CH₃F molecules by Willitsch et al.¹¹⁸ The ions were prepared in a linear Paul trap as a ‘Coulomb crystal’ and then the CH₃F molecules were introduced through a quadrupole-guide velocity selector to yield mean collision energies of 〈E_coll〉/k_B ≤ 1 K. The disappearance of ions as a result of reaction was followed by observing the decrease in their LIF signal and the voltage on the quadrupole was varied to investigate how the reaction cross-section depends on the collision energy. The rate constant within the middle of the range that was investigated was ca. 1.3(6) × 10⁻⁹ cm³ s⁻¹. This value compares with that in a room temperature sample of ca. 4.2(4) × 10⁻¹⁰ cm³ s⁻¹. The increase of a factor of ca. 3.3 in lowering the temperature through this range is roughly what one would expect on the basis of capture theories for a reaction between a monovalent ion and a molecule with a large electric dipole moment (1.85 D for CH₃F), so there is no apparent quantum effect – though the lowest temperature might still not be low enough for quantum effects to manifest themselves.

An alternative method to generate slowly moving molecules in a beam has been developed by Doyle and his co-workers.¹¹⁹ These experiments, making use of buffer gas cooled beams, are somewhat similar to those used to study rotational energy transfer performed by De Lucia et al. and referred to earlier.^14,15 Doyle et al. employ a cryogenically cooled reservoir into which a non-condensable gas (usually helium) is introduced, along with a much lower partial pressure of ‘test gas’. This latter species thermalizes with the buffer gas and forms an effusive beam which leaves through a small aperture in the cooled reservoir and contains a small fraction of the test gas. Compared with a supersonic beam source, a buffer gas cooled beam generally has a smaller forward velocity, but a comparable internal temperature. The method is versatile: thus, it has been applied to make beams of quite a large number of molecules (listed in Table 2 of ref. 118) and produces relatively intense beams, without achieving the very lowest temperatures.

A buffer gas-cooled beam of ND₃ has been combined with trapped OH radicals, generated using a Stark decelerator in combination with a magnetic trap, to examine cold collisions (ca. 5 K) between these two species:¹²⁰ the first such experiments on cold collisions between state-selected neutral species. Using REMPI (resonance-enhanced multiphoton ionisation) to observe ND₃ and LIF (laser-induced fluorescence) to observe OH, it was possible to estimate cross-sections for total (reactive + non-reactive) loss from the trap. This study appears to represent the state-of-the-art in respect of experimental studies of bimolecular reactions between neutral species at cold temperatures (below 5 K).

As a final example of experiments on chemical reactions performed under cold (ca. 1 K) conditions, we cite the study of Singh et al.¹²¹ on the reaction between atomic lithium and CaH. This study uses cryogenic helium buffer gas cooling, recalling the experiments of De Lucia ad co-workers on rotational energy transfer in CO.^14,15 The experiment is carried out in a 10 cm cubic copper cell cooled by a cryogen-free ⁴He refrigerator. Li atoms and CaH molecules are produced by laser ablation of solid targets of Li and CaH₂ and detected by laser absorption spectroscopy.¹²² To confirm that the loss of CaH was due to the two-body reaction between Li atoms and CaH a number of tests were conducted. Experiments were performed at helium densities from 0.2–1.2 × 10¹⁶ cm⁻³ and from 1.3–1.9 K. Within experimental error (estimated at a factor of 2), the derived rate coefficient was constant within this temperature range with a value of 3.6 × 10⁻¹⁰ cm³ s⁻¹.

5. Concluding remarks

From the material presented in the previous sections, it is clear that there are limitations both to the low temperatures that can be attained in gas-phase kinetics experiments and to the reactions that can be studied in such experiments. In this short final section, I identify some of these limitations and indicate how some of them might be overcome.

There are apparently no fundamental reasons why the CRESU technique could not be utilised to access temperatures very close to the absolute zero. This possibility has been discussed by Canosa et al.¹²³ However, to obtain such very low temperatures necessitates a very high Mach number at the exit of the Laval nozzle and that requires, in turn, a very high ratio between the pressure in the reservoir and that in the experimental chamber.¹²³ The preferred way to overcome the problems associated with the high pumping requirements that would be needed – but a technically demanding one – is to ‘chop’ the gas flow within the Laval nozzle. It is expected that the use of such a pulsed Laval source should allow temperatures as low as 1 K to be reached.¹²³ As well as lowering the temperatures that can then be attained, the use of such an apparatus will also reduce the consumption of expensive – for example, deuterated – gases in low temperature experiments, and thereby increase the range of reactions that can be studied at very low temperatures.

A limitation that CRESU experiments shares with many other kinetic measurements on elementary reactions is that it is not easy to determine product branching ratios in those cases where the reaction can yield more than one set of reaction products. However, significant progress towards this goal has been made in recent experiments from Leone's group.¹²⁴ They have coupled a pulsed Laval nozzle apparatus to a photoionisation mass spectrometer, for which the ionising radiation is provided by the tuneable vacuum ultraviolet output from the Advanced Light Source at Lawrence Berkeley National Laboratory. In these experiments, product ions are identified from the photoionisation mass spectra recorded with a quadrupole mass spectrometer. Rate constants for the reactions are derived by analysing time-resolved signals from individual product ions. In addition, branching ratios can be determined by comparing the signals from ions that are formed in different product channels.

Amongst other studies, Bouwman et al.¹²⁵ have investigated the reactions of ethynyl radicals with ethene and propene at 79 K. They have shown that, within experimental error, the reaction between C₂H radicals and C₂H₄ at 79 K proceeds exclusively to H₂C=CHCCH + H, whereas C₂H + C₃H₆ results in (85 ± 10)% C₄H₄ (vinylacetylene) + CH₃, and (15 ± 10)% C₅H₆ + H. The photoionisation method is able to distinguish between different isomers, as well as species that differ in their mass. One can anticipate that such experiments will provide a great deal of valuable information regarding the rates and products of low temperature reactions.

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Footnotes

† CRESU is an acronym for Cinétique de Réaction en Ecoulement Supersonique Uniforme or Reaction Kinetics in Uniform Supersonic Flows.

‡ In practice, the relative populations in different levels associated with a given degree of freedom are determined by kinetics, rather than thermodynamics, so that they do not necessarily obey the Boltzmann law. Moreover, the distributions associated with different degrees of freedom will not be in equilibrium.

§ By ‘reaction time’, we mean the time that it takes species in the centre of the flow to travel from the exit of the Laval nozzle to where relative concentrations of the radical are monitored.

¶ Of course, the rate of a chemical reaction depends both on its rate constant and the concentration of the reactants. In the present context, a rapid reaction is one for which the rate constant approaches that which might be estimated for all collisions between the reactants.

|| Later in this review, we shall distinguish between what we call ‘real barriers’ which have energies higher than that of the separated reactants, and ‘submerged barriers’ which are maxima along the MEP but which have energies below that of the separated reactants.

** Here, M represents the main species that is present in the gas mixture, and is frequently referred to as the ‘bath gas’ or the ‘third body’. In collisions with the energised adduct, M can remove energy from the adduct, thereby stabilising it.

†† I use the term ‘unnatural’ to emphasise that the lowest temperature achieved in laboratories are lower than those found anywhere else in the universe: the temperature of the cosmic background, as inferred from the distribution of the background cosmic radiation with frequency, is 2.718 K. As far as is known, these are no places in the universe where the temperature is less than 1 K.

‡‡ A summary of these methods is given in Table 1 of ref. 111.

§§ These are listed on page 06 of the review by Bell and Softley.¹¹¹

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